Subgroups — #31888519

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597053	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586987	C₂	1	1	yes	maximal, characteristic
4	#22586836	C₂	1	1	yes	maximal, characteristic
5	#22586242	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586236	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	—	2	1	no	—
9	#17986549	C₂²	1	1	yes	characteristic
10	#17986078	—	2	1	no	—
11	#17986078	C₄	1	2	yes	complemented
12	C₈×D₂₈₄	—	2	1	no	—
13	C₈×D₂₈₄	C₄	1	2	yes	complemented
14	#17986023	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
16	#17985965	—	2	1	no	—
17	C₂×C₈×Dic₇₁	—	2	1	no	—
18	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
19	#17985753	—	2	1	no	—
20	#17985744	C₄	1	2	yes	complemented
21	#17985742	C₄	1	2	yes	complemented
22	#17985734	C₄	1	4	yes	—
23	#431×C₇₁	C₂²	1	1	yes	characteristic
24	#431×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
26	#322×C₇₁	—	2	1	no	—
27	C₈²×C₇₁	—	2	1	no	—
28	C₂×#11765865	—	2	1	no	—
29	#14556134	—	4	2	no	—
30	#14556134	—	4	2	no	—
31	C₂×#11765849	—	2	1	no	—
32	C₂×#11765848	—	2	2	no	—
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₈×D₇₁	—	2	2	no	—
35	#14556128	C₂×C₄	1	1	yes	characteristic
36	#14556125	C₂×C₄	1	1	yes	characteristic
37	#14556116	—	2	1	no	—
38	#14556102	—	2	1	no	—
39	#14556102	C₂×C₄	1	2	yes	—
40	C₄×D₂₈₄	—	2	1	no	—
41	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₇₁	—	2	1	no	—
43	C₂×C₄×Dic₇₁	—	2	1	no	—
44	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
48	(C₂×D₇₁)⋊C₈	—	2	2	no	—
49	(C₂×D₇₁)⋊C₈	—	2	2	no	—
50	C₈×Dic₇₁	—	4	2	no	—
51	C₇₁⋊#95	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₁	—	2	1	no	—
56	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₇₁	—	2	1	no	—
58	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
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	#11765865	—	4	1	no	—
66	#11765865	—	4	1	no	—
67	#11765865	—	4	1	no	—
68	#11765865	—	4	1	no	—
69	C₂×Dic₁₄₂	—	2	1	no	—
70	C₂×D₂₈₄	—	2	1	no	—
71	C₂×D₂₈₄	—	2	1	no	—
72	C₂×D₂₈₄	C₄²	1	2	yes	—
73	C₂×C₂²⋊D₇₁	—	2	2	no	—
74	C₂²×Dic₇₁	—	2	2	no	—
75	C₂×C₄×D₇₁	—	2	2	no	—
76	#11765849	—	4	1	no	—
77	#11765849	—	4	1	no	—
78	#11765848	—	4	4	no	—
79	C₂×C₇₁⋊C₈	—	2	2	no	—
80	C₂×C₇₁⋊C₈	—	2	2	no	—
81	C₂×C₇₁⋊C₈	—	4	1	no	—
82	C₂×C₇₁⋊C₈	—	4	1	no	—
83	C₈×D₇₁	—	4	4	no	—
84	C₄⋊Dic₇₁	—	2	4	no	—
85	C₄⋊Dic₇₁	—	2	1	no	—
86	C₄⋊Dic₇₁	C₄²	1	2	yes	—
87	(C₂×D₇₁)⋊C₄	—	2	2	no	—
88	(C₂×D₇₁)⋊C₄	—	2	2	no	—
89	C₄×Dic₇₁	—	4	1	no	—
90	C₄×Dic₇₁	—	4	1	no	—
91	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₁	—	2	1	no	—
95	C₂×C₈×C₇₁	—	2	1	no	—
96	C₂×C₈×C₇₁	—	2	2	no	—
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₁	D₈	1	2	yes	—
103	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
104	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₁	#41	1	1	yes	characteristic
108	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	—	4	1	no	—
112	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
122	C₂²⋊D₇₁	—	4	2	no	—
123	C₂²⋊D₇₁	—	4	2	no	—
124	C₂×Dic₇₁	—	2	4	no	—
125	C₂×Dic₇₁	—	2	2	no	—
126	C₂×Dic₇₁	—	4	2	no	—
