Group #31800572 — subgroups

Subgroups — #31800572

Order 17728.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#17984562	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22579880	C₂	1	1	yes	maximal, characteristic, complemented
4	#22579878	C₂	1	1	yes	maximal, characteristic, complemented
5	#22579867	C₂	1	1	yes	maximal, characteristic, complemented
6	#22579862	C₂	1	4	yes	maximal, complemented
7	#22579857	C₂	1	4	yes	maximal, complemented
8	#22579837	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#17984562	—	2	4	no	—
11	C₂×Dic₅₅₄	C₂²	1	2	yes	—
12	C₂×Dic₅₅₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₅₄	C₂²	1	1	yes	characteristic
14	Q₈×D₂₇₇	—	2	4	no	—
15	C₂×C₂²⋊D₂₇₇	C₂²	1	2	yes	—
16	C₂²×Dic₂₇₇	C₂²	1	2	yes	—
17	C₂×C₄×D₂₇₇	C₂²	1	2	yes	—
18	C₂×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₇₇	C₂²	1	4	yes	—
20	#17984542	C₂²	1	2	yes	—
21	#17984542	C₂²	1	4	yes	—
22	C₂²⋊Dic₂₇₇	C₂²	1	4	yes	—
23	#17984540	C₂²	1	2	yes	—
24	C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₇₇	C₂²	1	4	yes	—
28	C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
29	C₂²×D₂₇₇	C₂³	1	1	yes	characteristic
30	Dic₅₅₄	—	2	2	no	—
31	Dic₅₅₄	—	2	2	no	—
32	Dic₅₅₄	—	2	4	no	—
33	C₂²⋊D₂₇₇	—	2	4	no	—
34	C₂×Dic₂₇₇	—	2	4	no	—
35	C₂×Dic₂₇₇	C₂³	1	2	yes	—
36	C₂×Dic₂₇₇	C₂³	1	2	yes	—
37	C₂×Dic₂₇₇	C₂³	1	2	yes	—
38	C₂×Dic₂₇₇	C₂³	1	1	yes	characteristic
39	C₄×D₂₇₇	—	2	4	no	—
40	C₄×D₂₇₇	D₄	1	4	yes	—
41	C₂³×C₂₇₇	C₂³	1	2	yes	—
42	Q₈×C₂₇₇	—	2	2	no	—
43	D₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
45	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
46	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
47	C₂²×C₂₇₇	—	2	2	no	—
48	C₂²×C₂₇₇	—	2	4	no	—
49	C₂²×C₂₇₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₂₇₇	—	2	2	no	—
51	C₂×D₂₇₇	C₂×D₄	1	2	yes	—
52	C₄×C₂₇₇	—	2	2	no	—
53	C₄×C₂₇₇	—	2	2	no	—
54	C₄×C₂₇₇	C₂×D₄	1	2	yes	—
55	Dic₂₇₇	—	2	2	no	—
56	Dic₂₇₇	—	2	2	no	—
57	Dic₂₇₇	—	2	2	no	—
58	Dic₂₇₇	C₂×D₄	1	2	yes	—
59	C₂×C₂₇₇	—	4	2	no	—
60	C₂×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₂₇₇	2₋¹⁺⁴	1	2	yes	—
62	D₂₇₇	—	2	2	no	—
63	C₂₇₇	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	277	1	no	maximal, Sylow
65	C₂×#41	—	277	1	no	—
66	C₂²×Q₈	—	277	1	no	—
67	C₄⋊Q₈	—	277	1	no	—
68	C₄⋊Q₈	—	277	1	no	—
69	#124	—	277	1	no	—
70	#124	D₂₇₇	1	1	yes	characteristic, complemented
71	#123	—	277	4	no	—
72	C₂²⋊Q₈	—	277	4	no	—
73	#117	—	277	1	no	—
74	#41	—	554	4	no	—
75	C₂×Q₈	—	277	2	no	—
76	C₂×Q₈	—	277	1	no	—
77	C₂×Q₈	—	277	1	no	—
78	C₂×Q₈	—	554	4	no	—
79	C₂×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
80	C₂×D₄	—	277	2	no	—
81	C₂×D₄	C₂×D₂₇₇	1	1	yes	characteristic
82	C₂²×C₄	—	277	1	no	—
83	C₂²×C₄	—	277	2	no	—
84	C₂²×C₄	—	277	2	no	—
85	C₄⋊C₄	—	277	2	no	—
86	C₄⋊C₄	—	277	4	no	—
87	C₄⋊C₄	—	277	4	no	—
88	C₂²⋊C₄	—	277	4	no	—
89	C₂²⋊C₄	—	277	2	no	—
90	C₂²⋊C₄	C₂×D₂₇₇	1	4	yes	—
91	C₄²	—	277	1	no	—
92	C₄²	C₂×D₂₇₇	1	1	yes	characteristic
93	C₂³	—	277	1	no	—
94	C₂³	C₂²×D₂₇₇	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	554	4	no	—
97	Q₈	—	554	2	no	—
98	Q₈	—	554	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	554	4	no	—
101	C₂×C₄	—	277	1	no	—
102	C₂×C₄	—	277	4	no	—
103	C₂×C₄	—	277	2	no	—
104	C₂×C₄	—	277	2	no	—
105	C₂×C₄	—	277	2	no	—
106	C₂×C₄	—	554	4	no	—
107	C₂×C₄	—	554	4	no	—
108	C₂×C₄	C₂²×D₂₇₇	1	2	yes	—
109	C₂×C₄	C₂²×D₂₇₇	1	2	yes	—
110	C₂×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	277	2	no	—
114	C₂²	—	554	2	no	—
115	C₂²	C₂³×D₂₇₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	277	2	no	—
119	C₄	—	554	2	no	—
120	C₄	—	554	2	no	—
121	C₄	—	554	2	no	—
122	C₄	D₄×D₂₇₇	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	554	2	no	—
125	C₂	C₂×D₄×D₂₇₇	1	1	yes	characteristic
126	C₂	#22579927	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984562

