Group #30770498 — subgroups

Subgroups — #30770498

Order 17152.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22562491	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562432	C₂	1	2	yes	maximal, complemented
3	#22559798	C₂	1	2	yes	maximal, complemented
4	#22558125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2246	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863768	—	2	2	no	—
8	#16863742	—	2	2	no	—
9	#16863224	—	2	2	no	—
10	#16863197	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₆₇	—	2	2	no	—
14	#493×C₆₇	C₂²	1	2	yes	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#11761616	—	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	#14523964	—	4	4	no	—
22	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₂×D₈×C₆₇	—	2	2	no	—
25	#127×C₆₇	—	2	2	no	—
26	D₁₆×C₆₇	—	4	4	no	—
27	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	2	no	—
30	#11761616	—	8	8	no	—
31	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₆₇⋊C₁₆	—	2	2	no	—
35	C₂×D₄×C₆₇	—	4	2	no	—
36	C₂×D₄×C₆₇	—	4	2	no	—
37	C₂×D₄×C₆₇	—	4	2	no	—
38	D₈×C₆₇	—	4	4	no	—
39	D₈×C₆₇	—	4	4	no	—
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₆₇	SD₁₆	1	4	yes	complemented
44	C₂³×C₆₇	—	8	2	no	—
45	D₄×C₆₇	—	8	2	no	—
46	D₄×C₆₇	—	8	2	no	—
47	D₄×C₆₇	—	8	4	no	—
48	D₄×C₆₇	—	8	4	no	—
49	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
51	C₂×Dic₆₇	—	8	2	no	—
52	Dic₁₃₄	—	8	4	no	—
53	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
54	C₈×C₆₇	#135	1	2	yes	—
55	C₆₇⋊C₈	—	2	4	no	—
56	C₂²×C₆₇	—	16	2	no	—
57	C₂²×C₆₇	—	16	2	no	—
58	C₂²×C₆₇	—	16	2	no	—
59	C₂²×C₆₇	#493	1	1	yes	characteristic
60	C₄×C₆₇	#492	1	1	yes	characteristic
61	C₄×C₆₇	#495	1	1	yes	characteristic
62	C₄×C₆₇	#509	1	4	yes	—
63	Dic₆₇	—	16	2	no	—
64	#12506	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	—	32	2	no	—
66	C₂×C₆₇	#1706	1	1	yes	characteristic
67	C₂×C₆₇	#2250	1	2	yes	—
68	#2254	—	67	1	no	—
69	#2246	D₆₇	1	1	yes	characteristic
70	#2240	—	67	2	no	—
71	#1706	—	67	2	no	—
72	#1311	—	67	1	no	—
73	C₆₇	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	2	2	no	—
75	#500	—	67	2	no	—
76	#493	C₂×D₆₇	1	2	yes	—
77	#492	—	134	2	no	—
78	#486	—	134	2	no	—
79	#367	—	134	2	no	—
80	#357	—	134	4	no	—
81	C₄×C₁₆	—	67	1	no	—
82	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
83	C₈²	—	67	1	no	—
84	C₂×SD₁₆	—	268	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	134	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	67	2	no	—
91	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
92	#107	—	134	2	no	—
93	#107	—	134	2	no	—
94	#102	—	268	4	no	—
95	C₄×C₈	—	67	2	no	—
96	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
97	C₂×Q₈	—	268	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	536	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	67	4	no	—
105	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₄⋊C₄	—	268	2	no	—
108	C₄⋊C₄	—	268	2	no	—
109	C₄²	D₄×D₆₇	1	1	yes	characteristic
110	C₁₆	—	134	2	no	—
111	C₁₆	#11761616	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	536	4	no	—
114	D₄	—	8	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	4	no	—
117	D₄	—	8	4	no	—
118	C₂×C₄	—	536	2	no	—
119	C₂×C₄	#14524039	1	1	yes	characteristic
120	C₂×C₄	D₈×D₆₇	1	2	yes	—
121	C₈	—	134	4	no	—
122	C₈	C₂×#11761616	1	2	yes	—
123	C₈	#14524062	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#16863771	1	1	yes	characteristic
128	C₄	—	1072	2	no	—
129	C₄	#16863768	1	1	yes	characteristic
130	C₄	#16863779	1	1	yes	characteristic
131	C₄	#16863830	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#22559798	1	1	yes	characteristic
134	C₂	#22562472	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562491

C₂

yes

maximal, characteristic, complemented

#22562432

C₂

yes

maximal, complemented

#22559798

C₂

yes

maximal, complemented

#22558125

C₂

yes

maximal, characteristic, complemented

C₆₇×#2246

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₆₇

—

#493×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761616

—

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523977

—

#14523964

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

#127×C₆₇

—

D₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

#11761616

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

complemented

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

#135

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12506

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1706

yes

characteristic

C₂×C₆₇

#2250

yes

—

#2254

—

#2246

D₆₇

yes

—

—

—

C₆₇

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#500

—

#493

C₂×D₆₇

yes

—

#492

—

134

—

#486

—

134

—

#367

—

134

—

#357

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×SD₁₆

—

268

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

134

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

134

—

#107

—

134

—

#102

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

268

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

536

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₆₇

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

107

C₄⋊C₄

—

268

—

108

C₄⋊C₄

—

268

—

109

C₄²

D₄×D₆₇

yes

characteristic

110

C₁₆

—

134

—

111

C₁₆

#11761616

yes

complemented

112

C₂³

—

113

Q₈

—

536

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

536

—

119

C₂×C₄

#14524039

yes

characteristic

120

C₂×C₄

D₈×D₆₇

yes

—

121

C₈

—

134

—

122

C₈

C₂×#11761616

yes

—

123

C₈

#14524062

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#16863771

yes

characteristic

128

C₄

—

1072

—

129

C₄

#16863768

yes

characteristic

130

C₄

#16863779

yes

characteristic

131

C₄

#16863830

yes

—

132

C₂

—

133

C₂

#22559798

yes

characteristic

134

C₂

#22562472

yes

—

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#2246	C₂
socle	C₂²×C₆₇	#493
hypercenter	#2246	D₆₇
hypocenter	C₆₇	#12506
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting, prime radical

C₆₇×#2246

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #22562491	C₂	semidirect
2. #22562432	C₂	semidirect
3. #22559798	C₂	semidirect
4. #22558125	C₂	semidirect
43. C₁₆×C₆₇	SD₁₆	semidirect
73. C₆₇	#12506	semidirect
111. C₁₆	#11761616	semidirect

H ≅ G/N

type

1. #22562491

C₂

semidirect

2. #22562432

C₂

semidirect

3. #22559798

C₂

semidirect

4. #22558125

C₂

semidirect

43. C₁₆×C₆₇

SD₁₆

semidirect

73. C₆₇

#12506

semidirect

111. C₁₆

#11761616

semidirect

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

subgroup	index
#22562491	2
#22562432	2
#22559798	2
#22558125	2
C₆₇×#2246	2
#12506	67

subgroup

index

C₆₇×#2246

Subgroups — #30770498

Conjugacy classes of subgroups

Complements

Maximal subgroups