Subgroups — #31913748

Order 18176.

426 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986036	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₇₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22588366	C₂	1	1	yes	maximal, characteristic
4	#22586432	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586431	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₁	C₂²	1	1	yes	characteristic
9	C₂×#14556086	—	2	1	no	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17986036	—	2	2	no	—
12	#17986036	C₄	1	4	yes	complemented
13	C₂×#14556062	C₂²	1	1	yes	characteristic
14	C₂×#14556062	C₂²	1	1	yes	characteristic
15	C₂×#14556059	—	2	1	no	—
16	#17985856	—	2	1	no	—
17	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
18	#17985761	—	2	1	no	—
19	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₈×D₇₁	—	2	1	no	—
25	C₂×C₈×D₇₁	—	2	1	no	—
26	C₂×C₈×D₇₁	—	2	1	no	—
27	C₂×C₈×D₇₁	—	2	1	no	—
28	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₇₁	D₄	1	2	yes	—
30	#14556086	—	2	4	no	—
31	#14556086	—	4	2	no	—
32	C₂×C₄⋊Dic₇₁	—	2	1	no	—
33	C₂×C₄⋊Dic₇₁	—	2	1	no	—
34	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
35	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
36	C₂×C₄×Dic₇₁	—	4	1	no	—
37	C₂×C₇₁⋊C₁₆	—	2	1	no	—
38	C₂×C₇₁⋊C₁₆	—	2	1	no	—
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	#14556062	—	2	1	no	—
41	#14556062	—	2	1	no	—
42	#14556062	—	2	1	no	—
43	#14556062	C₂×C₄	1	2	yes	—
44	#14556059	—	2	4	no	—
45	C₇₁⋊#95	—	4	1	no	—
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
49	C₂×C₁₆×C₇₁	—	2	1	no	—
50	C₂×C₁₆×C₇₁	—	2	1	no	—
51	C₂×C₁₆×C₇₁	—	2	1	no	—
52	#107×C₇₁	—	2	1	no	—
53	#107×C₇₁	—	2	1	no	—
54	#107×C₇₁	—	2	1	no	—
55	#107×C₇₁	C₂×C₄	1	2	yes	—
56	C₂³×D₇₁	D₈	1	1	yes	characteristic
57	C₂²×Dic₇₁	—	4	1	no	—
58	C₂²×Dic₇₁	—	4	1	no	—
59	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
60	C₂×C₄×D₇₁	—	2	1	no	—
61	C₂×C₄×D₇₁	—	2	1	no	—
62	C₂×C₄×D₇₁	—	2	1	no	—
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₇₁	SD₁₆	1	2	yes	—
66	C₂×C₇₁⋊C₈	—	2	1	no	—
67	C₂×C₇₁⋊C₈	—	2	1	no	—
68	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₈×D₇₁	—	2	2	no	—
70	C₈×D₇₁	—	2	2	no	—
71	C₈×D₇₁	—	2	2	no	—
72	C₈×D₇₁	—	2	2	no	—
73	C₄⋊Dic₇₁	—	2	2	no	—
74	C₄⋊Dic₇₁	—	4	1	no	—
75	C₄⋊Dic₇₁	—	4	1	no	—
76	C₄⋊Dic₇₁	—	4	1	no	—
77	(C₂×D₇₁)⋊C₄	—	4	4	no	—
78	(C₂×D₇₁)⋊C₄	—	8	2	no	—
79	C₄×Dic₇₁	—	4	4	no	—
80	C₇₁⋊C₁₆	—	4	1	no	—
81	C₂²×C₄×C₇₁	—	4	1	no	—
82	C₂²×C₄×C₇₁	—	4	1	no	—
83	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
89	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₄⋊C₄×C₇₁	—	4	1	no	—
92	C₄⋊C₄×C₇₁	—	4	1	no	—
93	C₄⋊C₄×C₇₁	—	4	1	no	—
94	C₁₆×C₇₁	—	4	1	no	—
95	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
96	C₂²×D₇₁	—	2	1	no	—
97	C₂²×D₇₁	—	2	1	no	—
98	C₂²×D₇₁	—	2	1	no	—
99	C₂²×D₇₁	—	2	1	no	—
100	C₂²×D₇₁	—	2	1	no	—
101	C₂²×D₇₁	—	2	1	no	—
102	C₂²×D₇₁	#102	1	2	yes	—
103	C₂×C₄×C₇₁	—	4	2	no	—
104	C₂×C₄×C₇₁	—	8	1	no	—
105	C₂×C₄×C₇₁	—	8	1	no	—
106	C₂×C₄×C₇₁	—	8	1	no	—
107	C₂×C₄×C₇₁	—	8	1	no	—
108	C₂×C₄×C₇₁	—	8	1	no	—
109	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	4	2	no	—
118	C₂×Dic₇₁	—	8	1	no	—
119	C₂×Dic₇₁	—	8	1	no	—
120	C₂×Dic₇₁	—	8	1	no	—
121	C₂×Dic₇₁	—	8	1	no	—
122	C₂×Dic₇₁	—	8	1	no	—
