Subgroups — #31890768

Order 18176.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597315	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586866	C₂	1	2	yes	maximal
4	#22586195	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986533	C₂²	1	1	yes	characteristic
7	#17986129	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic, complemented
9	#17986078	C₄	1	1	yes	characteristic, complemented
10	#17986070	—	2	1	no	—
11	C₈×D₂₈₄	C₄	1	1	yes	characteristic, complemented
12	C₈×C₂²⋊D₇₁	—	2	1	no	—
13	#17986058	—	2	1	no	—
14	#17986056	C₂²	1	1	yes	characteristic
15	C₂×#14556062	—	2	1	no	—
16	#17985953	—	2	1	no	—
17	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
18	#17985762	—	2	1	no	—
19	#17985737	—	2	2	no	—
20	#17985734	—	2	2	no	—
21	#432×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#335×C₇₁	—	2	1	no	—
24	#334×C₇₁	—	2	1	no	—
25	C₂×#11765848	—	2	1	no	—
26	C₂×C₈×D₇₁	—	2	1	no	—
27	#14556122	—	2	1	no	—
28	#14556120	Q₈	1	1	yes	characteristic
29	#14556116	Q₈	1	1	yes	characteristic
30	#14556111	—	2	1	no	—
31	#14556107	D₄	1	1	yes	characteristic
32	#14556102	D₄	1	1	yes	characteristic
33	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
34	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₇₁	—	2	1	no	—
36	#14556085	—	2	1	no	—
37	C₂×C₄⋊Dic₇₁	—	2	2	no	—
38	C₂×C₄×Dic₇₁	—	2	2	no	—
39	#14556062	—	2	4	no	—
40	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₇₁⋊C₈	—	2	1	no	—
43	Dic₇₁⋊C₈	—	2	1	no	—
44	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
45	(C₂×D₇₁)⋊C₈	—	2	1	no	—
46	(C₂×D₇₁)⋊C₈	—	2	1	no	—
47	#14556039	—	2	1	no	—
48	#14556038	—	2	1	no	—
49	C₈×Dic₇₁	—	2	1	no	—
50	C₇₁⋊#95	—	2	2	no	—
51	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₇₁	—	2	2	no	—
55	C₂²⋊C₈×C₇₁	—	2	2	no	—
56	#97×C₇₁	—	2	2	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₂₈₄	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	4	2	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	#11765848	—	4	2	no	—
67	C₂×C₇₁⋊C₈	—	2	1	no	—
68	C₂×C₇₁⋊C₈	—	2	1	no	—
69	C₈×D₇₁	—	4	2	no	—
70	C₄⋊Dic₇₁	—	2	4	no	—
71	C₄⋊Dic₇₁	—	4	2	no	—
72	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
73	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
74	Dic₇₁⋊C₄	—	2	1	no	—
75	Dic₇₁⋊C₄	—	2	1	no	—
76	Dic₇₁⋊C₄	—	2	1	no	—
77	Dic₇₁⋊C₄	—	2	1	no	—
78	C₂²⋊Dic₇₁	—	2	1	no	—
79	(C₂×D₇₁)⋊C₄	—	2	1	no	—
80	(C₂×D₇₁)⋊C₄	—	2	1	no	—
81	(C₂×D₇₁)⋊C₄	—	2	1	no	—
82	(C₂×D₇₁)⋊C₄	—	2	1	no	—
83	C₄×Dic₇₁	—	2	1	no	—
84	C₄×Dic₇₁	—	8	2	no	—
85	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₁	—	2	2	no	—
89	C₂×C₈×C₇₁	—	4	1	no	—
90	C₂×C₈×C₇₁	—	4	2	no	—
91	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₇₁	#41	1	1	yes	characteristic
99	C₂³×C₇₁	#121	1	1	yes	characteristic
100	C₂²×D₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	4	1	no	—
