Group #22565451 — subgroups

Subgroups — #22565451

Order 8576.

380 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863335	C₂	1	4	yes	maximal
2	#16863295	C₂	1	4	yes	maximal
3	#16863261	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
5	#384×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
10	C₆₇⋊#95	C₂²	1	4	yes	—
11	C₆₇⋊#95	C₂²	1	4	yes	—
12	C₆₇⋊#95	C₂²	1	8	yes	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₆₇	C₂³	1	4	yes	—
17	C₂²×Dic₆₇	C₂³	1	4	yes	—
18	C₄⋊Dic₆₇	—	2	4	no	—
19	C₄⋊Dic₆₇	—	2	4	no	—
20	C₄⋊Dic₆₇	—	4	4	no	—
21	C₄×Dic₆₇	—	2	8	no	—
22	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
25	C₄⋊C₄×C₆₇	—	2	8	no	—
26	C₄⋊C₄×C₆₇	—	4	2	no	—
27	C₄²×C₆₇	Q₈	1	4	yes	—
28	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	1	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	—	2	8	no	—
32	C₂×C₄×C₆₇	—	2	4	no	—
33	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₆₇	#41	1	4	yes	—
36	C₂×C₄×C₆₇	#41	1	4	yes	—
37	C₂×Dic₆₇	—	2	4	no	—
38	C₂×Dic₆₇	—	2	4	no	—
39	C₂×Dic₆₇	—	2	4	no	—
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	2	4	no	—
42	C₂×Dic₆₇	—	2	4	no	—
43	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂²×C₆₇	C₂×#41	1	2	yes	—
45	C₂²×C₆₇	C₂×#41	1	2	yes	—
46	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄×C₆₇	—	2	2	no	—
49	C₄×C₆₇	—	4	4	no	—
50	C₄×C₆₇	#125	1	8	yes	—
51	Dic₆₇	—	4	4	no	—
52	Dic₆₇	—	4	4	no	—
53	C₂×C₆₇	C₂×#125	1	2	yes	—
54	C₂×C₆₇	#544	1	1	yes	characteristic
55	C₂×C₆₇	#555	1	1	yes	characteristic
56	C₂×C₆₇	#556	1	2	yes	—
57	C₂×C₆₇	#559	1	1	yes	characteristic
58	#2571	—	67	1	no	maximal, Sylow
59	C₆₇	#2571	1	1	yes	characteristic, complemented, Sylow
60	#400	—	67	4	no	—
61	#389	—	67	4	no	—
62	#384	D₆₇	1	1	yes	characteristic
63	#382	—	67	4	no	—
64	C₄×C₄⋊C₄	—	67	2	no	—
65	C₂×C₄⋊C₄	—	67	4	no	—
66	C₂×C₄⋊C₄	—	67	2	no	—
67	C₂×C₄⋊C₄	—	67	2	no	—
68	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
69	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
70	C₂×C₄²	—	67	4	no	—
71	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
72	#95	—	67	4	no	—
73	#95	—	67	4	no	—
74	#95	—	67	8	no	—
75	C₂²×C₄	—	67	4	no	—
76	C₂²×C₄	—	67	4	no	—
77	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
78	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	268	4	no	—
83	C₄⋊C₄	—	134	4	no	—
84	C₄⋊C₄	—	134	4	no	—
85	C₄²	—	134	8	no	—
86	C₄²	Dic₁₃₄	1	4	yes	—
87	C₂³	C₂³×D₆₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	134	4	no	—
93	C₂×C₄	—	134	4	no	—
94	C₂×C₄	—	134	4	no	—
95	C₂×C₄	—	134	4	no	—
96	C₂×C₄	—	134	4	no	—
97	C₂×C₄	—	134	4	no	—
98	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
100	C₂×C₄	#11761627	1	4	yes	—
101	C₂×C₄	#11761629	1	4	yes	—
102	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
103	C₂²	C₂×#11761627	1	2	yes	—
104	C₂²	C₂×#11761629	1	2	yes	—
105	C₂²	#14524088	1	1	yes	characteristic
106	C₂²	#14524091	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	268	4	no	—
110	C₄	—	268	4	no	—
111	C₄	#14524034	1	8	yes	—
112	C₂	C₂×#14524034	1	2	yes	—
113	C₂	#16864027	1	1	yes	characteristic
114	C₂	#16864112	1	1	yes	characteristic
115	C₂	#16864120	1	2	yes	—
116	C₂	#16864132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863335

C₂

yes

maximal

#16863295

C₂

yes

maximal

#16863261

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal

#384×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#125

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×#125

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#400

—

#389

—

#384

D₆₇

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

Dic₁₃₄

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×Dic₁₃₄

yes

—

C₂×C₄

C₂×Dic₁₃₄

yes

—

100

C₂×C₄

#11761627

yes

—

101

C₂×C₄

#11761629

yes

—

102

C₂²

C₂²×Dic₁₃₄

yes

characteristic

103

C₂²

C₂×#11761627

yes

—

104

C₂²

C₂×#11761629

yes

—

105

C₂²

#14524088

yes

characteristic

106

C₂²

#14524091

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

268

—

110

C₄

—

268

—

111

C₄

#14524034

yes

—

112

C₂

C₂×#14524034

yes

—

113

C₂

#16864027

yes

characteristic

114

C₂

#16864112

yes

characteristic

115

C₂

#16864120

yes

—

116

C₂

#16864132

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#384×C₆₇	C₂
hypercenter	#384	D₆₇
hypocenter	C₆₇	#2571
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#384×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
59. C₆₇	#2571	semidirect

H ≅ G/N

type

59. C₆₇

#2571

semidirect

Subgroups — #22565451

Conjugacy classes of subgroups

Complements