Subgroups — #31895995

Order 18176.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589059	C₂	1	2	yes	maximal
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22586181	C₂	1	2	yes	maximal
5	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985754	—	2	2	no	—
11	#17985752	—	2	2	no	—
12	C₈×C₇₁⋊C₈	—	2	2	no	—
13	C₂×#107×C₇₁	C₄	1	1	yes	characteristic
14	C₂×#107×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
17	#340×C₇₁	C₄	1	2	yes	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	C₂²×C₇₁⋊C₈	—	2	1	no	—
20	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	4	4	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₁	—	2	2	no	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	#107×C₇₁	—	2	2	no	—
36	#107×C₇₁	C₈	1	4	yes	complemented
37	C₄×C₈×C₇₁	D₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	#95×C₇₁	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
51	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₇₁	—	2	2	no	—
53	C₂²×C₄×C₇₁	—	2	2	no	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	2	2	no	—
64	C₄⋊C₄×C₇₁	—	2	2	no	—
65	C₄⋊C₄×C₇₁	—	2	2	no	—
66	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
67	C₄²×C₇₁	—	4	2	no	—
68	C₄²×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₇₁	#41	1	1	yes	characteristic
73	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	4	no	—
77	C₂×C₄×C₇₁	—	4	2	no	—
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	—	4	2	no	—
80	C₂×C₄×C₇₁	—	4	2	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
96	C₈×C₇₁	—	2	2	no	—
97	C₈×C₇₁	C₄⋊C₈	1	4	yes	complemented
98	C₇₁⋊C₈	—	2	4	no	—
99	C₇₁⋊C₈	—	2	2	no	—
100	C₇₁⋊C₈	—	2	2	no	—
101	C₇₁⋊C₈	—	8	2	no	—
102	C₂²×C₇₁	#389	1	1	yes	characteristic
103	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
104	C₂²×C₇₁	#436	1	1	yes	characteristic
105	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
106	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
107	C₂²×C₇₁	#443	1	1	yes	characteristic
108	C₂²×C₇₁	#443	1	1	yes	characteristic
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	—	4	4	no	—
112	C₄×C₇₁	—	8	2	no	—
113	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₇₁	#424	1	1	yes	characteristic
115	C₄×C₇₁	C₈×Q₈	1	1	yes	characteristic
116	C₄×C₇₁	#446	1	1	yes	characteristic
117	C₄×C₇₁	#466	1	1	yes	characteristic
118	C₄×C₇₁	#467	1	1	yes	characteristic
119	C₄×C₇₁	#499	1	1	yes	characteristic
120	C₄×C₇₁	#500	1	1	yes	characteristic
121	#9594	—	71	1	no	maximal, Sylow
122	C₂×C₇₁	C₈×D₈	1	1	yes	characteristic
123	C₂×C₇₁	C₈×Q₁₆	1	1	yes	characteristic
124	C₂×C₇₁	#1589	1	1	yes	characteristic
125	C₂×C₇₁	#1591	1	1	yes	characteristic
126	C₂×C₇₁	#1939	1	1	yes	characteristic
127	C₂×C₇₁	#1942	1	1	yes	characteristic
128	C₂×C₇₁	#1945	1	1	yes	characteristic
129	#1939	—	71	2	no	—
130	C₄×#107	D₇₁	1	1	yes	characteristic
131	C₂×#335	—	71	1	no	—
132	C₂×C₈²	—	71	1	no	—
133	#1279	—	71	2	no	—
134	C₇₁	#9594	1	1	yes	characteristic, complemented, Sylow
135	C₂×#107	Dic₇₁	1	1	yes	characteristic
136	C₂×#107	Dic₇₁	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	71	2	no	—
138	C₂×C₄⋊C₈	—	142	2	no	—
139	C₂×C₄×C₈	—	71	1	no	—
140	C₂×C₄×C₈	—	71	1	no	—
141	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
142	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
143	#340	Dic₇₁	1	2	yes	—
144	#336	—	142	2	no	—
145	#335	—	142	2	no	—
