Subgroups — C₂×C₆₇×#3005 #30739607

Order 17152.

5098 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#137×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3005	C₂	1	16	yes	maximal, direct product factor
6	C₂×#452×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#449×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	—	2	8	no	—
12	C₂²×#41×C₆₇	—	2	8	no	—
13	C₂²×#41×C₆₇	—	2	2	no	—
14	C₂²×#41×C₆₇	—	2	4	no	—
15	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×#41×C₆₇	C₂²	1	4	yes	—
17	C₂²×#41×C₆₇	C₂²	1	8	yes	complemented
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×D₄×C₆₇	—	2	1	no	—
20	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₆₇	—	2	1	no	—
22	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#137×C₆₇	—	2	8	no	—
24	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
25	C₂×AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
26	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
27	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
28	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
29	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
30	C₂²×D₈×C₆₇	C₂²	1	2	yes	complemented
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#525×C₆₇	—	2	4	no	—
34	#525×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#123×C₆₇	—	2	1	no	—
36	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#120×C₆₇	—	2	1	no	—
39	C₂×C₄×D₄×C₆₇	—	2	2	no	—
40	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
41	#452×C₆₇	C₂²	1	32	yes	complemented
42	#449×C₆₇	C₂²	1	32	yes	complemented
43	#419×C₆₇	C₂²	1	8	yes	complemented
44	#418×C₆₇	C₂²	1	8	yes	complemented
45	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
46	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
47	#407×C₆₇	C₂²	1	8	yes	complemented
48	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
49	C₂⁵×C₆₇	—	2	1	no	—
50	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
51	C₂×#41×C₆₇	—	2	32	no	—
52	C₂×#41×C₆₇	—	2	64	no	—
53	C₂×#41×C₆₇	—	2	16	no	—
54	C₂×#41×C₆₇	—	2	32	no	—
55	C₂×#41×C₆₇	—	2	16	no	—
56	C₂×#41×C₆₇	—	2	32	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	2	16	no	—
59	C₂×#41×C₆₇	—	2	2	no	—
60	C₂×#41×C₆₇	—	2	4	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	C₂³	1	16	yes	—
63	C₂×#41×C₆₇	C₂³	1	4	yes	—
64	C₂²×Q₈×C₆₇	—	2	4	no	—
65	C₂²×Q₈×C₆₇	—	2	2	no	—
66	C₂²×Q₈×C₆₇	—	2	1	no	—
67	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
68	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
71	C₂²×D₄×C₆₇	—	2	16	no	—
72	C₂²×D₄×C₆₇	—	2	4	no	—
73	C₂²×D₄×C₆₇	—	2	8	no	—
74	C₂²×D₄×C₆₇	—	2	2	no	—
75	C₂²×D₄×C₆₇	—	2	1	no	—
76	C₂²×D₄×C₆₇	—	2	2	no	—
77	C₂²×D₄×C₆₇	—	2	1	no	—
78	C₂²×D₄×C₆₇	—	2	4	no	—
79	C₂²×D₄×C₆₇	D₄	1	8	yes	complemented
80	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
83	C₂³×C₄×C₆₇	—	2	16	no	—
84	C₂³×C₄×C₆₇	—	2	1	no	—
85	C₂³×C₄×C₆₇	—	2	4	no	—
86	C₂³×C₄×C₆₇	—	2	8	no	—
87	C₂³×C₄×C₆₇	—	2	4	no	—
88	C₂³×C₄×C₆₇	—	2	4	no	—
89	C₂³×C₄×C₆₇	—	2	1	no	—
90	C₂³×C₄×C₆₇	D₄	1	2	yes	—
91	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
92	C₂³×C₄×C₆₇	D₄	1	4	yes	—
93	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
