Subgroups — #30874641

Order 17152.

832 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863525	C₂	1	2	yes	maximal
2	C₄²×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
3	#22560930	C₂	1	2	yes	maximal
4	#22560929	C₂	1	2	yes	maximal
5	#22560924	C₂	1	2	yes	maximal
6	#22560360	C₂	1	1	yes	maximal, characteristic
7	#22560360	C₂	1	1	yes	maximal, characteristic
8	#22560064	C₂	1	1	yes	maximal, characteristic
9	#22560062	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1842	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	#16863525	—	2	8	no	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
19	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
20	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
22	C₂×#14523954	C₂²	1	2	yes	—
23	C₂×#14523953	C₂²	1	2	yes	—
24	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
25	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
26	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
27	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
28	C₄×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
29	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
33	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
34	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
36	#16863151	C₂²	1	2	yes	—
37	#16863150	C₂²	1	2	yes	—
38	#16863150	C₂²	1	2	yes	—
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
42	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₄³×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄×Dic₁₃₄	—	2	4	no	—
48	C₄×Dic₁₃₄	—	2	4	no	—
49	C₄×Dic₁₃₄	D₄	1	4	yes	—
50	C₄×Dic₁₃₄	Q₈	1	4	yes	—
51	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₆₇	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×Dic₆₇⋊C₄	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	C₂×C₄	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	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
73	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
74	Dic₆₇⋊C₈	—	2	4	no	—
75	Dic₆₇⋊C₈	—	2	4	no	—
76	Dic₆₇⋊C₈	—	4	2	no	—
77	Dic₆₇⋊C₈	—	4	2	no	—
78	#14523954	—	2	4	no	—
79	#14523953	—	2	4	no	—
80	C₈×Dic₆₇	—	2	4	no	—
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	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
85	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
86	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	—	2	4	no	—
97	C₄⋊C₈×C₆₇	—	2	2	no	—
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	4	2	no	—
100	C₂×Dic₁₃₄	—	2	2	no	—
101	C₂×Dic₁₃₄	—	2	2	no	—
102	C₂×Dic₁₃₄	C₄⋊C₄	1	4	yes	—
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	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	—	2	2	no	—
116	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
117	C₄⋊Dic₆₇	—	2	2	no	—
118	C₄⋊Dic₆₇	—	2	2	no	—
119	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
120	Dic₆₇⋊C₄	—	2	2	no	—
121	Dic₆₇⋊C₄	—	2	2	no	—
122	Dic₆₇⋊C₄	—	2	2	no	—
123	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
124	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
125	Dic₆₇⋊C₄	M₁₆	1	4	yes	—
126	C₄×Dic₆₇	—	2	2	no	—
127	C₄×Dic₆₇	—	2	2	no	—
128	C₄×Dic₆₇	—	2	2	no	—
129	C₄×Dic₆₇	—	2	1	no	—
130	C₄×Dic₆₇	—	2	1	no	—
131	C₄×Dic₆₇	—	2	1	no	—
132	C₄×Dic₆₇	—	2	1	no	—
133	C₄×Dic₆₇	—	2	2	no	—
134	C₄×Dic₆₇	—	2	2	no	—
135	C₄×Dic₆₇	—	4	1	no	—
136	C₄×Dic₆₇	—	4	1	no	—
137	C₄×Dic₆₇	M₁₆	1	4	yes	—
138	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
139	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
140	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
141	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	4	1	no	—
161	C₄²×C₆₇	C₂×D₄	1	2	yes	—
162	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
163	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
164	C₄²×C₆₇	#41	1	2	yes	—
165	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
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	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
180	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₆₇	#131	1	2	yes	—
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
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	—	2	1	no	—
