Group #47979345 — subgroups

Subgroups — #1956×C₂₇₇ #47979345

Order 35456.

332 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 160 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	C₂×C₄⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₂₇₇	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₇₇	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₇₇	—	2	2	no	—
8	C₄⋊Q₈×C₂₇₇	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
13	#103×C₂₇₇	—	2	8	no	—
14	C₄×C₈×C₂₇₇	—	2	2	no	—
15	C₂×Q₈×C₂₇₇	—	2	2	no	—
16	C₂×Q₈×C₂₇₇	—	2	4	no	—
17	C₂×Q₈×C₂₇₇	—	2	4	no	—
18	C₂×Q₈×C₂₇₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₇₇	—	2	2	no	—
20	C₂²×C₄×C₂₇₇	—	2	2	no	—
21	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
22	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₇₇	—	2	2	no	—
24	C₂×C₈×C₂₇₇	—	2	2	no	—
25	C₂×C₈×C₂₇₇	D₄	1	4	yes	—
26	C₄⋊C₄×C₂₇₇	—	2	8	no	—
27	C₄⋊C₄×C₂₇₇	—	2	4	no	—
28	C₄⋊C₄×C₂₇₇	—	4	4	no	—
29	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
30	C₄²×C₂₇₇	D₄	1	2	yes	—
31	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₂₇₇	—	2	8	no	—
33	Q₈×C₂₇₇	—	4	4	no	—
34	C₂×C₄×C₂₇₇	—	2	4	no	—
35	C₂×C₄×C₂₇₇	—	4	2	no	—
36	C₂×C₄×C₂₇₇	—	4	2	no	—
37	C₂×C₄×C₂₇₇	—	4	2	no	—
38	C₂×C₄×C₂₇₇	—	4	4	no	—
39	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₇₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₂₇₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	#41	1	2	yes	—
46	C₈×C₂₇₇	—	2	4	no	—
47	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₇₇	#124	1	1	yes	characteristic
49	C₂²×C₂₇₇	#127	1	1	yes	characteristic
50	C₂²×C₂₇₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₂₇₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₂₇₇	—	4	4	no	—
53	C₄×C₂₇₇	—	8	2	no	—
54	C₄×C₂₇₇	#103	1	8	yes	complemented
55	C₄×C₂₇₇	C₄×D₄	1	2	yes	—
56	C₄×C₂₇₇	#121	1	2	yes	—
57	C₂×C₂₇₇	#390	1	1	yes	characteristic
58	C₂×C₂₇₇	C₂×#103	1	2	yes	—
59	C₂×C₂₇₇	#487	1	2	yes	—
60	C₂×C₂₇₇	#492	1	1	yes	characteristic
61	C₂×C₂₇₇	#494	1	1	yes	characteristic
62	C₂₇₇	#1956	1	1	yes	characteristic, direct product factor, Sylow
63	#1956	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₄⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic
65	C₂×#103	C₂×C₂₇₇	1	4	yes	—
66	C₂×C₄×C₈	C₂×C₂₇₇	1	1	yes	characteristic
67	#384	C₂×C₂₇₇	1	1	yes	characteristic
68	C₂²×Q₈	C₂²×C₂₇₇	1	2	yes	—
69	C₂²×C₈	C₂²×C₂₇₇	1	2	yes	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₄×C₂₇₇	1	4	yes	complemented
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
75	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
76	#103	—	2	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	C₂×C₄×C₂₇₇	1	4	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	D₄×C₂₇₇	1	2	yes	—
85	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₄×C₂₇₇	1	4	yes	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	4	4	no	—
92	C₄²	C₂×C₄×C₂₇₇	1	2	yes	—
93	C₄²	D₄×C₂₇₇	1	2	yes	—
94	C₂³	C₂×D₄×C₂₇₇	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	Q₈	—	4	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	C₂²⋊C₄×C₂₇₇	1	4	yes	—
103	C₂×C₄	SD₁₆×C₂₇₇	1	4	yes	—
104	C₂×C₄	Q₁₆×C₂₇₇	1	4	yes	—
105	C₂×C₄	C₂²×C₄×C₂₇₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
107	C₂×C₄	C₂×D₄×C₂₇₇	1	1	yes	characteristic
108	C₂×C₄	#41×C₂₇₇	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×C₂²⋊C₄×C₂₇₇	1	1	yes	characteristic
111	C₂²	#124×C₂₇₇	1	1	yes	characteristic
112	C₂²	#127×C₂₇₇	1	1	yes	characteristic
113	C₂²	C₂×SD₁₆×C₂₇₇	1	2	yes	—
114	C₂²	C₂×Q₁₆×C₂₇₇	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	—	8	2	no	—
117	C₄	#103×C₂₇₇	1	8	yes	complemented
118	C₄	C₄×D₄×C₂₇₇	1	2	yes	—
119	C₄	#121×C₂₇₇	1	2	yes	—
120	C₂	#390×C₂₇₇	1	1	yes	characteristic
121	C₂	C₂×#103×C₂₇₇	1	2	yes	—
122	C₂	#487×C₂₇₇	1	2	yes	—
123	C₂	#492×C₂₇₇	1	1	yes	characteristic
124	C₂	#494×C₂₇₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#103×C₂₇₇

