Subgroups — #31897437

Order 18176.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556128	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22586179	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₄₂	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
9	#17985905	—	2	1	no	—
10	#17985897	—	2	1	no	—
11	#17985897	C₄	1	2	yes	—
12	#17985869	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₁	—	2	1	no	—
14	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
15	#17985754	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	—	2	1	no	—
18	C₄³×C₇₁	C₂²	1	1	yes	characteristic
19	#336×C₇₁	—	2	1	no	—
20	C₂²×Dic₁₄₂	—	2	1	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	C₈	1	4	yes	complemented
24	C₄×Dic₁₄₂	—	4	2	no	—
25	C₄×Dic₁₄₂	—	4	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	1	no	—
27	C₂×C₄⋊Dic₇₁	—	2	1	no	—
28	C₂×C₄⋊Dic₇₁	—	2	1	no	—
29	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
31	C₂×Dic₇₁⋊C₄	—	2	2	no	—
32	C₂×C₄×Dic₇₁	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
34	C₄×C₇₁⋊C₈	—	2	4	no	—
35	C₇₁⋊#95	—	2	2	no	—
36	C₇₁⋊#95	—	2	2	no	—
37	C₂²×C₈×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₁	—	2	2	no	—
46	C₄⋊C₈×C₇₁	Q₈	1	4	yes	complemented
47	C₄×C₈×C₇₁	—	4	2	no	—
48	C₂×Dic₁₄₂	—	2	4	no	—
49	C₂×Dic₁₄₂	—	4	2	no	—
50	C₂×Dic₁₄₂	—	4	2	no	—
51	C₂²×Dic₇₁	—	2	2	no	—
52	C₂²×Dic₇₁	—	2	2	no	—
53	C₂×C₇₁⋊C₈	—	2	4	no	—
54	C₂×C₇₁⋊C₈	—	4	2	no	—
55	C₂×C₇₁⋊C₈	—	4	2	no	—
56	C₄⋊Dic₇₁	—	2	4	no	—
57	C₄⋊Dic₇₁	—	2	2	no	—
58	C₄⋊Dic₇₁	—	2	4	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
62	Dic₇₁⋊C₄	—	4	4	no	—
63	C₄×Dic₇₁	—	4	2	no	—
64	C₄×Dic₇₁	—	4	2	no	—
65	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	—	2	4	no	—
73	C₂×C₈×C₇₁	—	4	2	no	—
74	C₂×C₈×C₇₁	—	4	2	no	—
75	C₄²×C₇₁	—	2	2	no	—
76	C₄²×C₇₁	—	2	2	no	—
77	C₄²×C₇₁	—	2	2	no	—
78	C₄²×C₇₁	—	4	1	no	—
79	C₄²×C₇₁	C₂×C₈	1	2	yes	—
80	C₄²×C₇₁	M₁₆	1	2	yes	—
81	C₄²×C₇₁	D₈	1	2	yes	—
82	C₄²×C₇₁	SD₁₆	1	2	yes	—
83	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₇₁	#41	1	2	yes	—
85	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	1	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₇₁	#102	1	2	yes	—
98	C₂×C₄×C₇₁	#102	1	2	yes	—
99	C₂×C₄×C₇₁	#104	1	2	yes	—
100	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#131	1	2	yes	—
107	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#137	1	2	yes	—
110	C₂×Dic₇₁	—	2	4	no	—
111	C₂×Dic₇₁	—	4	2	no	—
112	C₂×Dic₇₁	—	4	2	no	—
113	C₂×Dic₇₁	—	4	2	no	—
114	C₂×Dic₇₁	—	4	2	no	—
115	C₂×Dic₇₁	—	4	2	no	—
116	Dic₁₄₂	—	4	8	no	—
117	C₈×C₇₁	—	4	4	no	—
118	C₇₁⋊C₈	—	4	4	no	—
119	C₂²×C₇₁	#391	1	1	yes	characteristic
120	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₇₁	#408	1	1	yes	characteristic
122	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
123	C₂²×C₇₁	#419	1	1	yes	characteristic
124	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
125	C₂²×C₇₁	#421	1	1	yes	characteristic
126	C₄×C₇₁	—	2	1	no	—
127	C₄×C₇₁	—	2	1	no	—
128	C₄×C₇₁	—	2	2	no	—
129	C₄×C₇₁	—	2	2	no	—
130	C₄×C₇₁	—	4	1	no	—
131	C₄×C₇₁	D₄⋊C₈	1	4	yes	—
132	C₄×C₇₁	C₈×Q₈	1	2	yes	—
133	C₄×C₇₁	#446	1	2	yes	—
134	C₄×C₇₁	#474	1	2	yes	—
135	C₄×C₇₁	#475	1	2	yes	—
136	Dic₇₁	—	4	4	no	—
137	Dic₇₁	—	8	2	no	—
138	#9865	—	71	1	no	maximal, Sylow
139	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
140	C₂×C₇₁	#1479	1	1	yes	characteristic
141	C₂×C₇₁	#1480	1	1	yes	characteristic
142	C₂×C₇₁	#1485	1	1	yes	characteristic
143	C₂×C₇₁	#1984	1	1	yes	characteristic
144	C₂×C₇₁	#1986	1	1	yes	characteristic
145	C₂×C₇₁	#1988	1	1	yes	characteristic
146	C₄×C₄⋊Q₈	—	71	1	no	—
147	C₄×C₄⋊C₈	—	71	1	no	—
148	C₄×C₄⋊C₈	D₇₁	1	1	yes	characteristic
149	#1279	—	71	4	no	—
150	C₇₁	#9865	1	1	yes	characteristic, complemented, Sylow
151	C₂×C₄⋊Q₈	—	71	1	no	—
152	C₂×C₄×Q₈	—	142	1	no	—
153	C₂×C₄⋊C₈	—	142	1	no	—
