Group #20295253 — subgroups

Subgroups — #20295253

Order 6976.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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×#13182017	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812678	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812630	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812622	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812620	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	#13182017	—	2	2	no	—
10	C₂×#11724784	—	2	2	no	—
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₂×#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#13182008	—	2	1	no	—
15	C₂²⋊Dic₁₀₉	—	2	1	no	—
16	C₂×D₄×C₁₀₉	—	2	1	no	—
17	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
21	C₂³×C₁₀₉	—	2	1	no	—
22	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
23	D₄×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	—	2	1	no	—
25	D₄×C₁₀₉	—	4	2	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	—	2	1	no	—
33	Dic₂₁₈	—	2	1	no	—
34	Dic₂₁₈	—	2	1	no	—
35	#11724784	—	2	1	no	—
36	#11724784	D₄	1	2	yes	complemented
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	—	2	1	no	—
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	—	4	1	no	—
41	C₂²×C₁₀₉	—	4	1	no	—
42	C₂²×C₁₀₉	SD₁₆	1	2	yes	complemented
43	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	—	4	1	no	—
46	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₀₉	#41	1	1	yes	characteristic
48	Dic₁₀₉	—	4	1	no	—
49	Dic₁₀₉	—	4	1	no	—
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	—	8	1	no	—
52	C₂×C₁₀₉	#121	1	1	yes	characteristic
53	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₀₉	#135	1	1	yes	characteristic
55	C₁₀₉	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	109	1	no	maximal, Sylow
57	C₂×SD₁₆	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	C₂²⋊Q₈	—	109	1	no	—
60	#121	D₁₀₉	1	1	yes	characteristic
61	#106	—	109	1	no	—
62	#103	—	109	1	no	—
63	#102	—	109	1	no	—
64	C₂×Q₈	—	109	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
68	SD₁₆	—	218	2	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	218	2	no	—
72	C₄⋊C₄	—	109	1	no	—
73	C₄⋊C₄	—	218	1	no	—
74	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂²⋊C₄	—	2	1	no	—
76	C₂²⋊C₄	—	218	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
79	Q₈	—	218	1	no	—
80	Q₈	—	218	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	218	1	no	—
88	C₂×C₄	—	218	1	no	—
89	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₈	—	109	2	no	—
92	C₈	—	218	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	#13182017	1	2	yes	complemented
99	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	436	1	no	—
103	C₄	—	436	1	no	—
104	C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₄	#13182028	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	8	1	no	—
108	C₂	#16812671	1	1	yes	characteristic
109	C₂	C₂×#13182017	1	1	yes	characteristic
110	C₂	#16812719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182017

C₂

yes

maximal, characteristic, complemented

C₂²×#11724784

C₂

yes

maximal, characteristic, complemented

#16812678

C₂

yes

maximal, characteristic, complemented

#16812630

C₂

yes

maximal, characteristic, complemented

#16812622

C₂

yes

maximal, characteristic, complemented

#16812620

C₂

yes

maximal, characteristic, complemented

#121×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₂₁₈

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

C₂²⋊Dic₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

#11724784

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

SD₁₆

yes

complemented

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#121

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

characteristic, complemented, Sylow

#465

—

109

maximal, Sylow

C₂×SD₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²⋊Q₈

—

109

—

#121

D₁₀₉

yes

characteristic

#106

—

109

—

#103

—

109

—

#102

—

109

—

C₂×Q₈

—

109

—

C₂×D₄

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

SD₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

218

—

C₂³

—

C₂³

C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

218

—

Q₈

—

218

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13182017

yes

complemented

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

436

—

103

C₄

—

436

—

104

C₄

D₄×D₁₀₉

yes

characteristic

105

C₄

#13182028

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#16812671

yes

characteristic

109

C₂

C₂×#13182017

yes

characteristic

110

C₂

#16812719

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#121×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#121	D₁₀₉
hypocenter	C₁₀₉	#465
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#121×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂×#13182017	C₂	semidirect
2. C₂²×#11724784	C₂	semidirect
3. #16812678	C₂	semidirect
4. #16812630	C₂	semidirect
5. #16812622	C₂	semidirect
6. #16812620	C₂	semidirect
36. #11724784	D₄	semidirect
42. C₂²×C₁₀₉	SD₁₆	semidirect
55. C₁₀₉	#465	semidirect
98. C₂²	#13182017	semidirect

H ≅ G/N

type

1. C₂×#13182017

C₂

semidirect

2. C₂²×#11724784

C₂

semidirect

3. #16812678

C₂

semidirect

4. #16812630

C₂

semidirect

5. #16812622

C₂

semidirect

6. #16812620

C₂

semidirect

36. #11724784

D₄

semidirect

42. C₂²×C₁₀₉

SD₁₆

semidirect

55. C₁₀₉

#465

semidirect

98. C₂²

#13182017

semidirect

Subgroups — #20295253

Conjugacy classes of subgroups

Complements