106	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²⋊D₇₁	—	4	1	no	—
113	C₂²⋊D₇₁	—	4	1	no	—
114	C₂×Dic₇₁	—	2	1	no	—
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	4	4	no	—
118	C₂×Dic₇₁	—	8	2	no	—
119	C₂×Dic₇₁	—	8	2	no	—
120	D₂₈₄	—	4	1	no	—
121	C₄×D₇₁	—	4	1	no	—
122	C₄×D₇₁	—	4	1	no	—
123	Dic₁₄₂	—	4	1	no	—
124	C₈×C₇₁	—	2	1	no	—
125	C₈×C₇₁	—	2	1	no	—
126	C₈×C₇₁	—	2	1	no	—
127	C₈×C₇₁	—	8	2	no	—
128	C₈×C₇₁	#104	1	2	yes	—
129	C₇₁⋊C₈	—	4	1	no	—
130	C₇₁⋊C₈	—	4	1	no	—
131	C₂²×C₇₁	—	2	1	no	—
132	C₂²×C₇₁	—	2	1	no	—
133	C₂²×C₇₁	#380	1	1	yes	characteristic
134	C₂²×C₇₁	#472	1	1	yes	characteristic
135	C₂²×C₇₁	#473	1	1	yes	characteristic
136	C₂×D₇₁	—	4	1	no	—
137	C₂×D₇₁	—	4	1	no	—
138	C₂×D₇₁	—	4	1	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	2	1	no	—
144	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
145	C₄×C₇₁	#421	1	1	yes	characteristic
146	Dic₇₁	—	4	1	no	—
147	Dic₇₁	—	4	1	no	—
148	Dic₇₁	—	4	1	no	—
149	Dic₇₁	—	8	4	no	—
150	#8697	—	71	1	no	maximal, Sylow
151	C₂×C₇₁	—	2	1	no	—
152	C₂×C₇₁	—	2	1	no	—
153	C₂×C₇₁	#1806	1	1	yes	characteristic
154	C₂×C₇₁	#1812	1	1	yes	characteristic
155	C₂×C₇₁	#1816	1	1	yes	characteristic
156	D₇₁	—	8	1	no	—
157	#2966	—	71	1	no	—
158	C₄×#107	—	71	1	no	—
159	#1568	D₇₁	1	1	yes	characteristic, complemented
160	#1509	—	71	2	no	—
161	#1285	—	71	2	no	—
162	C₇₁	#8697	1	1	yes	characteristic, complemented, Sylow
163	#529	—	71	1	no	—
164	#446	—	71	1	no	—
165	C₈×Q₈	—	71	1	no	—
166	#436	—	71	1	no	—
167	#435	—	142	1	no	—
168	C₈×D₄	—	71	1	no	—
169	C₈×D₄	—	142	1	no	—
170	#433	—	142	1	no	—
171	#432	—	71	1	no	—
172	#432	C₂×D₇₁	1	2	yes	—
173	C₂×#107	—	142	1	no	—
174	#405	—	142	1	no	—
175	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	71	2	no	—
177	#340	—	142	1	no	—
178	#335	—	2	1	no	—
179	#334	—	2	1	no	—
180	#328	—	142	2	no	—
181	#327	—	142	2	no	—
182	C₂×M₁₆	—	142	1	no	—
183	C₂²×C₈	—	142	1	no	—
184	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
185	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
186	#126	—	142	1	no	—
187	#125	—	71	1	no	—
188	#124	—	71	1	no	—
189	#123	—	142	1	no	—
190	C₂²⋊Q₈	—	71	1	no	—
191	#121	—	71	1	no	—
192	C₄×Q₈	—	71	1	no	—
193	C₄×D₄	—	71	1	no	—
194	C₄×D₄	—	142	1	no	—
195	#117	—	142	1	no	—
196	C₂×C₄⋊C₄	—	142	2	no	—
197	C₂×C₄²	—	142	2	no	—
198	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
199	#107	—	142	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	71	1	no	—
202	C₄⋊C₈	—	71	1	no	—
203	C₄⋊C₈	—	142	1	no	—
204	C₄⋊C₈	—	142	1	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	—	142	1	no	—
207	C₂²⋊C₈	—	142	1	no	—
208	C₂²⋊C₈	—	142	1	no	—
209	#97	—	2	2	no	—
210	#97	—	142	1	no	—
211	#97	—	142	1	no	—
212	C₄×C₈	—	142	1	no	—
213	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
214	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
217	#95	—	142	2	no	—
218	C₂×Q₈	—	71	1	no	—
219	C₂×D₄	—	71	1	no	—
