Group #20295968 — subgroups

Subgroups — #20295968

Order 6976.

234 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊#89137	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#89137	C₂	1	2	yes	maximal
3	(C₂×#89137)⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#89137	C₂²	1	4	yes	—
6	C₄⋊#89137	—	2	4	no	—
7	C₂²⋊#89137	—	2	4	no	—
8	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
10	C₂×C₄×D₁₀₉	—	2	4	no	—
11	C₂×C₄×D₁₀₉	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄×D₁₀₉	C₄	1	1	yes	characteristic, complemented
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×D₁₀₉	—	2	2	no	—
18	C₂²×D₁₀₉	—	2	2	no	—
19	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×D₁₀₉	D₄	1	2	yes	—
21	C₂²×D₁₀₉	Q₈	1	2	yes	—
22	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
23	C₂×#89137	—	2	4	no	—
24	C₂×#89137	—	2	4	no	—
25	C₂×#89137	—	2	4	no	—
26	C₂×C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₀₉	—	2	2	no	—
33	C₄×D₁₀₉	—	2	2	no	—
34	C₄×D₁₀₉	—	2	2	no	—
35	C₄×D₁₀₉	D₄	1	4	yes	—
36	C₂²×C₁₀₉	—	2	1	no	—
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	C₄⋊C₄	1	2	yes	complemented
39	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	4	no	—
43	C₂×D₁₀₉	—	2	4	no	—
44	C₂×D₁₀₉	C₄⋊C₄	1	2	yes	complemented
45	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×D₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×D₁₀₉	#41	1	2	yes	—
49	#89137	—	4	4	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	complemented
52	Dic₁₀₉	—	2	1	no	—
53	Dic₁₀₉	C₂²⋊C₄	1	2	yes	complemented
54	C₂×C₁₀₉	—	2	2	no	—
55	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₀₉	#117	1	1	yes	characteristic
58	D₁₀₉	—	2	2	no	—
59	D₁₀₉	#121	1	2	yes	—
60	D₁₀₉	C₂²⋊Q₈	1	2	yes	—
61	C₁₀₉	#380	1	1	yes	characteristic, complemented, Sylow
62	#380	—	109	1	no	maximal, Sylow
63	C₂³×C₄	—	109	1	no	—
64	C₂×C₄⋊C₄	—	109	2	no	—
65	C₂×C₂²⋊C₄	—	109	2	no	—
66	#95	—	109	2	no	—
67	C₂⁴	—	109	1	no	—
68	C₂²×C₄	—	109	1	no	—
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	—	109	1	no	—
71	C₂²×C₄	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	109	4	no	—
74	C₂²×C₄	—	218	4	no	—
75	C₂²×C₄	#89137	1	1	yes	characteristic, complemented
76	C₄⋊C₄	—	218	4	no	—
77	C₂²⋊C₄	—	218	4	no	—
78	C₂³	—	109	1	no	—
79	C₂³	—	109	2	no	—
80	C₂³	—	109	2	no	—
81	C₂³	—	109	1	no	—
82	C₂³	—	218	2	no	—
83	C₂³	—	218	2	no	—
84	C₂³	C₂×#89137	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	4	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	218	2	no	—
90	C₂×C₄	—	218	2	no	—
91	C₂×C₄	—	218	2	no	—
92	C₂×C₄	—	218	4	no	—
93	C₂×C₄	—	218	4	no	—
94	C₂×C₄	—	218	4	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	C₂×#89137	1	1	yes	characteristic
97	C₂×C₄	C₂×#89137	1	1	yes	characteristic
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	109	2	no	—
101	C₂²	—	109	1	no	—
102	C₂²	—	109	1	no	—
103	C₂²	—	109	2	no	—
104	C₂²	—	109	2	no	—
105	C₂²	—	218	4	no	—
106	C₂²	—	218	4	no	—
107	C₂²	—	218	1	no	—
108	C₂²	—	218	1	no	—
109	C₂²	C₄⋊#89137	1	2	yes	complemented
110	C₂²	C₂²×#89137	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	109	2	no	—
113	C₄	—	436	4	no	—
114	C₄	—	218	1	no	—
115	C₄	C₂²⋊#89137	1	2	yes	complemented
116	C₂	—	2	2	no	—
117	C₂	—	109	2	no	—
118	C₂	—	109	2	no	—
119	C₂	—	218	2	no	—
120	C₂	C₂×C₂²⋊#89137	1	1	yes	characteristic
121	C₂	C₂×C₄⋊#89137	1	1	yes	characteristic
122	C₂	#16812778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#89137

