Subgroups — #30875718

Order 17152.

784 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22561143	C₂	1	2	yes	maximal
3	#22560767	C₂	1	2	yes	maximal
4	#22560764	C₂	1	2	yes	maximal
5	#22560360	C₂	1	1	yes	maximal, characteristic
6	#22560325	C₂	1	1	yes	maximal, characteristic
7	#22560070	C₂	1	2	yes	maximal
8	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
21	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
22	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
23	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
24	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
25	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
26	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
29	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
30	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
31	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863151	C₂²	1	2	yes	—
34	#16863150	C₂²	1	2	yes	—
35	#16863150	C₂²	1	2	yes	—
36	#16863150	C₂²	1	2	yes	—
37	#16863150	C₂²	1	2	yes	—
38	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×Dic₁₃₄	—	2	4	no	—
47	C₄×Dic₁₃₄	—	2	2	no	—
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×Dic₁₃₄	C₈	1	8	yes	complemented
50	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
72	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
73	Dic₆₇⋊C₈	—	2	4	no	—
74	Dic₆₇⋊C₈	—	4	2	no	—
75	C₄×C₆₇⋊C₈	—	4	2	no	—
76	C₈×Dic₆₇	—	2	4	no	—
77	C₈×Dic₆₇	—	4	2	no	—
78	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
79	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
80	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
81	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
82	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
83	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
85	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	—	2	2	no	—
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	—	2	2	no	—
97	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
98	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
112	C₄⋊Dic₆₇	—	2	2	no	—
113	C₄⋊Dic₆₇	—	2	2	no	—
114	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	Dic₆₇⋊C₄	—	2	2	no	—
117	Dic₆₇⋊C₄	—	2	2	no	—
118	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
119	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
120	Dic₆₇⋊C₄	M₁₆	1	4	yes	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	—	2	1	no	—
124	C₄×Dic₆₇	—	2	1	no	—
125	C₄×Dic₆₇	—	2	1	no	—
126	C₄×Dic₆₇	—	2	1	no	—
127	C₄×Dic₆₇	—	2	1	no	—
128	C₄×Dic₆₇	—	2	1	no	—
129	C₄×Dic₆₇	—	2	2	no	—
130	C₄×Dic₆₇	—	2	2	no	—
131	C₄×Dic₆₇	—	4	1	no	—
132	C₄×Dic₆₇	—	4	1	no	—
133	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
134	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
135	C₄×Dic₆₇	#41	1	2	yes	—
136	C₄×Dic₆₇	#41	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	4	1	no	—
157	C₄²×C₆₇	C₂×C₈	1	4	yes	—
158	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
159	C₄²×C₆₇	#41	1	2	yes	—
160	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
173	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
175	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
178	C₂×C₄×C₆₇	#131	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×Dic₆₇	—	2	1	no	—
186	C₂×Dic₆₇	—	2	1	no	—
187	C₂×Dic₆₇	—	2	1	no	—
188	C₂×Dic₆₇	—	2	1	no	—
189	C₂×Dic₆₇	—	2	1	no	—
190	C₂×Dic₆₇	—	2	1	no	—
191	C₂×Dic₆₇	—	2	1	no	—
192	C₂×Dic₆₇	—	2	1	no	—
193	C₂×Dic₆₇	—	2	1	no	—
194	C₂×Dic₆₇	—	2	1	no	—
195	C₂×Dic₆₇	—	2	1	no	—
196	C₂×Dic₆₇	—	2	1	no	—
197	C₂×Dic₆₇	—	2	1	no	—
198	C₂×Dic₆₇	—	2	1	no	—
199	C₂×Dic₆₇	—	2	1	no	—
200	C₂×Dic₆₇	—	2	1	no	—
201	C₂×Dic₆₇	#117	1	2	yes	—
202	C₂×Dic₆₇	#117	1	2	yes	—
203	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
204	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
