Subgroups — C₂×C₆₇×#3332 #30740403

Order 17152.

2426 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3332	C₂	1	16	yes	maximal, direct product factor
4	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#469×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#449×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#417×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂²×#41×C₆₇	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
18	#550×C₆₇	—	2	4	no	—
19	#550×C₆₇	C₂²	1	8	yes	complemented
20	#547×C₆₇	—	2	4	no	—
21	#547×C₆₇	C₂²	1	8	yes	complemented
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#127×C₆₇	—	2	1	no	—
24	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#123×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
27	C₂×#121×C₆₇	—	2	2	no	—
28	C₂×#121×C₆₇	C₂²	1	2	yes	—
29	C₂×#121×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	—	2	1	no	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	#486×C₆₇	C₂²	1	8	yes	complemented
36	#481×C₆₇	C₂²	1	16	yes	complemented
37	#475×C₆₇	C₂²	1	8	yes	complemented
38	#469×C₆₇	C₂²	1	16	yes	complemented
39	#459×C₆₇	C₂²	1	8	yes	complemented
40	#449×C₆₇	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	#436×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#106×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#417×C₆₇	C₂²	1	16	yes	complemented
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	4	4	no	—
59	C₂×#41×C₆₇	—	4	8	no	—
60	C₂×#41×C₆₇	C₂³	1	8	yes	—
61	C₂²×Q₈×C₆₇	—	2	1	no	—
62	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	—	2	2	no	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	1	no	—
67	C₂²×D₄×C₆₇	—	2	1	no	—
68	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	—	2	2	no	—
72	C₂³×C₄×C₆₇	—	2	2	no	—
73	C₂³×C₄×C₆₇	—	2	1	no	—
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₆₇	—	2	4	no	—
76	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂×D₈×C₆₇	—	4	4	no	—
78	C₂×D₈×C₆₇	C₂³	1	8	yes	—
79	C₂×M₁₆×C₆₇	—	4	4	no	—
80	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₇	—	2	2	no	—
85	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
86	#127×C₆₇	—	2	8	no	—
87	#127×C₆₇	—	2	2	no	—
88	#127×C₆₇	C₂³	1	4	yes	—
89	#123×C₆₇	—	2	16	no	—
90	C₂²⋊Q₈×C₆₇	—	2	8	no	—
91	#121×C₆₇	—	2	4	no	—
92	#121×C₆₇	—	2	4	no	—
93	#121×C₆₇	—	2	16	no	—
94	#121×C₆₇	C₂³	1	8	yes	—
95	#121×C₆₇	C₂³	1	8	yes	—
96	C₄×Q₈×C₆₇	—	2	2	no	—
97	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
98	C₄×D₄×C₆₇	—	2	8	no	—
99	C₄×D₄×C₆₇	—	2	2	no	—
100	C₄×D₄×C₆₇	D₄	1	4	yes	—
101	C₄×D₄×C₆₇	C₂³	1	4	yes	—
102	C₄×D₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
116	C₂×C₄²×C₆₇	—	2	1	no	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	#106×C₆₇	C₂³	1	8	yes	—
119	#106×C₆₇	C₂³	1	4	yes	—
120	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
121	#103×C₆₇	C₂³	1	8	yes	—
122	#103×C₆₇	C₂³	1	4	yes	—
123	#102×C₆₇	C₂³	1	8	yes	—
124	#102×C₆₇	C₂³	1	8	yes	—
125	#102×C₆₇	C₂³	1	4	yes	—
126	#102×C₆₇	C₂³	1	8	yes	—
127	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
128	C₄×C₈×C₆₇	C₂³	1	4	yes	—
129	C₂⁴×C₆₇	—	2	1	no	—
130	C₂⁴×C₆₇	—	2	2	no	—
131	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
132	#41×C₆₇	—	2	16	no	—
133	#41×C₆₇	—	4	8	no	—
134	#41×C₆₇	—	4	16	no	—
135	#41×C₆₇	—	4	32	no	—
136	C₂×Q₈×C₆₇	—	2	1	no	—
137	C₂×Q₈×C₆₇	—	2	4	no	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	—	4	4	no	—
140	C₂×Q₈×C₆₇	#41	1	4	yes	complemented
141	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
142	C₂×D₄×C₆₇	—	2	8	no	—
143	C₂×D₄×C₆₇	—	2	8	no	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	1	no	—
147	C₂×D₄×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	4	8	no	—
155	C₂×D₄×C₆₇	—	4	8	no	—
156	C₂×D₄×C₆₇	—	4	4	no	—
157	C₂×D₄×C₆₇	—	8	2	no	—
158	C₂×D₄×C₆₇	—	8	4	no	—
159	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
160	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
161	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	8	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	8	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	4	8	no	—