220	C₂×D₄	—	142	1	no	—
221	C₂²×C₄	—	284	2	no	—
222	C₂²×C₄	—	142	1	no	—
223	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
225	C₂²×C₄	#11765866	1	1	yes	characteristic
226	M₁₆	—	284	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	284	2	no	—
231	C₂×C₈	—	142	1	no	—
232	C₂×C₈	—	142	1	no	—
233	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
237	C₄⋊C₄	—	284	2	no	—
238	C₄⋊C₄	—	71	1	no	—
239	C₄⋊C₄	—	71	1	no	—
240	C₄⋊C₄	—	142	4	no	—
241	C₄⋊C₄	—	142	1	no	—
242	C₄⋊C₄	—	142	1	no	—
243	C₄⋊C₄	—	142	1	no	—
244	C₄⋊C₄	—	142	1	no	—
245	C₂²⋊C₄	—	142	1	no	—
246	C₂²⋊C₄	—	142	1	no	—
247	C₂²⋊C₄	—	142	1	no	—
248	C₂²⋊C₄	—	142	1	no	—
249	C₂²⋊C₄	—	142	1	no	—
250	C₄²	—	568	2	no	—
251	C₄²	—	142	1	no	—
252	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
254	C₄²	Q₈×D₇₁	1	1	yes	characteristic
255	C₄²	#11765864	1	1	yes	characteristic
256	C₂³	—	142	1	no	—
257	C₂³	#14556103	1	1	yes	characteristic
258	Q₈	—	284	1	no	—
259	D₄	—	284	1	no	—
260	D₄	—	284	1	no	—
261	D₄	—	284	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	284	1	no	—
268	C₂×C₄	—	284	1	no	—
269	C₂×C₄	—	284	4	no	—
270	C₂×C₄	—	568	2	no	—
271	C₂×C₄	—	568	2	no	—
272	C₂×C₄	—	142	1	no	—
273	C₂×C₄	—	142	1	no	—
274	C₂×C₄	—	142	1	no	—
275	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
277	C₂×C₄	#14556086	1	1	yes	characteristic
278	C₂×C₄	#14556099	1	1	yes	characteristic
279	C₂×C₄	#14556104	1	1	yes	characteristic
280	C₂×C₄	#14556108	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	8	2	no	—
285	C₈	—	284	1	no	—
286	C₈	—	284	1	no	—
287	C₈	#14556054	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	284	1	no	—
291	C₂²	—	284	1	no	—
292	C₂²	—	284	1	no	—
293	C₂²	#17985856	1	1	yes	characteristic
294	C₂²	#17986270	1	1	yes	characteristic
295	C₂²	#17986275	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	284	1	no	—
302	C₄	—	284	1	no	—
303	C₄	—	284	1	no	—
304	C₄	—	568	4	no	—
305	C₄	C₂×#14556054	1	1	yes	characteristic
306	C₄	#17986015	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	568	1	no	—
310	C₂	#22588357	1	1	yes	characteristic
311	C₂	#22588383	1	1	yes	characteristic
312	C₂	#22588401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985856
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1568	C₂
socle	C₂²×C₇₁	#380
prime radical	#14556102	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1568	D₇₁
hypocenter	C₇₁	#8697
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. C₄×#14556062	C₂	semidirect
4. #22586195	C₂	semidirect
5. C₇₁×#1568	C₂	semidirect
7. #17986129	C₄	semidirect
8. C₈×Dic₁₄₂	C₄	semidirect
9. #17986078	C₄	semidirect
11. C₈×D₂₈₄	C₄	semidirect
159. #1568	D₇₁	semidirect
162. C₇₁	#8697	semidirect

Maximal subgroups

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

subgroup	index
#22597315	2
C₄×#14556062	2
#22586866	2
#22586195	2
C₇₁×#1568	2
#8697	71