Subgroups — C₂×C₆₇×#3010 #30739613

Order 17152.

4910 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3010	C₂	1	16	yes	maximal, direct product factor
6	C₂×#449×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#448×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×2₋¹⁺⁴×C₆₇	—	2	4	no	—
12	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
13	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
14	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	8	yes	complemented
15	C₂²×#41×C₆₇	—	2	2	no	—
16	C₂²×#41×C₆₇	—	2	1	no	—
17	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
20	C₂³×Q₈×C₆₇	—	2	1	no	—
21	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂³×Q₈×C₆₇	C₂²	1	2	yes	complemented
23	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
25	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
26	C₂²×SD₁₆×C₆₇	—	2	8	no	—
27	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
28	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂²×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
30	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#535×C₆₇	—	2	4	no	—
34	#535×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#127×C₆₇	—	2	1	no	—
36	C₂×#124×C₆₇	—	2	1	no	—
37	C₂×#121×C₆₇	C₂²	1	4	yes	—
38	C₂×#120×C₆₇	—	2	2	no	—
39	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
40	#449×C₆₇	C₂²	1	32	yes	complemented
41	#448×C₆₇	C₂²	1	32	yes	complemented
42	#419×C₆₇	C₂²	1	8	yes	complemented
43	#416×C₆₇	C₂²	1	8	yes	complemented
44	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
46	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
47	#408×C₆₇	C₂²	1	8	yes	complemented
48	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
49	C₂⁵×C₆₇	—	2	1	no	—
50	2₋¹⁺⁴×C₆₇	—	2	64	no	—
51	2₋¹⁺⁴×C₆₇	—	2	32	no	—
52	2₋¹⁺⁴×C₆₇	—	2	32	no	—
53	C₂×#41×C₆₇	—	2	32	no	—
54	C₂×#41×C₆₇	—	2	16	no	—
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	2	16	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	—	2	4	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	—	2	4	no	—
64	C₂×#41×C₆₇	D₄	1	4	yes	—
65	C₂×#41×C₆₇	D₄	1	8	yes	complemented
66	C₂×#41×C₆₇	C₂³	1	4	yes	—
67	C₂×#41×C₆₇	C₂³	1	4	yes	—
68	C₂×#41×C₆₇	C₂³	1	8	yes	complemented
69	C₂²×Q₈×C₆₇	—	2	16	no	—
70	C₂²×Q₈×C₆₇	—	2	8	no	—
71	C₂²×Q₈×C₆₇	—	2	8	no	—
72	C₂²×Q₈×C₆₇	—	2	8	no	—
73	C₂²×Q₈×C₆₇	—	2	1	no	—
74	C₂²×Q₈×C₆₇	—	2	2	no	—
75	C₂²×Q₈×C₆₇	—	2	4	no	—
76	C₂²×Q₈×C₆₇	—	2	2	no	—
77	C₂²×Q₈×C₆₇	—	2	4	no	—
78	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
79	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
80	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
81	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
83	C₂²×Q₈×C₆₇	C₂³	1	8	yes	complemented
84	C₂²×D₄×C₆₇	—	2	1	no	—
85	C₂²×D₄×C₆₇	—	2	2	no	—
86	C₂²×D₄×C₆₇	—	2	4	no	—
87	C₂²×D₄×C₆₇	—	2	4	no	—
88	C₂²×D₄×C₆₇	—	2	4	no	—
89	C₂²×D₄×C₆₇	—	2	1	no	—
90	C₂²×D₄×C₆₇	—	2	2	no	—
91	C₂²×D₄×C₆₇	D₄	1	2	yes	—
92	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
93	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
94	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