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₇₇

C₂

yes

maximal, characteristic, complemented

#22579880

C₂

yes

maximal, characteristic, complemented

#22579878

C₂

yes

maximal, characteristic, complemented

#22579867

C₂

yes

maximal, characteristic, complemented

#22579862

C₂

yes

maximal, complemented

#22579857

C₂

yes

maximal, complemented

#22579837

C₂

yes

maximal, characteristic, complemented

#124×C₂₇₇

C₂

yes

maximal, characteristic, complemented

#17984562

—

C₂×Dic₅₅₄

C₂²

yes

—

C₂×Dic₅₅₄

C₂²

yes

characteristic

C₂×Dic₅₅₄

C₂²

yes

characteristic

Q₈×D₂₇₇

—

C₂×C₂²⋊D₂₇₇

C₂²

yes

—

C₂²×Dic₂₇₇

C₂²

yes

—

C₂×C₄×D₂₇₇

C₂²

yes

—

C₂×C₄×D₂₇₇

C₂²

yes

characteristic

C₄⋊Dic₂₇₇

C₂²

yes

—

#17984542

C₂²

yes

—

#17984542

C₂²

yes

—

C₂²⋊Dic₂₇₇

C₂²

yes

—

#17984540

C₂²

yes

—

C₄×Dic₂₇₇

C₂²

yes

characteristic

C₂×Q₈×C₂₇₇

C₂²

yes

characteristic

C₂×D₄×C₂₇₇

C₂²

yes

characteristic

C₂²⋊C₄×C₂₇₇

C₂²

yes

—

C₄²×C₂₇₇

C₂²

yes

characteristic

C₂²×D₂₇₇

C₂³

yes

characteristic

Dic₅₅₄

—

Dic₅₅₄

—

Dic₅₅₄

—

C₂²⋊D₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂³

yes

—

C₂×Dic₂₇₇

C₂³

yes

—

C₂×Dic₂₇₇

C₂³

yes

—

C₂×Dic₂₇₇

C₂³

yes

characteristic

C₄×D₂₇₇

—

C₄×D₂₇₇

D₄

yes

—

C₂³×C₂₇₇

C₂³

yes

—

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂³

yes

—

C₂×C₄×C₂₇₇

C₂³

yes

—

C₂×C₄×C₂₇₇

C₂³

yes

characteristic

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂⁴

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

C₂×D₄

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×D₄

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

—

Dic₂₇₇

C₂×D₄

yes

—

C₂×C₂₇₇

—

C₂×C₂₇₇

C₂²×D₄

yes

characteristic

C₂×C₂₇₇

2₋¹⁺⁴

yes

—

D₂₇₇

—

C₂₇₇

#536

yes

characteristic, complemented, Sylow

#536

—

277

maximal, Sylow

C₂×#41

—

277

—

C₂²×Q₈

—

277

—

C₄⋊Q₈

—

277

—

C₄⋊Q₈

—

277

—

#124

—

277

—

#124

D₂₇₇

yes

characteristic, complemented

#123

—

277

—

C₂²⋊Q₈

—

277

—

#117

—

277

—

#41

—

554

—

C₂×Q₈

—

277

—

C₂×Q₈

—

277

—

C₂×Q₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

C₂×D₂₇₇

yes

characteristic

C₂×D₄

—

277

—

C₂×D₄

C₂×D₂₇₇