127	C₂×Dic₇₁	—	4	2	no	—
128	D₂₈₄	—	4	2	no	—
129	D₂₈₄	—	4	1	no	—
130	D₂₈₄	—	4	1	no	—
131	D₂₈₄	—	4	2	no	—
132	C₄×D₇₁	—	4	2	no	—
133	C₄×D₇₁	—	4	2	no	—
134	Dic₁₄₂	—	4	1	no	—
135	Dic₁₄₂	—	4	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	—	4	2	no	—
139	C₈×C₇₁	—	4	2	no	—
140	C₈×C₇₁	#102	1	2	yes	—
141	C₈×C₇₁	#102	1	2	yes	—
142	C₇₁⋊C₈	—	4	2	no	—
143	C₇₁⋊C₈	—	4	2	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
148	C₂²×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
149	C₂×D₇₁	—	4	2	no	—
150	C₂×D₇₁	—	4	2	no	—
151	C₂×D₇₁	—	4	2	no	—
152	C₄×C₇₁	—	2	1	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
156	C₄×C₇₁	#416	1	1	yes	characteristic
157	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
158	C₄×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
159	C₄×C₇₁	#443	1	2	yes	—
160	Dic₇₁	—	4	4	no	—
161	Dic₇₁	—	4	2	no	—
162	#8251	—	71	1	no	maximal, Sylow
163	C₂×C₇₁	—	2	1	no	—
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	#1757	1	1	yes	characteristic
166	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
167	C₂×C₇₁	#1764	1	1	yes	characteristic
168	D₇₁	—	8	2	no	—
169	#2931	—	71	1	no	—
170	C₈×C₄⋊C₄	—	71	1	no	—
171	#1540	—	71	1	no	—
172	#1502	—	71	1	no	—
173	#1450	D₇₁	1	1	yes	characteristic, complemented
174	#1303	—	71	1	no	—
175	#1301	—	71	1	no	—
176	C₇₁	#8251	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	142	1	no	—
178	#532	—	71	1	no	—
179	#436	—	71	2	no	—
180	#436	—	142	1	no	—
181	C₈×D₄	—	71	2	no	—
182	C₈×D₄	—	142	1	no	—
183	#431	C₂×D₇₁	1	1	yes	characteristic
184	#431	C₂×D₇₁	1	1	yes	characteristic
185	#423	—	71	1	no	—
186	C₂×C₄⋊C₈	—	71	1	no	—
187	#408	—	142	1	no	—
188	C₂×C₄×C₈	—	142	1	no	—
189	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	71	1	no	—
191	#336	—	142	1	no	—
192	#332	—	71	2	no	—
193	#331	—	71	2	no	—
194	#327	—	71	4	no	—
195	#322	—	2	1	no	—
196	C₈²	—	2	1	no	—
197	C₂×#41	—	142	1	no	—
198	#131	—	284	2	no	—
199	#131	—	284	2	no	—
200	C₂×M₁₆	—	142	1	no	—
201	C₂×M₁₆	—	142	2	no	—
202	C₂²×C₈	—	142	1	no	—
203	C₂²×C₈	—	142	2	no	—
204	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
205	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
206	C₄⋊Q₈	—	71	1	no	—
207	#127	—	71	1	no	—
208	#124	—	142	1	no	—
209	#121	—	71	2	no	—
210	#121	—	142	1	no	—
211	C₄×D₄	—	71	2	no	—
212	C₄×D₄	—	142	1	no	—
213	C₂×C₄⋊C₄	—	142	1	no	—
214	C₂×C₄²	—	142	1	no	—
215	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	71	2	no	—
218	C₄⋊C₈	—	71	2	no	—
219	C₄⋊C₈	—	142	1	no	—
220	C₄⋊C₈	—	142	1	no	—
221	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	142	2	no	—
224	C₂²⋊C₈	—	142	2	no	—
225	C₂²⋊C₈	C₄×D₇₁	1	2	yes	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	284	2	no	—
229	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
230	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
233	#95	—	71	2	no	—
234	#41	—	284	1	no	—
235	#41	—	284	1	no	—
236	#41	—	284	1	no	—
237	#41	—	284	1	no	—
238	C₂×Q₈	—	142	1	no	—
239	C₂×D₄	—	71	2	no	—
240	C₂×D₄	—	142	2	no	—
241	C₂×D₄	—	142	1	no	—
242	C₂×D₄	—	142	1	no	—
243	C₂²×C₄	—	142	2	no	—
244	C₂²×C₄	—	142	2	no	—