95	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₆₇	—	2	1	no	—
97	C₂³×C₄×C₆₇	—	2	2	no	—
98	C₂³×C₄×C₆₇	—	2	1	no	—
99	C₂³×C₄×C₆₇	—	2	2	no	—
100	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
101	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
102	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
103	C₂×SD₁₆×C₆₇	—	2	16	no	—
104	C₂×SD₁₆×C₆₇	—	2	8	no	—
105	C₂×SD₁₆×C₆₇	—	2	32	no	—
106	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
107	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
108	C₂×D₈×C₆₇	—	2	8	no	—
109	C₂×D₈×C₆₇	C₂³	1	8	yes	—
110	C₂×M₁₆×C₆₇	—	2	4	no	—
111	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
112	C₂²×C₈×C₆₇	—	2	4	no	—
113	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
114	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
115	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
116	#127×C₆₇	—	2	8	no	—
117	#124×C₆₇	—	2	8	no	—
118	#121×C₆₇	—	2	8	no	—
119	#121×C₆₇	C₂³	1	16	yes	—
120	#120×C₆₇	—	2	16	no	—
121	#117×C₆₇	—	2	2	no	—
122	#117×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂×C₄²×C₆₇	—	2	1	no	—
127	#103×C₆₇	C₂³	1	8	yes	complemented
128	#103×C₆₇	C₂³	1	16	yes	complemented
129	#102×C₆₇	C₂³	1	8	yes	complemented
130	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
131	C₂⁴×C₆₇	—	2	1	no	—
132	C₂⁴×C₆₇	—	2	2	no	—
133	C₂⁴×C₆₇	—	2	4	no	—
134	C₂⁴×C₆₇	—	4	4	no	—
135	C₂⁴×C₆₇	—	8	4	no	—
136	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
137	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
138	#41×C₆₇	—	2	128	no	—
139	#41×C₆₇	—	2	64	no	—
140	#41×C₆₇	—	2	32	no	—
141	#41×C₆₇	—	2	16	no	—
142	#41×C₆₇	—	2	16	no	—
143	#41×C₆₇	—	2	16	no	—
144	#41×C₆₇	—	2	16	no	—
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	8	no	—
147	#41×C₆₇	—	2	8	no	—
148	#41×C₆₇	C₂×D₄	1	16	yes	complemented
149	C₂×Q₈×C₆₇	—	2	16	no	—
150	C₂×Q₈×C₆₇	—	2	64	no	—
151	C₂×Q₈×C₆₇	—	2	32	no	—
152	C₂×Q₈×C₆₇	—	2	16	no	—
153	C₂×Q₈×C₆₇	—	2	16	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	8	no	—
156	C₂×Q₈×C₆₇	—	2	16	no	—
157	C₂×Q₈×C₆₇	—	2	2	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	1	no	—
162	C₂×Q₈×C₆₇	—	2	8	no	—
163	C₂×Q₈×C₆₇	—	2	8	no	—
164	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
165	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
166	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
167	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	complemented
168	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
169	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
170	C₂×D₄×C₆₇	—	2	16	no	—
171	C₂×D₄×C₆₇	—	2	8	no	—
172	C₂×D₄×C₆₇	—	2	8	no	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	2	8	no	—
175	C₂×D₄×C₆₇	—	2	16	no	—
176	C₂×D₄×C₆₇	—	2	4	no	—
177	C₂×D₄×C₆₇	—	2	1	no	—
178	C₂×D₄×C₆₇	—	2	2	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	8	no	—
182	C₂×D₄×C₆₇	—	2	8	no	—
183	C₂×D₄×C₆₇	—	2	2	no	—
184	C₂×D₄×C₆₇	—	2	4	no	—
185	C₂×D₄×C₆₇	—	4	4	no	—
186	C₂×D₄×C₆₇	—	4	8	no	—
187	C₂×D₄×C₆₇	—	8	8	no	—
188	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
189	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