146	C₈²	—	142	2	no	—
147	C₂²×C₈	—	71	1	no	—
148	C₂²×C₈	—	71	1	no	—
149	C₂²×C₈	—	142	2	no	—
150	C₂²×C₈	—	142	1	no	—
151	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	#107	C₇₁⋊C₈	1	4	yes	complemented
159	C₄⋊C₈	—	284	4	no	—
160	C₄⋊C₈	—	142	2	no	—
161	C₄⋊C₈	—	142	2	no	—
162	C₄×C₈	—	71	2	no	—
163	C₄×C₈	—	142	1	no	—
164	C₄×C₈	—	142	1	no	—
165	C₄×C₈	—	142	1	no	—
166	C₄×C₈	Dic₁₄₂	1	2	yes	—
167	C₄×C₈	D₂₈₄	1	2	yes	—
168	#95	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
173	C₂²×C₄	#11765864	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	284	2	no	—
177	C₂×C₈	—	284	2	no	—
178	C₂×C₈	—	284	2	no	—
179	C₂×C₈	—	71	2	no	—
180	C₂×C₈	—	71	2	no	—
181	C₂×C₈	—	142	2	no	—
182	C₂×C₈	—	142	2	no	—
183	C₂×C₈	—	142	1	no	—
184	C₂×C₈	—	142	1	no	—
185	C₂×C₈	—	142	2	no	—
186	C₂×C₈	—	142	1	no	—
187	C₂×C₈	—	142	1	no	—
188	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
189	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
190	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
191	C₂×C₈	#11765849	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
199	C₄²	Q₈×D₇₁	1	1	yes	characteristic
200	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
201	C₄²	#11765866	1	1	yes	characteristic
202	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
213	C₂×C₄	#14556103	1	1	yes	characteristic
214	C₂×C₄	#14556108	1	1	yes	characteristic
215	C₂×C₄	#14556119	1	1	yes	characteristic
216	C₂×C₄	#14556127	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
218	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
219	C₂×C₄	#14556135	1	1	yes	characteristic
220	C₂×C₄	#14556135	1	1	yes	characteristic
221	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
223	C₂×C₄	#14556147	1	1	yes	characteristic
224	C₂×C₄	#14556151	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	568	2	no	—
227	C₈	—	142	4	no	—
228	C₈	—	142	2	no	—
229	C₈	—	142	2	no	—
230	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	complemented
231	C₂²	#17985898	1	1	yes	characteristic
232	C₂²	D₄×C₇₁⋊C₈	1	1	yes	characteristic
233	C₂²	#17986077	1	1	yes	characteristic
234	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
235	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
236	C₂²	#17986116	1	1	yes	characteristic
237	C₂²	#17986116	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
243	C₄	#17986026	1	1	yes	characteristic
244	C₄	Q₈×C₇₁⋊C₈	1	1	yes	characteristic
245	C₄	#17986130	1	1	yes	characteristic
246	C₄	#17986236	1	1	yes	characteristic
247	C₄	#17986241	1	1	yes	characteristic
248	C₄	#17986400	1	1	yes	characteristic
249	C₄	#17986403	1	1	yes	characteristic
250	C₂	D₈×C₇₁⋊C₈	1	1	yes	characteristic
251	C₂	Q₁₆×C₇₁⋊C₈	1	1	yes	characteristic
252	C₂	#22587225	1	1	yes	characteristic
253	C₂	#22587237	1	1	yes	characteristic
254	C₂	#22589059	1	1	yes	characteristic
255	C₂	#22589074	1	1	yes	characteristic
256	C₂	#22589089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#107×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#107	D₇₁
hypocenter	C₇₁	#9594
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
36. #107×C₇₁	C₈	semidirect
97. C₈×C₇₁	C₄⋊C₈	semidirect
134. C₇₁	#9594	semidirect
158. #107	C₇₁⋊C₈	semidirect
230. C₈	C₄⋊(C₇₁⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#22589059	2
C₂×#17985752	2
C₂×C₈×C₇₁⋊C₈	2
#22586181	2
C₄×#107×C₇₁	2
#9594	71