Subgroups — C₆₇×#61587 #30740838

Order 17152.

2430 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3332	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3274	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3205	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3192	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3103	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3006	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
11	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
13	#550×C₆₇	—	2	4	no	—
14	#550×C₆₇	C₂²	1	8	yes	complemented
15	#547×C₆₇	C₂²	1	4	yes	complemented
16	#546×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	4	yes	complemented
18	#537×C₆₇	—	2	1	no	—
19	#534×C₆₇	—	2	2	no	—
20	#532×C₆₇	C₂²	1	2	yes	—
21	C₂×#127×C₆₇	—	2	1	no	—
22	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×#121×C₆₇	—	2	2	no	—
24	C₂×#121×C₆₇	C₂²	1	4	yes	—
25	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄×D₄×C₆₇	—	2	1	no	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	#493×C₆₇	C₂²	1	4	yes	complemented
30	#486×C₆₇	C₂²	1	4	yes	complemented
31	#481×C₆₇	C₂²	1	8	yes	complemented
32	#481×C₆₇	C₂²	1	4	yes	complemented
33	#475×C₆₇	C₂²	1	4	yes	complemented
34	#469×C₆₇	C₂²	1	16	yes	complemented
35	#460×C₆₇	C₂²	1	4	yes	complemented
36	#459×C₆₇	C₂²	1	8	yes	complemented
37	#449×C₆₇	C₂²	1	16	yes	complemented
38	#448×C₆₇	C₂²	1	4	yes	complemented
39	#447×C₆₇	C₂²	1	8	yes	complemented
40	#447×C₆₇	C₂²	1	8	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	#436×C₆₇	C₂²	1	8	yes	complemented
43	#431×C₆₇	C₂²	1	2	yes	complemented
44	#417×C₆₇	C₂²	1	8	yes	complemented
45	#407×C₆₇	C₂²	1	4	yes	complemented
46	H(5,𝔽₂)×C₆₇	—	4	8	no	—
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	4	no	—
50	C₂×#41×C₆₇	—	4	8	no	—
51	C₂×#41×C₆₇	C₂³	1	8	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	2	2	no	—
56	C₂²×D₄×C₆₇	—	2	4	no	—
57	C₂²×D₄×C₆₇	—	2	1	no	—
58	C₂²×D₄×C₆₇	—	2	1	no	—
59	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	—	2	2	no	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	AGL(1,ℤ/8)×C₆₇	—	4	16	no	—
65	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
66	C₂×D₈×C₆₇	C₂³	1	16	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
68	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
69	#127×C₆₇	—	2	8	no	—
70	#127×C₆₇	—	2	2	no	—
71	#127×C₆₇	C₂³	1	4	yes	—
72	#127×C₆₇	C₂³	1	2	yes	—
73	#124×C₆₇	—	2	2	no	—
74	#123×C₆₇	—	2	8	no	—
75	C₂²⋊Q₈×C₆₇	—	2	4	no	—
76	#121×C₆₇	—	2	4	no	—
77	#121×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	4	no	—
79	#121×C₆₇	—	2	2	no	—
80	#121×C₆₇	—	2	4	no	—
81	#121×C₆₇	—	2	16	no	—
82	#121×C₆₇	C₂³	1	4	yes	—
83	#121×C₆₇	C₂³	1	8	yes	—
84	#121×C₆₇	C₂³	1	8	yes	—
85	#120×C₆₇	—	2	8	no	—
86	C₄×Q₈×C₆₇	—	2	2	no	—
87	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
88	C₄×D₄×C₆₇	—	2	8	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	D₄	1	4	yes	—
91	C₄×D₄×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
100	C₂×C₄²×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#106×C₆₇	C₂³	1	8	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
104	#103×C₆₇	C₂³	1	8	yes	—
105	#102×C₆₇	C₂³	1	16	yes	—
106	#102×C₆₇	C₂³	1	8	yes	—
107	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
109	C₄×C₈×C₆₇	C₂³	1	4	yes	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
113	#41×C₆₇	—	2	16	no	—
114	#41×C₆₇	—	4	8	no	—
115	#41×C₆₇	—	4	16	no	—
116	#41×C₆₇	—	4	16	no	—
117	#41×C₆₇	—	4	8	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
122	C₂×D₄×C₆₇	—	2	8	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	8	no	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	4	no	—