123	C₂×Dic₇₁	#102	1	2	yes	—
124	C₄×D₇₁	—	2	2	no	—
125	C₄×D₇₁	—	2	2	no	—
126	C₄×D₇₁	—	2	2	no	—
127	C₄×D₇₁	—	2	2	no	—
128	C₈×C₇₁	#117	1	1	yes	characteristic
129	C₈×C₇₁	#117	1	1	yes	characteristic
130	C₈×C₇₁	#123	1	1	yes	characteristic
131	C₈×C₇₁	#123	1	1	yes	characteristic
132	C₇₁⋊C₈	—	2	2	no	—
133	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
134	C₂²×C₇₁	#480	1	1	yes	characteristic
135	C₂²×C₇₁	#480	1	1	yes	characteristic
136	C₂²×C₇₁	#508	1	1	yes	characteristic
137	C₂²×C₇₁	#508	1	1	yes	characteristic
138	C₂²×C₇₁	#509	1	1	yes	characteristic
139	C₂²×C₇₁	#510	1	1	yes	characteristic
140	C₂×D₇₁	—	2	2	no	—
141	C₂×D₇₁	—	2	2	no	—
142	C₂×D₇₁	—	2	2	no	—
143	C₂×D₇₁	—	4	1	no	—
144	C₂×D₇₁	—	4	1	no	—
145	C₂×D₇₁	—	4	1	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₄×C₇₁	—	8	2	no	—
148	C₄×C₇₁	—	16	1	no	—
149	C₄×C₇₁	#381	1	1	yes	characteristic
150	C₄×C₇₁	#419	1	1	yes	characteristic
151	C₄×C₇₁	#483	1	1	yes	characteristic
152	C₄×C₇₁	#483	1	1	yes	characteristic
153	Dic₇₁	—	2	2	no	—
154	Dic₇₁	—	8	2	no	—
155	Dic₇₁	—	16	1	no	—
156	#12868	—	71	1	no	maximal, Sylow
157	C₂×C₇₁	#1823	1	1	yes	characteristic
158	C₂×C₇₁	#2138	1	1	yes	characteristic
159	C₂×C₇₁	#2139	1	1	yes	characteristic
160	C₂×C₇₁	#2234	1	1	yes	characteristic
161	C₂×C₇₁	#2234	1	1	yes	characteristic
162	C₂×C₇₁	#2237	1	1	yes	characteristic
163	C₂×C₇₁	#2237	1	1	yes	characteristic
164	D₇₁	—	4	2	no	—
165	C₂×#427	—	71	1	no	—
166	C₂×C₂²⋊C₁₆	—	71	1	no	—
167	#1808	—	71	1	no	—
168	#1380	—	71	1	no	—
169	#1380	—	71	1	no	—
170	#1380	—	71	1	no	—
171	#1380	D₇₁	1	1	yes	characteristic, complemented
172	C₇₁	#12868	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₈	—	71	1	no	—
174	C₂×#117	—	142	1	no	—
175	C₂²×C₁₆	—	71	1	no	—
176	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
177	#427	—	71	4	no	—
178	#427	—	142	2	no	—
179	C₂×#107	—	71	1	no	—
180	C₂×#107	—	71	1	no	—
181	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
182	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
183	C₂×#106	—	142	1	no	—
184	#380	—	142	1	no	—
185	C₂²⋊C₁₆	—	142	2	no	—
186	#340	—	142	1	no	—
187	C₂³×C₄	—	71	1	no	—
188	C₂²×C₈	—	71	1	no	—
189	C₂²×C₈	—	71	2	no	—
190	C₂²×C₈	—	71	2	no	—
191	C₂²×C₈	—	142	1	no	—
192	C₂²×C₈	—	142	1	no	—
193	C₂²×C₈	—	142	1	no	—
194	C₂²×C₈	—	142	1	no	—
195	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
196	#117	—	284	2	no	—
197	#117	—	142	4	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	142	1	no	—
201	C₂×C₄⋊C₄	—	142	1	no	—
202	C₂×C₂²⋊C₄	—	284	1	no	—
203	C₂×C₂²⋊C₄	—	284	1	no	—
204	C₂×C₄²	—	284	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	142	1	no	—
209	C₂×C₁₆	—	142	1	no	—
210	C₂×C₁₆	—	142	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	—	71	2	no	—
215	#107	—	142	1	no	—
216	#107	—	142	1	no	—
217	#107	—	142	1	no	—
218	#107	C₄×D₇₁	1	2	yes	—
219	#106	—	142	4	no	—
220	#95	—	284	1	no	—
221	C₂⁴	—	71	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	284	1	no	—
225	C₂²×C₄	—	284	1	no	—
226	C₂²×C₄	—	71	1	no	—
227	C₂²×C₄	—	71	2	no	—
228	C₂²×C₄	—	71	2	no	—
229	C₂²×C₄	—	142	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	—	142	1	no	—
232	C₂²×C₄	—	142	1	no	—
233	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
234	C₂×C₈	—	71	2	no	—