94	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
95	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
96	#137×C₆₇	—	2	32	no	—
97	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
98	C₂×Q₁₆×C₆₇	—	2	8	no	—
99	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	complemented
100	C₂×SD₁₆×C₆₇	—	2	8	no	—
101	C₂×SD₁₆×C₆₇	—	2	8	no	—
102	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
103	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
104	C₂×D₈×C₆₇	—	2	8	no	—
105	C₂×D₈×C₆₇	C₂³	1	8	yes	complemented
106	C₂×M₁₆×C₆₇	—	2	4	no	—
107	C₂×M₁₆×C₆₇	—	2	4	no	—
108	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
109	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
110	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
111	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
112	#123×C₆₇	—	2	8	no	—
113	C₂²⋊Q₈×C₆₇	—	2	4	no	—
114	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
115	#121×C₆₇	—	2	4	no	—
116	#121×C₆₇	C₂³	1	8	yes	complemented
117	#120×C₆₇	—	2	8	no	—
118	C₄×D₄×C₆₇	—	2	16	no	—
119	#117×C₆₇	—	2	2	no	—
120	#117×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂×C₄²×C₆₇	—	2	1	no	—
127	#103×C₆₇	C₂³	1	16	yes	complemented
128	#102×C₆₇	C₂³	1	16	yes	complemented
129	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
130	C₂⁴×C₆₇	—	2	4	no	—
131	C₂⁴×C₆₇	—	2	8	no	—
132	C₂⁴×C₆₇	—	2	4	no	—
133	C₂⁴×C₆₇	—	2	2	no	—
134	C₂⁴×C₆₇	—	2	1	no	—
135	C₂⁴×C₆₇	—	4	4	no	—
136	C₂⁴×C₆₇	—	4	8	no	—
137	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
138	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
140	#41×C₆₇	—	2	64	no	—
141	#41×C₆₇	—	2	128	no	—
142	#41×C₆₇	—	2	32	no	—
143	#41×C₆₇	—	2	32	no	—
144	#41×C₆₇	—	2	8	no	—
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	4	8	no	—
147	C₂×Q₈×C₆₇	—	2	8	no	—
148	C₂×Q₈×C₆₇	—	2	16	no	—
149	C₂×Q₈×C₆₇	—	2	4	no	—
150	C₂×Q₈×C₆₇	—	2	8	no	—
151	C₂×Q₈×C₆₇	—	2	4	no	—
152	C₂×Q₈×C₆₇	—	2	2	no	—
153	C₂×Q₈×C₆₇	—	2	1	no	—
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
157	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	complemented
158	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
159	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×D₄×C₆₇	—	2	32	no	—
161	C₂×D₄×C₆₇	—	2	64	no	—
162	C₂×D₄×C₆₇	—	2	16	no	—
163	C₂×D₄×C₆₇	—	2	8	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	2	8	no	—
166	C₂×D₄×C₆₇	—	2	16	no	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	2	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	8	no	—
171	C₂×D₄×C₆₇	—	2	2	no	—
172	C₂×D₄×C₆₇	—	2	2	no	—
173	C₂×D₄×C₆₇	—	2	1	no	—
174	C₂×D₄×C₆₇	—	4	8	no	—
175	C₂×D₄×C₆₇	—	4	4	no	—
176	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
177	C₂×D₄×C₆₇	C₂×D₄	1	16	yes	complemented
178	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₆₇	—	2	32	no	—
181	C₂²×C₄×C₆₇	—	2	64	no	—
182	C₂²×C₄×C₆₇	—	2	4	no	—
183	C₂²×C₄×C₆₇	—	2	8	no	—
184	C₂²×C₄×C₆₇	—	2	4	no	—
185	C₂²×C₄×C₆₇	—	2	2	no	—
186	C₂²×C₄×C₆₇	—	2	8	no	—
187	C₂²×C₄×C₆₇	—	2	16	no	—
188	C₂²×C₄×C₆₇	—	2	16	no	—
189	C₂²×C₄×C₆₇	—	2	4	no	—
190	C₂²×C₄×C₆₇	—	2	4	no	—
191	C₂²×C₄×C₆₇	—	2	8	no	—
192	C₂²×C₄×C₆₇	—	2	8	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	1	no	—
