Group #30770574 — subgroups

Subgroups — #30770574

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	#22562491	C₂	1	1	yes	maximal, characteristic
2	#22562489	C₂	1	2	yes	maximal
3	#22559826	C₂	1	2	yes	maximal
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863799	—	2	2	no	—
8	#16863775	—	2	2	no	—
9	#16863224	—	2	2	no	—
10	#16863223	—	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	#367×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	2	yes	—
23	C₄⋊Q₈×C₆₇	—	2	2	no	—
24	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
25	#107×C₆₇	—	2	2	no	—
26	#107×C₆₇	—	2	2	no	—
27	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₃₄	—	4	2	no	—
29	#11761618	—	8	8	no	—
30	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
31	C₄⋊Dic₆₇	—	4	2	no	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₆₇⋊C₁₆	Q₁₆	1	4	yes	complemented
34	C₂×Q₈×C₆₇	—	4	2	no	—
35	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₆₇	—	4	2	no	—
38	C₄⋊C₄×C₆₇	—	4	2	no	—
39	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₆₇	Q₁₆	1	4	yes	complemented
41	Q₈×C₆₇	—	8	4	no	—
42	C₂×C₄×C₆₇	—	8	2	no	—
43	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
45	C₂×Dic₆₇	—	8	2	no	—
46	Dic₁₃₄	—	8	4	no	—
47	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
48	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
49	C₆₇⋊C₈	SD₃₂	1	8	yes	—
50	C₂²×C₆₇	#493	1	1	yes	characteristic
51	C₄×C₆₇	—	16	2	no	—
52	C₄×C₆₇	#494	1	1	yes	characteristic
53	C₄×C₆₇	#494	1	1	yes	characteristic
54	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
55	Dic₆₇	—	16	2	no	—
56	#12523	—	67	1	no	maximal, Sylow
57	C₂×C₆₇	#1713	1	1	yes	characteristic
58	C₂×C₆₇	#2248	1	2	yes	—
59	#2254	—	67	2	no	—
60	#2254	—	67	1	no	—
61	#2254	D₆₇	1	1	yes	characteristic
62	#1713	—	67	2	no	—
63	C₈×C₁₆	—	67	1	no	—
64	C₆₇	#12523	1	1	yes	characteristic, complemented, Sylow
65	#500	—	67	2	no	—
66	#500	—	134	2	no	—
67	#500	C₂×D₆₇	1	2	yes	—
68	#494	—	134	2	no	—
69	#367	—	2	2	no	—
70	#367	—	134	4	no	—
71	#367	—	134	2	no	—
72	C₄×C₁₆	—	67	1	no	—
73	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
74	C₈²	—	67	1	no	—
75	C₂×Q₁₆	—	268	4	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	134	2	no	—
78	C₂×C₁₆	—	67	2	no	—
79	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#107	—	268	4	no	—
83	#107	—	134	2	no	—
84	#107	—	134	2	no	—
85	C₄×C₈	—	67	2	no	—
86	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	268	2	no	—
89	Q₁₆	—	536	8	no	—
90	C₂×C₈	—	67	4	no	—
91	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	268	2	no	—
96	C₄⋊C₄	—	268	2	no	—
97	C₄²	D₄×D₆₇	1	1	yes	characteristic
98	C₁₆	—	67	4	no	—
99	C₁₆	#11761618	1	4	yes	complemented
100	Q₈	—	8	4	no	—
101	Q₈	—	536	4	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	536	2	no	—
104	C₂×C₄	#14524039	1	1	yes	characteristic
105	C₂×C₄	D₈×D₆₇	1	2	yes	—
106	C₈	—	67	8	no	—
107	C₈	Q₁₆×D₆₇	1	2	yes	—
108	C₈	C₂×#11761618	1	2	yes	—
109	C₂²	#16863771	1	1	yes	characteristic
110	C₄	—	16	2	no	—
111	C₄	—	1072	2	no	—
112	C₄	#16863774	1	1	yes	characteristic
113	C₄	#16863775	1	1	yes	characteristic
114	C₄	SD₃₂×D₆₇	1	4	yes	—
115	C₂	#22559826	1	1	yes	characteristic
116	C₂	#22562463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562491

C₂

yes

maximal, characteristic

#22562489

C₂

yes

maximal

#22559826

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2254

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#367×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761618

—

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523977

—

#14523976

—

C₄×C₆₇⋊C₈

D₄

yes

—

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

—

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

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₆₇⋊C₈

SD₃₂

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×SD₃₂

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2254

—

#2254

—

#2254

D₆₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₆₇

#12523

yes

characteristic, complemented, Sylow

#500

—

#500

—

134

—

#500

C₂×D₆₇

yes

—

#494

—

134

—

#367

—

#367

—

134

—

#367

—

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

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

Q₁₆

—

536

—

C₂×C₈

—

C₂×C₈

D₄×D₆₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

D₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

#11761618

yes

complemented

100

Q₈

—

101

Q₈

—

536

—

102

C₂×C₄

—

103

C₂×C₄

—

536

—

104

C₂×C₄

#14524039

yes

characteristic

105

C₂×C₄

D₈×D₆₇

yes

—

106

C₈

—

107

C₈

Q₁₆×D₆₇

yes

—

108

C₈

C₂×#11761618

yes

—

109

C₂²

#16863771

yes

characteristic

110

C₄

—

111

C₄

—

1072

—

112

C₄

#16863774

yes

characteristic

113

C₄

#16863775

yes

characteristic

114

C₄

SD₃₂×D₆₇

yes

—

115

C₂

#22559826

yes

characteristic

116

C₂

#22562463

yes

—

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2254	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#2254	D₆₇
hypocenter	C₆₇	#12523
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2254

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
33. C₆₇⋊C₁₆	Q₁₆	semidirect
40. C₁₆×C₆₇	Q₁₆	semidirect
64. C₆₇	#12523	semidirect
99. C₁₆	#11761618	semidirect

H ≅ G/N

type

33. C₆₇⋊C₁₆

Q₁₆

semidirect

40. C₁₆×C₆₇

Q₁₆

semidirect

64. C₆₇

#12523

semidirect

99. C₁₆

#11761618

semidirect

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

subgroup	index
#22562491	2
#22562489	2
#22559826	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2254	2
#12523	67

subgroup

index

#22562491

#22562489

#22559826

C₁₆×C₆₇⋊C₈

C₆₇×#2254

#12523

Subgroups — #30770574

Conjugacy classes of subgroups

Complements

Maximal subgroups