Group #30856213 — subgroups

Subgroups — C₂×#22560363 #30856213

Order 17152.

1230 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 651 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	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	6	yes	maximal
2	#22560363	C₂	1	8	yes	maximal, direct product factor
3	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	24	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	24	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
8	#376×C₆₇	C₄	1	4	yes	—
9	#376×C₆₇	C₂²	1	4	yes	—
10	C₂×#95×C₆₇	C₄	1	4	yes	—
11	C₂²×C₆₇⋊C₈	—	2	24	no	—
12	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
13	C₄⋊(C₆₇⋊C₈)	—	2	48	no	—
14	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₆₇	D₄	1	12	yes	—
18	C₂×C₄²×C₆₇	Q₈	1	12	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
20	#95×C₆₇	C₂×C₄	1	16	yes	—
21	C₂×C₆₇⋊C₈	—	2	24	no	—
22	C₂×C₆₇⋊C₈	—	4	16	no	—
23	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	—	2	24	no	—
25	C₂²×C₄×C₆₇	C₄⋊C₄	1	24	yes	—
26	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
29	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
30	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₄²×C₆₇	C₄⋊C₄	1	24	yes	—
32	C₄²×C₆₇	C₂×D₄	1	12	yes	—
33	C₄²×C₆₇	C₂×Q₈	1	12	yes	—
34	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
35	C₂³×C₆₇	#131	1	8	yes	—
36	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	24	no	—
38	C₂×C₄×C₆₇	—	2	48	no	—
39	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
40	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
41	C₂×C₄×C₆₇	#127	1	4	yes	—
42	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
43	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₆₇	C₂²×Q₈	1	3	yes	—
45	C₆₇⋊C₈	—	4	8	no	—
46	C₂²×C₆₇	#384	1	4	yes	—
47	C₂²×C₆₇	#424	1	24	yes	—
48	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
49	C₂²×C₆₇	C₂×#131	1	4	yes	—
50	C₄×C₆₇	—	2	24	no	—
51	C₄×C₆₇	#384	1	4	yes	—
52	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
53	C₄×C₆₇	C₂×C₄⋊Q₈	1	3	yes	—
54	C₂×#1843	—	67	1	no	maximal, Sylow
55	C₂×C₆₇	#1843	1	8	yes	complemented
56	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
57	C₂×C₆₇	C₂×#424	1	6	yes	—
58	C₂²×C₄⋊C₈	—	67	6	no	—
59	C₂×#376	D₆₇	1	1	yes	characteristic
60	#1843	—	67	8	no	—
61	C₆₇	C₂×#1843	1	1	yes	characteristic, complemented, Sylow
62	C₂³×C₈	—	67	4	no	—
63	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
64	C₂×C₄⋊C₈	—	67	24	no	—
65	C₂×C₄⋊C₈	—	134	24	no	—
66	#376	Dic₆₇	1	4	yes	—
67	#376	C₂×D₆₇	1	4	yes	—
68	C₂×#95	Dic₆₇	1	4	yes	—
69	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
70	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
71	C₂²×C₈	—	67	8	no	—
72	C₂²×C₈	—	134	24	no	—
73	C₂×C₄²	Dic₁₃₄	1	12	yes	—
74	C₂×C₄²	D₂₆₈	1	12	yes	—
75	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
76	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
77	C₄⋊C₈	—	134	48	no	—
78	#95	C₂×Dic₆₇	1	16	yes	—
79	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
80	C₂²×C₄	—	2	24	no	—
81	C₂²×C₄	C₄⋊Dic₆₇	1	24	yes	—
82	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
83	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
85	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
86	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
87	C₂×C₈	—	268	16	no	—
88	C₂×C₈	—	134	24	no	—
89	C₄²	C₄⋊Dic₆₇	1	24	yes	—
90	C₄²	C₂×D₂₆₈	1	12	yes	—
91	C₄²	C₂×Dic₁₃₄	1	12	yes	—
92	C₂³	C₂×C₄⋊Dic₆₇	1	6	yes	—
93	C₂³	#14524050	1	8	yes	—
94	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	2	48	no	—
97	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
99	C₂×C₄	#14524040	1	4	yes	—
100	C₂×C₄	#14524043	1	12	yes	—
101	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
102	C₂×C₄	C₂²×Dic₁₃₄	1	3	yes	—
103	C₈	—	268	8	no	—
104	C₂²	#16863266	1	4	yes	—
105	C₂²	#16863423	1	24	yes	—
106	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
107	C₂²	C₂×#14524050	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	#16863266	1	4	yes	—
110	C₄	C₂×#14524040	1	1	yes	characteristic
111	C₄	C₂×#14524043	1	3	yes	—
112	C₂	#22560363	1	8	yes	direct product factor
113	C₂	C₂×#16863266	1	1	yes	characteristic
114	C₂	C₂×#16863423	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal

