Subgroups — #32016459

Order 18176.

622 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11765849	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986130	C₂	1	1	yes	maximal, characteristic
3	#22589115	C₂	1	2	yes	maximal
4	#22589115	C₂	1	1	yes	maximal, characteristic
5	#22589113	C₂	1	1	yes	maximal, characteristic
6	#22589113	C₂	1	2	yes	maximal
7	#22588480	C₂	1	2	yes	maximal
8	#22588477	C₂	1	1	yes	maximal, characteristic
9	#22588247	C₂	1	2	yes	maximal
10	C₄×#14556063	C₂	1	1	yes	maximal, characteristic
11	Q₈×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
12	#17986130	—	2	1	no	—
13	#17986130	—	2	2	no	—
14	#17986130	C₄	1	2	yes	—
15	#17986130	C₄	1	2	yes	—
16	#17986130	C₄	1	4	yes	—
17	Q₈×C₇₁⋊C₈	—	2	1	no	—
18	C₂×#14556063	C₂²	1	1	yes	characteristic
19	C₂×#14556063	C₂²	1	1	yes	characteristic
20	C₂×#14556063	C₂²	1	2	yes	—
21	C₂×#14556063	C₂²	1	2	yes	—
22	#17986023	C₂²	1	1	yes	characteristic
23	#17986023	C₂²	1	2	yes	—
24	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
26	C₄×#11765849	C₂²	1	1	yes	characteristic
27	C₄×#11765849	C₂²	1	2	yes	—
28	C₂×#14556039	C₂²	1	2	yes	—
29	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
30	#17985763	C₂²	1	2	yes	—
31	#17985763	C₂²	1	2	yes	—
32	#17985763	C₂²	1	1	yes	characteristic
33	#17985763	C₂²	1	2	yes	—
34	#17985763	C₂²	1	2	yes	—
35	#17985763	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
37	#446×C₇₁	—	2	1	no	—
38	#446×C₇₁	—	2	2	no	—
39	C₈×Q₈×C₇₁	—	2	1	no	—
40	#423×C₇₁	C₂²	1	1	yes	characteristic
41	#423×C₇₁	C₂²	1	2	yes	—
42	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
44	C₂×#11765849	C₂³	1	1	yes	characteristic
45	C₂×#11765849	C₂³	1	2	yes	—
46	C₂×#11765849	C₂³	1	1	yes	characteristic
47	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
50	#14556063	—	2	2	no	—
51	#14556063	—	2	4	no	—
52	#14556063	—	2	2	no	—
53	#14556063	—	2	4	no	—
54	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
55	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
56	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
57	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
58	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
61	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
62	#14556039	—	2	2	no	—
63	#14556039	—	2	1	no	—
64	#14556039	C₂×C₄	1	4	yes	—
65	#14556039	C₂×C₄	1	4	yes	—
66	C₄×C₇₁⋊C₈	—	2	1	no	—
67	C₄×C₇₁⋊C₈	—	2	2	no	—
68	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
69	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
70	C₂²×Q₈×C₇₁	Q₈	1	1	yes	characteristic
71	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
75	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
76	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
79	C₄×Q₈×C₇₁	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₇₁	Q₈	1	2	yes	—
82	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
84	C₄⋊C₈×C₇₁	—	2	1	no	—
85	C₄⋊C₈×C₇₁	—	2	2	no	—
86	C₄⋊C₈×C₇₁	—	2	2	no	—
87	C₄⋊C₈×C₇₁	—	2	1	no	—
88	#97×C₇₁	—	2	2	no	—
89	#97×C₇₁	—	2	1	no	—
90	C₄×C₈×C₇₁	—	2	1	no	—
91	C₄×C₈×C₇₁	—	2	2	no	—
92	#11765849	—	2	1	no	—
93	#11765849	—	2	1	no	—
94	#11765849	—	2	2	no	—
95	#11765849	—	2	1	no	—
96	#11765849	—	2	2	no	—
97	#11765849	#41	1	2	yes	—
98	C₂×C₇₁⋊C₈	—	2	1	no	—
99	C₂×C₇₁⋊C₈	—	2	1	no	—
100	C₂×C₇₁⋊C₈	—	2	2	no	—
101	C₂×C₇₁⋊C₈	—	2	1	no	—
102	C₂×C₇₁⋊C₈	—	2	2	no	—
103	C₂×C₇₁⋊C₈	—	2	1	no	—
104	C₂×C₇₁⋊C₈	—	2	1	no	—
105	C₂×C₇₁⋊C₈	—	2	1	no	—
106	C₂×C₇₁⋊C₈	—	2	2	no	—
107	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
110	C₂×Q₈×C₇₁	—	4	2	no	—
111	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
112	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
113	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
114	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
116	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
117	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
119	M₁₆×C₇₁	—	2	1	no	—
120	M₁₆×C₇₁	—	2	2	no	—
121	M₁₆×C₇₁	—	2	1	no	—
122	M₁₆×C₇₁	—	2	2	no	—
123	M₁₆×C₇₁	—	2	1	no	—
124	M₁₆×C₇₁	#41	1	2	yes	—
125	C₂×C₈×C₇₁	—	2	1	no	—
126	C₂×C₈×C₇₁	—	2	1	no	—
127	C₂×C₈×C₇₁	—	2	2	no	—
