Subgroups — #31891925

Order 18176.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597086	C₂	1	1	yes	maximal, characteristic
2	#22587116	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586990	C₂	1	1	yes	maximal, characteristic
4	#22586790	C₂	1	1	yes	maximal, characteristic
5	#22586178	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586174	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556136	—	2	1	no	—
9	#17986549	C₂²	1	1	yes	characteristic
10	#17986078	—	2	1	no	—
11	#17986072	—	2	2	no	—
12	C₈×D₂₈₄	—	2	1	no	—
13	#17986052	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#17985967	—	2	1	no	—
16	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	#17985751	—	2	1	no	—
18	#17985751	—	2	1	no	—
19	#17985746	C₄	1	1	yes	characteristic
20	#17985746	C₄	1	1	yes	characteristic
21	#17985742	C₄	1	1	yes	characteristic
22	#17985742	C₄	1	1	yes	characteristic
23	#17985726	—	2	2	no	—
24	C₂×#108×C₇₁	—	2	1	no	—
25	#423×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	#341×C₇₁	—	2	1	no	—
29	C₂×#11765865	—	2	1	no	—
30	#14556136	—	4	2	no	—
31	#14556136	—	4	2	no	—
32	C₂×#11765848	—	2	2	no	—
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₈×D₇₁	—	2	2	no	—
36	#14556128	C₂×C₄	1	1	yes	characteristic
37	#14556125	C₂×C₄	1	1	yes	characteristic
38	#14556116	—	2	1	no	—
39	#14556102	—	2	1	no	—
40	#14556102	Q₈	1	2	yes	—
41	C₄×D₂₈₄	—	2	1	no	—
42	C₄×D₂₈₄	D₄	1	2	yes	—
43	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
45	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
47	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
48	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
49	(C₂×D₇₁)⋊C₈	—	2	2	no	—
50	(C₂×D₇₁)⋊C₈	—	2	2	no	—
51	C₄×C₇₁⋊C₈	—	2	1	no	—
52	C₄×C₇₁⋊C₈	—	2	1	no	—
53	C₄×C₇₁⋊C₈	—	2	1	no	—
54	C₂×M₁₆×C₇₁	—	2	1	no	—
55	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₁	—	2	1	no	—
58	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	#108×C₇₁	—	4	2	no	—
60	C₄⋊C₈×C₇₁	—	2	1	no	—
61	C₄⋊C₈×C₇₁	—	2	1	no	—
62	C₄⋊C₈×C₇₁	—	2	1	no	—
63	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
65	#97×C₇₁	—	2	1	no	—
66	C₄×C₈×C₇₁	—	2	1	no	—
67	#11765865	—	4	1	no	—
68	#11765865	—	4	1	no	—
69	#11765865	—	4	1	no	—
70	#11765865	—	4	1	no	—
71	C₂×Dic₁₄₂	—	2	1	no	—
72	C₂×D₂₈₄	—	2	1	no	—
73	C₂×D₂₈₄	—	2	1	no	—
74	C₂×D₂₈₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₂²⋊D₇₁	—	2	2	no	—
76	C₂×C₄×D₇₁	—	2	2	no	—
77	#11765848	—	4	4	no	—
78	C₂×C₇₁⋊C₈	—	2	2	no	—
79	C₂×C₇₁⋊C₈	—	2	1	no	—
80	C₂×C₇₁⋊C₈	—	2	1	no	—
81	C₂×C₇₁⋊C₈	—	2	1	no	—
82	C₂×C₇₁⋊C₈	—	2	1	no	—
83	C₂×C₇₁⋊C₈	—	4	1	no	—
84	C₂×C₇₁⋊C₈	—	4	1	no	—
85	C₂×C₇₁⋊C₈	—	4	1	no	—
86	C₂×C₇₁⋊C₈	—	4	1	no	—
87	C₈×D₇₁	—	4	4	no	—
88	C₄⋊Dic₇₁	—	2	1	no	—
89	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
90	(C₂×D₇₁)⋊C₄	—	2	2	no	—
91	(C₂×D₇₁)⋊C₄	—	2	2	no	—
92	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
95	M₁₆×C₇₁	—	2	1	no	—
96	M₁₆×C₇₁	—	2	1	no	—
97	M₁₆×C₇₁	—	2	1	no	—
98	M₁₆×C₇₁	—	4	1	no	—
99	M₁₆×C₇₁	—	4	1	no	—
100	M₁₆×C₇₁	D₈	1	2	yes	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	—	2	1	no	—
103	C₂×C₈×C₇₁	—	2	1	no	—
104	C₂×C₈×C₇₁	—	2	1	no	—
105	C₂×C₈×C₇₁	—	4	1	no	—
106	C₂×C₈×C₇₁	—	4	1	no	—
107	C₂×C₈×C₇₁	—	4	1	no	—
108	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇₁	#41	1	1	yes	characteristic