172	C₂²×C₄×C₆₇	—	4	8	no	—
173	C₂²×C₄×C₆₇	—	4	4	no	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	SD₁₆×C₆₇	—	2	8	no	—
181	D₈×C₆₇	—	4	8	no	—
182	M₁₆×C₆₇	—	4	8	no	—
183	C₂×C₈×C₆₇	—	2	2	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	2	4	no	—
186	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
187	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
188	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
189	C₄⋊C₄×C₆₇	—	2	1	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	8	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	1	no	—
195	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
203	C₄²×C₆₇	—	2	4	no	—
204	C₄²×C₆₇	C₂×D₄	1	2	yes	—
205	C₄²×C₆₇	C₂⁴	1	2	yes	—
206	C₂³×C₆₇	—	2	4	no	—
207	C₂³×C₆₇	—	2	4	no	—
208	C₂³×C₆₇	—	2	4	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	4	4	no	—
212	C₂³×C₆₇	—	4	2	no	—
213	C₂³×C₆₇	—	4	2	no	—
214	C₂³×C₆₇	—	4	4	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
218	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	4	no	—
220	Q₈×C₆₇	—	2	2	no	—
221	Q₈×C₆₇	—	4	8	no	—
222	Q₈×C₆₇	C₂×#41	1	4	yes	complemented
223	D₄×C₆₇	—	2	8	no	—
224	D₄×C₆₇	—	2	2	no	—
225	D₄×C₆₇	—	2	4	no	—
226	D₄×C₆₇	—	2	2	no	—
227	D₄×C₆₇	—	2	4	no	—
228	D₄×C₆₇	—	4	16	no	—
229	D₄×C₆₇	—	4	16	no	—
230	D₄×C₆₇	—	4	2	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	8	4	no	—
234	D₄×C₆₇	—	8	8	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	8	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	4	16	no	—
246	C₂×C₄×C₆₇	—	4	16	no	—
247	C₂×C₄×C₆₇	—	4	2	no	—
248	C₂×C₄×C₆₇	—	4	4	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
257	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
258	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
259	C₈×C₆₇	—	2	2	no	—
260	C₈×C₆₇	—	2	2	no	—
261	C₈×C₆₇	—	2	4	no	—
262	C₂²×C₆₇	—	2	4	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	4	no	—
265	C₂²×C₆₇	—	4	8	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	4	2	no	—
269	C₂²×C₆₇	—	4	4	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	8	8	no	—
273	C₂²×C₆₇	—	8	4	no	—
274	C₂²×C₆₇	#546	1	2	yes	—
275	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
276	C₂²×C₆₇	#577	1	2	yes	—
277	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
278	C₄×C₆₇	—	2	4	no	—
279	C₄×C₆₇	—	2	4	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	4	2	no	—
282	C₄×C₆₇	—	4	4	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	#546	1	2	yes	—
285	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
286	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
287	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
288	C₂×#3332	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	4	4	no	—
290	C₂×C₆₇	—	8	4	no	—
291	C₂×C₆₇	—	8	2	no	—
292	C₂×C₆₇	#3332	1	4	yes	direct product factor
293	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
294	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
296	C₂×#550	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#547	C₂×C₆₇	1	1	yes	characteristic, complemented
298	#3332	C₂×C₆₇	1	16	yes	direct product factor
299	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#481	C₂×C₆₇	1	2	yes	complemented
301	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
302	C₂×#469	C₂×C₆₇	1	2	yes	complemented
303	C₂×#459	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₂×#449	C₂×C₆₇	1	2	yes	complemented
305	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₂×#417	C₂×C₆₇	1	2	yes	complemented
308	C₆₇	C₂×#3332	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²×#41	—	2	1	no	—
310	C₂²×#41	C₂²×C₆₇	1	2	yes	—
311	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
313	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
314	#550	—	2	4	no	—
315	#550	C₂²×C₆₇	1	8	yes	complemented
316	#547	—	2	4	no	—
317	#547	C₂²×C₆₇	1	8	yes	complemented
318	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#127	—	2	1	no	—
320	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#123	—	2	2	no	—