195	C₂²×C₄×C₆₇	—	2	4	no	—
196	C₂²×C₄×C₆₇	—	4	4	no	—
197	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
198	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
199	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
200	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
208	Q₁₆×C₆₇	—	2	16	no	—
209	SD₁₆×C₆₇	—	2	16	no	—
210	SD₁₆×C₆₇	—	2	16	no	—
211	D₈×C₆₇	—	2	16	no	—
212	M₁₆×C₆₇	—	2	8	no	—
213	M₁₆×C₆₇	—	2	8	no	—
214	C₂×C₈×C₆₇	—	2	4	no	—
215	C₂×C₈×C₆₇	—	2	4	no	—
216	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
217	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
218	C₄⋊C₄×C₆₇	—	2	8	no	—
219	C₄⋊C₄×C₆₇	—	2	2	no	—
220	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	2	8	no	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₄²×C₆₇	—	2	4	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	2	8	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	2	8	no	—
229	C₂³×C₆₇	—	2	16	no	—
230	C₂³×C₆₇	—	2	4	no	—
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	2	4	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	4	8	no	—
235	C₂³×C₆₇	—	4	16	no	—
236	C₂³×C₆₇	—	4	32	no	—
237	C₂³×C₆₇	—	4	16	no	—
238	C₂³×C₆₇	—	4	8	no	—
239	C₂³×C₆₇	—	4	2	no	—
240	C₂³×C₆₇	—	4	4	no	—
241	C₂³×C₆₇	#120	1	4	yes	—
242	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
243	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	8	no	—
246	Q₈×C₆₇	—	2	16	no	—
247	Q₈×C₆₇	—	2	4	no	—
248	Q₈×C₆₇	—	2	2	no	—
249	Q₈×C₆₇	—	2	2	no	—
250	Q₈×C₆₇	—	4	2	no	—
251	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
252	D₄×C₆₇	—	2	64	no	—
253	D₄×C₆₇	—	2	8	no	—
254	D₄×C₆₇	—	2	16	no	—
255	D₄×C₆₇	—	2	4	no	—
256	D₄×C₆₇	—	2	2	no	—
257	D₄×C₆₇	—	2	2	no	—
258	D₄×C₆₇	—	4	16	no	—
259	D₄×C₆₇	—	4	8	no	—
260	D₄×C₆₇	—	4	2	no	—
261	D₄×C₆₇	C₂²×D₄	1	8	yes	—
262	C₂×C₄×C₆₇	—	2	64	no	—
263	C₂×C₄×C₆₇	—	2	8	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	8	no	—
266	C₂×C₄×C₆₇	—	2	16	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	16	no	—
269	C₂×C₄×C₆₇	—	2	8	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	4	2	no	—
275	C₂×C₄×C₆₇	—	4	4	no	—
276	C₂×C₄×C₆₇	—	4	8	no	—
277	C₂×C₄×C₆₇	#120	1	4	yes	—
278	C₂×C₄×C₆₇	#120	1	8	yes	—
279	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
287	C₈×C₆₇	—	2	4	no	—
288	C₈×C₆₇	—	2	4	no	—
289	C₂²×C₆₇	—	2	8	no	—
290	C₂²×C₆₇	—	2	4	no	—
291	C₂²×C₆₇	—	2	4	no	—
292	C₂²×C₆₇	—	2	8	no	—
293	C₂²×C₆₇	—	2	4	no	—
294	C₂²×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	4	16	no	—
296	C₂²×C₆₇	—	4	32	no	—
297	C₂²×C₆₇	—	4	8	no	—
298	C₂²×C₆₇	—	4	16	no	—
299	C₂²×C₆₇	—	4	8	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	4	no	—
302	C₂²×C₆₇	—	8	4	no	—
303	C₂²×C₆₇	C₂×#120	1	4	yes	—
304	C₂²×C₆₇	C₂×#120	1	2	yes	—
305	C₂²×C₆₇	#575	1	4	yes	—
306	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₆₇	—	2	4	no	—
308	C₄×C₆₇	—	2	8	no	—
309	C₄×C₆₇	—	4	2	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	C₂×#120	1	4	yes	—
312	C₄×C₆₇	C₂×#120	1	2	yes	—
313	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
314	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
315	C₂×#3005	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₆₇	—	2	4	no	—