112	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×D₇₁	—	2	2	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	4	1	no	—
117	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
125	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
127	C₂²⋊D₇₁	—	4	2	no	—
128	C₂²⋊D₇₁	—	4	2	no	—
129	C₂×Dic₇₁	—	2	2	no	—
130	D₂₈₄	—	4	2	no	—
131	D₂₈₄	—	4	1	no	—
132	D₂₈₄	—	4	1	no	—
133	D₂₈₄	—	4	1	no	—
134	D₂₈₄	—	4	1	no	—
135	C₄×D₇₁	—	4	2	no	—
136	C₄×D₇₁	—	4	2	no	—
137	Dic₁₄₂	—	4	1	no	—
138	Dic₁₄₂	—	4	1	no	—
139	C₈×C₇₁	—	2	2	no	—
140	C₈×C₇₁	—	4	1	no	—
141	C₈×C₇₁	—	4	1	no	—
142	C₈×C₇₁	—	4	1	no	—
143	C₈×C₇₁	—	8	1	no	—
144	C₇₁⋊C₈	—	4	2	no	—
145	C₇₁⋊C₈	—	4	1	no	—
146	C₇₁⋊C₈	—	4	1	no	—
147	C₇₁⋊C₈	—	4	1	no	—
148	C₇₁⋊C₈	—	4	1	no	—
149	C₂²×C₇₁	—	2	1	no	—
150	C₂²×C₇₁	—	2	1	no	—
151	C₂²×C₇₁	#385	1	1	yes	characteristic
152	C₂²×C₇₁	#474	1	1	yes	characteristic
153	C₂²×C₇₁	#478	1	1	yes	characteristic
154	C₂×D₇₁	—	4	2	no	—
155	C₂×D₇₁	—	4	2	no	—
156	C₂×D₇₁	—	4	2	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	2	1	no	—
160	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
161	C₄×C₇₁	#440	1	1	yes	characteristic
162	C₄×C₇₁	#443	1	1	yes	characteristic
163	C₄×C₇₁	#444	1	1	yes	characteristic
164	C₄×C₇₁	#447	1	1	yes	characteristic
165	C₄×C₇₁	#452	1	1	yes	characteristic
166	Dic₇₁	—	4	2	no	—
167	#8930	—	71	1	no	maximal, Sylow
168	C₂×C₇₁	—	2	1	no	—
169	C₂×C₇₁	—	2	1	no	—
170	C₂×C₇₁	#1858	1	1	yes	characteristic
171	C₂×C₇₁	#1862	1	1	yes	characteristic
172	C₂×C₇₁	#1866	1	1	yes	characteristic
173	D₇₁	—	8	2	no	—
174	#2935	—	71	1	no	—
175	#1950	D₇₁	1	1	yes	characteristic, complemented
176	#1571	—	71	1	no	—
177	#1540	—	71	1	no	—
178	#1492	—	71	1	no	—
179	#1278	—	71	1	no	—
180	#1277	—	71	1	no	—
181	C₇₁	#8930	1	1	yes	characteristic, complemented, Sylow
182	C₂×#131	—	142	1	no	—
183	#532	—	71	1	no	—
184	#436	—	142	1	no	—
185	#435	—	142	2	no	—
186	C₈×D₄	—	142	1	no	—
187	#431	—	71	1	no	—
188	C₂×#108	—	2	1	no	—
189	#423	C₂×D₇₁	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	71	1	no	—
191	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
192	#408	—	142	1	no	—
193	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
194	C₂×C₄×C₈	—	71	1	no	—
195	#341	—	2	1	no	—
196	#335	—	142	1	no	—
197	#335	—	142	1	no	—
198	#332	—	71	1	no	—
199	#332	—	71	1	no	—
200	#331	—	71	1	no	—
201	#331	—	71	1	no	—
202	D₄⋊C₈	—	142	2	no	—
203	C₂×#41	—	142	1	no	—
204	#131	—	284	2	no	—
205	#131	—	284	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	142	2	no	—
208	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
209	C₂×M₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	142	2	no	—
212	C₂²×C₈	—	142	1	no	—
213	C₂²×C₈	—	142	1	no	—
214	C₄⋊Q₈	—	71	1	no	—
215	#127	—	71	1	no	—
216	#124	—	142	1	no	—
217	#121	—	71	2	no	—
218	#121	—	142	1	no	—
219	C₄×D₄	—	71	2	no	—
220	C₄×D₄	—	142	1	no	—
221	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
222	#108	—	4	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	71	1	no	—
227	C₄⋊C₈	—	71	1	no	—
228	C₄⋊C₈	—	71	1	no	—
229	C₄⋊C₈	—	71	1	no	—
230	C₄⋊C₈	—	142	1	no	—
231	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
232	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
233	C₂²⋊C₈	—	142	2	no	—
234	C₂²⋊C₈	—	142	2	no	—