#22560363

C₂

yes

maximal, direct product factor

C₂×#376×C₆₇

C₂

yes

maximal, characteristic

C₂³×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²×C₄²×C₆₇

C₂²

yes

—

#376×C₆₇

C₄

yes

—

#376×C₆₇

C₂²

yes

—

C₂×#95×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂³×C₄×C₆₇

C₂×C₄

yes

—

C₂³×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#95×C₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

—

C₂³×C₆₇

#131

yes

—

C₂³×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂²×Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#384

yes

—

C₂²×C₆₇

#424

yes

—

C₂²×C₆₇

C₂²×C₄⋊C₄

yes

—

C₂²×C₆₇

C₂×#131

yes

—

C₄×C₆₇

—

C₄×C₆₇

#384

yes

—

C₄×C₆₇

C₂×#127

yes

characteristic

C₄×C₆₇

C₂×C₄⋊Q₈

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂²×C₄⋊C₈

—

C₂×#376

D₆₇

yes

characteristic

#1843

—

C₆₇

C₂×#1843

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₄²

C₂×D₆₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

134

—

#376

Dic₆₇

yes

—

#376

C₂×D₆₇

yes

—

C₂×#95

Dic₆₇

yes

—

C₂³×C₄

C₂×Dic₆₇

yes

—

C₂³×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

134

—

C₂×C₄²

Dic₁₃₄

yes

—

C₂×C₄²

D₂₆₈

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

—

C₄⋊C₈

—

134

—

#95

C₂×Dic₆₇

yes

—

C₂⁴

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×D₂₆₈

yes

—

C₂²×C₄

C₂×Dic₁₃₄

yes

—

C₂²×C₄

C₂³×D₆₇

yes

characteristic

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₄²

C₄⋊Dic₆₇

yes

—

C₄²

C₂×D₂₆₈

yes

—

C₄²

C₂×Dic₁₃₄

yes

—

C₂³

C₂×C₄⋊Dic₆₇

yes

—

C₂³

#14524050

yes

—

C₂³

C₂³×Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

—

C₂×C₄

#14524040

yes

—

100

C₂×C₄

#14524043

yes

—

101

C₂×C₄

C₂²×D₂₆₈

yes

—

102

C₂×C₄

C₂²×Dic₁₃₄

yes

—

103

C₈

—

268

—

104

C₂²

#16863266

yes

—

105

C₂²

#16863423

yes

—

106

C₂²

C₂²×C₄⋊Dic₆₇

yes

—

107

C₂²

C₂×#14524050

yes

—

108

C₄

—

109

C₄

#16863266

yes

—

110

C₄

C₂×#14524040

yes

characteristic

111

C₄

C₂×#14524043

yes

—

112

C₂

#22560363

yes

direct product factor

113

C₂

C₂×#16863266

yes

characteristic

114

C₂

C₂×#16863423

yes

—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#376×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#376	D₆₇
hypocenter	C₆₇	C₂×#1843
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

subgroup

G/H

center

C₂³×C₄

C₂²×D₆₇

commutator

C₂³×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₂×#376×C₆₇

C₂

socle, prime radical

C₂⁴×C₆₇

C₂²×C₄

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³×C₄ ◁ C₂×#376

N	H ≅ G/N	type
2. #22560363	C₂	direct (N × H)
55. C₂×C₆₇	#1843	semidirect
61. C₆₇	C₂×#1843	semidirect
112. C₂	#22560363	direct (N × H)

H ≅ G/N

type

2. #22560363

C₂

direct (N × H)

55. C₂×C₆₇

#1843

semidirect

61. C₆₇

C₂×#1843

semidirect

112. C₂

#22560363

direct (N × H)

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
#22560363	2
C₂×#376×C₆₇	2
C₂×#1843	67

subgroup

index

C₂²×C₄⋊(C₆₇⋊C₈)

#22560363

C₂×#376×C₆₇

C₂×#1843

Subgroups — C₂×#22560363 #30856213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22560363 #30856213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560363 #30856213