yes

characteristic

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

277

—

C₄⋊C₄

—

277

—

C₂²⋊C₄

—

277

—

C₂²⋊C₄

—

277

—

C₂²⋊C₄

C₂×D₂₇₇

yes

—

C₄²

—

277

—

C₄²

C₂×D₂₇₇

yes

characteristic

C₂³

—

277

—

C₂³

C₂²×D₂₇₇

yes

—

Q₈

—

Q₈

—

554

—

Q₈

—

554

—

Q₈

—

554

—

D₄

—

100

D₄

—

554

—

101

C₂×C₄

—

277

—

102

C₂×C₄

—

277

—

103

C₂×C₄

—

277

—

104

C₂×C₄

—

277

—

105

C₂×C₄

—

277

—

106

C₂×C₄

—

554

—

107

C₂×C₄

—

554

—

108

C₂×C₄

C₂²×D₂₇₇

yes

—

109

C₂×C₄

C₂²×D₂₇₇

yes

—

110

C₂×C₄

C₂²×D₂₇₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

277

—

114

C₂²

—

554

—

115

C₂²

C₂³×D₂₇₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

277

—

119

C₄

—

554

—

120

C₄

—

554

—

121

C₄

—

554

—

122

C₄

D₄×D₂₇₇

yes

—

123

C₂

—

124

C₂

—

554

—

125

C₂

C₂×D₄×D₂₇₇

yes

characteristic

126

C₂

#22579927

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₇₇
commutator, socle, cosocle	C₂²×C₂₇₇	C₂⁴
Fitting	#124×C₂₇₇	C₂
prime radical	C₂×#17984562	C₂
hypercenter	#124	D₂₇₇
hypocenter	C₂₇₇	#536
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₇₇

commutator, socle, cosocle

C₂²×C₂₇₇

C₂⁴

Fitting

#124×C₂₇₇

C₂

C₂

D₂₇₇

C₂₇₇

G ▷ C₂²×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₂₇₇	C₂	semidirect
3. #22579880	C₂	semidirect
4. #22579878	C₂	semidirect
5. #22579867	C₂	semidirect
6. #22579862	C₂	semidirect
7. #22579857	C₂	semidirect
8. #22579837	C₂	semidirect
9. #124×C₂₇₇	C₂	semidirect
63. C₂₇₇	#536	semidirect
70. #124	D₂₇₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₂₇₇

C₂

semidirect

3. #22579880

C₂

semidirect

4. #22579878

C₂

semidirect

5. #22579867

C₂

semidirect

6. #22579862

C₂

semidirect

7. #22579857

C₂

semidirect

8. #22579837

C₂

semidirect

9. #124×C₂₇₇

C₂

semidirect

63. C₂₇₇

#536

semidirect

70. #124

D₂₇₇

semidirect

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

subgroup	index
C₂×#17984562	2
C₂×Q₈×D₂₇₇	2
#22579880	2
#22579878	2
#22579867	2
#22579862	2
#22579857	2
#22579837	2
#124×C₂₇₇	2
#536	277

subgroup

index

C₂×Q₈×D₂₇₇

#124×C₂₇₇

277

Subgroups — #31800572

Conjugacy classes of subgroups

Complements

Maximal subgroups