Subgroups — #31996492

Order 18176.

990 conjugacy classes of subgroups.

The subgroup lattice has 180 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	2	yes	maximal
2	C₄⋊C₄×C₇₁⋊C₈	C₂	1	6	yes	maximal
3	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	3	yes	maximal
4	#22588086	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₁⋊C₈	C₄	1	16	yes	complemented
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
10	#17985754	C₂²	1	3	yes	—
11	#17985754	C₂²	1	6	yes	—
12	#17985754	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₁	C₄	1	6	yes	—
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
17	C₄³×C₇₁	C₂²	1	3	yes	—
18	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
19	C₂²×C₇₁⋊C₈	C₂³	1	6	yes	—
20	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	24	yes	—
21	C₄×C₇₁⋊C₈	—	4	6	no	—
22	C₄×C₇₁⋊C₈	C₂×C₄	1	24	yes	—
23	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₇₁	C₈	1	16	yes	complemented
25	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
28	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
30	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	#95×C₇₁	C₂×C₄	1	6	yes	—
34	C₂×C₇₁⋊C₈	—	2	6	no	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	6	no	—
38	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
39	C₂×C₇₁⋊C₈	C₂²×C₄	1	12	yes	—
40	C₂×Q₈×C₇₁	C₄²	1	4	yes	—
41	C₂×Q₈×C₇₁	C₂×C₈	1	8	yes	—
42	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₇₁	C₄²	1	12	yes	—
48	C₄⋊C₄×C₇₁	C₂×C₈	1	24	yes	—
49	C₄²×C₇₁	—	2	3	no	—
50	C₄²×C₇₁	—	2	3	no	—
51	C₄²×C₇₁	—	2	2	no	—
52	C₄²×C₇₁	—	4	6	no	—
53	C₄²×C₇₁	C₂×C₈	1	24	yes	—
54	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
55	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
56	Q₈×C₇₁	C₄×C₈	1	16	yes	complemented
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	6	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	6	no	—
61	C₂×C₄×C₇₁	—	2	6	no	—
62	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
66	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
68	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
71	C₇₁⋊C₈	—	2	4	no	—
72	C₇₁⋊C₈	—	2	12	no	—
73	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
74	C₂²×C₇₁	#519	1	1	yes	characteristic
75	C₂²×C₇₁	#520	1	1	yes	characteristic
76	C₂²×C₇₁	C₂³×C₈	1	2	yes	—
77	C₂²×C₇₁	#568	1	2	yes	—
78	C₄×C₇₁	—	2	6	no	—
79	C₄×C₇₁	—	2	4	no	—
80	C₄×C₇₁	—	2	12	no	—
81	C₄×C₇₁	C₂×C₄×C₈	1	12	yes	—
82	C₄×C₇₁	#519	1	2	yes	—
83	C₄×C₇₁	#520	1	2	yes	—
84	#20569	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	#2276	1	1	yes	characteristic
86	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
87	C₂×C₇₁	#2875	1	1	yes	characteristic
88	C₂×C₇₁	#2982	1	2	yes	—
89	C₂×C₇₁	#2988	1	2	yes	—
90	C₂×C₈×Q₈	—	71	2	no	—
91	C₄²×Q₈	D₇₁	1	1	yes	characteristic
92	C₈×C₄⋊C₄	—	71	6	no	—
93	C₄×C₄⋊C₈	—	71	3	no	—
94	#1744	—	71	3	no	—
95	C₇₁	#20569	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	Dic₇₁	1	2	yes	—
97	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
98	C₈×Q₈	—	71	16	no	—
99	C₂×C₄⋊C₈	—	71	6	no	—
100	C₂×C₄×C₈	—	71	6	no	—
101	C₂×C₄×C₈	—	71	6	no	—
102	C₄×C₄⋊C₄	Dic₇₁	1	6	yes	—
103	C₄×C₄⋊C₄	C₂×D₇₁	1	3	yes	—
104	C₄³	C₂×D₇₁	1	3	yes	—
105	#336	—	71	1	no	—
106	#336	—	71	3	no	—
107	#336	—	71	6	no	—
108	C₂²×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
109	C₂²×C₈	—	71	2	no	—
110	C₂²×C₈	—	71	6	no	—
111	C₄×Q₈	C₇₁⋊C₈	1	16	yes	complemented
112	C₄×Q₈	C₄×D₇₁	1	8	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₇₁	1	6	yes	—
115	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
116	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
117	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
119	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
120	C₄⋊C₈	—	71	24	no	—
121	C₄×C₈	—	284	6	no	—
122	C₄×C₈	—	71	24	no	—
123	#95	C₂×Dic₇₁	1	6	yes	—
124	C₂×Q₈	C₄×Dic₇₁	1	4	yes	—
125	C₂×Q₈	C₂×C₇₁⋊C₈	1	8	yes	—
126	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
127	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
128	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
130	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
131	C₂×C₈	—	71	4	no	—
132	C₂×C₈	—	71	12	no	—
133	C₂×C₈	—	142	6	no	—
134	C₂×C₈	—	142	2	no	—
135	C₂×C₈	—	142	2	no	—
136	C₂×C₈	—	142	6	no	—
137	C₄⋊C₄	C₄×Dic₇₁	1	12	yes	—
138	C₄⋊C₄	C₂×C₇₁⋊C₈	1	24	yes	—
139	C₄²	—	2	3	no	—
140	C₄²	—	2	3	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	4	6	no	—
143	C₄²	C₂×C₇₁⋊C₈	1	24	yes	—
144	C₄²	C₂×C₄×D₇₁	1	12	yes	—
145	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
146	Q₈	C₄×C₇₁⋊C₈	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	6	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
153	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
154	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
155	C₂×C₄	C₂²×C₇₁⋊C₈	1	12	yes	—
156	C₂×C₄	C₂²×C₇₁⋊C₈	1	12	yes	—
157	C₂×C₄	C₂²×C₄×D₇₁	1	2	yes	—
158	C₂×C₄	#14556174	1	1	yes	characteristic
159	C₂×C₄	#14556175	1	1	yes	characteristic
160	C₂×C₄	#14556176	1	2	yes	—
161	C₈	—	142	4	no	—
162	C₈	—	142	12	no	—
163	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
164	C₂²	#17986477	1	1	yes	characteristic
165	C₂²	#17986482	1	1	yes	characteristic
166	C₂²	C₂³×C₇₁⋊C₈	1	2	yes	—
167	C₂²	#17986766	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	12	no	—
170	C₄	—	2	6	no	—
171	C₄	C₂×C₄×C₇₁⋊C₈	1	12	yes	—
172	C₄	#17986475	1	2	yes	—
173	C₄	#17986481	1	2	yes	—
174	C₂	#22590705	1	1	yes	characteristic
175	C₂	C₂²×C₄×C₇₁⋊C₈	1	1	yes	characteristic
176	C₂	#22596768	1	1	yes	characteristic
177	C₂	#22597421	1	2	yes	—
178	C₂	#22597451	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Fitting	C₄²×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂³×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄²×Q₈	D₇₁
hypocenter	C₇₁	#20569
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×C₇₁⋊C₈	C₄	semidirect
24. C₄×Q₈×C₇₁	C₈	semidirect
56. Q₈×C₇₁	C₄×C₈	semidirect
95. C₇₁	#20569	semidirect
111. C₄×Q₈	C₇₁⋊C₈	semidirect
146. Q₈	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
C₂×Q₈×C₇₁⋊C₈	2
C₄⋊C₄×C₇₁⋊C₈	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22588086	2
C₄²×Q₈×C₇₁	2
#20569	71