128	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
129	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
130	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
132	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
133	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
134	C₄⋊C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
135	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
137	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
138	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
139	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₇₁	C₂×D₄	1	2	yes	—
141	C₄²×C₇₁	C₂×D₄	1	2	yes	—
142	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
143	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
144	Q₈×C₇₁	—	2	2	no	—
145	Q₈×C₇₁	—	2	2	no	—
146	Q₈×C₇₁	—	4	2	no	—
147	C₂×C₄×C₇₁	—	2	2	no	—
148	C₂×C₄×C₇₁	—	4	1	no	—
149	C₂×C₄×C₇₁	—	4	2	no	—
150	C₂×C₄×C₇₁	—	4	2	no	—
151	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
157	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
165	C₈×C₇₁	—	2	2	no	—
166	C₈×C₇₁	—	4	1	no	—
167	C₈×C₇₁	—	4	2	no	—
168	C₇₁⋊C₈	—	2	2	no	—
169	C₇₁⋊C₈	—	2	4	no	—
170	C₇₁⋊C₈	—	2	2	no	—
171	C₇₁⋊C₈	—	2	2	no	—
172	C₇₁⋊C₈	—	4	1	no	—
173	C₇₁⋊C₈	—	4	2	no	—
174	C₂²×C₇₁	—	2	1	no	—
175	C₂²×C₇₁	—	2	1	no	—
176	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₇₁	#557	1	1	yes	characteristic
178	C₂²×C₇₁	#557	1	1	yes	characteristic
179	C₄×C₇₁	—	2	1	no	—
180	C₄×C₇₁	—	2	2	no	—
181	C₄×C₇₁	—	2	2	no	—
182	C₄×C₇₁	—	2	1	no	—
183	C₄×C₇₁	—	2	1	no	—
184	C₄×C₇₁	—	2	2	no	—
185	C₄×C₇₁	—	2	2	no	—
186	C₄×C₇₁	#430	1	2	yes	—
187	C₄×C₇₁	C₄×#41	1	1	yes	characteristic
188	C₄×C₇₁	#519	1	1	yes	characteristic
189	C₄×C₇₁	#548	1	1	yes	characteristic
190	C₄×C₇₁	#548	1	1	yes	characteristic
191	#22747	—	71	1	no	maximal, Sylow
192	C₂×C₇₁	—	2	1	no	—
193	C₂×C₇₁	—	2	1	no	—
194	C₂×C₇₁	#2351	1	1	yes	characteristic
195	C₂×C₇₁	C₂×#430	1	1	yes	characteristic
196	C₂×C₇₁	#2916	1	1	yes	characteristic
197	Q₈×M₁₆	—	71	1	no	—
198	Q₈×M₁₆	D₇₁	1	1	yes	characteristic
199	C₂×#446	—	71	1	no	—
200	#1950	—	71	2	no	—
201	#1950	—	71	1	no	—
202	#1950	—	71	2	no	—
203	#1950	—	71	1	no	—
204	#1838	—	71	2	no	—
205	#1837	—	71	1	no	—
206	#1778	—	71	2	no	—
207	C₄×#108	—	71	1	no	—
208	C₇₁	#22747	1	1	yes	characteristic, complemented, Sylow
209	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
210	#446	—	2	1	no	—
211	#446	—	2	2	no	—
212	#446	—	71	2	no	—
213	#446	—	71	2	no	—
214	#446	—	71	4	no	—
215	#446	—	142	1	no	—
216	#446	—	142	2	no	—
217	C₈×Q₈	—	2	1	no	—
218	C₈×Q₈	—	142	1	no	—
219	C₂×#108	—	71	1	no	—
220	C₂×#108	—	71	1	no	—
221	C₂×#108	—	71	2	no	—
222	C₂×#108	—	71	2	no	—
223	#423	—	71	1	no	—
224	#423	—	71	2	no	—
225	#423	C₂×D₇₁	1	1	yes	characteristic
226	#423	C₂×D₇₁	1	2	yes	—
227	C₂×C₄⋊C₈	—	71	1	no	—
228	C₂×C₄⋊C₈	—	71	2	no	—
229	C₄×M₁₆	—	71	1	no	—
230	C₄×M₁₆	—	71	2	no	—
231	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
232	C₄×M₁₆	C₂×D₇₁	1	2	yes	—
233	C₂×#97	—	71	2	no	—
234	C₂×C₄×C₈	—	71	1	no	—
235	#341	—	71	2	no	—
236	#341	—	71	2	no	—
237	#341	—	71	1	no	—
238	#341	—	71	2	no	—
239	#341	—	71	2	no	—
240	#341	—	71	1	no	—
241	C₂²×Q₈	Dic₁₄₂	1	1	yes	characteristic
242	C₂×M₁₆	—	71	1	no	—
243	C₂×M₁₆	—	71	2	no	—
244	C₂×M₁₆	—	71	1	no	—
245	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
246	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
248	C₂²×C₈	—	71	1	no	—
249	C₂²×C₈	—	71	1	no	—
250	C₂²×C₈	—	71	2	no	—
251	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
252	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
253	C₄×Q₈	C₄×D₇₁	1	2	yes	—
254	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
255	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
256	C₄×Q₈	C₂²⋊D₇₁	1	2	yes	—
257	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
258	C₂×C₄⋊C₄	Dic₁₄₂	1	2	yes	—
259	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
260	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
261	#108	—	142	2	no	—
262	#108	—	142	4	no	—
263	#108	—	142	2	no	—
264	#108	—	142	4	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	71	2	no	—
