Subgroups — #31996490

Order 18176.

990 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22588205	C₂	1	3	yes	maximal
4	C₈×C₄⋊Dic₇₁	C₂	1	3	yes	maximal
5	C₄⋊C₈×Dic₇₁	C₂	1	3	yes	maximal
6	#22588085	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₁	C₄	1	2	yes	—
9	Q₈×C₇₁⋊C₈	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
12	C₂×C₈×Dic₇₁	C₂²	1	6	yes	—
13	C₄⋊C₄×Dic₇₁	C₄	1	6	yes	—
14	C₄×C₄⋊Dic₇₁	C₂²	1	3	yes	—
15	C₄²×Dic₇₁	C₂²	1	3	yes	—
16	#17985753	C₂²	1	3	yes	—
17	#17985753	C₂²	1	3	yes	—
18	#17985753	C₂²	1	3	yes	—
19	#17985753	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₇₁	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
24	C₂²×C₇₁⋊C₈	C₂³	1	3	yes	—
25	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
26	Q₈×Dic₇₁	C₈	1	16	yes	complemented
27	C₂×C₄⋊Dic₇₁	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₇₁	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₇₁	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₇₁	C₂³	1	3	yes	—
32	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	12	yes	—
33	C₄×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
34	C₈×Dic₇₁	—	4	6	no	—
35	C₇₁⋊#95	C₂×C₄	1	6	yes	—
36	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₇₁	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
43	C₄⋊C₈×C₇₁	C₂×C₄	1	12	yes	—
44	C₄×C₈×C₇₁	C₂×C₄	1	12	yes	—
45	C₂²×Dic₇₁	C₂²×C₄	1	6	yes	—
46	C₂²×Dic₇₁	C₂²×C₄	1	2	yes	—
47	C₂×C₇₁⋊C₈	—	2	3	no	—
48	C₂×C₇₁⋊C₈	—	2	1	no	—
49	C₂×C₇₁⋊C₈	—	2	1	no	—
50	C₂×C₇₁⋊C₈	—	2	3	no	—
51	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
52	C₂×C₇₁⋊C₈	C₂²×C₄	1	6	yes	—
53	C₄⋊Dic₇₁	C₂×C₈	1	24	yes	—
54	C₄×Dic₇₁	—	2	3	no	—
55	C₄×Dic₇₁	—	2	1	no	—
56	C₄×Dic₇₁	—	2	1	no	—
57	C₄×Dic₇₁	—	2	3	no	—
58	C₄×Dic₇₁	—	4	6	no	—
59	C₄×Dic₇₁	C₂×C₈	1	24	yes	—
60	C₂×Q₈×C₇₁	C₄²	1	4	yes	—
61	C₂×Q₈×C₇₁	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₇₁	—	2	3	no	—
66	C₂×C₈×C₇₁	—	2	3	no	—
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
71	C₄⋊C₄×C₇₁	C₄²	1	12	yes	—
72	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
73	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
74	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₇₁	C₄×C₈	1	16	yes	complemented
76	C₂×C₄×C₇₁	—	2	6	no	—
77	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
79	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
80	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
85	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂×Dic₇₁	—	2	2	no	—
87	C₂×Dic₇₁	—	2	6	no	—
88	C₂×Dic₇₁	—	2	2	no	—
89	C₂×Dic₇₁	—	2	6	no	—
90	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
91	C₂×Dic₇₁	C₂²×C₈	1	12	yes	—
92	C₈×C₇₁	—	2	2	no	—
93	C₈×C₇₁	—	2	6	no	—
94	C₇₁⋊C₈	—	2	2	no	—
95	C₇₁⋊C₈	—	2	6	no	—
96	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₇₁	#519	1	1	yes	characteristic
98	C₂²×C₇₁	#520	1	1	yes	characteristic
99	C₂²×C₇₁	C₂³×C₈	1	2	yes	—
100	C₂²×C₇₁	#568	1	2	yes	—
101	C₄×C₇₁	—	2	6	no	—
102	C₄×C₇₁	C₂×C₄×C₈	1	12	yes	—
103	C₄×C₇₁	#519	1	1	yes	characteristic
104	C₄×C₇₁	#519	1	1	yes	characteristic
105	C₄×C₇₁	#520	1	1	yes	characteristic
106	C₄×C₇₁	#520	1	1	yes	characteristic
107	Dic₇₁	—	2	4	no	—
108	Dic₇₁	—	2	12	no	—
109	#20569	—	71	1	no	maximal, Sylow
110	C₂×C₇₁	#2276	1	1	yes	characteristic
111	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
112	C₂×C₇₁	#2875	1	1	yes	characteristic
113	C₂×C₇₁	#2982	1	2	yes	—
114	C₂×C₇₁	#2988	1	2	yes	—
115	C₂×C₈×Q₈	—	71	1	no	—
116	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
117	C₄²×Q₈	—	71	1	no	—
118	C₈×C₄⋊C₄	—	71	3	no	—
119	C₈×C₄⋊C₄	—	71	3	no	—
120	C₄×C₄⋊C₈	—	71	3	no	—
121	#1744	—	71	3	no	—
122	C₇₁	#20569	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄×Q₈	—	71	2	no	—
124	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
125	C₈×Q₈	—	71	8	no	—