191	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₆₇	—	2	16	no	—
194	C₂²×C₄×C₆₇	—	2	16	no	—
195	C₂²×C₄×C₆₇	—	2	8	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	8	no	—
198	C₂²×C₄×C₆₇	—	2	2	no	—
199	C₂²×C₄×C₆₇	—	2	1	no	—
200	C₂²×C₄×C₆₇	—	2	8	no	—
201	C₂²×C₄×C₆₇	—	2	4	no	—
202	C₂²×C₄×C₆₇	—	2	8	no	—
203	C₂²×C₄×C₆₇	—	2	4	no	—
204	C₂²×C₄×C₆₇	—	2	4	no	—
205	C₂²×C₄×C₆₇	—	2	2	no	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
208	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
211	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
213	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
215	SD₁₆×C₆₇	—	2	32	no	—
216	SD₁₆×C₆₇	—	2	16	no	—
217	D₈×C₆₇	—	2	16	no	—
218	M₁₆×C₆₇	—	2	8	no	—
219	C₂×C₈×C₆₇	—	2	4	no	—
220	C₂×C₈×C₆₇	—	2	8	no	—
221	C₂×C₈×C₆₇	—	2	4	no	—
222	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
223	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
224	C₄⋊C₄×C₆₇	—	2	2	no	—
225	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
226	C₂²⋊C₄×C₆₇	—	2	4	no	—
227	C₂²⋊C₄×C₆₇	—	2	16	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	4	no	—
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	2	8	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	2	4	no	—
235	C₂³×C₆₇	—	4	4	no	—
236	C₂³×C₆₇	—	4	8	no	—
237	C₂³×C₆₇	—	4	4	no	—
238	C₂³×C₆₇	—	4	8	no	—
239	C₂³×C₆₇	—	8	8	no	—
240	C₂³×C₆₇	—	8	16	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	32	no	—
246	Q₈×C₆₇	—	2	4	no	—
247	Q₈×C₆₇	—	2	8	no	—
248	Q₈×C₆₇	—	2	8	no	—
249	Q₈×C₆₇	—	2	16	no	—
250	Q₈×C₆₇	—	2	2	no	—
251	Q₈×C₆₇	—	2	4	no	—
252	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
253	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
254	D₄×C₆₇	—	2	32	no	—
255	D₄×C₆₇	—	2	16	no	—
256	D₄×C₆₇	—	2	16	no	—
257	D₄×C₆₇	—	2	8	no	—
258	D₄×C₆₇	—	2	4	no	—
259	D₄×C₆₇	—	2	2	no	—
260	D₄×C₆₇	—	2	4	no	—
261	D₄×C₆₇	—	2	4	no	—
262	D₄×C₆₇	—	4	4	no	—
263	D₄×C₆₇	—	4	8	no	—
264	D₄×C₆₇	—	8	16	no	—
265	D₄×C₆₇	C₂²×D₄	1	4	yes	—
266	C₂×C₄×C₆₇	—	2	32	no	—
267	C₂×C₄×C₆₇	—	2	32	no	—
268	C₂×C₄×C₆₇	—	2	8	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	16	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	16	no	—
275	C₂×C₄×C₆₇	—	2	8	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	4	4	no	—
280	C₂×C₄×C₆₇	#120	1	4	yes	—
281	C₂×C₄×C₆₇	#120	1	8	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
290	C₈×C₆₇	—	2	4	no	—
291	C₈×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	—	2	4	no	—
296	C₂²×C₆₇	—	2	8	no	—
297	C₂²×C₆₇	—	4	8	no	—
298	C₂²×C₆₇	—	4	8	no	—
299	C₂²×C₆₇	—	4	4	no	—
300	C₂²×C₆₇	—	4	4	no	—
301	C₂²×C₆₇	—	4	8	no	—
302	C₂²×C₆₇	—	8	16	no	—
303	C₂²×C₆₇	—	8	8	no	—
304	C₂²×C₆₇	C₂×#120	1	4	yes	—
305	C₂²×C₆₇	C₂×#120	1	2	yes	—
306	C₂²×C₆₇	#577	1	4	yes	—
307	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
308	C₄×C₆₇	—	2	8	no	—
309	C₄×C₆₇	—	2	4	no	—
310	C₄×C₆₇	—	2	4	no	—
311	C₄×C₆₇	—	4	4	no	—
312	C₄×C₆₇	C₂×#120	1	4	yes	—
313	C₄×C₆₇	C₂×#120	1	2	yes	—
314	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
315	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
316	C₂×#3010	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
317	C₂×C₆₇	—	2	4	no	—
