Subgroups — #22569304

Order 8576.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863527	C₂	1	2	yes	maximal, complemented
2	#16863527	C₂	1	4	yes	maximal, complemented
3	Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	#16863457	C₂	1	4	yes	maximal
5	#16863453	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
12	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	#14523954	C₂²	1	2	yes	—
15	#14523954	C₂²	1	4	yes	—
16	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
21	#125×C₆₇	C₄	1	2	yes	—
22	C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
23	C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
24	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	#117×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄²×C₆₇	C₂³	1	4	yes	—
48	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₆₇	—	2	4	no	—
50	Q₈×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	4	1	no	—
54	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₆₇⋊C₈	—	2	2	no	—
63	C₆₇⋊C₈	—	2	4	no	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₂²×C₆₇	—	2	1	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₇	—	2	1	no	—
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	2	4	no	—
72	C₄×C₆₇	—	2	1	no	—
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	—	2	4	no	—
75	C₄×C₆₇	#131	1	2	yes	—
76	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
77	C₂×C₆₇	—	4	1	no	—
78	C₂×C₆₇	#519	1	1	yes	characteristic
79	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
80	C₂×C₆₇	#568	1	1	yes	characteristic
81	#2985	—	67	1	no	maximal, Sylow
82	C₆₇	#2985	1	1	yes	characteristic, complemented, Sylow
83	#533	D₆₇	1	1	yes	characteristic
84	#446	—	67	2	no	—
85	#446	—	67	4	no	—
86	C₈×Q₈	—	67	2	no	—
87	#433	—	67	4	no	—
88	#432	—	67	1	no	—
89	#431	—	67	1	no	—
90	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
91	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
92	#125	Dic₆₇	1	2	yes	—
93	C₂²⋊Q₈	Dic₆₇	1	2	yes	—
94	C₂²⋊Q₈	Dic₆₇	1	2	yes	—
95	C₄×Q₈	C₂×D₆₇	1	2	yes	—
96	C₄×Q₈	C₂×D₆₇	1	2	yes	—
97	#117	C₂×D₆₇	1	2	yes	—
98	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
99	C₄⋊C₈	—	67	2	no	—
100	C₄⋊C₈	—	67	2	no	—
101	C₄⋊C₈	—	67	4	no	—
102	C₄⋊C₈	—	67	4	no	—
103	C₂²⋊C₈	—	67	2	no	—
104	C₂²⋊C₈	—	67	2	no	—
105	#97	—	67	2	no	—
106	#97	—	67	4	no	—
107	C₄×C₈	—	67	1	no	—
108	C₄×C₈	—	67	1	no	—
109	C₄×C₈	—	67	4	no	—
110	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
111	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
112	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
115	C₂×C₈	—	67	2	no	—
116	C₂×C₈	—	67	2	no	—
117	C₂×C₈	—	67	4	no	—
118	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
119	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
120	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
121	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
122	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
123	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
124	C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
125	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
126	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
127	C₄²	C₂²×D₆₇	1	1	yes	characteristic
128	C₄²	C₂²×D₆₇	1	4	yes	—
129	C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
141	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
142	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
144	C₈	—	134	2	no	—
145	C₈	—	134	2	no	—
146	C₈	—	134	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	#14524050	1	2	yes	—
158	C₄	#14524091	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#16863874	1	1	yes	characteristic
161	C₂	C₂×#14524050	1	1	yes	characteristic
162	C₂	#16864163	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#533×C₆₇	C₂
hypercenter	#533	D₆₇
hypocenter	C₆₇	#2985
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16863527	C₂	semidirect
2. #16863527	C₂	semidirect
3. Q₈×C₆₇⋊C₈	C₂	semidirect
82. C₆₇	#2985	semidirect