133	C₂×D₄×C₆₇	—	4	4	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	16	no	—
137	C₂×D₄×C₆₇	—	4	16	no	—
138	C₂×D₄×C₆₇	—	4	4	no	—
139	C₂×D₄×C₆₇	—	8	2	no	—
140	C₂×D₄×C₆₇	—	8	4	no	—
141	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	8	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	4	4	no	—
150	C₂²×C₄×C₆₇	—	4	8	no	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
157	SD₁₆×C₆₇	—	2	16	no	—
158	D₈×C₆₇	—	4	16	no	—
159	M₁₆×C₆₇	—	4	8	no	—
160	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
166	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	2	4	no	—
172	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
173	C₄²×C₆₇	—	2	4	no	—
174	C₄²×C₆₇	C₂×D₄	1	2	yes	—
175	C₄²×C₆₇	C₂⁴	1	2	yes	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	4	4	no	—
182	C₂³×C₆₇	—	4	4	no	—
183	C₂³×C₆₇	—	4	8	no	—
184	C₂³×C₆₇	—	4	4	no	—
185	C₂³×C₆₇	—	8	4	no	—
186	C₂³×C₆₇	—	8	4	no	—
187	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
188	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₆₇	—	2	4	no	—
190	Q₈×C₆₇	—	2	2	no	—
191	Q₈×C₆₇	—	4	4	no	—
192	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
193	D₄×C₆₇	—	2	8	no	—
194	D₄×C₆₇	—	2	8	no	—
195	D₄×C₆₇	—	2	8	no	—
196	D₄×C₆₇	—	4	16	no	—
197	D₄×C₆₇	—	4	4	no	—
198	D₄×C₆₇	—	4	16	no	—
199	D₄×C₆₇	—	4	8	no	—
200	D₄×C₆₇	—	4	8	no	—
201	D₄×C₆₇	—	4	4	no	—
202	D₄×C₆₇	—	8	2	no	—
203	D₄×C₆₇	—	8	4	no	—
204	D₄×C₆₇	—	8	4	no	—
205	D₄×C₆₇	—	8	8	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	8	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	16	no	—
216	C₂×C₄×C₆₇	—	4	8	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	8	no	—
219	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	complemented
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
225	C₈×C₆₇	—	2	8	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	2	4	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	4	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	8	4	no	—
238	C₂²×C₆₇	—	8	8	no	—
239	C₂²×C₆₇	—	16	8	no	—
240	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
249	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
250	#61587	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	—	8	4	no	—
254	C₂×C₆₇	—	8	4	no	—
255	C₂×C₆₇	#3484	1	1	yes	characteristic
256	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂×C₆₇	#3585	1	1	yes	characteristic
258	#3496	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#550	C₂×C₆₇	1	1	yes	characteristic, complemented
260	#3332	C₂×C₆₇	1	8	yes	complemented
261	#3274	C₂×C₆₇	1	8	yes	complemented
262	#3205	C₂×C₆₇	1	2	yes	complemented
263	#3192	C₂×C₆₇	1	4	yes	complemented
264	#3103	C₂×C₆₇	1	2	yes	complemented
265	#3006	C₂×C₆₇	1	4	yes	complemented
266	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₆₇	#61587	1	1	yes	characteristic, direct product factor, Sylow
268	C₂×H(5,𝔽₂)	—	2	1	no	—
269	C₂²×#41	C₂²×C₆₇	1	2	yes	complemented
270	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
271	#550	—	2	4	no	—
272	#550	C₂²×C₆₇	1	8	yes	complemented
273	#547	C₂²×C₆₇	1	4	yes	complemented
274	#546	—	2	4	no	—
275	D₄²	C₂²×C₆₇	1	4	yes	complemented
276	#537	—	2	1	no	—
277	#534	—	2	2	no	—
278	#532	C₂²×C₆₇	1	2	yes	—
279	C₂×#127	—	2	1	no	—
280	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
281	C₂×#121	—	2	2	no	—
282	C₂×#121	C₂²×C₆₇	1	4	yes	—
283	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
284	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
285	C₂×C₄×D₄	—	2	1	no	—
286	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
287	#493	C₂²×C₆₇	1	4	yes	complemented
288	#486	C₂²×C₆₇	1	4	yes	complemented
289	#481	C₂²×C₆₇	1	8	yes	complemented