318	C₂×C₆₇	—	4	4	no	—
319	C₂×C₆₇	—	4	4	no	—
320	C₂×C₆₇	—	8	4	no	—
321	C₂×C₆₇	#3010	1	4	yes	direct product factor
322	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
323	C₂×C₆₇	C₂×#577	1	2	yes	—
324	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic, complemented
326	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
327	C₂×#535	C₂×C₆₇	1	1	yes	characteristic, complemented
328	#3010	C₂×C₆₇	1	16	yes	direct product factor
329	C₂×#449	C₂×C₆₇	1	4	yes	complemented
330	C₂×#448	C₂×C₆₇	1	4	yes	complemented
331	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
333	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
334	C₆₇	C₂×#3010	1	1	yes	characteristic, direct product factor, Sylow
335	C₂×2₋¹⁺⁴	—	2	4	no	—
336	C₂×2₋¹⁺⁴	—	2	8	no	—
337	C₂×2₋¹⁺⁴	—	2	8	no	—
338	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	8	yes	complemented
339	C₂²×#41	—	2	2	no	—
340	C₂²×#41	—	2	1	no	—
341	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
342	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
343	C₂²×#41	C₂²×C₆₇	1	2	yes	complemented
344	C₂³×Q₈	—	2	1	no	—
345	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
346	C₂³×Q₈	C₂²×C₆₇	1	2	yes	complemented
347	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×AGL(1,ℤ/8)	—	2	8	no	—
349	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
350	C₂²×SD₁₆	—	2	8	no	—
351	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
352	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
353	C₂²×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
354	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
355	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
356	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
357	#535	—	2	4	no	—
358	#535	C₂²×C₆₇	1	8	yes	complemented
359	C₂×#127	—	2	1	no	—
360	C₂×#124	—	2	1	no	—
361	C₂×#121	C₂²×C₆₇	1	4	yes	—
362	C₂×#120	—	2	2	no	—
363	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
364	#449	C₂²×C₆₇	1	32	yes	complemented
365	#448	C₂²×C₆₇	1	32	yes	complemented
366	#419	C₂²×C₆₇	1	8	yes	complemented
367	#416	C₂²×C₆₇	1	8	yes	complemented
368	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
369	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
370	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
371	#408	C₂²×C₆₇	1	8	yes	complemented
372	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
373	C₂⁵	—	2	1	no	—
374	2₋¹⁺⁴	—	2	64	no	—
375	2₋¹⁺⁴	—	2	32	no	—
376	2₋¹⁺⁴	—	2	32	no	—
377	C₂×#41	—	2	32	no	—
378	C₂×#41	—	2	16	no	—
379	C₂×#41	—	2	8	no	—
380	C₂×#41	—	2	16	no	—
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	2	8	no	—
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	2	no	—
385	C₂×#41	—	2	4	no	—
386	C₂×#41	—	2	4	no	—
387	C₂×#41	—	2	4	no	—
388	C₂×#41	D₄×C₆₇	1	4	yes	—
389	C₂×#41	D₄×C₆₇	1	8	yes	complemented
390	C₂×#41	C₂³×C₆₇	1	4	yes	—
391	C₂×#41	C₂³×C₆₇	1	4	yes	—
392	C₂×#41	C₂³×C₆₇	1	8	yes	complemented
393	C₂²×Q₈	—	2	16	no	—
394	C₂²×Q₈	—	2	8	no	—
395	C₂²×Q₈	—	2	8	no	—
396	C₂²×Q₈	—	2	8	no	—
397	C₂²×Q₈	—	2	1	no	—
398	C₂²×Q₈	—	2	2	no	—
399	C₂²×Q₈	—	2	4	no	—
400	C₂²×Q₈	—	2	2	no	—
401	C₂²×Q₈	—	2	4	no	—