317	C₂×C₆₇	—	4	4	no	—
318	C₂×C₆₇	—	4	8	no	—
319	C₂×C₆₇	—	8	2	no	—
320	C₂×C₆₇	#3005	1	4	yes	direct product factor
321	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
322	C₂×C₆₇	C₂×#575	1	2	yes	—
323	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
324	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic, complemented
326	C₂×#525	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#3005	C₂×C₆₇	1	16	yes	direct product factor
328	C₂×#452	C₂×C₆₇	1	4	yes	complemented
329	C₂×#449	C₂×C₆₇	1	4	yes	complemented
330	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
333	C₆₇	C₂×#3005	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²×#41	—	2	8	no	—
335	C₂²×#41	—	2	8	no	—
336	C₂²×#41	—	2	2	no	—
337	C₂²×#41	—	2	4	no	—
338	C₂²×#41	C₂²×C₆₇	1	4	yes	—
339	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
340	C₂²×#41	C₂²×C₆₇	1	8	yes	complemented
341	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
342	C₂³×D₄	—	2	1	no	—
343	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂⁴×C₄	—	2	1	no	—
345	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×#137	—	2	8	no	—
347	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
348	C₂×AGL(1,ℤ/8)	—	2	8	no	—
349	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
350	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
351	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
352	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
353	C₂²×D₈	C₂²×C₆₇	1	2	yes	complemented
354	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
355	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
356	#525	—	2	4	no	—
357	#525	C₂²×C₆₇	1	8	yes	complemented
358	C₂×#123	—	2	1	no	—
359	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
360	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
361	C₂×#120	—	2	1	no	—
362	C₂×C₄×D₄	—	2	2	no	—
363	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
364	#452	C₂²×C₆₇	1	32	yes	complemented
365	#449	C₂²×C₆₇	1	32	yes	complemented
366	#419	C₂²×C₆₇	1	8	yes	complemented
367	#418	C₂²×C₆₇	1	8	yes	complemented
368	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
369	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
370	#407	C₂²×C₆₇	1	8	yes	complemented
371	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
372	C₂⁵	—	2	1	no	—
373	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
374	C₂×#41	—	2	32	no	—
375	C₂×#41	—	2	64	no	—
376	C₂×#41	—	2	16	no	—
377	C₂×#41	—	2	32	no	—
378	C₂×#41	—	2	16	no	—
379	C₂×#41	—	2	32	no	—
380	C₂×#41	—	2	8	no	—
381	C₂×#41	—	2	16	no	—
382	C₂×#41	—	2	2	no	—
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	C₂³×C₆₇	1	4	yes	—
386	C₂×#41	C₂³×C₆₇	1	16	yes	—
387	C₂²×Q₈	—	2	4	no	—
388	C₂²×Q₈	—	2	1	no	—
389	C₂²×Q₈	—	2	2	no	—
390	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
391	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
393	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
394	C₂²×D₄	—	2	16	no	—
395	C₂²×D₄	—	2	4	no	—
396	C₂²×D₄	—	2	8	no	—
397	C₂²×D₄	—	2	2	no	—
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	—	2	1	no	—
400	C₂²×D₄	—	2	2	no	—
401	C₂²×D₄	—	2	4	no	—
402	C₂²×D₄	D₄×C₆₇	1	8	yes	complemented