202	C₂×Dic₆₇	—	2	1	no	—
203	C₂×Dic₆₇	—	2	1	no	—
204	C₂×Dic₆₇	—	2	1	no	—
205	C₂×Dic₆₇	—	2	1	no	—
206	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
207	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
208	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
209	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
210	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
211	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
212	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
213	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
214	Dic₁₃₄	—	2	8	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₆₇⋊C₈	—	2	4	no	—
218	C₆₇⋊C₈	—	4	2	no	—
219	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
221	C₂²×C₆₇	#520	1	1	yes	characteristic
222	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
223	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
224	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
225	C₂²×C₆₇	#568	1	1	yes	characteristic
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	#446	1	4	yes	complemented
232	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
233	C₄×C₆₇	#550	1	1	yes	characteristic
234	C₄×C₆₇	#557	1	1	yes	characteristic
235	C₄×C₆₇	Q₈²	1	1	yes	characteristic
236	Dic₆₇	—	2	2	no	—
237	Dic₆₇	—	2	2	no	—
238	Dic₆₇	—	2	2	no	—
239	Dic₆₇	—	2	2	no	—
240	Dic₆₇	—	2	2	no	—
241	Dic₆₇	—	4	1	no	—
242	Dic₆₇	—	4	1	no	—
243	Dic₆₇	#423	1	4	yes	—
244	#22722	—	67	1	no	maximal, Sylow
245	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
247	C₂×C₆₇	#2906	1	1	yes	characteristic
248	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
249	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
250	C₂×C₆₇	#2989	1	1	yes	characteristic
251	C₂×C₆₇	#2990	1	1	yes	characteristic
252	C₂×#446	—	67	2	no	—
253	C₄²×Q₈	—	67	1	no	—
254	#1940	—	67	2	no	—
255	#1940	—	67	2	no	—
256	#1939	—	67	2	no	—
257	#1842	—	67	1	no	—
258	#1842	—	67	1	no	—
259	#1842	D₆₇	1	1	yes	characteristic
260	#1767	—	67	1	no	—
261	#1767	—	67	1	no	—
262	C₄×C₄⋊C₈	—	67	1	no	—
263	C₆₇	#22722	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₄×Q₈	—	67	1	no	—
265	C₂×C₄×Q₈	—	67	1	no	—
266	C₂×C₄×Q₈	—	67	1	no	—
267	#446	—	134	8	no	—
268	C₂×C₄⋊C₈	—	67	2	no	—
269	C₂×C₄⋊C₈	—	67	2	no	—
270	C₂×C₄⋊C₈	—	67	1	no	—
271	C₂×C₄⋊C₈	—	67	2	no	—
272	C₂×C₄⋊C₈	—	67	2	no	—
273	C₂×C₄⋊C₈	—	67	1	no	—
274	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
275	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
277	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
278	C₂×#97	—	67	2	no	—
279	C₂×#97	—	67	2	no	—
280	C₂×C₄×C₈	—	67	2	no	—
281	C₄×C₄⋊C₄	—	67	1	no	—
282	C₄×C₄⋊C₄	—	67	1	no	—
283	C₄×C₄⋊C₄	—	67	1	no	—
284	C₄×C₄⋊C₄	—	67	1	no	—
285	C₄×C₄⋊C₄	—	67	1	no	—
286	C₄×C₄⋊C₄	—	67	1	no	—
287	C₄×C₄⋊C₄	—	67	1	no	—
288	C₄×C₄⋊C₄	—	67	1	no	—
289	C₄×C₄⋊C₄	—	67	1	no	—
290	C₄³	—	67	1	no	—
291	C₄³	—	67	1	no	—
292	C₄³	C₂×D₆₇	1	1	yes	characteristic
293	#336	—	67	2	no	—
294	#336	—	67	2	no	—
295	#336	—	67	2	no	—
296	C₂²×Q₈	—	67	1	no	—
297	C₂²×C₈	—	67	2	no	—
298	C₂²×C₈	—	67	2	no	—
299	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
300	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
301	C₄×Q₈	—	67	4	no	—
302	C₄×Q₈	—	67	4	no	—
303	C₄×Q₈	—	134	4	no	—
304	C₄×Q₈	—	134	4	no	—
305	C₂×C₄⋊C₄	—	67	1	no	—
306	C₂×C₄⋊C₄	—	67	1	no	—
307	C₂×C₄⋊C₄	—	67	1	no	—
308	C₂×C₄⋊C₄	—	67	1	no	—
309	C₂×C₄⋊C₄	—	67	1	no	—
310	C₂×C₄⋊C₄	—	67	1	no	—
311	C₂×C₄⋊C₄	—	67	1	no	—
312	C₂×C₄⋊C₄	—	67	1	no	—
313	C₂×C₄⋊C₄	—	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	—	67	1	no	—
321	C₂×C₄²	—	67	1	no	—
322	C₂×C₄²	—	67	1	no	—
323	C₂×C₄²	—	67	1	no	—
324	C₂×C₄²	—	67	1	no	—
325	C₂×C₄²	—	67	1	no	—
326	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
327	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
328	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
329	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
330	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
332	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
333	C₄⋊C₈	—	2	4	no	—
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	—	4	2	no	—
337	C₄⋊C₈	—	268	2	no	—