402	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
403	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
404	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
405	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
406	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
407	C₂²×Q₈	C₂³×C₆₇	1	8	yes	complemented
408	C₂²×D₄	—	2	1	no	—
409	C₂²×D₄	—	2	2	no	—
410	C₂²×D₄	—	2	4	no	—
411	C₂²×D₄	—	2	4	no	—
412	C₂²×D₄	—	2	4	no	—
413	C₂²×D₄	—	2	1	no	—
414	C₂²×D₄	—	2	2	no	—
415	C₂²×D₄	D₄×C₆₇	1	2	yes	—
416	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
418	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
419	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	—	2	2	no	—
422	C₂³×C₄	—	2	1	no	—
423	C₂³×C₄	—	2	2	no	—
424	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
425	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
426	AGL(1,ℤ/8)	—	2	32	no	—
427	C₂×SD₁₆	—	2	16	no	—
428	C₂×SD₁₆	—	2	8	no	—
429	C₂×SD₁₆	—	2	32	no	—
430	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
431	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
432	C₂×D₈	—	2	8	no	—
433	C₂×D₈	C₂³×C₆₇	1	8	yes	—
434	C₂×M₁₆	—	2	4	no	—
435	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
436	C₂²×C₈	—	2	4	no	—
437	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
438	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
439	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
440	#127	—	2	8	no	—
441	#124	—	2	8	no	—
442	#121	—	2	8	no	—
443	#121	C₂³×C₆₇	1	16	yes	—
444	#120	—	2	16	no	—
445	#117	—	2	2	no	—
446	#117	C₂³×C₆₇	1	4	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	4	no	—
450	C₂×C₄²	—	2	1	no	—
451	#103	C₂³×C₆₇	1	8	yes	complemented
452	#103	C₂³×C₆₇	1	16	yes	complemented
453	#102	C₂³×C₆₇	1	8	yes	complemented
454	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
455	C₂⁴	—	2	1	no	—
456	C₂⁴	—	2	2	no	—
457	C₂⁴	—	2	4	no	—
458	C₂⁴	—	4	4	no	—
459	C₂⁴	—	8	4	no	—
460	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
461	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
462	#41	—	2	128	no	—
463	#41	—	2	64	no	—
464	#41	—	2	32	no	—
465	#41	—	2	16	no	—
466	#41	—	2	16	no	—
467	#41	—	2	16	no	—
468	#41	—	2	16	no	—
469	#41	—	2	8	no	—
470	#41	—	2	8	no	—
471	#41	—	2	8	no	—
472	#41	C₂×D₄×C₆₇	1	16	yes	complemented
473	C₂×Q₈	—	2	16	no	—
474	C₂×Q₈	—	2	64	no	—
475	C₂×Q₈	—	2	32	no	—
476	C₂×Q₈	—	2	16	no	—
477	C₂×Q₈	—	2	16	no	—
478	C₂×Q₈	—	2	4	no	—
479	C₂×Q₈	—	2	8	no	—
480	C₂×Q₈	—	2	16	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	8	no	—
487	C₂×Q₈	—	2	8	no	—
488	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
489	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
490	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
491	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	complemented
492	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
493	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
494	C₂×D₄	—	2	16	no	—
495	C₂×D₄	—	2	8	no	—
496	C₂×D₄	—	2	8	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	8	no	—
499	C₂×D₄	—	2	16	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	4	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	8	no	—