403	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
404	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
405	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
406	C₂³×C₄	—	2	16	no	—
407	C₂³×C₄	—	2	1	no	—
408	C₂³×C₄	—	2	4	no	—
409	C₂³×C₄	—	2	8	no	—
410	C₂³×C₄	—	2	4	no	—
411	C₂³×C₄	—	2	4	no	—
412	C₂³×C₄	—	2	1	no	—
413	C₂³×C₄	D₄×C₆₇	1	2	yes	—
414	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
415	C₂³×C₄	D₄×C₆₇	1	4	yes	—
416	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
417	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
419	#137	—	2	32	no	—
420	AGL(1,ℤ/8)	—	2	32	no	—
421	C₂×Q₁₆	—	2	8	no	—
422	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	complemented
423	C₂×SD₁₆	—	2	8	no	—
424	C₂×SD₁₆	—	2	8	no	—
425	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
426	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
427	C₂×D₈	—	2	8	no	—
428	C₂×D₈	C₂³×C₆₇	1	8	yes	complemented
429	C₂×M₁₆	—	2	4	no	—
430	C₂×M₁₆	—	2	4	no	—
431	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
432	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
433	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
434	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
435	#123	—	2	8	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
438	#121	—	2	4	no	—
439	#121	C₂³×C₆₇	1	8	yes	complemented
440	#120	—	2	8	no	—
441	C₄×D₄	—	2	16	no	—
442	#117	—	2	2	no	—
443	#117	C₂³×C₆₇	1	4	yes	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₄²	—	2	1	no	—
450	#103	C₂³×C₆₇	1	16	yes	complemented
451	#102	C₂³×C₆₇	1	16	yes	complemented
452	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	8	no	—
455	C₂⁴	—	2	2	no	—
456	C₂⁴	—	2	4	no	—
457	C₂⁴	—	2	1	no	—
458	C₂⁴	—	4	4	no	—
459	C₂⁴	—	4	8	no	—
460	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
461	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
463	#41	—	2	64	no	—
464	#41	—	2	128	no	—
465	#41	—	2	32	no	—
466	#41	—	2	32	no	—
467	#41	—	2	8	no	—
468	#41	—	2	8	no	—
469	#41	—	4	8	no	—
470	C₂×Q₈	—	2	8	no	—
471	C₂×Q₈	—	2	16	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	4	no	—
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	2	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
480	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	complemented
481	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
482	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
483	C₂×D₄	—	2	32	no	—
484	C₂×D₄	—	2	64	no	—
485	C₂×D₄	—	2	16	no	—
486	C₂×D₄	—	2	8	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	8	no	—
489	C₂×D₄	—	2	16	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	8	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	4	8	no	—
498	C₂×D₄	—	4	4	no	—
499	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
500	C₂×D₄	C₂×D₄×C₆₇	1	16	yes	complemented
501	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
502	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
503	C₂²×C₄	—	2	32	no	—
504	C₂²×C₄	—	2	64	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	8	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	8	no	—
510	C₂²×C₄	—	2	16	no	—
511	C₂²×C₄	—	2	16	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	8	no	—
