Group #22577902 — subgroups

Subgroups — C₂×#17984068 #22577902

Order 8768.

322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 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×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#17984068	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂×#14555487	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₇₄	—	2	4	no	—
9	C₂×Dic₂₇₄	C₂²	1	4	yes	complemented
10	C₂²×Dic₁₃₇	C₂²	1	2	yes	—
11	C₂²×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₃₇	—	2	4	no	—
13	C₂×C₄×D₁₃₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₃₇	C₂²	1	8	yes	complemented
15	#14555487	C₂²	1	8	yes	—
16	C₂²×C₄×C₁₃₇	C₂²	1	2	yes	—
17	C₂²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	complemented
19	C₂²×D₁₃₇	—	2	2	no	—
20	C₂²×D₁₃₇	—	2	2	no	—
21	C₂²×D₁₃₇	Q₈	1	4	yes	complemented
22	C₂²×D₁₃₇	C₂³	1	2	yes	—
23	Dic₂₇₄	—	2	8	no	—
24	C₂×Dic₁₃₇	—	2	2	no	—
25	C₂×Dic₁₃₇	—	2	4	no	—
26	C₂×Dic₁₃₇	C₂³	1	4	yes	—
27	C₂×Dic₁₃₇	C₂³	1	2	yes	—
28	C₄×D₁₃₇	—	2	8	no	—
29	C₂³×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	4	no	—
31	C₂×C₄×C₁₃₇	D₄	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂³	1	4	yes	—
33	C₂×C₄×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₁₃₇	C₂×D₄	1	2	yes	—
35	C₂²×C₁₃₇	C₂×Q₈	1	2	yes	—
36	C₂²×C₁₃₇	#41	1	2	yes	—
37	C₂²×C₁₃₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₃₇	—	2	8	no	—
39	C₂×D₁₃₇	—	2	2	no	—
40	C₂×D₁₃₇	—	2	2	no	—
41	C₂×D₁₃₇	C₂×Q₈	1	4	yes	complemented
42	C₄×C₁₃₇	—	2	4	no	—
43	C₄×C₁₃₇	C₂×D₄	1	4	yes	—
44	Dic₁₃₇	—	2	2	no	—
45	Dic₁₃₇	—	2	4	no	—
46	C₂×C₁₃₇	C₂²⋊Q₈	1	4	yes	complemented
47	C₂×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₃₇	C₂×#41	1	1	yes	characteristic
50	D₁₃₇	—	2	4	no	—
51	C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₂²⋊Q₈	—	137	1	no	maximal, Sylow
53	C₂²×Q₈	—	137	1	no	—
54	C₂³×C₄	—	137	1	no	—
55	C₂²⋊Q₈	—	137	8	no	—
56	C₂×C₄⋊C₄	—	137	2	no	—
57	C₂×C₄⋊C₄	D₁₃₇	1	1	yes	characteristic, complemented
58	C₂×C₂²⋊C₄	—	137	2	no	—
59	C₂⁴	—	137	1	no	—
60	C₂×Q₈	—	274	4	no	—
61	C₂×Q₈	—	137	4	no	—
62	C₂²×C₄	—	274	4	no	—
63	C₂²×C₄	—	137	2	no	—
64	C₂²×C₄	—	137	1	no	—
65	C₂²×C₄	—	137	4	no	—
66	C₂²×C₄	C₂×D₁₃₇	1	2	yes	—
67	C₂²×C₄	C₂×D₁₃₇	1	1	yes	characteristic
68	C₄⋊C₄	—	137	8	no	—
69	C₄⋊C₄	C₂×D₁₃₇	1	4	yes	complemented
70	C₂²⋊C₄	—	137	8	no	—
71	C₂³	—	274	2	no	—
72	C₂³	—	274	2	no	—
73	C₂³	—	137	4	no	—
74	C₂³	—	137	2	no	—
75	C₂³	C₂²×D₁₃₇	1	1	yes	characteristic
76	Q₈	—	274	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	274	2	no	—
79	C₂×C₄	—	274	4	no	—
80	C₂×C₄	—	274	8	no	—
81	C₂×C₄	—	137	4	no	—
82	C₂×C₄	—	137	2	no	—
83	C₂×C₄	D₅₄₈	1	4	yes	—
84	C₂×C₄	C₂²×D₁₃₇	1	4	yes	—
85	C₂×C₄	C₂²×D₁₃₇	1	2	yes	—
86	C₂²	—	274	8	no	—
87	C₂²	—	274	2	no	—
88	C₂²	—	274	2	no	—
89	C₂²	—	137	4	no	—
90	C₂²	C₂×D₅₄₈	1	2	yes	—
91	C₂²	Q₈×D₁₃₇	1	2	yes	—
92	C₂²	#14555509	1	2	yes	—
93	C₂²	C₂³×D₁₃₇	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	274	2	no	—
96	C₄	—	274	4	no	—
97	C₄	C₂×D₅₄₈	1	4	yes	—
98	C₂	—	274	4	no	—
99	C₂	#17984068	1	4	yes	direct product factor
100	C₂	C₂²×D₅₄₈	1	1	yes	characteristic
101	C₂	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
102	C₂	C₂×#14555509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₇₄