506	C₂×D₄	—	2	8	no	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	4	4	no	—
510	C₂×D₄	—	4	8	no	—
511	C₂×D₄	—	8	8	no	—
512	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
513	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
514	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
515	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
516	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
517	C₂²×C₄	—	2	16	no	—
518	C₂²×C₄	—	2	16	no	—
519	C₂²×C₄	—	2	8	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	8	no	—
524	C₂²×C₄	—	2	8	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	8	no	—
527	C₂²×C₄	—	2	4	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
531	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
532	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
539	SD₁₆	—	2	32	no	—
540	SD₁₆	—	2	16	no	—
541	D₈	—	2	16	no	—
542	M₁₆	—	2	8	no	—
543	C₂×C₈	—	2	4	no	—
544	C₂×C₈	—	2	8	no	—
545	C₂×C₈	—	2	4	no	—
546	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
547	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	16	no	—
552	C₄²	—	2	4	no	—
553	C₂³	—	2	4	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	8	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	4	4	no	—
560	C₂³	—	4	8	no	—
561	C₂³	—	4	4	no	—
562	C₂³	—	4	8	no	—
563	C₂³	—	8	8	no	—
564	C₂³	—	8	16	no	—
565	C₂³	#120×C₆₇	1	4	yes	—
566	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
567	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
568	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
569	Q₈	—	2	32	no	—
570	Q₈	—	2	4	no	—
571	Q₈	—	2	8	no	—
572	Q₈	—	2	8	no	—
573	Q₈	—	2	16	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	2	4	no	—
576	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
577	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
578	D₄	—	2	32	no	—
579	D₄	—	2	16	no	—
580	D₄	—	2	16	no	—
581	D₄	—	2	8	no	—
582	D₄	—	2	4	no	—
583	D₄	—	2	2	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	4	no	—
586	D₄	—	4	4	no	—
587	D₄	—	4	8	no	—
588	D₄	—	8	16	no	—
589	D₄	C₂²×D₄×C₆₇	1	4	yes	—
590	C₂×C₄	—	2	32	no	—
591	C₂×C₄	—	2	32	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	16	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	16	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	#120×C₆₇	1	4	yes	—
605	C₂×C₄	#120×C₆₇	1	8	yes	—
606	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
611	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
613	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
614	C₈	—	2	4	no	—
615	C₈	—	2	4	no	—
616	C₂²	—	2	8	no	—
617	C₂²	—	2	4	no	—
618	C₂²	—	2	8	no	—
619	C₂²	—	2	4	no	—
620	C₂²	—	2	4	no	—
621	C₂²	—	4	8	no	—
622	C₂²	—	4	8	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	8	no	—
625	C₂²	—	4	4	no	—
626	C₂²	—	8	16	no	—
627	C₂²	—	8	8	no	—
628	C₂²	C₂×#120×C₆₇	1	4	yes	—
629	C₂²	C₂×#120×C₆₇	1	2	yes	—
630	C₂²	#577×C₆₇	1	4	yes	—
631	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
632	C₄	—	2	8	no	—
633	C₄	—	2	4	no	—
634	C₄	—	2	4	no	—
635	C₄	—	4	4	no	—
636	C₄	C₂×#120×C₆₇	1	4	yes	—
637	C₄	C₂×#120×C₆₇	1	2	yes	—
