Subgroups — #31913935

Order 18176.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985784	C₂	1	1	yes	maximal, characteristic
4	#22588179	C₂	1	1	yes	maximal, characteristic
5	#22586429	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586417	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986415	—	2	2	no	—
10	#17986415	—	2	2	no	—
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556067	—	2	1	no	—
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	—	2	1	no	—
15	C₂×#14556054	C₄	1	1	yes	characteristic
16	#17985993	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985819	—	2	2	no	—
19	#17985784	—	2	2	no	—
20	#17985763	C₄	1	1	yes	characteristic
21	#17985758	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
24	#405×C₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₁	—	2	1	no	—
26	C₄⋊C₁₆×C₇₁	—	2	1	no	—
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556134	—	2	1	no	—
29	#14556134	—	2	1	no	—
30	#14556134	D₄	1	2	yes	—
31	#14556134	Q₈	1	2	yes	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765848	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₁	—	2	1	no	—
36	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
37	#14556068	—	2	1	no	—
38	#14556068	D₄	1	2	yes	—
39	#14556067	—	2	4	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	—	2	1	no	—
42	C₂×C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×C₇₁⋊C₁₆	—	2	1	no	—
44	C₁₆×D₇₁	—	2	4	no	—
45	#14556054	—	2	2	no	—
46	#14556051	C₈	1	1	yes	characteristic
47	#14556051	C₈	1	1	yes	characteristic
48	#14556048	C₈	1	1	yes	characteristic
49	#14556048	C₈	1	1	yes	characteristic
50	C₈×Dic₇₁	—	2	1	no	—
51	C₈×Dic₇₁	—	2	1	no	—
52	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
55	M₃₂×C₇₁	—	2	1	no	—
56	M₃₂×C₇₁	—	2	1	no	—
57	C₂×C₁₆×C₇₁	—	2	1	no	—
58	C₂×C₁₆×C₇₁	—	2	1	no	—
59	C₂×C₁₆×C₇₁	—	2	1	no	—
60	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
61	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
62	#97×C₇₁	—	2	1	no	—
63	C₄×C₈×C₇₁	—	2	1	no	—
64	#11765865	—	2	1	no	—
65	#11765865	—	2	1	no	—
66	#11765865	C₄⋊C₄	1	2	yes	—
67	#11765865	M₁₆	1	2	yes	—
68	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
69	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₂²⋊D₇₁	—	2	1	no	—
71	C₂²×Dic₇₁	—	2	1	no	—
72	C₂×C₄×D₇₁	—	2	1	no	—
73	#11765849	—	2	1	no	—
74	#11765849	C₄⋊C₄	1	2	yes	—
75	#11765848	—	2	2	no	—
76	#11765848	—	2	2	no	—
77	C₂×C₇₁⋊C₈	—	2	1	no	—
78	C₂×C₇₁⋊C₈	—	2	1	no	—
79	C₂×C₇₁⋊C₈	—	2	1	no	—
80	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
81	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
82	C₈×D₇₁	—	2	2	no	—
83	C₈×D₇₁	—	2	2	no	—
84	C₄×Dic₇₁	—	2	1	no	—
85	C₄×Dic₇₁	—	2	1	no	—
86	C₇₁⋊C₁₆	—	2	2	no	—
87	C₇₁⋊C₁₆	—	4	1	no	—
88	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	M₁₆×C₇₁	—	2	1	no	—
90	M₁₆×C₇₁	—	2	1	no	—
91	C₂×C₈×C₇₁	—	2	1	no	—
92	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
98	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₇₁	—	2	1	no	—
101	C₄²×C₇₁	—	2	1	no	—
102	C₁₆×C₇₁	—	2	1	no	—
103	C₁₆×C₇₁	—	2	1	no	—
104	C₁₆×C₇₁	—	4	1	no	—
105	C₂³×C₇₁	#131	1	1	yes	characteristic
106	C₂²×D₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
115	C₂²⋊D₇₁	—	2	2	no	—
116	C₂²⋊D₇₁	—	2	2	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	2	no	—
119	C₂×Dic₇₁	—	4	1	no	—
120	C₂×Dic₇₁	—	4	1	no	—
121	D₂₈₄	—	2	1	no	—
122	D₂₈₄	C₄⋊C₈	1	2	yes	—
123	C₄×D₇₁	—	2	2	no	—
124	C₄×D₇₁	—	2	2	no	—
125	Dic₁₄₂	—	2	1	no	—
126	Dic₁₄₂	C₄⋊C₈	1	2	yes	—
127	C₈×C₇₁	—	4	1	no	—
128	C₈×C₇₁	#120	1	1	yes	characteristic
129	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
130	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₇₁	#123	1	1	yes	characteristic
132	C₇₁⋊C₈	—	2	2	no	—
133	C₇₁⋊C₈	—	2	1	no	—
134	C₇₁⋊C₈	—	2	1	no	—
135	C₂²×C₇₁	—	2	1	no	—
136	C₂²×C₇₁	—	2	1	no	—
137	C₂²×C₇₁	#424	1	1	yes	characteristic
138	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₇₁	#435	1	1	yes	characteristic
140	C₂×D₇₁	—	2	2	no	—
141	C₂×D₇₁	—	4	1	no	—
142	C₂×D₇₁	—	4	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	#385	1	1	yes	characteristic
146	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
148	C₄×C₇₁	#435	1	1	yes	characteristic
149	Dic₇₁	—	2	2	no	—
150	Dic₇₁	—	4	2	no	—
151	#12917	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	—	2	1	no	—
153	C₂×C₇₁	—	2	1	no	—
154	C₂×C₇₁	#1850	1	1	yes	characteristic
155	C₂×C₇₁	#2170	1	1	yes	characteristic
156	C₂×C₇₁	#2171	1	1	yes	characteristic
157	D₇₁	—	4	2	no	—
158	C₂×#504	—	71	1	no	—
159	C₂×#364	—	71	1	no	—