270	C₄⋊C₈	—	71	2	no	—
271	C₄⋊C₈	—	71	4	no	—
272	C₄⋊C₈	—	71	4	no	—
273	C₄⋊C₈	—	142	1	no	—
274	C₄⋊C₈	—	142	2	no	—
275	C₄⋊C₈	—	142	2	no	—
276	C₄⋊C₈	—	142	1	no	—
277	#97	—	2	2	no	—
278	#97	—	2	1	no	—
279	#97	—	71	4	no	—
280	#97	—	71	4	no	—
281	#97	—	142	2	no	—
282	#97	—	142	1	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	71	2	no	—
286	C₄×C₈	—	71	2	no	—
287	C₄×C₈	—	142	1	no	—
288	C₄×C₈	—	142	2	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	Dic₇₁⋊C₄	1	1	yes	characteristic
291	C₂×Q₈	Dic₇₁⋊C₄	1	1	yes	characteristic
292	C₂×Q₈	Dic₇₁⋊C₄	1	2	yes	—
293	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
294	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
295	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
296	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
297	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
298	M₁₆	—	2	1	no	—
299	M₁₆	—	2	1	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	71	2	no	—
304	M₁₆	—	142	1	no	—
305	M₁₆	—	142	2	no	—
306	M₁₆	—	142	2	no	—
307	M₁₆	—	142	1	no	—
308	M₁₆	—	142	1	no	—
309	M₁₆	#11765866	1	2	yes	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	71	2	no	—
314	C₂×C₈	—	71	2	no	—
315	C₂×C₈	—	71	4	no	—
316	C₂×C₈	—	142	1	no	—
317	C₂×C₈	—	142	2	no	—
318	C₂×C₈	—	142	1	no	—
319	C₂×C₈	—	142	1	no	—
320	C₂×C₈	—	142	2	no	—
321	C₂×C₈	—	142	1	no	—
322	C₂×C₈	—	142	2	no	—
323	C₂×C₈	—	142	1	no	—
324	C₂×C₈	—	142	1	no	—
325	C₄⋊C₄	Dic₇₁⋊C₄	1	2	yes	—
326	C₄⋊C₄	Dic₇₁⋊C₄	1	2	yes	—
327	C₄⋊C₄	Dic₇₁⋊C₄	1	2	yes	—
328	C₄⋊C₄	Dic₇₁⋊C₄	1	2	yes	—
329	C₄⋊C₄	Dic₇₁⋊C₄	1	2	yes	—
330	C₄⋊C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
331	C₄⋊C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
332	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
333	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
334	C₄²	C₂×C₄×D₇₁	1	2	yes	—
335	C₄²	C₂×C₄×D₇₁	1	2	yes	—
336	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
337	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
338	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
339	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
340	C₂³	C₂²×Dic₁₄₂	1	1	yes	characteristic
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	Q₈	—	4	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
349	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
350	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
351	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
352	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
353	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
354	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
355	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
358	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
359	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
360	C₂×C₄	#14556175	1	1	yes	characteristic
361	C₂×C₄	#14556176	1	1	yes	characteristic
362	C₈	—	2	2	no	—
363	C₈	—	4	1	no	—
364	C₈	—	4	2	no	—
365	C₈	—	284	1	no	—
366	C₈	—	284	2	no	—
367	C₈	—	142	2	no	—
368	C₈	—	142	2	no	—
369	C₈	—	142	4	no	—
370	C₈	—	142	2	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	C₂²×Dic₇₁⋊C₄	1	1	yes	characteristic
374	C₂²	#17986725	1	1	yes	characteristic
375	C₂²	#17986728	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	#17986047	1	2	yes	—
384	C₄	C₄×#11765866	1	1	yes	characteristic
385	C₄	#17986475	1	1	yes	characteristic
386	C₄	#17986656	1	1	yes	characteristic
387	C₄	#17986660	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	#22591258	1	1	yes	characteristic
391	C₂	C₂×#17986047	1	1	yes	characteristic
392	C₂	#22596949	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	Q₈×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₁	C₂²×Q₈
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	Q₈×M₁₆	D₇₁
hypocenter	C₇₁	#22747
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
208. C₇₁	#22747	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11765849	2
C₂×#17986130	2
#22589115	2
#22589113	2
#22588480	2
#22588477	2
#22588247	2
C₄×#14556063	2
Q₈×M₁₆×C₇₁	2
#22747	71