Group #22580857 — subgroups

Subgroups — #22580857

Order 8896.

186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂²×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₃₉⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic
7	#95×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₇₈	—	2	2	no	—
9	C₂×Dic₂₇₈	—	2	2	no	—
10	C₂×Dic₂₇₈	C₄	1	4	yes	complemented
11	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
15	Dic₁₃₉⋊C₄	—	2	2	no	—
16	C₄×Dic₁₃₉	—	2	2	no	—
17	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
20	Dic₂₇₈	—	2	8	no	—
21	C₂×Dic₁₃₉	—	2	1	no	—
22	C₂×Dic₁₃₉	—	2	1	no	—
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₂×Dic₁₃₉	—	2	1	no	—
25	C₂×Dic₁₃₉	—	2	1	no	—
26	C₂×Dic₁₃₉	—	2	1	no	—
27	C₂×Dic₁₃₉	—	2	1	no	—
28	C₂×Dic₁₃₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₃₉	D₄	1	2	yes	—
31	C₂×Dic₁₃₉	D₄	1	2	yes	—
32	C₂×Dic₁₃₉	Q₈	1	2	yes	complemented
33	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	—	2	1	no	—
35	C₂×C₄×C₁₃₉	—	2	1	no	—
36	C₂×C₄×C₁₃₉	—	2	1	no	—
37	C₂×C₄×C₁₃₉	—	2	1	no	—
38	C₂×C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₄×C₁₃₉	—	2	1	no	—
40	C₂×C₄×C₁₃₉	—	2	1	no	—
41	C₂×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₁₃₉	Q₈	1	2	yes	complemented
43	C₂²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₁₃₉	#41	1	1	yes	characteristic
49	C₂²×C₁₃₉	#41	1	1	yes	characteristic
50	C₄×C₁₃₉	—	2	2	no	—
51	C₄×C₁₃₉	—	2	2	no	—
52	C₄×C₁₃₉	—	4	1	no	—
53	Dic₁₃₉	—	2	2	no	—
54	Dic₁₃₉	—	2	2	no	—
55	Dic₁₃₉	—	4	1	no	—
56	Dic₁₃₉	C₂²⋊C₄	1	4	yes	—
57	C₂×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₃₉	#124	1	1	yes	characteristic
63	C₂×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₁₃₉	#391	1	1	yes	characteristic, complemented, Sylow
65	#391	—	139	1	no	maximal, Sylow
66	C₂²×Q₈	—	139	1	no	—
67	C₂×C₄⋊C₄	—	139	1	no	—
68	C₂×C₄²	—	139	1	no	—
69	#95	—	139	1	no	—
70	#95	—	139	1	no	—
71	#95	—	139	1	no	—
72	#95	D₁₃₉	1	1	yes	characteristic
73	C₂×Q₈	—	278	2	no	—
74	C₂×Q₈	—	278	2	no	—
75	C₂×Q₈	—	139	4	no	—
76	C₂²×C₄	—	139	1	no	—
77	C₂²×C₄	—	139	1	no	—
78	C₂²×C₄	—	139	1	no	—
79	C₂²×C₄	—	139	1	no	—
80	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
83	C₄⋊C₄	—	278	2	no	—
84	C₄²	—	278	2	no	—
85	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
86	Q₈	—	278	8	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	278	1	no	—
95	C₂×C₄	—	278	1	no	—
96	C₂×C₄	—	278	1	no	—
97	C₂×C₄	—	278	1	no	—
98	C₂×C₄	—	278	1	no	—
99	C₂×C₄	—	278	1	no	—
100	C₂×C₄	—	278	1	no	—
101	C₂×C₄	—	139	2	no	—
102	C₂×C₄	—	139	2	no	—
103	C₂×C₄	—	139	2	no	—
104	C₂×C₄	—	139	2	no	—
105	C₂×C₄	—	139	2	no	—
106	C₂×C₄	C₄×D₁₃₉	1	2	yes	—
107	C₂×C₄	Dic₂₇₈	1	2	yes	complemented
108	C₂²	C₂×C₄×D₁₃₉	1	1	yes	characteristic
109	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
110	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
111	C₂²	Q₈×D₁₃₉	1	1	yes	characteristic
112	C₂²	C₂×Dic₂₇₈	1	1	yes	characteristic
113	C₂²	#14555784	1	1	yes	characteristic
114	C₂²	#14555785	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	1	no	—
118	C₄	—	278	2	no	—
119	C₄	—	278	2	no	—
120	C₄	—	556	1	no	—
121	C₄	—	139	4	no	—
122	C₂	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
123	C₂	#17984807	1	1	yes	characteristic
124	C₂	C₄×Dic₂₇₈	1	1	yes	characteristic
125	C₂	#17984819	1	1	yes	characteristic
126	C₂	#17984820	1	1	yes	characteristic
127	C₂	#17984828	1	1	yes	characteristic
128	C₂	#17984841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₇₈