245	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
247	C₂²×C₄	#11765865	1	1	yes	characteristic
248	M₁₆	—	284	1	no	—
249	M₁₆	—	284	1	no	—
250	M₁₆	—	284	4	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	284	1	no	—
256	C₂×C₈	—	284	1	no	—
257	C₂×C₈	—	284	4	no	—
258	C₂×C₈	—	142	2	no	—
259	C₂×C₈	—	142	2	no	—
260	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
261	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
262	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	D₅₆₈	1	2	yes	—
265	C₂×C₈	#11765854	1	2	yes	—
266	C₄⋊C₄	—	71	2	no	—
267	C₄⋊C₄	—	142	4	no	—
268	C₄⋊C₄	—	142	1	no	—
269	C₂²⋊C₄	—	142	2	no	—
270	C₂²⋊C₄	—	142	2	no	—
271	C₄²	—	284	1	no	—
272	C₄²	—	284	1	no	—
273	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
276	C₄²	#11765865	1	1	yes	characteristic
277	C₂³	—	142	2	no	—
278	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
279	Q₈	—	284	1	no	—
280	Q₈	—	284	1	no	—
281	D₄	—	284	2	no	—
282	D₄	—	284	2	no	—
283	D₄	—	284	2	no	—
284	D₄	—	284	1	no	—
285	D₄	—	284	1	no	—
286	D₄	—	284	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	284	2	no	—
290	C₂×C₄	—	284	2	no	—
291	C₂×C₄	—	284	2	no	—
292	C₂×C₄	—	284	2	no	—
293	C₂×C₄	—	142	4	no	—
294	C₂×C₄	—	142	2	no	—
295	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
296	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	#14556085	1	1	yes	characteristic
298	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
301	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
302	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
303	C₂×C₄	#14556150	1	1	yes	characteristic
304	C₂×C₄	#14556150	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	284	2	no	—
310	C₈	—	284	2	no	—
311	C₈	#14556050	1	2	yes	—
312	C₈	#14556050	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	284	2	no	—
316	C₂²	—	284	2	no	—
317	C₂²	—	284	2	no	—
318	C₂²	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
319	C₂²	C₄×D₅₆₈	1	1	yes	characteristic
320	C₂²	C₄×#11765854	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	284	4	no	—
325	C₄	—	284	2	no	—
326	C₄	C₂×#14556050	1	1	yes	characteristic
327	C₄	#17985996	1	1	yes	characteristic
328	C₄	C₄×D₅₆₈	1	1	yes	characteristic
329	C₄	C₄×#11765854	1	1	yes	characteristic
330	C₄	#17986114	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	568	2	no	—
334	C₂	#22588141	1	1	yes	characteristic
335	C₂	C₄×#14556050	1	1	yes	characteristic
336	C₂	#22588179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	#17986549	C₂²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#8251
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊Dic₇₁	C₂	semidirect
5. #22586242	C₂	semidirect
6. #22586236	C₂	semidirect
7. C₇₁×#1450	C₂	semidirect
11. #17986078	C₄	semidirect
13. C₈×D₂₈₄	C₄	semidirect
20. #17985744	C₄	semidirect
21. #17985742	C₄	semidirect
173. #1450	D₇₁	semidirect
176. C₇₁	#8251	semidirect

Maximal subgroups

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

subgroup	index
#22597053	2
C₈×C₄⋊Dic₇₁	2
#22586987	2
#22586836	2
#22586242	2
#22586236	2
C₇₁×#1450	2
#8251	71