322	C₂×C₂²⋊Q₈	—	2	1	no	—
323	C₂×#121	—	2	2	no	—
324	C₂×#121	C₂²×C₆₇	1	2	yes	—
325	C₂×#121	C₂²×C₆₇	1	2	yes	—
326	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×C₄×D₄	—	2	1	no	—
328	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
330	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
331	#486	C₂²×C₆₇	1	8	yes	complemented
332	#481	C₂²×C₆₇	1	16	yes	complemented
333	#475	C₂²×C₆₇	1	8	yes	complemented
334	#469	C₂²×C₆₇	1	16	yes	complemented
335	#459	C₂²×C₆₇	1	8	yes	complemented
336	#449	C₂²×C₆₇	1	16	yes	complemented
337	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
338	#436	C₂²×C₆₇	1	8	yes	complemented
339	C₂×#106	C₂²×C₆₇	1	2	yes	—
340	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
342	#417	C₂²×C₆₇	1	16	yes	complemented
343	C₂×#103	C₂²×C₆₇	1	2	yes	—
344	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×#102	C₂²×C₆₇	1	2	yes	—
346	C₂×#102	C₂²×C₆₇	1	2	yes	—
347	C₂×#102	C₂²×C₆₇	1	2	yes	—
348	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
350	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×#41	—	2	8	no	—
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	8	no	—
354	C₂×#41	—	4	4	no	—
355	C₂×#41	—	4	8	no	—
356	C₂×#41	C₂³×C₆₇	1	8	yes	—
357	C₂²×Q₈	—	2	1	no	—
358	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	2	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
365	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
367	C₂³×C₄	—	2	2	no	—
368	C₂³×C₄	—	2	2	no	—
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
371	C₂×SD₁₆	—	2	4	no	—
372	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
373	C₂×D₈	—	4	4	no	—
374	C₂×D₈	C₂³×C₆₇	1	8	yes	—
375	C₂×M₁₆	—	4	4	no	—
376	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
379	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
382	#127	—	2	8	no	—
383	#127	—	2	2	no	—
384	#127	C₂³×C₆₇	1	4	yes	—
385	#123	—	2	16	no	—
386	C₂²⋊Q₈	—	2	8	no	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	16	no	—
390	#121	C₂³×C₆₇	1	8	yes	—
391	#121	C₂³×C₆₇	1	8	yes	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	D₄×C₆₇	1	4	yes	—
397	C₄×D₄	C₂³×C₆₇	1	4	yes	—
398	C₄×D₄	C₂³×C₆₇	1	4	yes	—
399	C₂×C₄⋊C₄	—	2	2	no	—
400	C₂×C₄⋊C₄	—	2	4	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
414	#106	C₂³×C₆₇	1	4	yes	—
415	#106	C₂³×C₆₇	1	8	yes	—
416	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
417	#103	C₂³×C₆₇	1	4	yes	—
418	#103	C₂³×C₆₇	1	8	yes	—
419	#102	C₂³×C₆₇	1	8	yes	—
420	#102	C₂³×C₆₇	1	8	yes	—
421	#102	C₂³×C₆₇	1	4	yes	—
422	#102	C₂³×C₆₇	1	8	yes	—
423	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
424	C₄×C₈	C₂³×C₆₇	1	4	yes	—
425	C₂⁴	—	2	1	no	—
426	C₂⁴	—	2	2	no	—
427	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
428	#41	—	2	16	no	—
429	#41	—	4	8	no	—
430	#41	—	4	16	no	—
431	#41	—	4	32	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	#41×C₆₇	1	4	yes	complemented
437	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	4	8	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	8	2	no	—
454	C₂×D₄	—	8	4	no	—
455	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
456	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
457	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	4	8	no	—
468	C₂²×C₄	—	4	8	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
473	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
476	SD₁₆	—	2	8	no	—
477	D₈	—	4	8	no	—
478	M₁₆	—	4	8	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
483	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
484	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
492	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
493	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
494	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
499	C₄²	—	2	4	no	—
500	C₄²	C₂×D₄×C₆₇	1	2	yes	—
501	C₄²	C₂⁴×C₆₇	1	2	yes	—
502	C₂³	—	2	4	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	4	4	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	4	no	—
512	C₂³	—	4	2	no	—