290	#481	C₂²×C₆₇	1	4	yes	complemented
291	#475	C₂²×C₆₇	1	4	yes	complemented
292	#469	C₂²×C₆₇	1	16	yes	complemented
293	#460	C₂²×C₆₇	1	4	yes	complemented
294	#459	C₂²×C₆₇	1	8	yes	complemented
295	#449	C₂²×C₆₇	1	16	yes	complemented
296	#448	C₂²×C₆₇	1	4	yes	complemented
297	#447	C₂²×C₆₇	1	8	yes	complemented
298	#447	C₂²×C₆₇	1	8	yes	complemented
299	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
300	#436	C₂²×C₆₇	1	8	yes	complemented
301	#431	C₂²×C₆₇	1	2	yes	complemented
302	#417	C₂²×C₆₇	1	8	yes	complemented
303	#407	C₂²×C₆₇	1	4	yes	complemented
304	H(5,𝔽₂)	—	4	8	no	—
305	C₂×#41	—	2	4	no	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	4	no	—
308	C₂×#41	—	4	8	no	—
309	C₂×#41	C₂³×C₆₇	1	8	yes	—
310	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
311	C₂²×D₄	—	2	2	no	—
312	C₂²×D₄	—	2	4	no	—
313	C₂²×D₄	—	2	2	no	—
314	C₂²×D₄	—	2	4	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
319	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
322	AGL(1,ℤ/8)	—	4	16	no	—
323	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
324	C₂×D₈	C₂³×C₆₇	1	16	yes	—
325	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
326	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
327	#127	—	2	8	no	—
328	#127	—	2	2	no	—
329	#127	C₂³×C₆₇	1	4	yes	—
330	#127	C₂³×C₆₇	1	2	yes	—
331	#124	—	2	2	no	—
332	#123	—	2	8	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	#121	—	2	4	no	—
335	#121	—	2	4	no	—
336	#121	—	2	4	no	—
337	#121	—	2	2	no	—
338	#121	—	2	4	no	—
339	#121	—	2	16	no	—
340	#121	C₂³×C₆₇	1	4	yes	—
341	#121	C₂³×C₆₇	1	8	yes	—
342	#121	C₂³×C₆₇	1	8	yes	—
343	#120	—	2	8	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
346	C₄×D₄	—	2	8	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	D₄×C₆₇	1	4	yes	—
349	C₄×D₄	C₂³×C₆₇	1	4	yes	—
350	C₄×D₄	C₂³×C₆₇	1	4	yes	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	4	no	—
357	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
360	#106	C₂³×C₆₇	1	8	yes	—
361	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
362	#103	C₂³×C₆₇	1	8	yes	—
363	#102	C₂³×C₆₇	1	16	yes	—
364	#102	C₂³×C₆₇	1	8	yes	—
365	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
366	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
367	C₄×C₈	C₂³×C₆₇	1	4	yes	—
368	C₂⁴	—	2	2	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
371	#41	—	2	16	no	—
372	#41	—	4	8	no	—
373	#41	—	4	16	no	—
374	#41	—	4	16	no	—
375	#41	—	4	8	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	8	no	—
393	C₂×D₄	—	4	8	no	—
394	C₂×D₄	—	4	16	no	—
395	C₂×D₄	—	4	16	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂×D₄	—	8	4	no	—
399	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
400	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
401	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	4	4	no	—
408	C₂²×C₄	—	4	8	no	—
409	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
412	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
415	SD₁₆	—	2	16	no	—
416	D₈	—	4	16	no	—
417	M₁₆	—	4	8	no	—
418	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
424	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
425	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
431	C₄²	—	2	4	no	—
432	C₄²	C₂×D₄×C₆₇	1	2	yes	—
433	C₄²	C₂⁴×C₆₇	1	2	yes	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	4	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	4	no	—
443	C₂³	—	8	4	no	—
444	C₂³	—	8	4	no	—
445	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
446	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
447	Q₈	—	2	4	no	—
448	Q₈	—	2	2	no	—
449	Q₈	—	4	4	no	—
450	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	complemented
451	D₄	—	2	8	no	—
