Subgroups — C₆₇×#61562 #30740813

Order 17152.

2370 conjugacy classes of subgroups.

The subgroup lattice has 492 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
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3298	C₂	1	8	yes	maximal, complemented
4	D₄×SD₁₆×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3206	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3187	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3103	C₂	1	2	yes	maximal, complemented
8	C₂×#448×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
11	C₂³×Q₈×C₆₇	C₂²	1	2	yes	complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
13	D₄×Q₈×C₆₇	—	2	4	no	—
14	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
15	#547×C₆₇	C₂²	1	4	yes	—
16	#546×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	4	yes	—
18	#537×C₆₇	—	2	1	no	—
19	#534×C₆₇	—	2	2	no	—
20	#532×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#121×C₆₇	C₂²	1	4	yes	—
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	#492×C₆₇	C₂²	1	4	yes	complemented
30	#487×C₆₇	C₂²	1	4	yes	complemented
31	#483×C₆₇	C₂²	1	8	yes	complemented
32	#481×C₆₇	C₂²	1	4	yes	complemented
33	#475×C₆₇	C₂²	1	4	yes	complemented
34	#465×C₆₇	C₂²	1	16	yes	complemented
35	#461×C₆₇	C₂²	1	8	yes	complemented
36	#460×C₆₇	C₂²	1	4	yes	complemented
37	#450×C₆₇	C₂²	1	8	yes	complemented
38	#448×C₆₇	C₂²	1	16	yes	complemented
39	#448×C₆₇	C₂²	1	4	yes	complemented
40	#448×C₆₇	C₂²	1	8	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
43	#431×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#103×C₆₇	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₈×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	C₂²×Q₈×C₆₇	—	2	2	no	—
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	4	8	no	—
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×D₄×C₆₇	—	2	4	no	—
56	C₂²×D₄×C₆₇	—	2	4	no	—
57	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
59	C₂³×C₄×C₆₇	—	2	2	no	—
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×SD₁₆×C₆₇	—	4	16	no	—
62	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
63	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
65	C₄⋊Q₈×C₆₇	—	2	8	no	—
66	C₄⋊Q₈×C₆₇	—	2	2	no	—
67	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
68	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
69	#127×C₆₇	C₂³	1	2	yes	—
70	#124×C₆₇	—	2	2	no	—
71	#123×C₆₇	—	2	8	no	—
72	C₂²⋊Q₈×C₆₇	—	2	4	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	—	2	16	no	—
75	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
76	#121×C₆₇	—	2	4	no	—
77	#121×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	2	no	—
79	#121×C₆₇	—	2	4	no	—
80	#121×C₆₇	C₂³	1	4	yes	—
81	#121×C₆₇	C₂³	1	8	yes	—
82	#120×C₆₇	—	2	8	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
85	C₄×D₄×C₆₇	—	2	8	no	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	D₄	1	4	yes	—
88	C₄×D₄×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	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	#103×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	4	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
112	#41×C₆₇	—	4	16	no	—
113	#41×C₆₇	—	4	8	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	8	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	16	no	—
120	C₂×Q₈×C₆₇	—	4	8	no	—
121	C₂×Q₈×C₆₇	—	4	4	no	—
122	C₂×Q₈×C₆₇	—	4	8	no	—
123	C₂×Q₈×C₆₇	—	4	16	no	—
124	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
125	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
126	C₂×D₄×C₆₇	—	2	8	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	4	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	16	no	—
133	C₂×D₄×C₆₇	—	4	16	no	—
134	C₂×D₄×C₆₇	SD₁₆	1	8	yes	complemented
