Subgroups — #31891945

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	#22597083	C₂	1	1	yes	maximal, characteristic
2	#22589116	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587110	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586990	C₂	1	1	yes	maximal, characteristic
5	#22586792	C₂	1	1	yes	maximal, characteristic
6	#22586177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	—	2	1	no	—
9	#17986549	C₂²	1	1	yes	characteristic
10	#17986076	—	2	1	no	—
11	#17986069	—	2	2	no	—
12	#17986064	—	2	1	no	—
13	#17986052	C₂²	1	1	yes	characteristic
14	C₂×#14556064	—	2	1	no	—
15	#17986023	C₂²	1	1	yes	characteristic
16	#17985965	—	2	1	no	—
17	C₄×#11765849	C₂²	1	1	yes	characteristic
18	#17985764	—	2	1	no	—
19	#17985749	—	2	1	no	—
20	#17985749	—	2	1	no	—
21	#17985746	C₄	1	1	yes	characteristic
22	#17985746	C₄	1	1	yes	characteristic
23	#17985742	C₄	1	1	yes	characteristic
24	#17985742	C₄	1	1	yes	characteristic
25	#17985728	—	2	2	no	—
26	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×#11765865	—	2	1	no	—
30	#14556134	—	4	2	no	—
31	#14556134	—	4	2	no	—
32	C₂×#11765849	—	2	1	no	—
33	C₂×#11765849	D₄	1	1	yes	characteristic
34	C₂×#11765849	D₄	1	1	yes	characteristic
35	C₂×#11765848	—	2	2	no	—
36	C₂×C₈×D₇₁	—	2	2	no	—
37	#14556128	C₂×C₄	1	1	yes	characteristic
38	#14556125	C₂×C₄	1	1	yes	characteristic
39	#14556116	—	2	1	no	—
40	#14556102	—	2	1	no	—
41	#14556102	Q₈	1	2	yes	—
42	C₄×D₂₈₄	—	2	1	no	—
43	C₄×D₂₈₄	D₄	1	2	yes	—
44	#14556064	—	4	2	no	—
45	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
47	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
50	(C₂×D₇₁)⋊C₈	—	2	2	no	—
51	(C₂×D₇₁)⋊C₈	—	2	2	no	—
52	#14556039	—	2	1	no	—
53	C₄×C₇₁⋊C₈	—	2	1	no	—
54	C₄×C₇₁⋊C₈	—	2	1	no	—
55	C₂²×C₈×C₇₁	—	2	1	no	—
56	C₂²×C₈×C₇₁	—	2	1	no	—
57	C₂²×C₈×C₇₁	—	2	1	no	—
58	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₁	—	2	1	no	—
60	C₄⋊C₈×C₇₁	—	2	1	no	—
61	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
64	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
65	C₄×C₈×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	#11765849	—	2	1	no	—
78	#11765849	—	2	1	no	—
79	#11765849	—	2	1	no	—
80	#11765849	—	4	1	no	—
81	#11765849	—	4	1	no	—
82	#11765849	SD₁₆	1	2	yes	—
83	#11765848	—	4	4	no	—
84	C₂×C₇₁⋊C₈	—	2	1	no	—
85	C₂×C₇₁⋊C₈	—	2	1	no	—
86	C₂×C₇₁⋊C₈	—	2	1	no	—
87	C₂×C₇₁⋊C₈	—	2	1	no	—
88	C₂×C₇₁⋊C₈	—	4	1	no	—
89	C₈×D₇₁	—	4	4	no	—
90	C₄⋊Dic₇₁	—	2	1	no	—
91	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
92	(C₂×D₇₁)⋊C₄	—	2	2	no	—
93	(C₂×D₇₁)⋊C₄	—	2	2	no	—
94	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	—	2	1	no	—
99	C₂×C₈×C₇₁	—	2	1	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	—	4	1	no	—
103	C₂×C₈×C₇₁	—	4	1	no	—
104	C₂×C₈×C₇₁	—	4	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×SD₁₆	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	AGL(1,ℤ/8)	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	—	4	2	no	—
140	C₈×C₇₁	—	4	1	no	—
141	C₈×C₇₁	—	4	1	no	—
142	C₈×C₇₁	—	4	1	no	—
143	C₈×C₇₁	—	4	1	no	—
144	C₇₁⋊C₈	—	2	2	no	—
145	C₇₁⋊C₈	—	4	1	no	—
146	C₇₁⋊C₈	—	4	1	no	—
147	C₇₁⋊C₈	—	4	1	no	—
148	C₇₁⋊C₈	—	8	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	#476	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×SD₁₆	1	1	yes	characteristic
161	C₄×C₇₁	#442	1	1	yes	characteristic
162	C₄×C₇₁	#443	1	1	yes	characteristic
163	C₄×C₇₁	#444	1	1	yes	characteristic
164	C₄×C₇₁	#449	1	1	yes	characteristic
165	C₄×C₇₁	#450	1	1	yes	characteristic
166	Dic₇₁	—	4	2	no	—
167	#8932	—	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	#1864	1	1	yes	characteristic
172	C₂×C₇₁	#1866	1	1	yes	characteristic
