Group #47979012 — subgroups

Subgroups — #1623×C₂₇₇ #47979012

Order 35456.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	Q₈²×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#492×C₂₇₇	C₂	1	1	yes	maximal, characteristic
3	#460×C₂₇₇	C₂	1	2	yes	maximal
4	#333×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊C₈×C₂₇₇	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₂₇₇	—	2	2	no	—
7	C₂×SD₁₆×C₂₇₇	—	2	2	no	—
8	C₄⋊Q₈×C₂₇₇	—	2	2	no	—
9	C₄⋊Q₈×C₂₇₇	—	2	2	no	—
10	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
11	#127×C₂₇₇	C₂²	1	1	yes	characteristic
12	C₄×Q₈×C₂₇₇	—	2	2	no	—
13	C₄×Q₈×C₂₇₇	C₂²	1	2	yes	—
14	C₄⋊C₈×C₂₇₇	C₂²	1	2	yes	—
15	#102×C₂₇₇	—	2	2	no	—
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₇₇	—	2	2	no	—
18	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
19	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
20	C₂×D₄×C₂₇₇	—	2	2	no	—
21	C₂×D₄×C₂₇₇	—	2	1	no	—
22	SD₁₆×C₂₇₇	—	4	4	no	—
23	SD₁₆×C₂₇₇	—	4	4	no	—
24	C₂×C₈×C₂₇₇	—	2	1	no	—
25	C₂×C₈×C₂₇₇	—	2	2	no	—
26	C₄⋊C₄×C₂₇₇	—	2	4	no	—
27	C₄⋊C₄×C₂₇₇	—	2	2	no	—
28	C₄⋊C₄×C₂₇₇	—	2	2	no	—
29	C₄⋊C₄×C₂₇₇	D₄	1	2	yes	—
30	C₄²×C₂₇₇	—	2	2	no	—
31	C₄²×C₂₇₇	—	2	2	no	—
32	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₂³×C₂₇₇	—	4	1	no	—
34	Q₈×C₂₇₇	—	2	2	no	—
35	Q₈×C₂₇₇	—	4	2	no	—
36	Q₈×C₂₇₇	—	8	2	no	—
37	Q₈×C₂₇₇	SD₁₆	1	4	yes	complemented
38	D₄×C₂₇₇	—	4	2	no	—
39	D₄×C₂₇₇	—	4	2	no	—
40	D₄×C₂₇₇	—	4	2	no	—
41	C₂×C₄×C₂₇₇	—	2	2	no	—
42	C₂×C₄×C₂₇₇	—	2	2	no	—
43	C₂×C₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
46	C₈×C₂₇₇	—	2	2	no	—
47	C₈×C₂₇₇	—	4	2	no	—
48	C₂²×C₂₇₇	—	8	2	no	—
49	C₂²×C₂₇₇	—	8	1	no	—
50	C₂²×C₂₇₇	#120	1	1	yes	characteristic
51	C₄×C₂₇₇	—	2	4	no	—
52	C₄×C₂₇₇	—	2	1	no	—
53	C₄×C₂₇₇	—	4	2	no	—
54	C₄×C₂₇₇	—	4	2	no	—
55	C₄×C₂₇₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₂₇₇	#137	1	2	yes	—
57	C₂×C₂₇₇	—	16	1	no	—
58	C₂×C₂₇₇	#448	1	2	yes	—
59	C₂×C₂₇₇	#453	1	1	yes	characteristic
60	C₂₇₇	#1623	1	1	yes	characteristic, direct product factor, Sylow
61	#1623	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	Q₈²	C₂×C₂₇₇	1	1	yes	characteristic, complemented
63	#492	C₂×C₂₇₇	1	1	yes	characteristic
64	#460	C₂×C₂₇₇	1	2	yes	—
65	#333	C₂×C₂₇₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	C₂×C₂₇₇	1	2	yes	complemented
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	2	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂²×C₂₇₇	1	1	yes	characteristic
72	#127	C₂²×C₂₇₇	1	1	yes	characteristic
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂²×C₂₇₇	1	2	yes	—
75	C₄⋊C₈	C₂²×C₂₇₇	1	2	yes	—
76	#102	—	2	2	no	—
77	C₄×C₈	C₂²×C₂₇₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	D₄×C₂₇₇	1	2	yes	—
80	C₂×Q₈	D₄×C₂₇₇	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	4	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	D₄×C₂₇₇	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂³×C₂₇₇	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	SD₁₆×C₂₇₇	1	4	yes	complemented
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×D₄×C₂₇₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
107	C₈	—	2	2	no	—
108	C₈	—	4	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#120×C₂₇₇	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×SD₁₆×C₂₇₇	1	2	yes	—
117	C₄	#137×C₂₇₇	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#448×C₂₇₇	1	2	yes	—
120	C₂	#453×C₂₇₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₂₇₇