135	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	8	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	4	4	no	—
144	C₂²×C₄×C₆₇	—	4	8	no	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
151	SD₁₆×C₆₇	—	2	16	no	—
152	SD₁₆×C₆₇	—	4	16	no	—
153	C₂×C₈×C₆₇	—	4	8	no	—
154	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
155	C₄⋊C₄×C₆₇	—	2	4	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	8	no	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
170	C₄²×C₆₇	—	2	4	no	—
171	C₄²×C₆₇	C₂×D₄	1	2	yes	—
172	C₄²×C₆₇	C₂⁴	1	2	yes	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	—	4	2	no	—
176	C₂³×C₆₇	—	4	8	no	—
177	C₂³×C₆₇	—	8	4	no	—
178	C₂³×C₆₇	—	8	4	no	—
179	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	complemented
180	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	4	no	—
183	Q₈×C₆₇	—	2	2	no	—
184	Q₈×C₆₇	—	2	8	no	—
185	Q₈×C₆₇	—	4	16	no	—
186	Q₈×C₆₇	—	4	4	no	—
187	Q₈×C₆₇	—	4	4	no	—
188	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
189	D₄×C₆₇	—	2	8	no	—
190	D₄×C₆₇	—	4	16	no	—
191	D₄×C₆₇	—	4	8	no	—
192	D₄×C₆₇	—	4	4	no	—
193	D₄×C₆₇	—	8	2	no	—
194	D₄×C₆₇	—	8	4	no	—
195	D₄×C₆₇	C₂×SD₁₆	1	16	yes	complemented
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	8	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	4	4	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	16	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	—	4	8	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	complemented
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
217	C₈×C₆₇	—	2	8	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	4	8	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	4	no	—
226	C₂²×C₆₇	—	16	8	no	—
227	C₂²×C₆₇	C₂²×SD₁₆	1	8	yes	—
228	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
229	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	4	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	2	4	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	4	4	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	C₂²×SD₁₆	1	4	yes	—
238	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
239	#61562	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₆₇	—	2	2	no	—
241	C₂×C₆₇	—	2	8	no	—
242	C₂×C₆₇	—	8	4	no	—
243	C₂×C₆₇	#3484	1	1	yes	characteristic
244	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
245	C₂×C₆₇	#3586	1	1	yes	characteristic
246	#3496	C₂×C₆₇	1	1	yes	characteristic
247	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#3298	C₂×C₆₇	1	8	yes	complemented
249	D₄×SD₁₆	C₂×C₆₇	1	8	yes	complemented
250	#3206	C₂×C₆₇	1	2	yes	complemented
251	#3187	C₂×C₆₇	1	4	yes	complemented
252	#3103	C₂×C₆₇	1	2	yes	complemented
253	C₂×#448	C₂×C₆₇	1	4	yes	complemented
254	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₆₇	#61562	1	1	yes	characteristic, direct product factor, Sylow
256	C₂×H(5,𝔽₂)	—	2	1	no	—
257	C₂³×Q₈	C₂²×C₆₇	1	2	yes	complemented
258	C₂²×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
259	D₄×Q₈	—	2	4	no	—
260	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
261	#547	C₂²×C₆₇	1	4	yes	—
262	#546	—	2	4	no	—
263	D₄²	C₂²×C₆₇	1	4	yes	—
264	#537	—	2	1	no	—
265	#534	—	2	2	no	—
266	#532	C₂²×C₆₇	1	2	yes	—
267	C₂×C₄⋊Q₈	—	2	1	no	—
268	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
269	C₂×C₂²⋊Q₈	—	2	2	no	—
270	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
271	C₂×#121	C₂²×C₆₇	1	4	yes	—
272	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
273	C₂×C₄×D₄	—	2	1	no	—
274	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
275	#492	C₂²×C₆₇	1	4	yes	complemented
276	#487	C₂²×C₆₇	1	4	yes	complemented
277	#483	C₂²×C₆₇	1	8	yes	complemented
278	#481	C₂²×C₆₇	1	4	yes	complemented
279	#475	C₂²×C₆₇	1	4	yes	complemented
280	#465	C₂²×C₆₇	1	16	yes	complemented
281	#461	C₂²×C₆₇	1	8	yes	complemented
282	#460	C₂²×C₆₇	1	4	yes	complemented