C₂

yes

maximal, characteristic

C₂×Dic₁₃₉⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₉

C₂

yes

maximal, characteristic

C₁₃₉⋊#95

C₂

yes

maximal, characteristic

C₁₃₉⋊#95

C₂

yes

maximal, characteristic

C₁₃₉⋊#95

C₂

yes

maximal, characteristic

#95×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₄

yes

complemented

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

D₄

yes

—

C₂×Dic₁₃₉

D₄

yes

—

C₂×Dic₁₃₉

Q₈

yes

complemented

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₄

yes

—

C₂×C₄×C₁₃₉

Q₈

yes

complemented

C₂²×C₁₃₉

C₂²×C₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×Q₈

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₂²⋊C₄

yes

—

C₂×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₃₉

C₄×Q₈

yes

characteristic

C₂×C₁₃₉

C₄×Q₈

yes

characteristic

C₂×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₃₉

#124

yes

characteristic

C₂×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₁₃₉

#391

yes

characteristic, complemented, Sylow

#391

—

139

maximal, Sylow

C₂²×Q₈

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

—

139

—

#95

—

139

—

#95

—

139

—

#95

—

139

—

#95

D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄²

—

278

—

C₂³

C₂²×D₁₃₉

yes

characteristic

Q₈

—

278

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

100

C₂×C₄

—

278

—

101

C₂×C₄

—

139

—

102

C₂×C₄

—

139

—

103

C₂×C₄

—

139

—

104

C₂×C₄

—

139

—

105

C₂×C₄

—

139

—

106

C₂×C₄

C₄×D₁₃₉

yes

—

107

C₂×C₄

Dic₂₇₈

yes

complemented

108

C₂²

C₂×C₄×D₁₃₉

yes

characteristic

109

C₂²

D₄×D₁₃₉

yes

characteristic

110

C₂²

D₄×D₁₃₉

yes

characteristic

111

C₂²

Q₈×D₁₃₉

yes

characteristic

112

C₂²

C₂×Dic₂₇₈

yes

characteristic

113

C₂²

#14555784

yes

characteristic

114

C₂²

#14555785

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

278

—

119

C₄

—

278

—

120

C₄

—

556

—

121

C₄

—

139

—

122

C₂

C₂²⋊C₄×D₁₃₉

yes

characteristic

123

C₂

#17984807

yes

characteristic

124

C₂

C₄×Dic₂₇₈

yes

characteristic

125

C₂

#17984819

yes

characteristic

126

C₂

#17984820

yes

characteristic

127

C₂

#17984828

yes

characteristic

128

C₂

#17984841

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂²×C₄
Fitting	#95×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂³
hypercenter	#95	D₁₃₉
hypocenter	C₁₃₉	#391
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂²×C₄

Fitting

#95×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
10. C₂×Dic₂₇₈	C₄	semidirect
32. C₂×Dic₁₃₉	Q₈	semidirect
42. C₂×C₄×C₁₃₉	Q₈	semidirect
64. C₁₃₉	#391	semidirect
107. C₂×C₄	Dic₂₇₈	semidirect

H ≅ G/N

type

10. C₂×Dic₂₇₈

C₄

semidirect

32. C₂×Dic₁₃₉

Q₈

semidirect

42. C₂×C₄×C₁₃₉

Q₈

semidirect

64. C₁₃₉

#391

semidirect

107. C₂×C₄

Dic₂₇₈

semidirect

Subgroups — #22580857

Conjugacy classes of subgroups

Complements