Group #31807377 — subgroups

Subgroups — #31807377

Order 17792.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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×#17984842	C₂	1	1	yes	maximal, characteristic
2	#22580819	C₂	1	1	yes	maximal, characteristic
3	#22580716	C₂	1	4	yes	maximal
4	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	—	2	1	no	—
6	#17984842	—	2	2	no	—
7	#17984842	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
11	C₁₃₉⋊#95	—	2	1	no	—
12	C₁₃₉⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	#97×C₁₃₉	—	2	2	no	—
16	C₂×Dic₂₇₈	—	2	4	no	—
17	C₂×Dic₂₇₈	—	4	4	no	—
18	C₂²×Dic₁₃₉	—	2	2	no	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	4	no	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	—	2	4	no	—
23	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	Q₈	1	4	yes	complemented
30	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₃₉	D₄	1	2	yes	—
32	Dic₂₇₈	—	4	8	no	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₂×Dic₁₃₉	—	4	4	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₂×Dic₁₃₉	—	4	2	no	—
37	C₂×Dic₁₃₉	—	4	2	no	—
38	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	—	2	2	no	—
40	C₂×C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₃₉	#41	1	2	yes	—
47	C₈×C₁₃₉	—	2	4	no	—
48	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₃₉	#124	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₃₉	#137	1	2	yes	—
53	C₄×C₁₃₉	—	2	2	no	—
54	C₄×C₁₃₉	—	2	2	no	—
55	C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
56	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
57	Dic₁₃₉	—	4	4	no	—
58	Dic₁₃₉	—	8	2	no	—
59	C₂×C₁₃₉	#391	1	1	yes	characteristic
60	C₂×C₁₃₉	#418	1	1	yes	characteristic
61	C₂×C₁₃₉	#419	1	1	yes	characteristic
62	C₂×C₁₃₉	#489	1	2	yes	—
63	C₂×C₁₃₉	#501	1	2	yes	—
64	C₁₃₉	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	139	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	—	139	1	no	—
67	C₂×#97	D₁₃₉	1	1	yes	characteristic
68	#382	—	139	1	no	—
69	#340	—	139	4	no	—
70	C₂²×Q₈	—	278	1	no	—
71	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
72	C₄⋊Q₈	—	278	2	no	—
73	C₄⋊Q₈	—	139	4	no	—
74	C₂×C₄⋊C₄	—	278	1	no	—
75	C₂×C₄⋊C₄	—	139	2	no	—
76	C₂×C₄⋊C₄	—	139	2	no	—
77	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	278	1	no	—
80	#95	—	278	1	no	—
81	C₂×Q₈	—	278	4	no	—
82	C₂×Q₈	—	556	4	no	—
83	C₂²×C₄	—	278	2	no	—
84	C₂²×C₄	—	278	2	no	—
85	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
86	C₂²×C₄	D₅₅₆	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₂₇₈	1	4	yes	complemented
91	C₄⋊C₄	—	278	4	no	—
92	C₄⋊C₄	—	278	2	no	—
93	C₄⋊C₄	—	278	4	no	—
94	C₄⋊C₄	—	139	4	no	—
95	C₄²	C₄×D₁₃₉	1	2	yes	—
96	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
97	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
98	Q₈	—	556	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	278	4	no	—
102	C₂×C₄	—	556	4	no	—
103	C₂×C₄	—	556	2	no	—
104	C₂×C₄	—	556	2	no	—
105	C₂×C₄	—	556	2	no	—
106	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
111	C₂×C₄	#14555785	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
114	C₂²	#17984830	1	1	yes	characteristic
115	C₂²	#17984842	1	1	yes	characteristic
116	C₂²	#17984870	1	2	yes	—
117	C₂²	#17984875	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	556	4	no	—
121	C₄	—	1112	2	no	—
122	C₄	C₄×Dic₂₇₈	1	2	yes	—
123	C₄	#17984821	1	2	yes	—
124	C₂	#22580859	1	1	yes	characteristic
125	C₂	#22580959	1	1	yes	characteristic
126	C₂	#22580963	1	1	yes	characteristic
127	C₂	#22581313	1	2	yes	—
128	C₂	#22581362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984842

C₂

yes

maximal, characteristic

#22580819

C₂

yes

maximal, characteristic

#22580716

C₂

yes

maximal

C₂×#97×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

—

#17984842

—

#17984842

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₁₃₉⋊#95

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

Q₈

yes

complemented

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

#124

yes

characteristic

C₂²×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×Q₈

yes

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₂×C₁₃₉

#501

yes

—

C₁₃₉

#1993

yes

characteristic, complemented, Sylow

#1993

—

139

maximal, Sylow

C₂×C₄⋊Q₈

—

139

—

C₂×#97

D₁₃₉

yes

characteristic

#382

—

139

—

#340

—

139

—

C₂²×Q₈

—

278

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#97

—

#95

—

278

—

#95

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

556

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

characteristic

C₂²×C₄

D₅₅₆

yes

characteristic

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₇₈

yes

complemented

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

C₄×D₁₃₉

yes

—

C₄²

C₂²⋊D₁₃₉

yes

—

C₂³

C₂×D₅₅₆

yes

characteristic

Q₈

—

556

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

556

—

103

C₂×C₄

—

556

—

104

C₂×C₄

—

556

—

105

C₂×C₄

—

556

—

106

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

110

C₂×C₄

C₂×Dic₂₇₈

yes

—

111

C₂×C₄

#14555785

yes

—

112

C₈

—

113

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

114

C₂²

#17984830

yes

characteristic

115

C₂²

#17984842

yes

characteristic

116

C₂²

#17984870

yes

—

117

C₂²

#17984875

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

556

—

121

C₄

—

1112

—

122

C₄

C₄×Dic₂₇₈

yes

—

123

C₄

#17984821

yes

—

124

C₂

#22580859

yes

characteristic

125

C₂

#22580959

yes

characteristic

126

C₂

#22580963

yes

characteristic

127

C₂

#22581313

yes

—

128

C₂

#22581362

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#97×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	#1993
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#97×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×#97

D₁₃₉

hypocenter

C₁₃₉

#1993

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #17984842	C₄	semidirect
29. C₂×C₈×C₁₃₉	Q₈	semidirect
64. C₁₃₉	#1993	semidirect
90. C₂×C₈	Dic₂₇₈	semidirect

H ≅ G/N

type

7. #17984842

C₄

semidirect

29. C₂×C₈×C₁₃₉

Q₈

semidirect

64. C₁₃₉

#1993

semidirect

90. C₂×C₈

Dic₂₇₈

semidirect

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

subgroup	index
C₂×#17984842	2
#22580819	2
#22580716	2
C₂×#97×C₁₃₉	2
#1993	139

subgroup

index

C₂×#97×C₁₃₉

139

Subgroups — #31807377

Conjugacy classes of subgroups

Complements

Maximal subgroups