283	#450	C₂²×C₆₇	1	8	yes	complemented
284	#448	C₂²×C₆₇	1	4	yes	complemented
285	#448	C₂²×C₆₇	1	8	yes	complemented
286	#448	C₂²×C₆₇	1	16	yes	complemented
287	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
288	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
289	#431	C₂²×C₆₇	1	2	yes	complemented
290	C₂×#103	C₂²×C₆₇	1	8	yes	complemented
291	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
292	H(5,𝔽₂)	—	4	8	no	—
293	C₂×#41	—	2	4	no	—
294	C₂×#41	—	2	2	no	—
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	2	4	no	—
297	C₂²×Q₈	—	4	8	no	—
298	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
300	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
301	C₂²×D₄	—	2	4	no	—
302	C₂²×D₄	—	2	4	no	—
303	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
304	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
307	C₂×SD₁₆	—	4	16	no	—
308	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
309	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
311	C₄⋊Q₈	—	2	8	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
314	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
315	#127	C₂³×C₆₇	1	2	yes	—
316	#124	—	2	2	no	—
317	#123	—	2	8	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	2	16	no	—
321	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
322	#121	—	2	4	no	—
323	#121	—	2	4	no	—
324	#121	—	2	2	no	—
325	#121	—	2	4	no	—
326	#121	C₂³×C₆₇	1	4	yes	—
327	#121	C₂³×C₆₇	1	8	yes	—
328	#120	—	2	8	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
331	C₄×D₄	—	2	8	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	D₄×C₆₇	1	4	yes	—
334	C₄×D₄	C₂³×C₆₇	1	4	yes	—
335	C₄×D₄	C₂³×C₆₇	1	4	yes	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	4	no	—
345	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
348	#106	C₂³×C₆₇	1	8	yes	—
349	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
350	#103	C₂³×C₆₇	1	8	yes	—
351	#103	C₂³×C₆₇	1	16	yes	—
352	#102	C₂³×C₆₇	1	8	yes	—
353	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
354	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
355	C₄×C₈	C₂³×C₆₇	1	4	yes	—
356	C₂⁴	—	2	4	no	—
357	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
358	#41	—	4	16	no	—
359	#41	—	4	8	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	8	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	16	no	—
366	C₂×Q₈	—	4	8	no	—
367	C₂×Q₈	—	4	4	no	—
368	C₂×Q₈	—	4	8	no	—
369	C₂×Q₈	—	4	16	no	—
370	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
371	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	—	4	16	no	—
379	C₂×D₄	—	4	16	no	—
380	C₂×D₄	SD₁₆×C₆₇	1	8	yes	complemented
381	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
382	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	—	4	8	no	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
397	SD₁₆	—	2	16	no	—
398	SD₁₆	—	4	16	no	—
399	C₂×C₈	—	4	8	no	—
400	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
409	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
410	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	4	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	4	no	—
415	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
416	C₄²	—	2	4	no	—
417	C₄²	C₂×D₄×C₆₇	1	2	yes	—
418	C₄²	C₂⁴×C₆₇	1	2	yes	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	8	no	—
423	C₂³	—	8	4	no	—
424	C₂³	—	8	4	no	—
425	C₂³	C₂×SD₁₆×C₆₇	1	8	yes	complemented
426	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
427	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
428	Q₈	—	2	4	no	—
429	Q₈	—	2	2	no	—
430	Q₈	—	2	8	no	—
431	Q₈	—	4	16	no	—
432	Q₈	—	4	4	no	—
433	Q₈	—	4	4	no	—
434	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	complemented
435	D₄	—	2	8	no	—
436	D₄	—	4	16	no	—
437	D₄	—	4	8	no	—