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₇

C₂

yes

maximal, characteristic

#17984068

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₃₇

C₂

yes

maximal, complemented

C₂×#14555487

C₂

yes

maximal

C₂×C₄⋊C₄×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂²

yes

complemented

C₂²×Dic₁₃₇

C₂²

yes

—

C₂²×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂²

yes

—

C₄⋊Dic₁₃₇

C₂²

yes

complemented

#14555487

C₂²

yes

—

C₂²×C₄×C₁₃₇

C₂²

yes

—

C₂²×C₄×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂²

yes

complemented

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

Q₈

yes

complemented

C₂²×D₁₃₇

C₂³

yes

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

C₂³

yes

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂³

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

D₄

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₁₃₇

#41

yes

—

C₂²×C₁₃₇

C₂⁴

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₂×Q₈

yes

complemented

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×D₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

C₂²⋊Q₈

yes

complemented

C₂×C₁₃₇

C₂²×D₄

yes

characteristic

C₂×C₁₃₇

C₂²×Q₈

yes

characteristic

C₂×C₁₃₇

C₂×#41

yes

characteristic

D₁₃₇

—

C₁₃₇

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

137

maximal, Sylow

C₂²×Q₈

—

137

—

C₂³×C₄

—

137

—

C₂²⋊Q₈

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

D₁₃₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

137

—

C₂⁴

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×D₁₃₇

yes

—

C₂²×C₄

C₂×D₁₃₇

yes

characteristic

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×D₁₃₇

yes

complemented

C₂²⋊C₄

—

137

—

C₂³

—

274

—

C₂³

—

274

—

C₂³

—

137

—

C₂³

—

137

—

C₂³

C₂²×D₁₃₇

yes

characteristic

Q₈

—

274

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

D₅₄₈

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂×C₄

C₂²×D₁₃₇

yes

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂×D₅₄₈

yes

—

C₂²

Q₈×D₁₃₇

yes

—

C₂²

#14555509

yes

—

C₂²

C₂³×D₁₃₇

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

274

—

C₄

C₂×D₅₄₈

yes

—

C₂

—

274

—

C₂

#17984068

yes

direct product factor

100

C₂

C₂²×D₅₄₈

yes

characteristic

101

C₂

C₂×Q₈×D₁₃₇

yes

characteristic

102

C₂

C₂×#14555509

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₃₇
commutator	C₂²×C₁₃₇	C₂⁴
Frattini	C₂²	C₂³×D₁₃₇
Fitting	C₂×C₄⋊C₄×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₃₇
hypocenter	C₁₃₇	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₃₇

commutator

C₂²×C₁₃₇

C₂⁴

Frattini

C₂²

C₂³×D₁₃₇

Fitting

C₂×C₄⋊C₄×C₁₃₇

C₂

socle

C₂³×C₁₃₇

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₃₇

hypocenter

C₁₃₇

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂²×Dic₂₇₄	C₂	semidirect
3. #17984068	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₃₇	C₂	semidirect
6. C₂×C₄⋊C₄×C₁₃₇	C₂	semidirect
9. C₂×Dic₂₇₄	C₂²	semidirect
14. C₄⋊Dic₁₃₇	C₂²	semidirect
18. C₄⋊C₄×C₁₃₇	C₂²	semidirect
21. C₂²×D₁₃₇	Q₈	semidirect
41. C₂×D₁₃₇	C₂×Q₈	semidirect
46. C₂×C₁₃₇	C₂²⋊Q₈	semidirect
51. C₁₃₇	C₂×C₂²⋊Q₈	semidirect
57. C₂×C₄⋊C₄	D₁₃₇	semidirect
69. C₄⋊C₄	C₂×D₁₃₇	semidirect
99. C₂	#17984068	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₂₇₄

C₂

semidirect

3. #17984068

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₃₇

C₂

semidirect

6. C₂×C₄⋊C₄×C₁₃₇

C₂

semidirect

9. C₂×Dic₂₇₄

C₂²

semidirect

14. C₄⋊Dic₁₃₇

C₂²

semidirect

18. C₄⋊C₄×C₁₃₇

C₂²

semidirect

21. C₂²×D₁₃₇

Q₈

semidirect

41. C₂×D₁₃₇

C₂×Q₈

semidirect

46. C₂×C₁₃₇

C₂²⋊Q₈

semidirect

51. C₁₃₇

C₂×C₂²⋊Q₈

semidirect

57. C₂×C₄⋊C₄

D₁₃₇

semidirect

69. C₄⋊C₄

C₂×D₁₃₇

semidirect

99. C₂

#17984068

direct (N × H)

Subgroups — C₂×#17984068 #22577902

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17984068 #22577902

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17984068 #22577902