Group #31807205 — subgroups

Subgroups — #31807205

Order 17792.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂×#17984765	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	#17984842	—	2	2	no	—
8	#17984842	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
10	#17984765	—	2	8	no	—
11	C₄×C₁₃₉⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	2	no	—
16	C₂×Dic₂₇₈	—	2	4	no	—
17	C₂×Dic₂₇₈	—	2	4	no	—
18	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₁₃₉	—	2	8	no	—
24	C₂²×C₄×C₁₃₉	—	2	2	no	—
25	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₉	—	2	4	no	—
28	C₄⋊C₄×C₁₃₉	—	4	4	no	—
29	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₃₉	D₄	1	2	yes	—
31	Dic₂₇₈	—	2	8	no	—
32	Dic₂₇₈	—	4	4	no	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₂×Dic₁₃₉	—	4	2	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₁₃₉⋊C₈	—	2	4	no	—
37	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	—	4	2	no	—
39	C₂×C₄×C₁₃₉	—	4	4	no	—
40	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₃₉	SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₃₉	Q₁₆	1	4	yes	—
43	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	#41	1	2	yes	—
47	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₃₉	#124	1	1	yes	characteristic
49	C₂²×C₁₃₉	#127	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₃₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₃₉	—	8	2	no	—
53	C₄×C₁₃₉	#103	1	8	yes	complemented
54	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
55	C₄×C₁₃₉	#121	1	2	yes	—
56	Dic₁₃₉	—	4	4	no	—
57	C₂×C₁₃₉	#390	1	1	yes	characteristic
58	C₂×C₁₃₉	C₂×#103	1	2	yes	—
59	C₂×C₁₃₉	#487	1	2	yes	—
60	C₂×C₁₃₉	#492	1	1	yes	characteristic
61	C₂×C₁₃₉	#494	1	1	yes	characteristic
62	C₁₃₉	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	139	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	139	1	no	—
65	C₂×#103	—	139	4	no	—
66	C₂×C₄×C₈	—	139	1	no	—
67	#384	D₁₃₉	1	1	yes	characteristic
68	C₂²×Q₈	—	139	2	no	—
69	C₂²×C₈	—	139	2	no	—
70	C₄⋊Q₈	—	278	2	no	—
71	C₄⋊Q₈	—	139	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	278	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
75	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
76	#103	—	278	8	no	—
77	C₄×C₈	—	278	2	no	—
78	C₂×Q₈	—	278	2	no	—
79	C₂×Q₈	—	278	4	no	—
80	C₂×Q₈	—	278	4	no	—
81	C₂×Q₈	—	139	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	278	2	no	—
84	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
86	C₂×C₈	—	278	2	no	—
87	C₂×C₈	—	278	2	no	—
88	C₂×C₈	—	139	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	278	8	no	—
92	C₄²	C₄×D₁₃₉	1	2	yes	—
93	C₄²	D₅₅₆	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
95	Q₈	—	278	8	no	—
96	Q₈	—	556	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	278	4	no	—
100	C₂×C₄	—	556	2	no	—
101	C₂×C₄	—	556	2	no	—
102	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	4	yes	—
103	C₂×C₄	#14555773	1	4	yes	—
104	C₂×C₄	#14555775	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
107	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
108	C₂×C₄	#14555786	1	2	yes	—
109	C₈	—	278	4	no	—
110	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
111	C₂²	#17984831	1	1	yes	characteristic
112	C₂²	#17984838	1	1	yes	characteristic
113	C₂²	C₂×#14555773	1	2	yes	—
114	C₂²	C₂×#14555775	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	556	4	no	—
117	C₄	#17984765	1	8	yes	complemented
118	C₄	#17984804	1	2	yes	—
119	C₄	#17984817	1	2	yes	—
120	C₂	#22580854	1	1	yes	characteristic
121	C₂	C₂×#17984765	1	2	yes	—
122	C₂	#22581305	1	2	yes	—
123	C₂	#22581325	1	1	yes	characteristic
124	C₂	#22581332	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984842

C₂

yes

maximal, characteristic

C₂×#17984765

C₂

yes

maximal

C₂×C₄×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

#384×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984842

—

#17984842

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

—

#17984765

—

C₄×C₁₃₉⋊C₈

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

Q₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#103

yes

complemented

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

C₂×#103

yes

—

C₂×C₁₃₉

#487

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1956

—

139

maximal, Sylow

C₂×C₄⋊Q₈

—

139

—

C₂×#103

—

139

—

C₂×C₄×C₈

—

139

—

#384

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

C₄×C₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄²

C₄×D₁₃₉

yes

—

C₄²

D₅₅₆

yes

—

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

Q₈

—

278

—

Q₈

—

556

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

100

C₂×C₄

—

556

—

101

C₂×C₄

—

556

—

102

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

103

C₂×C₄

#14555773

yes

—

104

C₂×C₄

#14555775

yes

—

105

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

106

C₂×C₄

D₄×D₁₃₉

yes

—

107

C₂×C₄

C₂×D₅₅₆

yes

characteristic

108

C₂×C₄

#14555786

yes

—

109

C₈

—

278

—

110

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

111

C₂²

#17984831

yes

characteristic

112

C₂²

#17984838

yes

characteristic

113

C₂²

C₂×#14555773

yes

—

114

C₂²

C₂×#14555775

yes

—

115

C₄

—

116

C₄

—

556

—

117

C₄

#17984765

yes

complemented

118

C₄

#17984804

yes

—

119

C₄

#17984817

yes

—

120

C₂

#22580854

yes

characteristic

121

C₂

C₂×#17984765

yes

—

122

C₂

#22581305

yes

—

123

C₂

#22581325

yes

characteristic

124

C₂

#22581332

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#384×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#384	D₁₃₉
hypocenter	C₁₃₉	#1956
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#384×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #17984842	C₄	semidirect
53. C₄×C₁₃₉	#103	semidirect
62. C₁₃₉	#1956	semidirect
117. C₄	#17984765	semidirect

H ≅ G/N

type

8. #17984842

C₄

semidirect

53. C₄×C₁₃₉

#103

semidirect

62. C₁₃₉

#1956

semidirect

117. C₄

#17984765

semidirect

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

subgroup	index
C₂×#17984842	2
C₂×#17984765	2
C₂×C₄×C₁₃₉⋊C₈	2
#384×C₁₃₉	2
#1956	139

subgroup

index

C₂×#17984842

C₂×#17984765

C₂×C₄×C₁₃₉⋊C₈

#384×C₁₃₉

#1956

139

Subgroups — #31807205

Conjugacy classes of subgroups

Complements

Maximal subgroups