C₂

yes

maximal, characteristic, complemented

#492×C₂₇₇

C₂

yes

maximal, characteristic

#460×C₂₇₇

C₂

yes

maximal

#333×C₂₇₇

C₂

yes

maximal, characteristic, complemented

Q₈⋊C₈×C₂₇₇

C₂

yes

maximal, complemented

C₂×SD₁₆×C₂₇₇

—

C₂×SD₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

#127×C₂₇₇

C₂²

yes

characteristic

C₄×Q₈×C₂₇₇

—

C₄×Q₈×C₂₇₇

C₂²

yes

—

C₄⋊C₈×C₂₇₇

C₂²

yes

—

#102×C₂₇₇

—

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

SD₁₆×C₂₇₇

—

SD₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₄²×C₂₇₇

—

C₄²×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂³×C₂₇₇

—

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

Q₈×C₂₇₇

SD₁₆

yes

complemented

D₄×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₈×C₂₇₇

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×SD₁₆

yes

—

C₄×C₂₇₇

#137

yes

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#448

yes

—

C₂×C₂₇₇

#453

yes

characteristic

C₂₇₇

#1623

yes

characteristic, direct product factor, Sylow

#1623

C₂₇₇

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₂₇₇

yes

characteristic, complemented

#492

C₂×C₂₇₇

yes

characteristic

#460

C₂×C₂₇₇

yes

—

#333

C₂×C₂₇₇

yes

characteristic, complemented

Q₈⋊C₈

C₂×C₂₇₇

yes

complemented

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₂₇₇

yes

characteristic

#127

C₂²×C₂₇₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂²×C₂₇₇

yes

—

C₄⋊C₈

C₂²×C₂₇₇

yes

—

#102

—

C₄×C₈

C₂²×C₂₇₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₂₇₇

yes

—

C₂×Q₈

D₄×C₂₇₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₇₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂³×C₂₇₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

SD₁₆×C₂₇₇

yes

complemented

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×D₄×C₂₇₇

yes

characteristic

106

C₂×C₄

C₂×D₄×C₂₇₇

yes

—

107

C₈

—

108

C₈

—

109

C₂²

—

110

C₂²

—

111

C₂²

#120×C₂₇₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×SD₁₆×C₂₇₇

yes

—

117

C₄

#137×C₂₇₇

yes

—

118

C₂

—

119

C₂

#448×C₂₇₇

yes

—

120

C₂

#453×C₂₇₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₇₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₇₇
prime radical	#127×C₂₇₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇₇ ◁ C₄²×C₂₇₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₇₇

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₂₇₇

prime radical

#127×C₂₇₇

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₇₇ ◁ C₄²×C₂₇₇ ◁ G

N	H ≅ G/N	type
1. Q₈²×C₂₇₇	C₂	semidirect
4. #333×C₂₇₇	C₂	semidirect
5. Q₈⋊C₈×C₂₇₇	C₂	semidirect
37. Q₈×C₂₇₇	SD₁₆	semidirect
60. C₂₇₇	#1623	direct (N × H)
61. #1623	C₂₇₇	direct (N × H)
62. Q₈²	C₂×C₂₇₇	semidirect
65. #333	C₂×C₂₇₇	semidirect
66. Q₈⋊C₈	C₂×C₂₇₇	semidirect
98. Q₈	SD₁₆×C₂₇₇	semidirect

H ≅ G/N

type

1. Q₈²×C₂₇₇

C₂

semidirect

4. #333×C₂₇₇

C₂

semidirect

5. Q₈⋊C₈×C₂₇₇

C₂

semidirect

37. Q₈×C₂₇₇

SD₁₆

semidirect

60. C₂₇₇

#1623

direct (N × H)

61. #1623

C₂₇₇

direct (N × H)

62. Q₈²

C₂×C₂₇₇

semidirect

65. #333

C₂×C₂₇₇

semidirect

66. Q₈⋊C₈

C₂×C₂₇₇

semidirect

98. Q₈

SD₁₆×C₂₇₇

semidirect

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

subgroup	index
Q₈²×C₂₇₇	2
#492×C₂₇₇	2
#460×C₂₇₇	2
#333×C₂₇₇	2
Q₈⋊C₈×C₂₇₇	2
#1623	277

subgroup

index

Q₈²×C₂₇₇

#492×C₂₇₇

#460×C₂₇₇

#333×C₂₇₇

Q₈⋊C₈×C₂₇₇

#1623

277

Subgroups — #1623×C₂₇₇ #47979012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1623×C277 #47979012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1623×C₂₇₇ #47979012