235	C₂×C₈	—	142	1	no	—
236	C₂×C₈	—	142	1	no	—
237	C₂×C₈	—	142	2	no	—
238	C₂×C₈	—	142	2	no	—
239	C₂×C₈	—	142	2	no	—
240	C₂×C₈	—	142	2	no	—
241	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
242	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
243	C₂×C₈	#11765864	1	1	yes	characteristic
244	C₂×C₈	#11765865	1	1	yes	characteristic
245	C₂×C₈	#11765865	1	1	yes	characteristic
246	C₂×C₈	#11765866	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	284	1	no	—
252	C₄⋊C₄	—	284	1	no	—
253	C₄⋊C₄	—	284	1	no	—
254	C₄⋊C₄	—	142	2	no	—
255	C₂²⋊C₄	—	284	4	no	—
256	C₂²⋊C₄	—	568	2	no	—
257	C₄²	—	284	4	no	—
258	C₁₆	—	4	1	no	—
259	C₁₆	—	284	1	no	—
260	C₂³	—	71	2	no	—
261	C₂³	—	142	1	no	—
262	C₂³	—	142	1	no	—
263	C₂³	—	142	1	no	—
264	C₂³	—	142	1	no	—
265	C₂³	—	142	1	no	—
266	C₂³	—	142	1	no	—
267	C₂³	D₈×D₇₁	1	1	yes	characteristic
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	284	2	no	—
275	C₂×C₄	—	568	1	no	—
276	C₂×C₄	—	568	1	no	—
277	C₂×C₄	—	568	1	no	—
278	C₂×C₄	—	568	1	no	—
279	C₂×C₄	—	568	1	no	—
280	C₂×C₄	—	71	2	no	—
281	C₂×C₄	—	142	1	no	—
282	C₂×C₄	—	142	1	no	—
283	C₂×C₄	—	142	2	no	—
284	C₂×C₄	—	142	2	no	—
285	C₂×C₄	—	142	2	no	—
286	C₂×C₄	—	142	2	no	—
287	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
288	C₂×C₄	#14556110	1	1	yes	characteristic
289	C₂×C₄	#14556113	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
291	C₂×C₄	#14556157	1	1	yes	characteristic
292	C₂×C₄	#14556160	1	1	yes	characteristic
293	C₈	—	142	2	no	—
294	C₈	#14556085	1	1	yes	characteristic
295	C₈	#14556086	1	1	yes	characteristic
296	C₈	#14556110	1	1	yes	characteristic
297	C₈	#14556113	1	1	yes	characteristic
298	C₂²	—	284	1	no	—
299	C₂²	—	284	1	no	—
300	C₂²	—	284	1	no	—
301	C₂²	—	284	1	no	—
302	C₂²	—	142	2	no	—
303	C₂²	—	142	2	no	—
304	C₂²	—	142	2	no	—
305	C₂²	#102×D₇₁	1	1	yes	characteristic
306	C₂²	#17986311	1	1	yes	characteristic
307	C₂²	#17986314	1	1	yes	characteristic
308	C₂²	#17986430	1	1	yes	characteristic
309	C₂²	#17986432	1	1	yes	characteristic
310	C₂²	#17986433	1	1	yes	characteristic
311	C₂²	#17986438	1	1	yes	characteristic
312	C₄	—	8	2	no	—
313	C₄	—	16	1	no	—
314	C₄	—	568	2	no	—
315	C₄	—	1136	1	no	—
316	C₄	—	142	2	no	—
317	C₄	#17985861	1	1	yes	characteristic
318	C₄	#17986006	1	1	yes	characteristic
319	C₄	#17986328	1	1	yes	characteristic
320	C₄	#17986331	1	1	yes	characteristic
321	C₂	—	284	2	no	—
322	C₂	#22588425	1	1	yes	characteristic
323	C₂	#22590046	1	1	yes	characteristic
324	C₂	#22590049	1	1	yes	characteristic
325	C₂	#22590530	1	1	yes	characteristic
326	C₂	#22590534	1	1	yes	characteristic
327	C₂	#22590547	1	1	yes	characteristic
328	C₂	#22590551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1380	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂³×D₇₁	D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1380	D₇₁
hypocenter	C₇₁	#12868
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. #22586432	C₂	semidirect
5. #22586431	C₂	semidirect
6. #22586431	C₂	semidirect
7. C₇₁×#1380	C₂	semidirect
12. #17986036	C₄	semidirect
171. #1380	D₇₁	semidirect
172. C₇₁	#12868	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986036	2
C₂×(C₂×D₇₁)⋊C₁₆	2
#22588366	2
#22586432	2
#22586431	2
C₇₁×#1380	2
#12868	71