338	C₄⋊C₈	—	268	2	no	—
339	C₄⋊C₈	—	67	4	no	—
340	C₄⋊C₈	—	134	4	no	—
341	C₄⋊C₈	—	134	4	no	—
342	C₄⋊C₈	—	134	2	no	—
343	#97	—	134	4	no	—
344	#97	—	134	4	no	—
345	C₄×C₈	—	134	4	no	—
346	#95	—	67	1	no	—
347	#95	—	67	1	no	—
348	#95	—	67	1	no	—
349	#95	—	67	1	no	—
350	#95	—	67	1	no	—
351	#95	—	67	1	no	—
352	C₂×Q₈	—	67	4	no	—
353	C₂×Q₈	—	134	2	no	—
354	C₂×Q₈	—	134	2	no	—
355	C₂²×C₄	—	67	1	no	—
356	C₂²×C₄	—	67	1	no	—
357	C₂²×C₄	—	67	1	no	—
358	C₂²×C₄	—	67	1	no	—
359	C₂²×C₄	—	67	1	no	—
360	C₂²×C₄	—	67	1	no	—
361	C₂²×C₄	—	67	1	no	—
362	C₂²×C₄	—	67	1	no	—
363	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	67	4	no	—
377	C₂×C₈	—	134	2	no	—
378	C₂×C₈	—	134	2	no	—
379	C₂×C₈	—	134	2	no	—
380	C₂×C₈	—	134	2	no	—
381	C₂×C₈	—	134	2	no	—
382	C₄⋊C₄	—	67	4	no	—
383	C₄⋊C₄	—	67	4	no	—
384	C₄⋊C₄	—	67	4	no	—
385	C₄⋊C₄	—	67	4	no	—
386	C₄⋊C₄	—	134	2	no	—
387	C₄⋊C₄	—	134	2	no	—
388	C₄⋊C₄	—	134	2	no	—
389	C₄⋊C₄	—	134	2	no	—
390	C₄⋊C₄	—	134	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	4	1	no	—
397	C₄²	—	268	1	no	—
398	C₄²	—	268	1	no	—
399	C₄²	—	67	4	no	—
400	C₄²	—	67	2	no	—
401	C₄²	—	67	2	no	—
402	C₄²	—	67	2	no	—
403	C₄²	—	67	2	no	—
404	C₄²	—	134	2	no	—
405	C₄²	—	134	2	no	—
406	C₄²	—	134	2	no	—
407	C₄²	—	134	2	no	—
408	C₄²	—	134	2	no	—
409	C₄²	—	134	1	no	—
410	C₄²	—	134	1	no	—
411	C₄²	—	134	1	no	—
412	C₄²	—	134	1	no	—
413	C₄²	D₄×D₆₇	1	2	yes	—
414	C₄²	Q₈×D₆₇	1	2	yes	—
415	C₄²	Q₈×D₆₇	1	2	yes	—
416	C₄²	#11761627	1	2	yes	—
417	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
418	Q₈	—	134	8	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	67	2	no	—
431	C₂×C₄	—	67	2	no	—
432	C₂×C₄	—	67	2	no	—
433	C₂×C₄	—	67	2	no	—
434	C₂×C₄	—	67	2	no	—
435	C₂×C₄	—	67	2	no	—
436	C₂×C₄	—	67	2	no	—
437	C₂×C₄	—	67	2	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	—	134	1	no	—
445	C₂×C₄	—	134	1	no	—
446	C₂×C₄	—	134	1	no	—
447	C₂×C₄	—	134	1	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	—	134	1	no	—
450	C₂×C₄	—	134	1	no	—
451	C₂×C₄	—	134	1	no	—
452	C₂×C₄	—	134	1	no	—
453	C₂×C₄	—	134	1	no	—
454	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
455	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
456	C₂×C₄	#14524008	1	2	yes	—
457	C₂×C₄	#14524008	1	2	yes	—
458	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
459	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
460	C₂×C₄	#14524049	1	2	yes	—
461	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
465	C₂×C₄	#14524088	1	1	yes	characteristic
466	C₂×C₄	#14524091	1	1	yes	characteristic
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	—	268	2	no	—
470	C₈	—	134	4	no	—
471	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
472	C₂²	C₂×#14524008	1	1	yes	characteristic
473	C₂²	#16863877	1	1	yes	characteristic
474	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
475	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
476	C₂²	C₂×#14524049	1	1	yes	characteristic
477	C₂²	#16864162	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	1	no	—
483	C₄	—	268	1	no	—
484	C₄	—	268	1	no	—
485	C₄	—	67	4	no	—
486	C₄	—	134	2	no	—
487	C₄	—	134	2	no	—
488	C₄	—	134	2	no	—
489	C₄	—	134	2	no	—
490	C₄	—	134	2	no	—
491	C₄	#16863525	1	4	yes	complemented
492	C₄	#16864061	1	1	yes	characteristic
493	C₄	#16864066	1	1	yes	characteristic
494	C₄	#16864121	1	1	yes	characteristic
495	C₄	#16864126	1	1	yes	characteristic
496	C₂	#22563121	1	1	yes	characteristic
497	C₂	#423×D₆₇	1	1	yes	characteristic
498	C₂	#22568774	1	1	yes	characteristic
499	C₂	C₂×#16863525	1	1	yes	characteristic
500	C₂	#22569160	1	1	yes	characteristic
501	C₂	#22569323	1	1	yes	characteristic
502	C₂	#22569331	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1842	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#22722
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
73. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
231. C₄×C₆₇	#446	semidirect
263. C₆₇	#22722	semidirect
491. C₄	#16863525	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863525	2
C₄²×Dic₁₃₄	2
#22560930	2
#22560929	2
#22560924	2
#22560360	2
#22560064	2
#22560062	2
C₄⋊C₈×Dic₆₇	2
C₆₇×#1842	2
#22722	67