Group #30760937 — subgroups

Subgroups — #30760937

Order 17152.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#22560366	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×#14523975	—	2	6	no	—
6	C₂×#14523965	—	2	6	no	—
7	#16863266	C₂²	1	3	yes	—
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	6	no	—
14	#14523975	—	4	12	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	12	no	—
27	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₆₇	—	2	6	no	—
29	C₂×C₈×C₆₇	—	2	6	no	—
30	C₂×C₈×C₆₇	SD₁₆	1	12	yes	—
31	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₆₇	#41	1	3	yes	—
33	C₂³×C₆₇	#127	1	1	yes	characteristic
34	C₂²×D₆₇	—	4	2	no	—
35	C₂²×D₆₇	—	8	3	no	—
36	C₂²×D₆₇	—	8	3	no	—
37	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₆₇	#123	1	3	yes	—
39	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
40	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	6	yes	—
41	C₂×Dic₆₇	—	8	3	no	—
42	C₂×Dic₆₇	—	8	6	no	—
43	D₂₆₈	—	4	12	no	—
44	D₂₆₈	—	8	6	no	—
45	C₈×C₆₇	—	2	12	no	—
46	C₂²×C₆₇	#399	1	1	yes	characteristic
47	C₂²×C₆₇	#492	1	3	yes	—
48	C₂²×C₆₇	#496	1	3	yes	—
49	C₂×D₆₇	—	8	6	no	—
50	C₂×D₆₇	—	16	1	no	—
51	C₂×D₆₇	—	16	3	no	—
52	C₄×C₆₇	#476	1	6	yes	—
53	C₄×C₆₇	#481	1	6	yes	—
54	Dic₆₇	—	16	3	no	—
55	#10683	—	67	1	no	maximal, Sylow
56	C₂×C₆₇	#1705	1	1	yes	characteristic
57	C₂×C₆₇	#1715	1	3	yes	—
58	C₂×C₆₇	#2094	1	3	yes	—
59	D₆₇	—	16	2	no	—
60	#1957	—	67	3	no	—
61	#1844	—	67	3	no	—
62	#1273	D₆₇	1	1	yes	characteristic, complemented
63	C₆₇	#10683	1	1	yes	characteristic, complemented, Sylow
64	C₂×#127	—	67	1	no	—
65	C₂×#106	—	134	6	no	—
66	C₂×#102	—	134	6	no	—
67	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
68	#384	—	67	3	no	—
69	C₂²×D₄	—	134	3	no	—
70	C₂²×C₈	D₂₆₈	1	6	yes	—
71	#127	—	67	2	no	—
72	#127	—	134	3	no	—
73	C₂×C₄⋊C₄	—	134	3	no	—
74	C₂×C₄⋊C₄	—	134	6	no	—
75	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
76	#106	—	268	12	no	—
77	#102	—	268	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	268	1	no	—
80	C₂×D₄	—	268	3	no	—
81	C₂×D₄	—	268	6	no	—
82	C₂×D₄	—	268	6	no	—
83	C₂×D₄	—	134	6	no	—
84	C₂²×C₄	—	268	3	no	—
85	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#11761617	1	12	yes	—
89	C₄⋊C₄	—	268	6	no	—
90	C₄⋊C₄	—	268	12	no	—
91	C₄²	Q₈×D₆₇	1	1	yes	characteristic
92	C₄²	#11761627	1	3	yes	—
93	C₂³	—	268	2	no	—
94	C₂³	—	536	3	no	—
95	C₂³	—	536	3	no	—
96	C₂³	#14524040	1	1	yes	characteristic
97	D₄	—	268	12	no	—
98	D₄	—	536	6	no	—
99	C₂×C₄	—	536	3	no	—
100	C₂×C₄	—	536	6	no	—
101	C₂×C₄	#14524023	1	3	yes	—
102	C₂×C₄	#14524027	1	3	yes	—
103	C₂×C₄	C₂×#11761617	1	6	yes	—
104	C₂×C₄	#14524071	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	536	6	no	—
107	C₂²	—	1072	1	no	—
108	C₂²	—	1072	3	no	—
109	C₂²	#16863333	1	1	yes	characteristic
110	C₂²	#16863769	1	3	yes	—
111	C₂²	#16863785	1	3	yes	—
112	C₄	—	1072	3	no	—
113	C₄	#16863687	1	6	yes	—
114	C₄	#16863715	1	6	yes	—
115	C₂	—	1072	2	no	—
116	C₂	#22559796	1	1	yes	characteristic
117	C₂	#22559837	1	3	yes	—
118	C₂	#22561741	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561015

C₂

yes

maximal

#22560366

C₂

yes

maximal, complemented

C₆₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

#14524040

—

#14524040

Q₈

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523975

—

#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

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

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

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₂²×C₆₇

#399

yes

characteristic

C₂²×C₆₇

#492

yes

—

C₂²×C₆₇

#496

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#476

yes

—

C₄×C₆₇

#481

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2094

yes

—

D₆₇

—

#1957

—

#1844

—

#1273

D₆₇

yes

characteristic, complemented

C₆₇

#10683

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#106

—

134

—

C₂×#102

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

#384

—

C₂²×D₄

—

134

—

C₂²×C₈

D₂₆₈

yes

—

#127

—

#127

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#106

—

268

—

#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

#11761617

yes

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

Q₈×D₆₇

yes

characteristic

C₄²

#11761627

yes

—

C₂³

—

268

—

C₂³

—

536

—

C₂³

—

536

—

C₂³

#14524040

yes

characteristic

D₄

—

268

—

D₄

—

536

—

C₂×C₄

—

536

—

100

C₂×C₄

—

536

—

101

C₂×C₄

#14524023

yes

—

102

C₂×C₄

#14524027

yes

—

103

C₂×C₄

C₂×#11761617

yes

—

104

C₂×C₄

#14524071

yes

—

105

C₈

—

106

C₂²

—

536

—

107

C₂²

—

1072

—

108

C₂²

—

1072

—

109

C₂²

#16863333

yes

characteristic

110

C₂²

#16863769

yes

—

111

C₂²

#16863785

yes

—

112

C₄

—

1072

—

113

C₄

#16863687

yes

—

114

C₄

#16863715

yes

—

115

C₂

—

1072

—

116

C₂

#22559796

yes

characteristic

117

C₂

#22559837

yes

—

118

C₂

#22561741

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	#10683
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. #22560366	C₂	semidirect
3. C₆₇×#1273	C₂	semidirect
62. #1273	D₆₇	semidirect
63. C₆₇	#10683	semidirect

H ≅ G/N

type

2. #22560366

C₂

semidirect

3. C₆₇×#1273

C₂

semidirect

62. #1273

D₆₇

semidirect

63. C₆₇

#10683

semidirect

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

subgroup	index
#22561015	2
#22560366	2
C₆₇×#1273	2
#10683	67

subgroup

index

#22561015

#22560366

C₆₇×#1273

#10683

Subgroups — #30760937

Conjugacy classes of subgroups

Complements

Maximal subgroups