C₂

yes

maximal

C₂×C₄×C₈×C₂₇₇

C₂

yes

maximal, characteristic

#384×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₇₇

C₂²

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₄⋊Q₈×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#103×C₂₇₇

—

C₄×C₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

D₄

yes

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

SD₁₆

yes

—

C₂×C₄×C₂₇₇

Q₁₆

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

C₂×SD₁₆

yes

—

C₂²×C₂₇₇

C₂×Q₁₆

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#103

yes

complemented

C₄×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

#121

yes

—

C₂×C₂₇₇

#390

yes

characteristic

C₂×C₂₇₇

C₂×#103

yes

—

C₂×C₂₇₇

#487

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, direct product factor, Sylow

#1956

C₂₇₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₂₇₇

yes

characteristic

C₂×#103

C₂×C₂₇₇

yes

—

C₂×C₄×C₈

C₂×C₂₇₇

yes

characteristic

#384

C₂×C₂₇₇

yes

characteristic

C₂²×Q₈

C₂²×C₂₇₇

yes

—

C₂²×C₈

C₂²×C₂₇₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₇₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₇₇

yes

—

C₂×C₄²

C₂²×C₂₇₇

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₇₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₇₇

yes

—

C₂²×C₄

C₂³×C₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₇₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₇₇

yes

—

C₄²

D₄×C₂₇₇

yes

—

C₂³

C₂×D₄×C₂₇₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²⋊C₄×C₂₇₇

yes

—

103

C₂×C₄

SD₁₆×C₂₇₇

yes

—

104

C₂×C₄

Q₁₆×C₂₇₇

yes

—

105

C₂×C₄

C₂²×C₄×C₂₇₇

yes

characteristic

106

C₂×C₄

C₂×D₄×C₂₇₇

yes

—

107

C₂×C₄

C₂×D₄×C₂₇₇

yes

characteristic

108

C₂×C₄

#41×C₂₇₇

yes

—

109

C₈

—

110

C₂²

C₂×C₂²⋊C₄×C₂₇₇

yes

characteristic

111

C₂²

#124×C₂₇₇

yes

characteristic

112

C₂²

#127×C₂₇₇

yes

characteristic

113

C₂²

C₂×SD₁₆×C₂₇₇

yes

—

114

C₂²

C₂×Q₁₆×C₂₇₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

#103×C₂₇₇

yes

complemented

118

C₄

C₄×D₄×C₂₇₇

yes

—

119

C₄

#121×C₂₇₇

yes

—

120

C₂

#390×C₂₇₇

yes

characteristic

121

C₂

C₂×#103×C₂₇₇

yes

—

122

C₂

#487×C₂₇₇

yes

—

123

C₂

#492×C₂₇₇

yes

characteristic

124

C₂

#494×C₂₇₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₇₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₇₇
Frattini, cosocle	C₂²×C₄	C₂³×C₂₇₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₇ ◁ C₂×C₄²×C₂₇₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₇₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₇₇

Frattini, cosocle

C₂²×C₄

C₂³×C₂₇₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₇₇ ◁ C₂×C₄²×C₂₇₇ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₂₇₇	C₄	semidirect
54. C₄×C₂₇₇	#103	semidirect
62. C₂₇₇	#1956	direct (N × H)
63. #1956	C₂₇₇	direct (N × H)
71. C₄⋊Q₈	C₄×C₂₇₇	semidirect
117. C₄	#103×C₂₇₇	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₂₇₇

C₄

semidirect

54. C₄×C₂₇₇

#103

semidirect

62. C₂₇₇

#1956

direct (N × H)

63. #1956

C₂₇₇

direct (N × H)

71. C₄⋊Q₈

C₄×C₂₇₇

semidirect

117. C₄

#103×C₂₇₇

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₂₇₇	2
C₂×#103×C₂₇₇	2
C₂×C₄×C₈×C₂₇₇	2
#384×C₂₇₇	2
#1956	277

subgroup

index

C₂×C₄⋊Q₈×C₂₇₇

C₂×#103×C₂₇₇

C₂×C₄×C₈×C₂₇₇

#384×C₂₇₇

#1956

277

Subgroups — #1956×C₂₇₇ #47979345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1956×C277 #47979345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1956×C₂₇₇ #47979345