452	D₄	—	2	8	no	—
453	D₄	—	2	8	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	4	no	—
456	D₄	—	4	16	no	—
457	D₄	—	4	8	no	—
458	D₄	—	4	4	no	—
459	D₄	—	4	8	no	—
460	D₄	—	8	4	no	—
461	D₄	—	8	4	no	—
462	D₄	—	8	2	no	—
463	D₄	—	8	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	complemented
478	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
481	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
482	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
483	C₈	—	2	8	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	4	no	—
496	C₂²	—	8	8	no	—
497	C₂²	—	16	8	no	—
498	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
499	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	4	no	—
502	C₄	—	2	4	no	—
503	C₄	—	2	1	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
507	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
508	C₂	—	2	2	no	—
509	C₂	—	4	4	no	—
510	C₂	—	8	4	no	—
511	C₂	—	8	4	no	—
512	C₂	C₆₇×#3484	1	1	yes	characteristic
513	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
514	C₂	C₆₇×#3585	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3496	C₂	semidirect
2. C₂×#550×C₆₇	C₂	semidirect
3. C₆₇×#3332	C₂	semidirect
4. C₆₇×#3274	C₂	semidirect
5. C₆₇×#3205	C₂	semidirect
6. C₆₇×#3192	C₂	semidirect
7. C₆₇×#3103	C₂	semidirect
8. C₆₇×#3006	C₂	semidirect
9. C₆₇×#2975	C₂	semidirect
11. C₂²×#41×C₆₇	C₂²	semidirect
12. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. #550×C₆₇	C₂²	semidirect
15. #547×C₆₇	C₂²	semidirect
17. D₄²×C₆₇	C₂²	semidirect
22. C₂×#127×C₆₇	C₂²	semidirect
25. C₂×#121×C₆₇	C₂²	semidirect
26. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
29. #493×C₆₇	C₂²	semidirect
30. #486×C₆₇	C₂²	semidirect
31. #481×C₆₇	C₂²	semidirect
32. #481×C₆₇	C₂²	semidirect
33. #475×C₆₇	C₂²	semidirect
34. #469×C₆₇	C₂²	semidirect
35. #460×C₆₇	C₂²	semidirect
36. #459×C₆₇	C₂²	semidirect
37. #449×C₆₇	C₂²	semidirect
38. #448×C₆₇	C₂²	semidirect
39. #447×C₆₇	C₂²	semidirect
40. #447×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
43. #431×C₆₇	C₂²	semidirect
44. #417×C₆₇	C₂²	semidirect
45. #407×C₆₇	C₂²	semidirect
192. Q₈×C₆₇	H(5,𝔽₂)	semidirect
219. C₂×C₄×C₆₇	AGL(1,ℤ/8)	semidirect
250. #61587	C₆₇	direct (N × H)
258. #3496	C₂×C₆₇	semidirect
259. C₂×#550	C₂×C₆₇	semidirect
260. #3332	C₂×C₆₇	semidirect
261. #3274	C₂×C₆₇	semidirect
262. #3205	C₂×C₆₇	semidirect
263. #3192	C₂×C₆₇	semidirect
264. #3103	C₂×C₆₇	semidirect
265. #3006	C₂×C₆₇	semidirect
266. #2975	C₂×C₆₇	semidirect
267. C₆₇	#61587	direct (N × H)
269. C₂²×#41	C₂²×C₆₇	semidirect
270. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
272. #550	C₂²×C₆₇	semidirect
273. #547	C₂²×C₆₇	semidirect
275. D₄²	C₂²×C₆₇	semidirect
280. C₂×#127	C₂²×C₆₇	semidirect
283. C₂×#121	C₂²×C₆₇	semidirect
284. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
287. #493	C₂²×C₆₇	semidirect
288. #486	C₂²×C₆₇	semidirect
289. #481	C₂²×C₆₇	semidirect
290. #481	C₂²×C₆₇	semidirect
291. #475	C₂²×C₆₇	semidirect
292. #469	C₂²×C₆₇	semidirect
293. #460	C₂²×C₆₇	semidirect
294. #459	C₂²×C₆₇	semidirect
295. #449	C₂²×C₆₇	semidirect
296. #448	C₂²×C₆₇	semidirect
297. #447	C₂²×C₆₇	semidirect
298. #447	C₂²×C₆₇	semidirect
299. C₄×SD₁₆	C₂²×C₆₇	semidirect
300. #436	C₂²×C₆₇	semidirect
301. #431	C₂²×C₆₇	semidirect
302. #417	C₂²×C₆₇	semidirect
303. #407	C₂²×C₆₇	semidirect
450. Q₈	H(5,𝔽₂)×C₆₇	semidirect
477. C₂×C₄	AGL(1,ℤ/8)×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3496	2
C₂×#550×C₆₇	2
C₆₇×#3332	2
C₆₇×#3274	2
C₆₇×#3205	2
C₆₇×#3192	2
C₆₇×#3103	2
C₆₇×#3006	2
C₆₇×#2975	2
#61587	67

Subgroups — C67×#61587 #30740838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61587 #30740838