Group #30760906 — subgroups

Subgroups — #30760906

Order 17152.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22561015	C₂	1	3	yes	maximal
2	#22560337	C₂	1	3	yes	maximal, complemented
3	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524040	C₂²	1	1	yes	characteristic
5	C₂×#14523965	—	2	6	no	—
6	#16863266	C₂²	1	3	yes	—
7	#16863159	—	2	6	no	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₂²×D₂₆₈	—	2	3	no	—
10	#14524040	—	2	3	no	—
11	#14524040	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	—	2	3	no	—
13	C₂×C₄⋊Dic₆₇	—	2	3	no	—
14	C₂×C₄⋊Dic₆₇	—	2	3	no	—
15	#14523965	—	4	12	no	—
16	C₂²×C₈×C₆₇	D₄	1	6	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₇	—	2	6	no	—
19	C₂³×D₆₇	—	4	1	no	—
20	C₂×D₂₆₈	—	2	6	no	—
21	C₂×D₂₆₈	—	4	6	no	—
22	C₂×D₂₆₈	—	4	6	no	—
23	C₂×D₂₆₈	—	4	3	no	—
24	C₂²×Dic₆₇	—	4	3	no	—
25	C₄⋊Dic₆₇	—	4	6	no	—
26	C₄⋊Dic₆₇	—	4	6	no	—
27	C₄⋊Dic₆₇	—	4	6	no	—
28	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
29	C₂×C₈×C₆₇	—	2	6	no	—
30	C₂×C₈×C₆₇	—	2	6	no	—
31	C₂×C₈×C₆₇	—	2	6	no	—
32	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₆₇	#41	1	3	yes	—
34	C₂³×C₆₇	#127	1	1	yes	characteristic
35	C₂²×D₆₇	—	4	2	no	—
36	C₂²×D₆₇	—	8	3	no	—
37	C₂²×D₆₇	—	8	3	no	—
38	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₆₇	#123	1	3	yes	—
40	C₂×C₄×C₆₇	#135	1	6	yes	—
41	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₆₇	—	8	3	no	—
43	C₂×Dic₆₇	—	8	3	no	—
44	C₂×Dic₆₇	—	8	3	no	—
45	D₂₆₈	—	4	12	no	—
46	D₂₆₈	—	8	6	no	—
47	C₈×C₆₇	—	4	6	no	—
48	C₂²×C₆₇	#399	1	1	yes	characteristic
49	C₂²×C₆₇	#495	1	3	yes	—
50	C₂²×C₆₇	#496	1	3	yes	—
51	C₂×D₆₇	—	8	6	no	—
52	C₂×D₆₇	—	16	1	no	—
53	C₂×D₆₇	—	16	3	no	—
54	C₄×C₆₇	#478	1	6	yes	—
55	C₄×C₆₇	#482	1	6	yes	—
56	Dic₆₇	—	16	3	no	—
57	#10678	—	67	1	no	maximal, Sylow
58	C₂×C₆₇	#1711	1	1	yes	characteristic
59	C₂×C₆₇	#1717	1	3	yes	—
60	C₂×C₆₇	#2092	1	3	yes	—
61	D₆₇	—	16	2	no	—
62	#1957	—	67	3	no	—
63	#1834	—	67	3	no	—
64	#1273	D₆₇	1	1	yes	characteristic, complemented
65	C₆₇	#10678	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	67	1	no	—
67	C₂×#102	—	134	6	no	—
68	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
69	#384	—	67	3	no	—
70	#340	—	134	6	no	—
71	C₂²×D₄	—	134	3	no	—
72	C₂²×C₈	D₂₆₈	1	6	yes	—
73	#127	—	67	2	no	—
74	#127	—	134	3	no	—
75	C₂×C₄⋊C₄	—	134	3	no	—
76	C₂×C₄⋊C₄	—	134	3	no	—
77	C₂×C₄⋊C₄	—	134	3	no	—
78	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
79	#102	—	268	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	268	1	no	—
82	C₂×D₄	—	268	3	no	—
83	C₂×D₄	—	268	6	no	—
84	C₂×D₄	—	268	6	no	—
85	C₂×D₄	—	134	6	no	—
86	C₂²×C₄	—	268	3	no	—
87	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	268	6	no	—
92	C₄⋊C₄	—	268	6	no	—
93	C₄⋊C₄	—	268	6	no	—
94	C₄²	Q₈×D₆₇	1	1	yes	characteristic
95	C₄²	#11761627	1	3	yes	—
96	C₂³	—	268	2	no	—
97	C₂³	—	536	3	no	—
98	C₂³	—	536	3	no	—
99	C₂³	#14524040	1	1	yes	characteristic
100	D₄	—	268	12	no	—
101	D₄	—	536	6	no	—
102	C₂×C₄	—	536	3	no	—
103	C₂×C₄	—	536	3	no	—
104	C₂×C₄	—	536	3	no	—
105	C₂×C₄	#14524023	1	3	yes	—
106	C₂×C₄	#14524027	1	3	yes	—
107	C₂×C₄	#14524065	1	6	yes	—
108	C₂×C₄	#14524071	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	536	6	no	—
111	C₂²	—	1072	1	no	—
112	C₂²	—	1072	3	no	—
113	C₂²	#16863333	1	1	yes	characteristic
114	C₂²	#16863780	1	3	yes	—
115	C₂²	#16863785	1	3	yes	—
116	C₄	—	1072	3	no	—
117	C₄	#16863697	1	6	yes	—
118	C₄	#16863721	1	6	yes	—
119	C₂	—	1072	2	no	—
120	C₂	#22559821	1	1	yes	characteristic
121	C₂	#22559848	1	3	yes	—
122	C₂	#22561729	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561015

