Group #22626271 — subgroups

Subgroups — C₂×#17989510 #22626271

Order 9344.

168 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 82 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×#14556665	C₂	1	2	yes	maximal
2	#17989510	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
4	#14556665	—	2	2	no	—
5	#14556665	C₂²	1	4	yes	—
6	C₂×#11924031	—	2	2	no	—
7	C₂²×#10596831	C₄	1	1	yes	characteristic
8	C₂²×#10596831	C₂²	1	1	yes	characteristic
9	C₄×#10596831	C₄	1	2	yes	—
10	C₄×#10596831	C₂²	1	2	yes	—
11	C₂×C₄²×C₇₃	C₄	1	1	yes	characteristic
12	#11924031	—	2	4	no	—
13	C₂×#10596831	C₈	1	2	yes	—
14	C₂×#10596831	C₈	1	2	yes	—
15	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
16	C₂×#10596831	C₂×C₄	1	2	yes	—
17	C₂×#10596831	D₄	1	1	yes	characteristic
18	C₂×#10596831	D₄	1	1	yes	characteristic
19	C₂×#10596831	Q₈	1	1	yes	characteristic
20	C₂×#10596831	Q₈	1	1	yes	characteristic
21	C₂×#10596831	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	—	2	1	no	—
23	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₃	—	2	1	no	—
25	C₄²×C₇₃	C₂×C₄	1	2	yes	—
26	C₂³×C₇₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₃	—	2	4	no	—
28	C₂×C₄×C₇₃	—	2	2	no	—
29	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇₃	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	#10596831	C₄⋊C₄	1	1	yes	characteristic
36	#10596831	C₄⋊C₄	1	1	yes	characteristic
37	#10596831	C₂×C₈	1	4	yes	—
38	#10596831	C₂×D₄	1	1	yes	characteristic
39	#10596831	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	2	no	—
41	C₂²×C₇₃	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₇₃	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₇₃	—	2	4	no	—
45	C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₇₃	—	2	1	no	—
50	C₂×C₇₃	—	2	1	no	—
51	C₂×C₇₃	#364	1	2	yes	complemented
52	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×#364	—	73	1	no	maximal, Sylow
54	C₇₃	C₂×#364	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	73	2	no	—
56	C₂×C₄×C₈	—	73	1	no	—
57	#364	—	73	4	no	—
58	C₂²×C₈	—	73	1	no	—
59	C₂²×C₈	—	73	1	no	—
60	C₂×C₄²	#64408	1	1	yes	characteristic
61	M₃₂	—	73	4	no	—
62	M₃₂	—	146	2	no	—
63	C₂×C₁₆	—	146	2	no	—
64	C₄×C₈	—	73	2	no	—
65	C₄×C₈	—	73	2	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
68	C₂×C₈	—	73	1	no	—
69	C₂×C₈	—	73	1	no	—
70	C₂×C₈	—	73	1	no	—
71	C₂×C₈	—	73	1	no	—
72	C₂×C₈	—	73	2	no	—
73	C₂×C₈	—	73	2	no	—
74	C₂×C₈	—	73	2	no	—
75	C₂×C₈	—	73	1	no	—
76	C₂×C₈	—	73	1	no	—
77	C₄²	—	2	1	no	—
78	C₄²	C₂×#64408	1	2	yes	—
79	C₁₆	—	146	4	no	—
80	C₂³	#11924037	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
84	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
85	C₂×C₄	#11924036	1	1	yes	characteristic
86	C₂×C₄	#11924036	1	1	yes	characteristic
87	C₂×C₄	#11924037	1	1	yes	characteristic
88	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
89	C₈	—	73	4	no	—
90	C₈	—	73	1	no	—
91	C₈	—	73	1	no	—
92	C₈	—	73	1	no	—
93	C₈	—	73	1	no	—
94	C₂²	—	2	2	no	—
95	C₂²	#14556658	1	1	yes	characteristic
96	C₂²	#14556658	1	1	yes	characteristic
97	C₂²	C₂×#11924037	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	#14556658	1	1	yes	characteristic
100	C₄	#14556658	1	1	yes	characteristic
101	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
102	C₄	C₂×#11924036	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#17989510	1	2	yes	direct product factor
106	C₂	C₂×#14556658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556665

C₂

yes

maximal

#17989510

C₂

yes

maximal, direct product factor

C₂×C₄×#10596831

C₂

yes

maximal, characteristic

#14556665

—

#14556665

C₂²

yes

—

C₂×#11924031

—

C₂²×#10596831

C₄

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₄×#10596831

C₄

yes

—

C₄×#10596831

C₂²

yes

—

C₂×C₄²×C₇₃

C₄

yes

characteristic

#11924031

—

C₂×#10596831

C₈

yes

—

C₂×#10596831

C₈

yes

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₂×#10596831

Q₈

yes

characteristic

C₂×#10596831

Q₈

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₂×C₈

yes

—

#10596831

C₂×D₄

yes

characteristic

#10596831

C₂×Q₈

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊C₈

yes

characteristic

C₂²×C₇₃

C₄⋊C₈

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#364

yes

complemented

C₂×C₇₃

C₂×C₄⋊C₈

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#64408

yes

characteristic

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

C₂×#64408

yes

—

C₁₆

—

146

—

C₂³

#11924037

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#64408

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

#14556658

yes

characteristic

100

C₄

#14556658

yes

characteristic

101

C₄

C₂×C₄⋊#64408

yes

characteristic

102

C₄

C₂×#11924036

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#17989510

yes

direct product factor

106

C₂

C₂×#14556658

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×C₄²×C₇₃	C₄
socle	C₂²×C₇₃	C₄⋊C₈
hypercenter	C₂×C₄²	#64408
hypocenter	C₇₃	C₂×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₄⋊#64408

commutator

C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×C₄²×C₇₃

C₄

socle

C₂²×C₇₃

C₄⋊C₈

C₂×C₄²

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. #17989510	C₂	direct (N × H)
51. C₂×C₇₃	#364	semidirect
54. C₇₃	C₂×#364	semidirect
105. C₂	#17989510	direct (N × H)

H ≅ G/N

type

2. #17989510

C₂

direct (N × H)

51. C₂×C₇₃

#364

semidirect

54. C₇₃

C₂×#364

semidirect

105. C₂

#17989510

direct (N × H)

Subgroups — C₂×#17989510 #22626271

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989510 #22626271

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989510 #22626271