173	D₇₁	—	8	2	no	—
174	#2935	—	71	1	no	—
175	#1950	—	71	1	no	—
176	#1570	—	71	1	no	—
177	#1540	—	71	1	no	—
178	#1492	—	71	1	no	—
179	#1278	—	71	1	no	—
180	#1277	D₇₁	1	1	yes	characteristic, complemented
181	C₇₁	#8932	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	—	142	1	no	—
189	#423	—	71	1	no	—
190	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
192	#408	—	142	1	no	—
193	C₄×M₁₆	—	71	1	no	—
194	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
195	#341	—	142	1	no	—
196	#334	—	142	1	no	—
197	#334	—	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	—	71	1	no	—
207	C₂×M₁₆	—	71	1	no	—
208	C₂×M₁₆	—	142	2	no	—
209	C₂×M₁₆	—	142	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	142	2	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	—	284	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	71	1	no	—
226	C₄⋊C₈	—	71	1	no	—
227	C₄⋊C₈	—	142	1	no	—
228	C₄⋊C₈	—	142	1	no	—
229	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
230	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
231	C₄⋊C₈	D₂₈₄	1	1	yes	characteristic
232	C₄⋊C₈	D₂₈₄	1	1	yes	characteristic
233	C₂²⋊C₈	—	142	2	no	—
234	C₂²⋊C₈	—	142	2	no	—
235	C₂²⋊C₈	—	142	1	no	—
236	#97	—	142	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	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	D₄×D₇₁	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
253	C₂²×C₄	#11765865	1	1	yes	characteristic
254	M₁₆	—	284	4	no	—
255	M₁₆	—	284	1	no	—
256	M₁₆	—	284	1	no	—
257	M₁₆	—	71	2	no	—
258	M₁₆	—	142	1	no	—
259	M₁₆	—	142	1	no	—
260	M₁₆	—	142	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	284	4	no	—
273	C₂×C₈	—	284	1	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	C₂×D₂₈₄	1	1	yes	characteristic
285	C₄²	#11765864	1	1	yes	characteristic
286	C₂³	—	142	2	no	—
287	C₂³	#14556106	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	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
306	C₂×C₄	#14556098	1	1	yes	characteristic
307	C₂×C₄	#14556108	1	1	yes	characteristic
308	C₂×C₄	#14556110	1	1	yes	characteristic
309	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
310	C₂×C₄	#14556151	1	1	yes	characteristic
311	C₂×C₄	#14556152	1	1	yes	characteristic
312	C₂×C₄	#14556155	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	284	1	no	—
319	C₈	—	284	1	no	—
320	C₈	—	284	1	no	—
321	C₈	—	568	1	no	—
322	C₈	—	142	2	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	#17985873	1	1	yes	characteristic
329	C₂²	#17986288	1	1	yes	characteristic
330	C₂²	#17986302	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	#17986084	1	1	yes	characteristic
336	C₄	#17986108	1	1	yes	characteristic
337	C₄	#17986117	1	1	yes	characteristic
338	C₄	#17986122	1	1	yes	characteristic
339	C₄	#17986147	1	1	yes	characteristic
340	C₄	#17986151	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	568	2	no	—
344	C₂	#22588561	1	1	yes	characteristic
345	C₂	#22588602	1	1	yes	characteristic
346	C₂	#22588618	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556098
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1277	C₂
socle	C₂²×C₇₁	#385
prime radical	#17986549	C₂²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1277	D₇₁
hypocenter	C₇₁	#8932
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
2. #22589116	C₂	semidirect
3. #22587110	C₂	semidirect
6. #22586177	C₂	semidirect
7. C₇₁×#1277	C₂	semidirect
180. #1277	D₇₁	semidirect
181. C₇₁	#8932	semidirect

Maximal subgroups

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

subgroup	index
#22597083	2
#22589116	2
#22587110	2
#22586990	2
#22586792	2
#22586177	2
C₇₁×#1277	2
#8932	71