638	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
639	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
640	C₂	—	2	4	no	—
641	C₂	—	4	4	no	—
642	C₂	—	4	4	no	—
643	C₂	—	8	4	no	—
644	C₂	C₆₇×#3010	1	4	yes	direct product factor
645	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
646	C₂	C₂×#577×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 ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
3. C₂³×SD₁₆×C₆₇	C₂	semidirect
4. C₂×#535×C₆₇	C₂	semidirect
5. C₆₇×#3010	C₂	direct (N × H)
6. C₂×#449×C₆₇	C₂	semidirect
7. C₂×#448×C₆₇	C₂	semidirect
8. C₂×#419×C₆₇	C₂	semidirect
9. C₂×#416×C₆₇	C₂	semidirect
10. C₂×#408×C₆₇	C₂	semidirect
14. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
19. C₂²×#41×C₆₇	C₂²	semidirect
22. C₂³×Q₈×C₆₇	C₂²	semidirect
25. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
29. C₂²×SD₁₆×C₆₇	C₂²	semidirect
34. #535×C₆₇	C₂²	semidirect
40. #449×C₆₇	C₂²	semidirect
41. #448×C₆₇	C₂²	semidirect
42. #419×C₆₇	C₂²	semidirect
43. #416×C₆₇	C₂²	semidirect
44. C₂×#103×C₆₇	C₂²	semidirect
45. C₂×#103×C₆₇	C₂²	semidirect
46. C₂×#102×C₆₇	C₂²	semidirect
47. #408×C₆₇	C₂²	semidirect
48. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
65. C₂×#41×C₆₇	D₄	semidirect
68. C₂×#41×C₆₇	C₂³	semidirect
79. C₂²×Q₈×C₆₇	D₄	semidirect
83. C₂²×Q₈×C₆₇	C₂³	semidirect
127. #103×C₆₇	C₂³	semidirect
128. #103×C₆₇	C₂³	semidirect
129. #102×C₆₇	C₂³	semidirect
130. C₂²⋊C₈×C₆₇	C₂³	semidirect
148. #41×C₆₇	C₂×D₄	semidirect
167. C₂×Q₈×C₆₇	C₂×D₄	semidirect
316. C₂×#3010	C₆₇	direct (N × H)
321. C₂×C₆₇	#3010	direct (N × H)
324. C₂²×2₋¹⁺⁴	C₂×C₆₇	semidirect
325. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
326. C₂³×SD₁₆	C₂×C₆₇	semidirect
327. C₂×#535	C₂×C₆₇	semidirect
328. #3010	C₂×C₆₇	direct (N × H)
329. C₂×#449	C₂×C₆₇	semidirect
330. C₂×#448	C₂×C₆₇	semidirect
331. C₂×#419	C₂×C₆₇	semidirect
332. C₂×#416	C₂×C₆₇	semidirect
333. C₂×#408	C₂×C₆₇	semidirect
334. C₆₇	C₂×#3010	direct (N × H)
338. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
343. C₂²×#41	C₂²×C₆₇	semidirect
346. C₂³×Q₈	C₂²×C₆₇	semidirect
349. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
353. C₂²×SD₁₆	C₂²×C₆₇	semidirect
358. #535	C₂²×C₆₇	semidirect
364. #449	C₂²×C₆₇	semidirect
365. #448	C₂²×C₆₇	semidirect
366. #419	C₂²×C₆₇	semidirect
367. #416	C₂²×C₆₇	semidirect
368. C₂×#103	C₂²×C₆₇	semidirect
369. C₂×#103	C₂²×C₆₇	semidirect
370. C₂×#102	C₂²×C₆₇	semidirect
371. #408	C₂²×C₆₇	semidirect
372. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
389. C₂×#41	D₄×C₆₇	semidirect
392. C₂×#41	C₂³×C₆₇	semidirect
403. C₂²×Q₈	D₄×C₆₇	semidirect
407. C₂²×Q₈	C₂³×C₆₇	semidirect
451. #103	C₂³×C₆₇	semidirect
452. #103	C₂³×C₆₇	semidirect
453. #102	C₂³×C₆₇	semidirect
454. C₂²⋊C₈	C₂³×C₆₇	semidirect
472. #41	C₂×D₄×C₆₇	semidirect
491. C₂×Q₈	C₂×D₄×C₆₇	semidirect
644. C₂	C₆₇×#3010	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂³×SD₁₆×C₆₇	2
C₂×#535×C₆₇	2
C₆₇×#3010	2
C₂×#449×C₆₇	2
C₂×#448×C₆₇	2
C₂×#419×C₆₇	2
C₂×#416×C₆₇	2
C₂×#408×C₆₇	2
C₂×#3010	67

Subgroups — C2×C67×#3010 #30739613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3010 #30739613