Group #30770567 — subgroups

Subgroups — #30770567

Order 17152.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22562484	C₂	1	1	yes	maximal, characteristic
2	#22562482	C₂	1	2	yes	maximal
3	#22559826	C₂	1	2	yes	maximal
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863799	—	2	2	no	—
8	#16863775	—	2	2	no	—
9	#16863222	—	2	2	no	—
10	#16863221	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#366×C₆₇	—	2	2	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#11761618	—	4	4	no	—
17	#14524043	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
19	#14523977	—	2	2	no	—
20	#14523977	—	2	2	no	—
21	#14523976	—	4	4	no	—
22	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₆₇	—	2	2	no	—
25	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
26	#107×C₆₇	—	2	2	no	—
27	#107×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	2	no	—
30	#11761618	—	8	8	no	—
31	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
32	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₆₇⋊C₁₆	Q₁₆	1	4	yes	complemented
36	C₂×Q₈×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₆₇	—	4	2	no	—
40	C₄⋊C₄×C₆₇	—	4	2	no	—
41	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₆₇	Q₁₆	1	4	yes	complemented
43	Q₈×C₆₇	—	8	4	no	—
44	C₂×C₄×C₆₇	—	8	2	no	—
45	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₆₇	—	8	2	no	—
49	Dic₁₃₄	—	8	4	no	—
50	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
51	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
52	C₆₇⋊C₈	D₁₆	1	4	yes	—
53	C₆₇⋊C₈	Q₃₂	1	4	yes	—
54	C₂²×C₆₇	#493	1	1	yes	characteristic
55	C₄×C₆₇	—	16	2	no	—
56	C₄×C₆₇	#494	1	1	yes	characteristic
57	C₄×C₆₇	#494	1	1	yes	characteristic
58	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
59	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
60	Dic₆₇	—	16	2	no	—
61	#12521	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	#1713	1	1	yes	characteristic
63	C₂×C₆₇	#2246	1	1	yes	characteristic
64	C₂×C₆₇	#2247	1	1	yes	characteristic
65	#2252	—	67	2	no	—
66	#2252	—	67	1	no	—
67	#2252	D₆₇	1	1	yes	characteristic
68	#1713	—	67	2	no	—
69	C₈×C₁₆	—	67	1	no	—
70	C₆₇	#12521	1	1	yes	characteristic, complemented, Sylow
71	#500	—	67	2	no	—
72	#500	—	134	2	no	—
73	#500	C₂×D₆₇	1	2	yes	—
74	#494	—	134	2	no	—
75	#366	—	2	2	no	—
76	#366	—	134	4	no	—
77	#366	—	134	2	no	—
78	C₄×C₁₆	—	67	1	no	—
79	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
80	C₈²	—	67	1	no	—
81	C₂×Q₁₆	—	268	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	134	2	no	—
84	C₂×C₁₆	—	67	2	no	—
85	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	268	4	no	—
89	#107	—	134	2	no	—
90	#107	—	134	2	no	—
91	C₄×C₈	—	67	1	no	—
92	C₄×C₈	—	67	1	no	—
93	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	268	2	no	—
96	Q₁₆	—	536	8	no	—
97	C₂×C₈	—	67	2	no	—
98	C₂×C₈	—	67	2	no	—
99	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	268	2	no	—
104	C₄⋊C₄	—	268	2	no	—
105	C₄²	D₄×D₆₇	1	1	yes	characteristic
106	C₁₆	—	67	4	no	—
107	C₁₆	#11761618	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	536	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	536	2	no	—
112	C₂×C₄	#14524039	1	1	yes	characteristic
113	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
114	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
115	C₈	—	67	4	no	—
116	C₈	—	67	4	no	—
117	C₈	Q₁₆×D₆₇	1	2	yes	—
118	C₈	C₂×#11761618	1	2	yes	—
119	C₂²	#16863771	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	1072	2	no	—
122	C₄	#16863774	1	1	yes	characteristic
123	C₄	#16863775	1	1	yes	characteristic
124	C₄	D₁₆×D₆₇	1	2	yes	—
125	C₄	Q₃₂×D₆₇	1	2	yes	—
126	C₂	#22559826	1	1	yes	characteristic
127	C₂	#22562457	1	1	yes	characteristic
128	C₂	#22562460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562484

C₂

yes

maximal, characteristic

#22562482

C₂

yes

maximal

#22559826

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2252

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#366×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761618

—

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523977

—

#14523976

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

#11761618

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

Q₁₆

yes

complemented

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₆₇⋊C₈

D₁₆

yes

—

C₆₇⋊C₈

Q₃₂

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×D₁₆

yes

—

C₄×C₆₇

C₂×Q₃₂

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₆₇

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

134

—

#500

C₂×D₆₇

yes

—

#494

—

134

—

#366

—

#366

—

134

—

#366

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×Q₁₆

—

268

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

#107

—

#107

—

268

—

#107

—

134

—

#107

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

Q₁₆

—

536

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₆₇

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

268

—

104

C₄⋊C₄

—

268

—

105

C₄²

D₄×D₆₇

yes

characteristic

106

C₁₆

—

107

C₁₆

#11761618

yes

complemented

108

Q₈

—

109

Q₈

—

536

—

110

C₂×C₄

—

111

C₂×C₄

—

536

—

112

C₂×C₄

#14524039

yes

characteristic

113

C₂×C₄

D₈×D₆₇

yes

characteristic

114

C₂×C₄

D₈×D₆₇

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₆₇

yes

—

118

C₈

C₂×#11761618

yes

—

119

C₂²

#16863771

yes

characteristic

120

C₄

—

121

C₄

—

1072

—

122

C₄

#16863774

yes

characteristic

123

C₄

#16863775

yes

characteristic

124

C₄

D₁₆×D₆₇

yes

—

125

C₄

Q₃₂×D₆₇

yes

—

126

C₂

#22559826

yes

characteristic

127

C₂

#22562457

yes

characteristic

128

C₂

#22562460

yes

characteristic

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2252	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#12521
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2252

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
35. C₆₇⋊C₁₆	Q₁₆	semidirect
42. C₁₆×C₆₇	Q₁₆	semidirect
70. C₆₇	#12521	semidirect
107. C₁₆	#11761618	semidirect

H ≅ G/N

type

35. C₆₇⋊C₁₆

Q₁₆

semidirect

42. C₁₆×C₆₇

Q₁₆

semidirect

70. C₆₇

#12521

semidirect

107. C₁₆

#11761618

semidirect

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

subgroup	index
#22562484	2
#22562482	2
#22559826	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2252	2
#12521	67

subgroup

index

#22562484

#22562482

#22559826

C₁₆×C₆₇⋊C₈

C₆₇×#2252

#12521

Subgroups — #30770567

Conjugacy classes of subgroups

Complements

Maximal subgroups