Group #30760513 — subgroups

Subgroups — #30760513

Order 17152.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22559000	C₂	1	3	yes	maximal, complemented
2	#22558878	C₂	1	3	yes	maximal
3	C₆₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524040	C₂²	1	1	yes	characteristic
5	#16863449	C₂²	1	3	yes	—
6	#16863148	—	2	3	no	—
7	#16863146	—	2	3	no	—
8	#16863139	—	2	3	no	—
9	#16863139	—	2	3	no	—
10	#423×C₆₇	C₂²	1	3	yes	—
11	C₂²×D₂₆₈	—	2	3	no	—
12	#14524040	—	2	3	no	—
13	#14524040	Q₈	1	2	yes	—
14	C₄⋊(C₆₇⋊C₈)	D₄	1	3	yes	—
15	C₄⋊(C₆₇⋊C₈)	D₄	1	3	yes	—
16	C₂²⋊(C₆₇⋊C₈)	—	2	3	no	—
17	C₄×C₆₇⋊C₈	—	2	3	no	—
18	C₂×M₁₆×C₆₇	—	2	3	no	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	—	2	3	no	—
21	C₄⋊C₈×C₆₇	—	2	3	no	—
22	C₂³×D₆₇	—	4	1	no	—
23	C₂×D₂₆₈	—	2	6	no	—
24	C₂×D₂₆₈	—	4	3	no	—
25	C₂×D₂₆₈	—	8	3	no	—
26	C₂×D₂₆₈	—	8	3	no	—
27	C₂×C₆₇⋊C₈	—	2	3	no	—
28	C₂×C₆₇⋊C₈	—	2	3	no	—
29	C₂²×C₄×C₆₇	#41	1	3	yes	—
30	M₁₆×C₆₇	—	8	3	no	—
31	C₂×C₈×C₆₇	—	4	3	no	—
32	C₄²×C₆₇	C₂×D₄	1	3	yes	—
33	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
34	C₂³×C₆₇	#126	1	1	yes	characteristic
35	C₂²×D₆₇	—	4	2	no	—
36	C₂²×D₆₇	—	8	6	no	—
37	C₂×C₄×C₆₇	—	2	3	no	—
38	C₂×C₄×C₆₇	—	2	3	no	—
39	C₂×C₄×C₆₇	#121	1	3	yes	—
40	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
41	D₂₆₈	—	8	6	no	—
42	D₂₆₈	—	8	3	no	—
43	D₂₆₈	—	8	3	no	—
44	C₈×C₆₇	—	8	3	no	—
45	C₆₇⋊C₈	—	4	3	no	—
46	C₆₇⋊C₈	—	4	3	no	—
47	C₂²×C₆₇	—	2	3	no	—
48	C₂²×C₆₇	#396	1	1	yes	characteristic
49	C₂×D₆₇	—	8	6	no	—
50	C₂×D₆₇	—	16	4	no	—
51	C₄×C₆₇	—	2	3	no	—
52	C₄×C₆₇	#471	1	3	yes	—
53	C₄×C₆₇	#472	1	3	yes	—
54	#10611	—	67	1	no	maximal, Sylow
55	C₂×C₆₇	—	4	1	no	—
56	C₂×C₆₇	#2063	1	3	yes	—
57	D₆₇	—	16	2	no	—
58	#1574	—	67	3	no	—
59	#1545	—	67	3	no	—
60	#1309	D₆₇	1	1	yes	characteristic, complemented
61	C₆₇	#10611	1	1	yes	characteristic, complemented, Sylow
62	C₂×#127	—	67	1	no	—
63	#431	—	67	3	no	—
64	#423	C₂×D₆₇	1	3	yes	—
65	#335	—	134	3	no	—
66	#334	—	134	3	no	—
67	#331	—	134	3	no	—
68	#331	—	134	3	no	—
69	C₂²×D₄	—	134	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	67	2	no	—
72	#127	—	134	3	no	—
73	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	67	3	no	—
77	C₄⋊C₈	—	67	3	no	—
78	C₂²⋊C₈	—	134	3	no	—
79	C₄×C₈	—	134	3	no	—
80	C₂⁴	—	268	1	no	—
81	C₂×D₄	—	268	3	no	—
82	C₂×D₄	—	536	3	no	—
83	C₂×D₄	—	536	3	no	—
84	C₂×D₄	—	134	6	no	—
85	C₂²×C₄	#11761628	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	134	3	no	—
89	C₂×C₈	—	134	3	no	—
90	C₄²	D₄×D₆₇	1	3	yes	—
91	C₄²	Q₈×D₆₇	1	1	yes	characteristic
92	C₂³	—	268	2	no	—
93	C₂³	—	536	6	no	—
94	C₂³	#14524037	1	1	yes	characteristic
95	D₄	—	536	6	no	—
96	D₄	—	536	3	no	—
97	D₄	—	536	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#14524016	1	3	yes	—
101	C₂×C₄	#14524021	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	268	3	no	—
104	C₈	—	268	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	536	6	no	—
107	C₂²	—	1072	4	no	—
108	C₂²	#16863320	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#16863662	1	3	yes	—
111	C₄	#16863665	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	1072	2	no	—
114	C₂	#22561570	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559000