438	D₄	—	4	4	no	—
439	D₄	—	8	4	no	—
440	D₄	—	8	2	no	—
441	D₄	C₂×SD₁₆×C₆₇	1	16	yes	complemented
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	16	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	complemented
458	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
463	C₈	—	2	8	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	16	8	no	—
473	C₂²	C₂²×SD₁₆×C₆₇	1	8	yes	—
474	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
475	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	2	4	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	4	no	—
480	C₄	—	4	4	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	4	no	—
483	C₄	C₂²×SD₁₆×C₆₇	1	4	yes	—
484	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
485	C₂	—	2	2	no	—
486	C₂	—	2	8	no	—
487	C₂	—	8	4	no	—
488	C₂	C₆₇×#3484	1	1	yes	characteristic
489	C₂	C₂³×SD₁₆×C₆₇	1	1	yes	characteristic
490	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3496	C₂
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
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3298	C₂	semidirect
4. D₄×SD₁₆×C₆₇	C₂	semidirect
5. C₆₇×#3206	C₂	semidirect
6. C₆₇×#3187	C₂	semidirect
7. C₆₇×#3103	C₂	semidirect
8. C₂×#448×C₆₇	C₂	semidirect
9. C₆₇×#2972	C₂	semidirect
11. C₂³×Q₈×C₆₇	C₂²	semidirect
12. C₂²×SD₁₆×C₆₇	C₂²	semidirect
14. D₄×Q₈×C₆₇	C₂²	semidirect
22. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
24. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
26. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
29. #492×C₆₇	C₂²	semidirect
30. #487×C₆₇	C₂²	semidirect
31. #483×C₆₇	C₂²	semidirect
32. #481×C₆₇	C₂²	semidirect
33. #475×C₆₇	C₂²	semidirect
34. #465×C₆₇	C₂²	semidirect
35. #461×C₆₇	C₂²	semidirect
36. #460×C₆₇	C₂²	semidirect
37. #450×C₆₇	C₂²	semidirect
38. #448×C₆₇	C₂²	semidirect
39. #448×C₆₇	C₂²	semidirect
40. #448×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. C₈×D₄×C₆₇	C₂²	semidirect
43. #431×C₆₇	C₂²	semidirect
44. C₂×#103×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
134. C₂×D₄×C₆₇	SD₁₆	semidirect
179. C₂³×C₆₇	C₂×SD₁₆	semidirect
188. Q₈×C₆₇	H(5,𝔽₂)	semidirect
195. D₄×C₆₇	C₂×SD₁₆	semidirect
211. C₂×C₄×C₆₇	C₂×SD₁₆	semidirect
239. #61562	C₆₇	direct (N × H)
247. C₂×D₄×Q₈	C₂×C₆₇	semidirect
248. #3298	C₂×C₆₇	semidirect
249. D₄×SD₁₆	C₂×C₆₇	semidirect
250. #3206	C₂×C₆₇	semidirect
251. #3187	C₂×C₆₇	semidirect
252. #3103	C₂×C₆₇	semidirect
253. C₂×#448	C₂×C₆₇	semidirect
254. #2972	C₂×C₆₇	semidirect
255. C₆₇	#61562	direct (N × H)
257. C₂³×Q₈	C₂²×C₆₇	semidirect
258. C₂²×SD₁₆	C₂²×C₆₇	semidirect
260. D₄×Q₈	C₂²×C₆₇	semidirect
268. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
270. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
272. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
275. #492	C₂²×C₆₇	semidirect
276. #487	C₂²×C₆₇	semidirect
277. #483	C₂²×C₆₇	semidirect
278. #481	C₂²×C₆₇	semidirect
279. #475	C₂²×C₆₇	semidirect
280. #465	C₂²×C₆₇	semidirect
281. #461	C₂²×C₆₇	semidirect
282. #460	C₂²×C₆₇	semidirect
283. #450	C₂²×C₆₇	semidirect
284. #448	C₂²×C₆₇	semidirect
285. #448	C₂²×C₆₇	semidirect
286. #448	C₂²×C₆₇	semidirect
287. C₄×SD₁₆	C₂²×C₆₇	semidirect
288. C₈×D₄	C₂²×C₆₇	semidirect
289. #431	C₂²×C₆₇	semidirect
290. C₂×#103	C₂²×C₆₇	semidirect
291. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
380. C₂×D₄	SD₁₆×C₆₇	semidirect
425. C₂³	C₂×SD₁₆×C₆₇	semidirect
434. Q₈	H(5,𝔽₂)×C₆₇	semidirect
441. D₄	C₂×SD₁₆×C₆₇	semidirect
457. C₂×C₄	C₂×SD₁₆×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×D₄×Q₈×C₆₇	2
C₆₇×#3298	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3206	2
C₆₇×#3187	2
C₆₇×#3103	2
C₂×#448×C₆₇	2
C₆₇×#2972	2
#61562	67

Subgroups — C67×#61562 #30740813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61562 #30740813