C₂

yes

maximal

#22560337

C₂

yes

maximal, complemented

C₆₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

characteristic

C₂×#14523965

—

#16863266

C₂²

yes

—

#16863159

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

#14524040

—

#14524040

Q₈

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523965

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₂²×C₆₇

#399

yes

characteristic

C₂²×C₆₇

#495

yes

—

C₂²×C₆₇

#496

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#478

yes

—

C₄×C₆₇

#482

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2092

yes

—

D₆₇

—

#1957

—

#1834

—

#1273

D₆₇

yes

characteristic, complemented

C₆₇

#10678

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

#384

—

#340

—

134

—

C₂²×D₄

—

134

—

C₂²×C₈

D₂₆₈

yes

—

#127

—

#127

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#102

—

268

—

C₄×C₈

—

C₂⁴

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×D₂₆₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

Q₈×D₆₇

yes

characteristic

C₄²

#11761627

yes

—

C₂³

—

268

—

C₂³

—

536

—

C₂³

—

536

—

C₂³

#14524040

yes

characteristic

100

D₄

—

268

—

101

D₄

—

536

—

102

C₂×C₄

—

536

—

103

C₂×C₄

—

536

—

104

C₂×C₄

—

536

—

105

C₂×C₄

#14524023

yes

—

106

C₂×C₄

#14524027

yes

—

107

C₂×C₄

#14524065

yes

—

108

C₂×C₄

#14524071

yes

—

109

C₈

—

110

C₂²

—

536

—

111

C₂²

—

1072

—

112

C₂²

—

1072

—

113

C₂²

#16863333

yes

characteristic

114

C₂²

#16863780

yes

—

115

C₂²

#16863785

yes

—

116

C₄

—

1072

—

117

C₄

#16863697

yes

—

118

C₄

#16863721

yes

—

119

C₂

—

1072

—

120

C₂

#22559821

yes

characteristic

121

C₂

#22559848

yes

—

122

C₂

#22561729

yes

—

	subgroup	G/H
center	C₂³	#14524040
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1273	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#10678
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14524040

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1273

C₂

C₂³×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #22560337	C₂	semidirect
3. C₆₇×#1273	C₂	semidirect
64. #1273	D₆₇	semidirect
65. C₆₇	#10678	semidirect

H ≅ G/N

type

2. #22560337

C₂

semidirect

3. C₆₇×#1273

C₂

semidirect

64. #1273

D₆₇

semidirect

65. C₆₇

#10678

semidirect

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

subgroup	index
#22561015	2
#22560337	2
C₆₇×#1273	2
#10678	67

subgroup

index

#22561015

#22560337

C₆₇×#1273

#10678

Subgroups — #30760906

Conjugacy classes of subgroups

Complements

Maximal subgroups