160	#2151	D₇₁	1	1	yes	characteristic, complemented
161	C₂×#350	—	71	1	no	—
162	#1764	—	71	1	no	—
163	#1379	—	71	1	no	—
164	#1374	—	71	1	no	—
165	C₇₁	#12917	1	1	yes	characteristic, complemented, Sylow
166	C₂×#131	—	71	1	no	—
167	#504	—	142	2	no	—
168	#504	—	142	2	no	—
169	C₂×M₃₂	—	71	1	no	—
170	C₂×M₃₂	—	142	1	no	—
171	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
172	C₂²×C₁₆	—	71	1	no	—
173	C₂²×C₁₆	—	142	1	no	—
174	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
175	C₂×#104	—	71	1	no	—
176	#416	—	71	1	no	—
177	#405	C₂×D₇₁	1	1	yes	characteristic
178	C₂×C₄×C₈	—	71	1	no	—
179	#364	—	142	2	no	—
180	C₄⋊C₁₆	—	2	1	no	—
181	C₄⋊C₁₆	—	2	1	no	—
182	#350	—	142	2	no	—
183	#341	—	71	1	no	—
184	#339	—	71	1	no	—
185	C₂×#41	—	71	1	no	—
186	#131	—	71	2	no	—
187	#131	—	71	2	no	—
188	#131	—	142	1	no	—
189	#131	—	142	1	no	—
190	C₂×M₁₆	—	71	1	no	—
191	C₂×M₁₆	—	142	1	no	—
192	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
193	C₂²×C₈	—	71	1	no	—
194	C₂²×C₈	—	142	1	no	—
195	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
196	#117	C₄×D₇₁	1	1	yes	characteristic
197	C₂×C₄²	—	71	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	71	2	no	—
201	M₃₂	—	142	4	no	—
202	M₃₂	—	142	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	142	4	no	—
207	C₂×C₁₆	—	142	1	no	—
208	C₂×C₁₆	—	142	1	no	—
209	C₂×C₁₆	—	142	1	no	—
210	C₂×C₁₆	—	142	1	no	—
211	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
212	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
213	#104	—	142	2	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#102	—	71	1	no	—
217	#102	—	71	1	no	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	142	1	no	—
221	C₄×C₈	—	142	1	no	—
222	#41	—	71	2	no	—
223	#41	—	71	2	no	—
224	#41	—	142	1	no	—
225	#41	—	142	1	no	—
226	C₂×Q₈	—	71	1	no	—
227	C₂×D₄	—	71	1	no	—
228	C₂×D₄	—	142	1	no	—
229	C₂²×C₄	—	142	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	71	2	no	—
235	M₁₆	—	142	1	no	—
236	M₁₆	—	142	2	no	—
237	M₁₆	—	142	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	71	1	no	—
241	C₂×C₈	—	142	1	no	—
242	C₂×C₈	—	142	1	no	—
243	C₂×C₈	—	142	1	no	—
244	C₂×C₈	—	142	2	no	—
245	C₂×C₈	—	142	2	no	—
246	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
247	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
250	C₂×C₈	#11765865	1	1	yes	characteristic
251	C₂×C₈	#11765866	1	1	yes	characteristic
252	C₄⋊C₄	C₈×D₇₁	1	1	yes	characteristic
253	C₄⋊C₄	C₈×D₇₁	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	142	1	no	—
257	C₄²	—	142	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	284	1	no	—
262	C₁₆	—	142	2	no	—
263	C₂³	—	142	1	no	—
264	C₂³	#14556134	1	1	yes	characteristic
265	Q₈	—	71	2	no	—
266	Q₈	—	142	1	no	—
267	D₄	—	71	2	no	—
268	D₄	—	142	2	no	—
269	D₄	—	142	2	no	—
270	D₄	—	142	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	284	1	no	—
274	C₂×C₄	—	284	1	no	—
275	C₂×C₄	—	142	2	no	—
276	C₂×C₄	—	142	1	no	—
277	C₂×C₄	—	142	2	no	—
278	C₂×C₄	—	142	2	no	—
279	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
281	C₂×C₄	#14556090	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
284	C₂×C₄	#14556136	1	1	yes	characteristic
285	C₈	—	4	1	no	—
286	C₈	—	142	1	no	—
287	C₈	—	142	1	no	—
288	C₈	—	142	2	no	—
289	C₈	#14556096	1	1	yes	characteristic
290	C₈	#14556104	1	1	yes	characteristic
291	C₈	#14556106	1	1	yes	characteristic
292	C₈	#14556113	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	284	1	no	—
296	C₂²	—	284	1	no	—
297	C₂²	—	142	2	no	—
298	C₂²	#17986024	1	1	yes	characteristic
299	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
300	C₂²	#17986069	1	1	yes	characteristic
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	284	2	no	—
304	C₄	—	142	2	no	—
305	C₄	#17985870	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
307	C₄	#17986064	1	1	yes	characteristic
308	C₄	#17986068	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	284	2	no	—
312	C₂	#22588516	1	1	yes	characteristic
313	C₂	#22590175	1	1	yes	characteristic
314	C₂	#22590180	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2151	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2151	D₇₁
hypocenter	C₇₁	#12917
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. C₂×#17985819	C₂	semidirect
5. #22586429	C₂	semidirect
6. #22586417	C₂	semidirect
7. C₇₁×#2151	C₂	semidirect
160. #2151	D₇₁	semidirect
165. C₇₁	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986415	2
C₂×#17985819	2
C₂×#17985784	2
#22588179	2
#22586429	2
#22586417	2
C₇₁×#2151	2
#12917	71