Subgroups — #32151144

Order 18176.

532 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₇₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	#22590172	C₂	1	6	yes	maximal, complemented
3	#22590139	C₂	1	3	yes	maximal, complemented
4	#22590096	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	6	yes	—
7	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	6	yes	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	3	yes	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	6	yes	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	6	yes	—
12	C₄×#41×C₇₁	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₇₁	C₄	1	3	yes	—
16	C₈×D₄×C₇₁	C₂²	1	3	yes	—
17	#431×C₇₁	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
19	#14556068	—	4	6	no	—
20	C₂×C₇₁⋊C₁₆	C₂³	1	6	yes	—
21	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
22	C₂×#41×C₇₁	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
25	C₄×Q₈×C₇₁	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₇₁	C₈	1	3	yes	—
28	C₄×D₄×C₇₁	C₂×C₄	1	3	yes	—
29	#117×C₇₁	C₈	1	3	yes	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₇₁	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₇₁	C₂³	1	3	yes	—
38	C₇₁⋊C₁₆	—	2	6	no	—
39	C₇₁⋊C₁₆	—	2	2	no	—
40	#41×C₇₁	—	2	4	no	—
41	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₇₁	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₇₁	—	2	3	no	—
47	C₂×C₈×C₇₁	—	2	3	no	—
48	C₂×C₈×C₇₁	—	4	3	no	—
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
53	C₄⋊C₄×C₇₁	C₂×C₈	1	3	yes	—
54	C₄⋊C₄×C₇₁	C₂×C₈	1	3	yes	—
55	C₂²⋊C₄×C₇₁	C₂×C₈	1	6	yes	—
56	C₄²×C₇₁	C₂×C₈	1	3	yes	—
57	C₄²×C₇₁	C₂×C₈	1	3	yes	—
58	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
60	C₂³×C₇₁	C₂²×C₈	1	3	yes	—
61	Q₈×C₇₁	M₃₂	1	4	yes	complemented
62	D₄×C₇₁	—	2	6	no	—
63	C₂×C₄×C₇₁	—	2	3	no	—
64	C₂×C₄×C₇₁	—	2	3	no	—
65	C₂×C₄×C₇₁	—	2	6	no	—
66	C₂×C₄×C₇₁	—	2	6	no	—
67	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₈×C₇₁	—	2	3	no	—
74	C₈×C₇₁	—	2	1	no	—
75	C₈×C₇₁	—	2	3	no	—
76	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
77	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
78	C₂²×C₇₁	—	2	3	no	—
79	C₂²×C₇₁	—	2	3	no	—
80	C₂²×C₇₁	—	2	3	no	—
81	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	3	no	—
84	C₄×C₇₁	—	2	3	no	—
85	C₄×C₇₁	C₂×M₃₂	1	6	yes	—
86	C₄×C₇₁	#504	1	2	yes	—
87	C₄×C₇₁	#520	1	1	yes	characteristic
88	C₄×C₇₁	#568	1	1	yes	characteristic
89	#33505	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	4	3	no	—
91	C₂×C₇₁	#2988	1	1	yes	characteristic
92	C₂×C₇₁	C₂²×M₃₂	1	1	yes	characteristic
93	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
94	C₈×#41	D₇₁	1	1	yes	characteristic
95	Q₈×C₁₆	—	71	2	no	—
96	#2169	—	71	6	no	—
97	#2162	—	71	3	no	—
98	#2150	—	71	3	no	—
99	C₇₁	#33505	1	1	yes	characteristic, complemented, Sylow
100	C₄×#41	Dic₇₁	1	1	yes	characteristic
101	C₂×M₃₂	—	71	6	no	—
102	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
103	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
104	C₈×D₄	Dic₇₁	1	3	yes	—
105	C₈×D₄	C₂×D₇₁	1	3	yes	—
106	#431	Dic₇₁	1	3	yes	—
107	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
108	C₄⋊C₁₆	—	71	3	no	—
109	C₄⋊C₁₆	—	71	6	no	—
110	C₄⋊C₁₆	—	71	1	no	—
111	C₂²⋊C₁₆	—	71	6	no	—
112	C₄×C₁₆	—	71	6	no	—
113	C₂×#41	C₇₁⋊C₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
115	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
116	C₄×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
117	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₄×D₄	C₇₁⋊C₈	1	3	yes	—
119	C₄×D₄	C₂×Dic₇₁	1	3	yes	—
120	#117	C₇₁⋊C₈	1	3	yes	—
121	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
122	M₃₂	—	284	6	no	—
123	C₂×C₁₆	—	71	2	no	—
124	C₂×C₁₆	—	71	6	no	—
125	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
126	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
127	C₂²⋊C₈	C₂×Dic₇₁	1	6	yes	—
128	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
129	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
130	C₄×C₈	C₂²×D₇₁	1	3	yes	—
131	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
132	#41	—	2	4	no	—
133	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
134	C₂×D₄	C₂×C₇₁⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
136	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
137	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	—	2	3	no	—
140	C₂×C₈	—	4	3	no	—
141	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
143	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
144	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
145	C₄⋊C₄	C₂×C₇₁⋊C₈	1	3	yes	—
146	C₄⋊C₄	C₂×C₇₁⋊C₈	1	3	yes	—
147	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	6	yes	—
148	C₄²	C₂×C₇₁⋊C₈	1	3	yes	—
149	C₄²	C₂×C₇₁⋊C₈	1	3	yes	—
150	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
151	C₄²	C₂²×Dic₇₁	1	3	yes	—
152	C₁₆	—	142	6	no	—
153	C₁₆	—	142	2	no	—
154	C₂³	C₂²×C₇₁⋊C₈	1	3	yes	—
155	Q₈	#14556068	1	4	yes	complemented
156	D₄	—	2	6	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₇₁⋊C₈	1	3	yes	—
165	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
167	C₈	—	2	3	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	3	no	—
170	C₈	#14556174	1	1	yes	characteristic
171	C₈	#14556175	1	1	yes	characteristic
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	—	2	3	no	—
175	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂×#14556068	1	6	yes	—
180	C₄	#17986416	1	2	yes	—
181	C₄	#17986482	1	1	yes	characteristic
182	C₄	#17986766	1	1	yes	characteristic
183	C₂	—	4	3	no	—
184	C₂	#22597451	1	1	yes	characteristic
185	C₂	C₂²×#14556068	1	1	yes	characteristic
186	C₂	C₂×#17986416	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₈×#41×C₇₁	C₂
prime radical	C₂×#41×C₇₁	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₈×#41	D₇₁
hypocenter	C₇₁	#33505
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₇₁⋊C₁₆	C₂	semidirect
2. #22590172	C₂	semidirect
3. #22590139	C₂	semidirect
4. #22590096	C₂	semidirect
61. Q₈×C₇₁	M₃₂	semidirect
99. C₇₁	#33505	semidirect
155. Q₈	#14556068	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₇₁⋊C₁₆	2
#22590172	2
#22590139	2
#22590096	2
C₈×#41×C₇₁	2
#33505	71