235	C₂²⋊C₈	—	142	1	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	142	1	no	—
239	C₄×C₈	—	142	1	no	—
240	C₄×C₈	—	142	1	no	—
241	#41	—	284	1	no	—
242	#41	—	284	1	no	—
243	#41	—	284	1	no	—
244	#41	—	284	1	no	—
245	C₂×Q₈	—	142	1	no	—
246	C₂×D₄	—	71	2	no	—
247	C₂×D₄	—	142	2	no	—
248	C₂×D₄	—	142	1	no	—
249	C₂×D₄	—	142	1	no	—
250	C₂²×C₄	—	142	2	no	—
251	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
253	C₂²×C₄	#11765866	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	284	4	no	—
260	M₁₆	#11765850	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	284	4	no	—
269	C₂×C₈	—	284	1	no	—
270	C₂×C₈	—	284	1	no	—
271	C₂×C₈	—	284	1	no	—
272	C₂×C₈	—	284	1	no	—
273	C₂×C₈	—	142	2	no	—
274	C₂×C₈	—	142	1	no	—
275	C₂×C₈	—	142	1	no	—
276	C₂×C₈	—	142	1	no	—
277	C₂×C₈	—	142	1	no	—
278	C₄⋊C₄	—	71	2	no	—
279	C₄⋊C₄	—	142	1	no	—
280	C₂²⋊C₄	—	142	2	no	—
281	C₂²⋊C₄	—	142	2	no	—
282	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
283	C₄²	D₄×D₇₁	1	1	yes	characteristic
284	C₄²	D₄×D₇₁	1	1	yes	characteristic
285	C₄²	#11765865	1	1	yes	characteristic
286	C₂³	—	142	2	no	—
287	C₂³	#14556104	1	1	yes	characteristic
288	Q₈	—	284	1	no	—
289	Q₈	—	284	1	no	—
290	D₄	—	284	2	no	—
291	D₄	—	284	2	no	—
292	D₄	—	284	2	no	—
293	D₄	—	284	1	no	—
294	D₄	—	284	1	no	—
295	D₄	—	284	1	no	—
296	D₄	—	284	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	284	2	no	—
301	C₂×C₄	—	284	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
305	C₂×C₄	#14556090	1	1	yes	characteristic
306	C₂×C₄	#14556096	1	1	yes	characteristic
307	C₂×C₄	#14556106	1	1	yes	characteristic
308	C₂×C₄	#14556113	1	1	yes	characteristic
309	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
310	C₂×C₄	#14556148	1	1	yes	characteristic
311	C₂×C₄	#14556153	1	1	yes	characteristic
312	C₂×C₄	#14556159	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	8	1	no	—
318	C₈	—	284	2	no	—
319	C₈	—	284	1	no	—
320	C₈	—	284	1	no	—
321	C₈	—	284	1	no	—
322	C₈	—	284	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	284	2	no	—
326	C₂²	—	284	2	no	—
327	C₂²	—	284	2	no	—
328	C₂²	#17985870	1	1	yes	characteristic
329	C₂²	#17986276	1	1	yes	characteristic
330	C₂²	#17986296	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	284	2	no	—
335	C₄	C₄×#11765850	1	1	yes	characteristic
336	C₄	#17986097	1	1	yes	characteristic
337	C₄	#17986115	1	1	yes	characteristic
338	C₄	#17986120	1	1	yes	characteristic
339	C₄	#17986132	1	1	yes	characteristic
340	C₄	#17986159	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	568	2	no	—
344	C₂	#22588558	1	1	yes	characteristic
345	C₂	#22588586	1	1	yes	characteristic
346	C₂	#22588612	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556096
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1950	C₂
socle	C₂²×C₇₁	#385
prime radical	#17986549	C₂²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1950	D₇₁
hypocenter	C₇₁	#8930
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
2. #22587116	C₂	semidirect
5. #22586178	C₂	semidirect
6. #22586174	C₂	semidirect
7. C₇₁×#1950	C₂	semidirect
175. #1950	D₇₁	semidirect
181. C₇₁	#8930	semidirect

Maximal subgroups

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

subgroup	index
#22597086	2
#22587116	2
#22586990	2
#22586790	2
#22586178	2
#22586174	2
C₇₁×#1950	2
#8930	71