205	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
206	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
207	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
208	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
209	Dic₁₃₄	—	2	8	no	—
210	C₈×C₆₇	—	4	2	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₆₇⋊C₈	—	2	4	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂²×C₆₇	#520	1	1	yes	characteristic
217	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
218	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
220	C₂²×C₆₇	#568	1	1	yes	characteristic
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	C₈×Q₈	1	4	yes	—
227	C₄×C₆₇	#554	1	1	yes	characteristic
228	C₄×C₆₇	#556	1	1	yes	characteristic
229	C₄×C₆₇	#557	1	1	yes	characteristic
230	C₄×C₆₇	#559	1	1	yes	characteristic
231	Dic₆₇	—	2	2	no	—
232	Dic₆₇	—	2	2	no	—
233	Dic₆₇	—	2	2	no	—
234	Dic₆₇	—	2	2	no	—
235	Dic₆₇	—	2	2	no	—
236	Dic₆₇	—	4	1	no	—
237	Dic₆₇	—	4	1	no	—
238	Dic₆₇	#431	1	4	yes	—
239	#22821	—	67	1	no	maximal, Sylow
240	C₂×C₆₇	#2355	1	1	yes	characteristic
241	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
242	C₂×C₆₇	#2925	1	1	yes	characteristic
243	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
244	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
245	C₂×C₆₇	#2988	1	1	yes	characteristic
246	C₂×C₆₇	#2990	1	1	yes	characteristic
247	C₄²×Q₈	—	67	1	no	—
248	#1984	—	67	2	no	—
249	#1916	—	67	2	no	—
250	#1916	—	67	2	no	—
251	#1842	—	67	1	no	—
252	#1831	—	67	1	no	—
253	#1831	D₆₇	1	1	yes	characteristic
254	C₈×C₄⋊C₄	—	67	2	no	—
255	C₄×C₄⋊C₈	—	67	1	no	—
256	C₄×C₄⋊C₈	—	67	1	no	—
257	C₄×C₄⋊C₈	—	67	1	no	—
258	C₆₇	#22821	1	1	yes	characteristic, complemented, Sylow
259	C₂×C₄×Q₈	—	67	1	no	—
260	C₂×C₄×Q₈	—	67	1	no	—
261	C₂×C₄×Q₈	—	67	1	no	—
262	C₂×C₄⋊C₈	—	67	2	no	—
263	C₂×C₄⋊C₈	—	67	1	no	—
264	C₂×C₄⋊C₈	—	67	2	no	—
265	C₂×C₄⋊C₈	—	67	1	no	—
266	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
268	C₂×C₄×C₈	—	67	2	no	—
269	C₂×C₄×C₈	—	67	2	no	—
270	C₂×C₄×C₈	—	67	2	no	—
271	C₄×C₄⋊C₄	—	67	1	no	—
272	C₄×C₄⋊C₄	—	67	1	no	—
273	C₄×C₄⋊C₄	—	67	1	no	—
274	C₄×C₄⋊C₄	—	67	1	no	—
275	C₄×C₄⋊C₄	—	67	1	no	—
276	C₄×C₄⋊C₄	—	67	1	no	—
277	C₄×C₄⋊C₄	—	67	1	no	—
278	C₄×C₄⋊C₄	—	67	1	no	—
279	C₄×C₄⋊C₄	—	67	1	no	—
280	C₄³	—	67	1	no	—
281	C₄³	—	67	1	no	—
282	C₄³	C₂×D₆₇	1	1	yes	characteristic
283	#336	—	67	2	no	—
284	#336	—	67	2	no	—
285	#336	—	67	2	no	—
286	#336	—	67	2	no	—
287	#336	—	67	2	no	—
288	#336	C₂×D₆₇	1	2	yes	—
289	#336	C₂×D₆₇	1	2	yes	—
290	C₂²×Q₈	—	67	1	no	—
291	C₂²×C₈	—	67	2	no	—
292	C₂²×C₈	—	67	2	no	—
293	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
294	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
295	C₄×Q₈	—	67	8	no	—
296	C₄×Q₈	—	134	2	no	—
297	C₄×Q₈	—	134	2	no	—
298	C₄×Q₈	—	134	4	no	—
299	C₂×C₄⋊C₄	—	67	1	no	—
300	C₂×C₄⋊C₄	—	67	1	no	—
301	C₂×C₄⋊C₄	—	67	1	no	—
302	C₂×C₄⋊C₄	—	67	1	no	—
303	C₂×C₄⋊C₄	—	67	1	no	—
304	C₂×C₄⋊C₄	—	67	1	no	—
305	C₂×C₄⋊C₄	—	67	1	no	—
306	C₂×C₄⋊C₄	—	67	1	no	—
307	C₂×C₄⋊C₄	—	67	1	no	—
308	C₂×C₄²	—	67	1	no	—
309	C₂×C₄²	—	67	1	no	—
310	C₂×C₄²	—	67	1	no	—
311	C₂×C₄²	—	67	1	no	—
312	C₂×C₄²	—	67	1	no	—
313	C₂×C₄²	—	67	1	no	—
314	C₂×C₄²	—	67	1	no	—
315	C₂×C₄²	—	67	1	no	—
316	C₂×C₄²	—	67	1	no	—
317	C₂×C₄²	—	67	1	no	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
324	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
326	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	268	2	no	—
330	C₄⋊C₈	—	67	4	no	—
331	C₄⋊C₈	—	134	4	no	—
332	C₄⋊C₈	—	134	2	no	—
333	C₄×C₈	—	268	2	no	—
334	C₄×C₈	—	268	2	no	—
335	C₄×C₈	—	134	4	no	—
336	#95	—	67	1	no	—
337	#95	—	67	1	no	—
338	#95	—	67	1	no	—
339	#95	—	67	1	no	—
340	#95	—	67	1	no	—
341	#95	—	67	1	no	—
342	C₂×Q₈	—	67	4	no	—
343	C₂×Q₈	—	134	2	no	—
344	C₂×Q₈	—	134	2	no	—
345	C₂²×C₄	—	67	1	no	—
346	C₂²×C₄	—	67	1	no	—