154	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
155	C₂×C₄×C₈	—	2	1	no	—
156	C₂×C₄×C₈	—	71	2	no	—
157	#391	—	142	1	no	—
158	#389	—	71	2	no	—
159	#389	—	142	1	no	—
160	#384	—	71	1	no	—
161	C₄×C₄⋊C₄	—	71	2	no	—
162	C₄×C₄⋊C₄	—	142	1	no	—
163	C₄³	C₂×D₇₁	1	1	yes	characteristic
164	#336	—	2	1	no	—
165	#336	—	142	1	no	—
166	C₂²×Q₈	—	142	1	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	142	2	no	—
169	C₄⋊Q₈	—	71	4	no	—
170	C₄⋊Q₈	—	142	2	no	—
171	C₄×Q₈	—	284	2	no	—
172	C₄×Q₈	—	284	2	no	—
173	C₂×C₄⋊C₄	—	71	2	no	—
174	C₂×C₄⋊C₄	—	71	2	no	—
175	C₂×C₄⋊C₄	—	142	1	no	—
176	C₂×C₄⋊C₄	—	142	2	no	—
177	C₂×C₄⋊C₄	—	142	1	no	—
178	C₂×C₄⋊C₄	—	142	1	no	—
179	C₂×C₄²	—	142	2	no	—
180	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
182	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
183	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	142	4	no	—
189	C₄⋊C₈	Dic₁₄₂	1	4	yes	complemented
190	C₄×C₈	—	4	2	no	—
191	C₄×C₈	—	142	4	no	—
192	#95	—	142	2	no	—
193	#95	—	142	2	no	—
194	C₂×Q₈	—	284	2	no	—
195	C₂×Q₈	—	284	2	no	—
196	C₂×Q₈	—	142	4	no	—
197	C₂²×C₄	—	142	2	no	—
198	C₂²×C₄	—	142	2	no	—
199	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	284	2	no	—
210	C₂×C₈	—	284	2	no	—
211	C₂×C₈	—	142	4	no	—
212	C₄⋊C₄	—	284	2	no	—
213	C₄⋊C₄	—	284	4	no	—
214	C₄⋊C₄	—	284	2	no	—
215	C₄⋊C₄	—	71	4	no	—
216	C₄⋊C₄	—	142	4	no	—
217	C₄⋊C₄	—	142	4	no	—
218	C₄⋊C₄	—	142	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	284	2	no	—
224	C₄²	—	284	2	no	—
225	C₄²	C₈×D₇₁	1	2	yes	—
226	C₄²	#11765848	1	2	yes	—
227	C₄²	#11765850	1	2	yes	—
228	C₄²	#11765852	1	2	yes	—
229	C₄²	C₂×Dic₁₄₂	1	2	yes	—
230	C₄²	#11765865	1	2	yes	—
231	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
232	Q₈	—	284	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	284	2	no	—
243	C₂×C₄	—	284	2	no	—
244	C₂×C₄	—	284	2	no	—
245	C₂×C₄	—	284	2	no	—
246	C₂×C₄	—	284	2	no	—
247	C₂×C₄	—	142	4	no	—
248	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
249	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
250	C₂×C₄	#14556048	1	2	yes	—
251	C₂×C₄	#14556048	1	2	yes	—
252	C₂×C₄	#14556054	1	2	yes	—
253	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
254	C₂×C₄	#14556107	1	2	yes	—
255	C₂×C₄	#14556116	1	1	yes	characteristic
256	C₂×C₄	#14556128	1	1	yes	characteristic
257	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
258	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
259	C₂×C₄	#14556134	1	2	yes	—
260	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
261	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
262	C₂×C₄	#14556158	1	2	yes	—
263	C₈	—	4	4	no	—
264	C₈	—	284	4	no	—
265	C₂²	#17985905	1	1	yes	characteristic
266	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
267	C₂²	#17985965	1	1	yes	characteristic
268	C₂²	C₂×#14556048	1	1	yes	characteristic
269	C₂²	#17986007	1	1	yes	characteristic
270	C₂²	C₂×#14556054	1	1	yes	characteristic
271	C₂²	#17986015	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	284	4	no	—
278	C₄	—	568	2	no	—
279	C₄	#17985726	1	4	yes	—
280	C₄	C₈×Dic₁₄₂	1	2	yes	—
281	C₄	#17986129	1	2	yes	—
282	C₄	#17986279	1	2	yes	—
283	C₄	#17986284	1	2	yes	—
284	C₂	C₂×#17985726	1	1	yes	characteristic
285	C₂	#22586734	1	1	yes	characteristic
286	C₂	#22586738	1	1	yes	characteristic
287	C₂	#22586760	1	1	yes	characteristic
288	C₂	#22589276	1	1	yes	characteristic
289	C₂	#22589284	1	1	yes	characteristic
290	C₂	#22589292	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×C₄⋊C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#9865
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
23. #14556128	C₈	semidirect
46. C₄⋊C₈×C₇₁	Q₈	semidirect
150. C₇₁	#9865	semidirect
189. C₄⋊C₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556128	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22586179	2
C₄×C₄⋊C₈×C₇₁	2
#9865	71