C₂

yes

maximal, complemented

#22558878

C₂

yes

maximal

C₆₇×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

characteristic

#16863449

C₂²

yes

—

—

—

—

—

#423×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

#14524040

—

#14524040

Q₈

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#126

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#396

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#471

yes

—

C₄×C₆₇

#472

yes

—

#10611

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2063

yes

—

D₆₇

—

#1574

—

#1545

—

#1309

D₆₇

yes

characteristic, complemented

C₆₇

#10611

yes

characteristic, complemented, Sylow

C₂×#127

—

#431

—

#423

C₂×D₆₇

yes

—

#335

—

134

—

#334

—

134

—

#331

—

134

—

#331

—

134

—

C₂²×D₄

—

134

—

C₂×M₁₆

—

#127

—

#127

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

134

—

C₄×C₈

—

134

—

C₂⁴

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

536

—

C₂×D₄

—

536

—

C₂×D₄

—

134

—

C₂²×C₄

#11761628

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄²

D₄×D₆₇

yes

—

C₄²

Q₈×D₆₇

yes

characteristic

C₂³

—

268

—

C₂³

—

536

—

C₂³

#14524037

yes

characteristic

D₄

—

536

—

D₄

—

536

—

D₄

—

536

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#14524016

yes

—

101

C₂×C₄

#14524021

yes

—

102

C₈

—

103

C₈

—

268

—

104

C₈

—

268

—

105

C₂²

—

106

C₂²

—

536

—

107

C₂²

—

1072

—

108

C₂²

#16863320

yes

characteristic

109

C₄

—

110

C₄

#16863662

yes

—

111

C₄

#16863665

yes

—

112

C₂

—

113

C₂

—

1072

—

114

C₂

#22561570

yes

—

	subgroup	G/H
center	C₂²	#16863320
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1309	C₂
socle	C₂²×C₆₇	#396
prime radical	C₂×#14524040	C₂²
hypercenter	#1309	D₆₇
hypocenter	C₆₇	#10611
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16863320

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1309

C₂

C₂²×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #22559000	C₂	semidirect
3. C₆₇×#1309	C₂	semidirect
60. #1309	D₆₇	semidirect
61. C₆₇	#10611	semidirect

H ≅ G/N

type

1. #22559000

C₂

semidirect

3. C₆₇×#1309

C₂

semidirect

60. #1309

D₆₇

semidirect

61. C₆₇

#10611

semidirect

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

subgroup	index
#22559000	2
#22558878	2
C₆₇×#1309	2
#10611	67

subgroup

index

#22559000

#22558878

C₆₇×#1309

#10611

Subgroups — #30760513

Conjugacy classes of subgroups

Complements

Maximal subgroups