C₂

yes

maximal, complemented

C₂×C₂²⋊#89137

C₂

yes

maximal

(C₂×#89137)⋊C₄

C₂

yes

maximal, complemented

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂²×#89137

C₂²

yes

—

C₄⋊#89137

—

C₂²⋊#89137

—

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₄

yes

characteristic, complemented

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₄

yes

characteristic, complemented

C₂×C₄×D₁₀₉

C₄

yes

characteristic, complemented

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₄

yes

characteristic, complemented

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

D₄

yes

—

C₂²×D₁₀₉

Q₈

yes

—

C₂²×D₁₀₉

C₂³

yes

—

—

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₄

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₄

yes

complemented

C₂×D₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

C₂×D₄

yes

—

C₂×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×D₁₀₉

#41

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#117

yes

characteristic

D₁₀₉

—

D₁₀₉

#121

yes

—

D₁₀₉

C₂²⋊Q₈

yes

—

C₁₀₉

#380

yes

characteristic, complemented, Sylow

#380

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

#95

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

#89137

yes

characteristic, complemented

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

C₂²

—

C₂²

—

100

C₂²

—

109

—

101

C₂²

—

109

—

102

C₂²

—

109

—

103

C₂²

—

109

—

104

C₂²

—

109

—

105

C₂²

—

218

—

106

C₂²

—

218

—

107

C₂²

—

218

—

108

C₂²

—

218

—

109

C₂²

C₄⋊#89137

yes

complemented

110

C₂²

C₂²×#89137

yes

characteristic

111

C₄

—

112

C₄

—

109

—

113

C₄

—

436

—

114

C₄

—

218

—

115

C₄

C₂²⋊#89137

yes

complemented

116

C₂

—

117

C₂

—

109

—

118

C₂

—

109

—

119

C₂

—

218

—

120

C₂

C₂×C₂²⋊#89137

yes

characteristic

121

C₂

C₂×C₄⋊#89137

yes

characteristic

122

C₂

#16812778

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂²×C₄×C₁₀₉	C₄
hypercenter	C₂²×C₄	#89137
hypocenter	C₁₀₉	#380
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂²×C₄×C₁₀₉

C₄

C₂²×C₄

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂²×C₄

N	H ≅ G/N	type
1. C₂×C₄⋊#89137	C₂	semidirect
3. (C₂×#89137)⋊C₄	C₂	semidirect
9. C₂²×Dic₁₀₉	C₄	semidirect
11. C₂×C₄×D₁₀₉	C₄	semidirect
12. C₂×C₄×D₁₀₉	C₄	semidirect
15. C₂²×C₄×C₁₀₉	C₄	semidirect
38. C₂²×C₁₀₉	C₄⋊C₄	semidirect
44. C₂×D₁₀₉	C₄⋊C₄	semidirect
51. C₄×C₁₀₉	C₂²⋊C₄	semidirect
53. Dic₁₀₉	C₂²⋊C₄	semidirect
61. C₁₀₉	#380	semidirect
75. C₂²×C₄	#89137	semidirect
109. C₂²	C₄⋊#89137	semidirect
115. C₄	C₂²⋊#89137	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#89137

C₂

semidirect

3. (C₂×#89137)⋊C₄

C₂

semidirect

9. C₂²×Dic₁₀₉

C₄

semidirect

11. C₂×C₄×D₁₀₉

C₄

semidirect

12. C₂×C₄×D₁₀₉

C₄

semidirect

15. C₂²×C₄×C₁₀₉

C₄

semidirect

38. C₂²×C₁₀₉

C₄⋊C₄

semidirect

44. C₂×D₁₀₉

C₄⋊C₄

semidirect

51. C₄×C₁₀₉

C₂²⋊C₄

semidirect

53. Dic₁₀₉

C₂²⋊C₄

semidirect

61. C₁₀₉

#380

semidirect

75. C₂²×C₄

#89137

semidirect

109. C₂²

C₄⋊#89137

semidirect

115. C₄

C₂²⋊#89137

semidirect

Subgroups — #20295968

Conjugacy classes of subgroups

Complements