Subgroups — #30866259

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22560984	C₂	1	4	yes	maximal, complemented
4	#22560982	C₂	1	4	yes	maximal, complemented
5	#22560245	C₂	1	4	yes	maximal
6	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863453	—	2	1	no	—
9	#16863453	—	2	1	no	—
10	#16863449	—	2	1	no	—
11	#16863449	—	2	1	no	—
12	C₂×#14523978	C₂²	1	8	yes	—
13	#16863420	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
17	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	#16863360	C₂²	1	2	yes	—
21	C₄×#11761612	C₂²	1	2	yes	—
22	#16863160	C₂²	1	8	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#432×C₆₇	—	2	1	no	—
25	#432×C₆₇	—	2	1	no	—
26	#431×C₆₇	—	2	1	no	—
27	#431×C₆₇	—	2	1	no	—
28	#423×C₆₇	C₂²	1	2	yes	—
29	#407×C₆₇	C₂²	1	2	yes	—
30	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂×#11761612	C₂³	1	4	yes	—
32	C₂²×C₆₇⋊C₈	—	4	4	no	—
33	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
36	#14523978	—	4	8	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
41	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
43	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
44	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
45	#14523954	—	2	2	no	—
46	C₄×C₆₇⋊C₈	—	2	2	no	—
47	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	—	2	2	no	—
64	C₄⋊C₈×C₆₇	—	2	2	no	—
65	C₂²⋊C₈×C₆₇	—	2	2	no	—
66	C₂²⋊C₈×C₆₇	—	2	2	no	—
67	#97×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	—	2	2	no	—
69	#11761612	—	2	4	no	—
70	#11761612	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	2	8	no	—
72	C₂×C₆₇⋊C₈	—	2	4	no	—
73	C₂×C₆₇⋊C₈	—	4	4	no	—
74	C₂×C₆₇⋊C₈	—	4	4	no	—
75	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	4	2	no	—
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₆₇	—	2	4	no	—
100	M₁₆×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	—	2	2	no	—
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	#108	1	4	yes	complemented
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
156	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
157	C₈×C₆₇	—	4	4	no	—
158	C₆₇⋊C₈	—	2	8	no	—
159	C₆₇⋊C₈	—	4	4	no	—
160	C₂²×C₆₇	—	2	8	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	C₂×#108	1	2	yes	—
168	C₂²×C₆₇	#430	1	2	yes	—
169	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₆₇	#544	1	2	yes	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	C₂×#108	1	2	yes	—
180	C₄×C₆₇	C₂×#108	1	2	yes	—
181	C₄×C₆₇	#518	1	1	yes	characteristic
182	C₄×C₆₇	#519	1	1	yes	characteristic
183	C₄×C₆₇	#537	1	2	yes	—
184	#21969	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	—	2	2	no	—
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	#2316	1	1	yes	characteristic
189	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
190	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
191	#2920	—	67	1	no	—
192	#2920	D₆₇	1	1	yes	characteristic
193	C₂²×C₄⋊C₈	—	67	1	no	—
194	#1950	—	67	4	no	—
195	#1950	—	67	4	no	—
196	#1810	—	67	4	no	—
197	C₆₇	#21969	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₈	—	67	2	no	—
199	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
200	#432	—	2	1	no	—
201	#432	—	2	1	no	—
202	#432	—	134	1	no	—
203	#432	—	134	1	no	—
204	#431	—	2	1	no	—
205	#431	—	2	1	no	—
206	#431	—	134	1	no	—
207	#431	—	134	1	no	—
208	C₂×#108	—	67	8	no	—
209	#423	—	67	2	no	—
210	#423	C₂×D₆₇	1	2	yes	—
211	C₂×C₄⋊C₈	—	67	2	no	—
212	C₂×C₄⋊C₈	—	67	2	no	—
213	C₂×C₄⋊C₈	—	67	2	no	—
214	C₂×C₄⋊C₈	—	67	2	no	—
215	C₂×C₄⋊C₈	—	67	2	no	—
216	C₂×C₄⋊C₈	—	67	2	no	—
217	#407	—	67	2	no	—
218	#407	C₂×D₆₇	1	2	yes	—
219	C₄×M₁₆	—	67	2	no	—
220	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
221	#341	—	67	8	no	—
222	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
223	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
224	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
225	C₂×M₁₆	—	67	4	no	—
226	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
227	C₂²×C₈	—	268	4	no	—
228	C₂²×C₈	—	67	4	no	—
229	C₂²×C₈	—	67	4	no	—
230	C₂²×C₈	—	67	4	no	—
231	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
232	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
243	#108	—	268	8	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	67	4	no	—
247	C₄⋊C₈	—	67	4	no	—
248	C₄⋊C₈	—	67	4	no	—
249	C₄⋊C₈	—	67	4	no	—
250	C₄⋊C₈	—	134	2	no	—
251	C₄⋊C₈	—	134	2	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	134	2	no	—
255	C₂²⋊C₈	—	134	2	no	—
256	#97	—	2	2	no	—
257	#97	—	134	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	134	2	no	—
260	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
267	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
272	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
278	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
279	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
280	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
281	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
282	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	134	4	no	—
286	M₁₆	—	134	4	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	268	4	no	—
289	C₂×C₈	—	268	4	no	—
290	C₂×C₈	—	67	8	no	—
291	C₂×C₈	—	134	4	no	—
292	C₂×C₈	—	134	8	no	—
293	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
294	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
295	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
296	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
297	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
298	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
299	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
300	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
307	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₂³	—	2	2	no	—
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
313	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
314	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	#14523978	1	4	yes	complemented
334	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
335	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
336	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
337	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
338	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
339	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
340	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
341	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
342	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
343	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
346	C₂×C₄	#14524089	1	2	yes	—
347	C₂×C₄	#14524090	1	2	yes	—
348	C₈	—	4	4	no	—
349	C₈	—	268	4	no	—
350	C₈	—	134	8	no	—
351	C₂²	—	2	8	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	C₂×#14523978	1	2	yes	—
359	C₂²	#16863444	1	2	yes	—
360	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
361	C₂²	#16864024	1	2	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	C₂×#14523978	1	2	yes	—
371	C₄	C₂×#14523978	1	2	yes	—
372	C₄	#16863870	1	1	yes	characteristic
373	C₄	#16863873	1	1	yes	characteristic
374	C₄	#16863974	1	2	yes	—
375	C₂	—	2	2	no	—
376	C₂	—	2	2	no	—
377	C₂	—	2	2	no	—
378	C₂	#22562825	1	1	yes	characteristic
379	C₂	C₂²×#14523978	1	1	yes	characteristic
380	C₂	C₂×#16863444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2920	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2920	D₆₇
hypocenter	C₆₇	#21969
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #22560984	C₂	semidirect
4. #22560982	C₂	semidirect
142. C₂×C₄×C₆₇	#108	semidirect
197. C₆₇	#21969	semidirect
333. C₂×C₄	#14523978	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
#22560984	2
#22560982	2
#22560245	2
C₆₇×#2920	2
#21969	67