347	C₂²×C₄	—	67	1	no	—
348	C₂²×C₄	—	67	1	no	—
349	C₂²×C₄	—	67	1	no	—
350	C₂²×C₄	—	67	1	no	—
351	C₂²×C₄	—	67	1	no	—
352	C₂²×C₄	—	67	1	no	—
353	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	67	4	no	—
367	C₂×C₈	—	134	2	no	—
368	C₂×C₈	—	134	2	no	—
369	C₂×C₈	—	134	2	no	—
370	C₂×C₈	—	134	2	no	—
371	C₂×C₈	—	134	2	no	—
372	C₄⋊C₄	—	67	4	no	—
373	C₄⋊C₄	—	67	4	no	—
374	C₄⋊C₄	—	67	4	no	—
375	C₄⋊C₄	—	67	4	no	—
376	C₄⋊C₄	—	134	2	no	—
377	C₄⋊C₄	—	134	2	no	—
378	C₄⋊C₄	—	134	2	no	—
379	C₄⋊C₄	—	134	2	no	—
380	C₄⋊C₄	—	134	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	4	1	no	—
388	C₄²	—	268	1	no	—
389	C₄²	—	268	1	no	—
390	C₄²	—	67	4	no	—
391	C₄²	—	67	4	no	—
392	C₄²	—	67	2	no	—
393	C₄²	—	67	2	no	—
394	C₄²	—	134	2	no	—
395	C₄²	—	134	2	no	—
396	C₄²	—	134	2	no	—
397	C₄²	—	134	2	no	—
398	C₄²	—	134	1	no	—
399	C₄²	—	134	1	no	—
400	C₄²	—	134	1	no	—
401	C₄²	—	134	1	no	—
402	C₄²	—	134	1	no	—
403	C₄²	—	134	1	no	—
404	C₄²	C₈×D₆₇	1	4	yes	—
405	C₄²	Q₈×D₆₇	1	2	yes	—
406	C₄²	#11761627	1	2	yes	—
407	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
408	Q₈	—	134	8	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	67	2	no	—
421	C₂×C₄	—	67	2	no	—
422	C₂×C₄	—	67	2	no	—
423	C₂×C₄	—	67	2	no	—
424	C₂×C₄	—	67	2	no	—
425	C₂×C₄	—	67	2	no	—
426	C₂×C₄	—	67	2	no	—
427	C₂×C₄	—	67	2	no	—
428	C₂×C₄	—	134	1	no	—
429	C₂×C₄	—	134	1	no	—
430	C₂×C₄	—	134	1	no	—
431	C₂×C₄	—	134	1	no	—
432	C₂×C₄	—	134	1	no	—
433	C₂×C₄	—	134	1	no	—
434	C₂×C₄	—	134	1	no	—
435	C₂×C₄	—	134	1	no	—
436	C₂×C₄	—	134	1	no	—
437	C₂×C₄	—	134	1	no	—
438	C₂×C₄	—	134	1	no	—
439	C₂×C₄	—	134	1	no	—
440	C₂×C₄	—	134	1	no	—
441	C₂×C₄	—	134	1	no	—
442	C₂×C₄	—	134	1	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	#14524008	1	2	yes	—
445	C₂×C₄	#14524008	1	2	yes	—
446	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
447	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
448	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
449	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
450	C₂×C₄	#14524051	1	2	yes	—
451	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
452	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
453	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
455	C₂×C₄	#14524089	1	1	yes	characteristic
456	C₂×C₄	#14524090	1	1	yes	characteristic
457	C₈	—	4	2	no	—
458	C₈	—	4	2	no	—
459	C₈	—	268	2	no	—
460	C₈	—	134	4	no	—
461	C₂²	#117×D₆₇	1	1	yes	characteristic
462	C₂²	C₂×#14524008	1	1	yes	characteristic
463	C₂²	#16863876	1	1	yes	characteristic
464	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
465	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
466	C₂²	C₂×#14524051	1	1	yes	characteristic
467	C₂²	#16864162	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	4	1	no	—
473	C₄	—	268	1	no	—
474	C₄	—	268	1	no	—
475	C₄	—	67	4	no	—
476	C₄	—	134	2	no	—
477	C₄	—	134	2	no	—
478	C₄	—	134	2	no	—
479	C₄	—	134	2	no	—
480	C₄	—	134	2	no	—
481	C₄	#16863520	1	4	yes	—
482	C₄	#16864102	1	1	yes	characteristic
483	C₄	#16864115	1	1	yes	characteristic
484	C₄	#16864123	1	1	yes	characteristic
485	C₄	#16864128	1	1	yes	characteristic
486	C₂	#22563161	1	1	yes	characteristic
487	C₂	#431×D₆₇	1	1	yes	characteristic
488	C₂	#22568876	1	1	yes	characteristic
489	C₂	C₂×#16863520	1	1	yes	characteristic
490	C₂	#22569160	1	1	yes	characteristic
491	C₂	#22569317	1	1	yes	characteristic
492	C₂	#22569336	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#22821
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
49. C₄×Dic₁₃₄	C₈	semidirect
72. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
258. C₆₇	#22821	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₃₄	2
#22561143	2
#22560767	2
#22560764	2
#22560360	2
#22560325	2
#22560070	2
C₄⋊C₈×Dic₆₇	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₆₇×#1831	2
#22821	67