515	C₂²×C₄	—	2	8	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	4	4	no	—
520	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
522	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
523	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
527	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
528	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
531	Q₁₆	—	2	16	no	—
532	SD₁₆	—	2	16	no	—
533	SD₁₆	—	2	16	no	—
534	D₈	—	2	16	no	—
535	M₁₆	—	2	8	no	—
536	M₁₆	—	2	8	no	—
537	C₂×C₈	—	2	4	no	—
538	C₂×C₈	—	2	4	no	—
539	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
540	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	2	8	no	—
546	C₂²⋊C₄	—	2	8	no	—
547	C₄²	—	2	4	no	—
548	C₂³	—	2	4	no	—
549	C₂³	—	2	8	no	—
550	C₂³	—	2	4	no	—
551	C₂³	—	2	8	no	—
552	C₂³	—	2	16	no	—
553	C₂³	—	2	4	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	4	8	no	—
558	C₂³	—	4	16	no	—
559	C₂³	—	4	16	no	—
560	C₂³	—	4	32	no	—
561	C₂³	—	4	8	no	—
562	C₂³	—	4	2	no	—
563	C₂³	—	4	4	no	—
564	C₂³	#120×C₆₇	1	4	yes	—
565	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
566	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
567	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
568	Q₈	—	2	8	no	—
569	Q₈	—	2	16	no	—
570	Q₈	—	2	4	no	—
571	Q₈	—	2	2	no	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
575	D₄	—	2	64	no	—
576	D₄	—	2	8	no	—
577	D₄	—	2	16	no	—
578	D₄	—	2	4	no	—
579	D₄	—	2	2	no	—
580	D₄	—	2	2	no	—
581	D₄	—	4	16	no	—
582	D₄	—	4	8	no	—
583	D₄	—	4	2	no	—
584	D₄	C₂²×D₄×C₆₇	1	8	yes	—
585	C₂×C₄	—	2	64	no	—
586	C₂×C₄	—	2	8	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	8	no	—
589	C₂×C₄	—	2	16	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	16	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	4	no	—
599	C₂×C₄	—	4	8	no	—
600	C₂×C₄	#120×C₆₇	1	4	yes	—
601	C₂×C₄	#120×C₆₇	1	8	yes	—
602	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
603	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
604	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
609	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
610	C₈	—	2	4	no	—
611	C₈	—	2	4	no	—
612	C₂²	—	2	4	no	—
613	C₂²	—	2	8	no	—
614	C₂²	—	2	4	no	—
615	C₂²	—	2	4	no	—
616	C₂²	—	2	4	no	—
617	C₂²	—	2	8	no	—
618	C₂²	—	4	16	no	—
619	C₂²	—	4	32	no	—
620	C₂²	—	4	8	no	—
621	C₂²	—	4	8	no	—
622	C₂²	—	4	16	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	8	4	no	—
626	C₂²	C₂×#120×C₆₇	1	4	yes	—
627	C₂²	C₂×#120×C₆₇	1	2	yes	—
628	C₂²	#575×C₆₇	1	4	yes	—
629	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
630	C₄	—	2	4	no	—
631	C₄	—	2	8	no	—
632	C₄	—	4	4	no	—
633	C₄	—	4	2	no	—
634	C₄	C₂×#120×C₆₇	1	4	yes	—
635	C₄	C₂×#120×C₆₇	1	2	yes	—
636	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
637	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
638	C₂	—	2	4	no	—
639	C₂	—	4	4	no	—
640	C₂	—	4	8	no	—
641	C₂	—	8	2	no	—
642	C₂	C₆₇×#3005	1	4	yes	direct product factor
643	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
644	C₂	C₂×#575×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×#41×C₆₇	C₂	semidirect
2. C₂²×#137×C₆₇	C₂	semidirect
3. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
4. C₂×#525×C₆₇	C₂	semidirect
5. C₆₇×#3005	C₂	direct (N × H)
6. C₂×#452×C₆₇	C₂	semidirect
7. C₂×#449×C₆₇	C₂	semidirect
8. C₂×#419×C₆₇	C₂	semidirect
9. C₂×#418×C₆₇	C₂	semidirect
10. C₂×#407×C₆₇	C₂	semidirect
17. C₂²×#41×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
26. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
27. C₂²×Q₁₆×C₆₇	C₂²	semidirect
28. C₂²×SD₁₆×C₆₇	C₂²	semidirect
29. C₂²×SD₁₆×C₆₇	C₂²	semidirect
30. C₂²×D₈×C₆₇	C₂²	semidirect
34. #525×C₆₇	C₂²	semidirect
36. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
37. C₂×#121×C₆₇	C₂²	semidirect
41. #452×C₆₇	C₂²	semidirect
42. #449×C₆₇	C₂²	semidirect
43. #419×C₆₇	C₂²	semidirect
44. #418×C₆₇	C₂²	semidirect
45. C₂×#103×C₆₇	C₂²	semidirect
46. C₂×#102×C₆₇	C₂²	semidirect
47. #407×C₆₇	C₂²	semidirect
48. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
67. C₂²×Q₈×C₆₇	D₄	semidirect
79. C₂²×D₄×C₆₇	D₄	semidirect
99. C₂×Q₁₆×C₆₇	C₂³	semidirect
102. C₂×SD₁₆×C₆₇	C₂³	semidirect
103. C₂×SD₁₆×C₆₇	C₂³	semidirect
105. C₂×D₈×C₆₇	C₂³	semidirect
114. C₂²⋊Q₈×C₆₇	C₂³	semidirect
116. #121×C₆₇	C₂³	semidirect
127. #103×C₆₇	C₂³	semidirect
128. #102×C₆₇	C₂³	semidirect
129. C₂²⋊C₈×C₆₇	C₂³	semidirect
157. C₂×Q₈×C₆₇	C₂×D₄	semidirect
177. C₂×D₄×C₆₇	C₂×D₄	semidirect
315. C₂×#3005	C₆₇	direct (N × H)
320. C₂×C₆₇	#3005	direct (N × H)
323. C₂³×#41	C₂×C₆₇	semidirect
324. C₂²×#137	C₂×C₆₇	semidirect
325. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
326. C₂×#525	C₂×C₆₇	semidirect
327. #3005	C₂×C₆₇	direct (N × H)
328. C₂×#452	C₂×C₆₇	semidirect
329. C₂×#449	C₂×C₆₇	semidirect
330. C₂×#419	C₂×C₆₇	semidirect
331. C₂×#418	C₂×C₆₇	semidirect
332. C₂×#407	C₂×C₆₇	semidirect
333. C₆₇	C₂×#3005	direct (N × H)
340. C₂²×#41	C₂²×C₆₇	semidirect
347. C₂×#137	C₂²×C₆₇	semidirect
349. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
350. C₂²×Q₁₆	C₂²×C₆₇	semidirect
351. C₂²×SD₁₆	C₂²×C₆₇	semidirect
352. C₂²×SD₁₆	C₂²×C₆₇	semidirect
353. C₂²×D₈	C₂²×C₆₇	semidirect
357. #525	C₂²×C₆₇	semidirect
359. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
360. C₂×#121	C₂²×C₆₇	semidirect
364. #452	C₂²×C₆₇	semidirect
365. #449	C₂²×C₆₇	semidirect
366. #419	C₂²×C₆₇	semidirect
367. #418	C₂²×C₆₇	semidirect
368. C₂×#103	C₂²×C₆₇	semidirect
369. C₂×#102	C₂²×C₆₇	semidirect
370. #407	C₂²×C₆₇	semidirect
371. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
390. C₂²×Q₈	D₄×C₆₇	semidirect
402. C₂²×D₄	D₄×C₆₇	semidirect
422. C₂×Q₁₆	C₂³×C₆₇	semidirect
425. C₂×SD₁₆	C₂³×C₆₇	semidirect
426. C₂×SD₁₆	C₂³×C₆₇	semidirect
428. C₂×D₈	C₂³×C₆₇	semidirect
437. C₂²⋊Q₈	C₂³×C₆₇	semidirect
439. #121	C₂³×C₆₇	semidirect
450. #103	C₂³×C₆₇	semidirect
451. #102	C₂³×C₆₇	semidirect
452. C₂²⋊C₈	C₂³×C₆₇	semidirect
480. C₂×Q₈	C₂×D₄×C₆₇	semidirect
500. C₂×D₄	C₂×D₄×C₆₇	semidirect
642. C₂	C₆₇×#3005	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#41×C₆₇	2
C₂²×#137×C₆₇	2
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#525×C₆₇	2
C₆₇×#3005	2
C₂×#452×C₆₇	2
C₂×#449×C₆₇	2
C₂×#419×C₆₇	2
C₂×#418×C₆₇	2
C₂×#407×C₆₇	2
C₂×#3005	67

Subgroups — C2×C67×#3005 #30739607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3005 #30739607