Group #30770544 — subgroups

Subgroups — #30770544

Order 17152.

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

normal subgroups only

This group has 310 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	#22562462	C₂	1	1	yes	maximal, characteristic
2	#22562461	C₂	1	2	yes	maximal
3	#22559825	C₂	1	2	yes	maximal
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	—	2	2	no	—
7	C₂×#14523989	—	2	4	no	—
8	#16863799	—	2	4	no	—
9	#16863776	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₆₇	—	2	2	no	—
13	#494×C₆₇	C₂²	1	2	yes	—
14	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₆₈	—	2	2	no	—
16	C₂×Dic₂₆₈	—	2	2	no	—
17	#14524043	—	2	2	no	—
18	Dic₅₃₆	—	4	4	no	—
19	#14523989	—	4	8	no	—
20	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
21	#14523976	—	4	8	no	—
22	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₆₇	—	2	2	no	—
24	C₂×Q₁₆×C₆₇	—	2	2	no	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	Q₃₂×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	Dic₂₆₈	—	4	4	no	—
31	Dic₂₆₈	—	4	4	no	—
32	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
33	C₄⋊Dic₆₇	—	4	4	no	—
34	C₆₇⋊C₁₆	D₈	1	4	yes	—
35	C₂×Q₈×C₆₇	—	4	2	no	—
36	Q₁₆×C₆₇	—	4	4	no	—
37	Q₁₆×C₆₇	—	4	4	no	—
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	4	4	no	—
41	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₆₇	D₈	1	4	yes	—
43	Q₈×C₆₇	—	8	2	no	—
44	Q₈×C₆₇	—	8	2	no	—
45	C₂×C₄×C₆₇	—	8	2	no	—
46	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₆₇	—	8	2	no	—
49	Dic₁₃₄	—	8	2	no	—
50	Dic₁₃₄	—	8	2	no	—
51	C₈×C₆₇	C₂×D₈	1	2	yes	—
52	C₈×C₆₇	C₂×D₈	1	2	yes	—
53	C₆₇⋊C₈	Q₃₂	1	8	yes	complemented
54	C₂²×C₆₇	#493	1	1	yes	characteristic
55	C₄×C₆₇	—	16	2	no	—
56	C₄×C₆₇	#493	1	1	yes	characteristic
57	C₄×C₆₇	#493	1	1	yes	characteristic
58	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
59	Dic₆₇	—	16	2	no	—
60	#12516	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1712	1	1	yes	characteristic
62	C₂×C₆₇	#2247	1	2	yes	—
63	#2247	—	67	2	no	—
64	#2247	—	67	1	no	—
65	#2247	D₆₇	1	1	yes	characteristic
66	#1713	—	67	2	no	—
67	C₈×C₁₆	—	67	1	no	—
68	C₆₇	#12516	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	2	2	no	—
70	C₂×Q₃₂	—	134	4	no	—
71	C₂×Q₃₂	—	134	2	no	—
72	#500	—	134	4	no	—
73	#494	—	67	2	no	—
74	#494	C₂×D₆₇	1	2	yes	—
75	C₄×C₁₆	—	67	1	no	—
76	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
77	C₈²	—	67	1	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	134	2	no	—
81	C₂×Q₁₆	—	134	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	134	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	268	8	no	—
86	Q₃₂	—	268	4	no	—
87	C₂×C₁₆	—	67	2	no	—
88	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
89	#107	—	268	8	no	—
90	C₄×C₈	—	67	2	no	—
91	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	268	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	268	4	no	—
97	Q₁₆	—	268	4	no	—
98	C₂×C₈	—	67	4	no	—
99	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	268	4	no	—
103	C₄²	D₄×D₆₇	1	1	yes	characteristic
104	C₁₆	—	67	4	no	—
105	C₁₆	#11761613	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	536	2	no	—
109	Q₈	—	536	2	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	2	yes	—
114	C₈	—	67	8	no	—
115	C₈	D₈×D₆₇	1	2	yes	—
116	C₈	C₂×#11761613	1	2	yes	—
117	C₂²	#16863771	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1072	2	no	—
120	C₄	#16863771	1	1	yes	characteristic
121	C₄	#16863772	1	1	yes	characteristic
122	C₄	Q₃₂×D₆₇	1	4	yes	—
123	C₂	#22559823	1	1	yes	characteristic
124	C₂	#22562460	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562462

C₂

yes

maximal, characteristic

#22562461

C₂

yes

maximal

#22559825

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2247

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

C₂×#14523989

—

#16863799

—

#16863776

C₂²

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₆₇

—

#494×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

Dic₅₃₆

—

#14523989

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523976

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

D₈

yes

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

D₈

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₈

yes

—

C₈×C₆₇

C₂×D₈

yes

—

C₆₇⋊C₈

Q₃₂

yes

complemented

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×Q₃₂

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2247

—

#2247

—

#2247

D₆₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₆₇

#12516

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

134

—

C₂×Q₃₂

—

134

—

#500

—

134

—

#494

—

#494

C₂×D₆₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

Q₃₂

—

Q₃₂

—

268

—

Q₃₂

—

268

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

268

—

Q₁₆

—

268

—

C₂×C₈

—

C₂×C₈

D₄×D₆₇

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

268

—

103

C₄²

D₄×D₆₇

yes

characteristic

104

C₁₆

—

105

C₁₆

#11761613

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

536

—

109

Q₈

—

536

—

110

C₂×C₄

—

111

C₂×C₄

—

536

—

112

C₂×C₄

#14524039

yes

characteristic

113

C₂×C₄

D₈×D₆₇

yes

—

114

C₈

—

115

C₈

D₈×D₆₇

yes

—

116

C₈

C₂×#11761613

yes

—

117

C₂²

#16863771

yes

characteristic

118

C₄

—

119

C₄

—

1072

—

120

C₄

#16863771

yes

characteristic

121

C₄

#16863772

yes

characteristic

122

C₄

Q₃₂×D₆₇

yes

—

123

C₂

#22559823

yes

characteristic

124

C₂

#22562460

yes

—

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2247	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#2247	D₆₇
hypocenter	C₆₇	#12516
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2247

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. C₆₇⋊C₈	Q₃₂	semidirect
68. C₆₇	#12516	semidirect

H ≅ G/N

type

53. C₆₇⋊C₈

Q₃₂

semidirect

68. C₆₇

#12516

semidirect

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

subgroup	index
#22562462	2
#22562461	2
#22559825	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2247	2
#12516	67

subgroup

index

#22562462

#22562461

#22559825

C₁₆×C₆₇⋊C₈

C₆₇×#2247

#12516

Subgroups — #30770544

Conjugacy classes of subgroups

Complements

Maximal subgroups