513	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
514	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
515	Q₈	—	2	4	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	4	8	no	—
518	Q₈	C₂×#41×C₆₇	1	4	yes	complemented
519	D₄	—	2	8	no	—
520	D₄	—	2	2	no	—
521	D₄	—	2	4	no	—
522	D₄	—	2	2	no	—
523	D₄	—	2	4	no	—
524	D₄	—	4	16	no	—
525	D₄	—	4	16	no	—
526	D₄	—	4	4	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	8	no	—
529	D₄	—	8	4	no	—
530	D₄	—	8	8	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	4	16	no	—
542	C₂×C₄	—	4	16	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	8	no	—
547	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
548	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
552	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
553	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
554	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
555	C₈	—	2	2	no	—
556	C₈	—	2	2	no	—
557	C₈	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	4	8	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	8	8	no	—
569	C₂²	—	8	4	no	—
570	C₂²	#546×C₆₇	1	2	yes	—
571	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
572	C₂²	#577×C₆₇	1	2	yes	—
573	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	4	no	—
576	C₄	—	2	4	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	4	no	—
579	C₄	—	4	2	no	—
580	C₄	#546×C₆₇	1	2	yes	—
581	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
582	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
583	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
584	C₂	—	4	4	no	—
585	C₂	—	8	2	no	—
586	C₂	—	8	4	no	—
587	C₂	C₆₇×#3332	1	4	yes	direct product factor
588	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
589	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
590	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	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 ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#550×C₆₇	C₂	semidirect
2. C₂×#547×C₆₇	C₂	semidirect
3. C₆₇×#3332	C₂	direct (N × H)
4. C₂×#486×C₆₇	C₂	semidirect
5. C₂×#481×C₆₇	C₂	semidirect
6. C₂×#475×C₆₇	C₂	semidirect
7. C₂×#469×C₆₇	C₂	semidirect
8. C₂×#459×C₆₇	C₂	semidirect
9. C₂×#449×C₆₇	C₂	semidirect
10. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
11. C₂×#436×C₆₇	C₂	semidirect
12. C₂×#417×C₆₇	C₂	semidirect
19. #550×C₆₇	C₂²	semidirect
21. #547×C₆₇	C₂²	semidirect
35. #486×C₆₇	C₂²	semidirect
36. #481×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
38. #469×C₆₇	C₂²	semidirect
39. #459×C₆₇	C₂²	semidirect
40. #449×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
46. #417×C₆₇	C₂²	semidirect
140. C₂×Q₈×C₆₇	#41	semidirect
222. Q₈×C₆₇	C₂×#41	semidirect
288. C₂×#3332	C₆₇	direct (N × H)
292. C₂×C₆₇	#3332	direct (N × H)
296. C₂×#550	C₂×C₆₇	semidirect
297. C₂×#547	C₂×C₆₇	semidirect
298. #3332	C₂×C₆₇	direct (N × H)
299. C₂×#486	C₂×C₆₇	semidirect
300. C₂×#481	C₂×C₆₇	semidirect
301. C₂×#475	C₂×C₆₇	semidirect
302. C₂×#469	C₂×C₆₇	semidirect
303. C₂×#459	C₂×C₆₇	semidirect
304. C₂×#449	C₂×C₆₇	semidirect
305. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
306. C₂×#436	C₂×C₆₇	semidirect
307. C₂×#417	C₂×C₆₇	semidirect
308. C₆₇	C₂×#3332	direct (N × H)
315. #550	C₂²×C₆₇	semidirect
317. #547	C₂²×C₆₇	semidirect
331. #486	C₂²×C₆₇	semidirect
332. #481	C₂²×C₆₇	semidirect
333. #475	C₂²×C₆₇	semidirect
334. #469	C₂²×C₆₇	semidirect
335. #459	C₂²×C₆₇	semidirect
336. #449	C₂²×C₆₇	semidirect
337. C₄×SD₁₆	C₂²×C₆₇	semidirect
338. #436	C₂²×C₆₇	semidirect
342. #417	C₂²×C₆₇	semidirect
436. C₂×Q₈	#41×C₆₇	semidirect
518. Q₈	C₂×#41×C₆₇	semidirect
587. C₂	C₆₇×#3332	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#550×C₆₇	2
C₂×#547×C₆₇	2
C₆₇×#3332	2
C₂×#486×C₆₇	2
C₂×#481×C₆₇	2
C₂×#475×C₆₇	2
C₂×#469×C₆₇	2
C₂×#459×C₆₇	2
C₂×#449×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#436×C₆₇	2
C₂×#417×C₆₇	2
C₂×#3332	67

Subgroups — C2×C67×#3332 #30740403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3332 #30740403