126	C₈×Q₈	Dic₇₁	1	8	yes	complemented
127	C₂×C₄⋊C₈	—	71	3	no	—
128	C₂×C₄⋊C₈	C₂×D₇₁	1	3	yes	—
129	C₂×C₄×C₈	—	71	6	no	—
130	C₂×C₄×C₈	—	71	3	no	—
131	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
132	C₄×C₄⋊C₄	—	71	6	no	—
133	C₄×C₄⋊C₄	—	71	3	no	—
134	C₄³	—	71	3	no	—
135	#336	—	71	1	no	—
136	#336	—	71	3	no	—
137	#336	—	71	3	no	—
138	#336	—	71	3	no	—
139	C₂²×Q₈	C₄×D₇₁	1	1	yes	characteristic
140	C₂²×C₈	—	71	1	no	—
141	C₂²×C₈	—	71	3	no	—
142	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
144	C₄×Q₈	—	71	16	no	—
145	C₄×Q₈	C₄×D₇₁	1	4	yes	—
146	C₄×Q₈	C₂×Dic₇₁	1	4	yes	—
147	C₂×C₄⋊C₄	—	71	6	no	—
148	C₂×C₄⋊C₄	C₄×D₇₁	1	3	yes	—
149	C₂×C₄²	—	71	6	no	—
150	C₂×C₄²	—	71	6	no	—
151	C₂×C₄²	—	71	1	no	—
152	C₂×C₄²	—	71	3	no	—
153	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
154	C₄⋊C₈	—	71	12	no	—
155	C₄⋊C₈	C₂×Dic₇₁	1	12	yes	—
156	C₄×C₈	—	284	6	no	—
157	C₄×C₈	—	71	12	no	—
158	C₄×C₈	C₂×Dic₇₁	1	12	yes	—
159	#95	—	71	6	no	—
160	C₂×Q₈	C₄×Dic₇₁	1	4	yes	—
161	C₂×Q₈	C₈×D₇₁	1	8	yes	—
162	C₂²×C₄	—	71	6	no	—
163	C₂²×C₄	—	71	2	no	—
164	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
165	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
166	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
167	C₂×C₈	—	2	3	no	—
168	C₂×C₈	—	2	3	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	71	2	no	—
172	C₂×C₈	—	71	6	no	—
173	C₂×C₈	—	142	3	no	—
174	C₂×C₈	—	142	1	no	—
175	C₂×C₈	—	142	1	no	—
176	C₂×C₈	—	142	3	no	—
177	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
178	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
179	C₄⋊C₄	—	71	24	no	—
180	C₄⋊C₄	C₄×Dic₇₁	1	12	yes	—
181	C₄²	—	284	6	no	—
182	C₄²	—	71	24	no	—
183	C₄²	—	142	3	no	—
184	C₄²	—	142	1	no	—
185	C₄²	—	142	1	no	—
186	C₄²	—	142	3	no	—
187	C₄²	C₂×C₄×D₇₁	1	6	yes	—
188	C₄²	C₂²×Dic₇₁	1	6	yes	—
189	C₂³	C₂²×C₄×D₇₁	1	1	yes	characteristic
190	Q₈	C₈×Dic₇₁	1	16	yes	complemented
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	71	4	no	—
193	C₂×C₄	—	71	12	no	—
194	C₂×C₄	—	142	2	no	—
195	C₂×C₄	—	142	6	no	—
196	C₂×C₄	—	142	2	no	—
197	C₂×C₄	—	142	6	no	—
198	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
199	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
200	C₂×C₄	C₂×C₈×D₇₁	1	12	yes	—
201	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
202	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
203	C₂×C₄	#14556174	1	1	yes	characteristic
204	C₂×C₄	#14556175	1	1	yes	characteristic
205	C₂×C₄	#14556176	1	1	yes	characteristic
206	C₂×C₄	#14556176	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	6	no	—
209	C₈	—	142	2	no	—
210	C₈	—	142	6	no	—
211	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
212	C₂²	#17986475	1	1	yes	characteristic
213	C₂²	#17986481	1	1	yes	characteristic
214	C₂²	C₂²×C₈×D₇₁	1	2	yes	—
215	C₂²	#17986764	1	2	yes	—
216	C₄	—	2	6	no	—
217	C₄	—	142	4	no	—
218	C₄	—	142	12	no	—
219	C₄	C₂×C₈×Dic₇₁	1	12	yes	—
220	C₄	#17986475	1	1	yes	characteristic
221	C₄	#17986477	1	1	yes	characteristic
222	C₄	#17986481	1	1	yes	characteristic
223	C₄	#17986482	1	1	yes	characteristic
224	C₂	#22590705	1	1	yes	characteristic
225	C₂	C₂²×C₈×Dic₇₁	1	1	yes	characteristic
226	C₂	#22596766	1	1	yes	characteristic
227	C₂	#22597419	1	2	yes	—
228	C₂	#22597450	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Fitting	C₂×C₈×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂³×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	#20569
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×C₇₁⋊C₈	C₄	semidirect
21. C₈×Q₈×C₇₁	C₄	semidirect
26. Q₈×Dic₇₁	C₈	semidirect
75. Q₈×C₇₁	C₄×C₈	semidirect
122. C₇₁	#20569	semidirect
126. C₈×Q₈	Dic₇₁	semidirect
190. Q₈	C₈×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₁⋊C₈	2
C₄×Q₈×Dic₇₁	2
#22588205	2
C₈×C₄⋊Dic₇₁	2
C₄⋊C₈×Dic₇₁	2
#22588085	2
C₂×C₈×Q₈×C₇₁	2
#20569	71