Group #34013314 — subgroups

Subgroups — #34013314

Order 18688.

414 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 236 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⋊#11924031	C₂	1	6	yes	maximal
2	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924031	C₂²	1	4	yes	—
4	C₄⋊#11924031	—	2	12	no	—
5	C₂×C₄×#10596831	C₄	1	3	yes	—
6	C₂×C₄×#10596831	C₂²	1	3	yes	—
7	C₄³×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924031	—	2	12	no	—
9	C₂²×#10596831	C₂×C₄	1	3	yes	—
10	C₂²×#10596831	C₂³	1	1	yes	characteristic
11	C₄×#10596831	—	2	6	no	—
12	C₄×#10596831	D₄	1	6	yes	—
13	C₄×#10596831	Q₈	1	6	yes	—
14	C₂×C₄²×C₇₃	C₈	1	3	yes	—
15	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
16	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
17	#11924031	—	4	4	no	—
18	C₂×#10596831	—	2	6	no	—
19	C₂×#10596831	C₄⋊C₄	1	6	yes	—
20	C₂×#10596831	C₂×D₄	1	3	yes	—
21	C₂×#10596831	C₂×Q₈	1	3	yes	—
22	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
23	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
24	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₇₃	—	2	6	no	—
26	C₄²×C₇₃	C₁₆	1	4	yes	complemented
27	C₄²×C₇₃	C₄⋊C₄	1	6	yes	—
28	C₄²×C₇₃	M₁₆	1	6	yes	—
29	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇₃	—	2	6	no	—
31	C₂×C₄×C₇₃	C₄⋊C₈	1	6	yes	—
32	C₂×C₄×C₇₃	C₄⋊C₈	1	6	yes	—
33	C₂×C₄×C₇₃	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₇₃	M₃₂	1	6	yes	—
35	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
36	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
37	#10596831	—	2	6	no	—
38	#10596831	#127	1	1	yes	characteristic
39	#10596831	C₄⋊Q₈	1	3	yes	—
40	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
41	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
42	C₂²×C₇₃	C₂×M₃₂	1	3	yes	—
43	C₄×C₇₃	—	2	6	no	—
44	C₄×C₇₃	C₄⋊C₁₆	1	12	yes	complemented
45	C₄×C₇₃	#384	1	1	yes	characteristic
46	C₄×C₇₃	#423	1	3	yes	—
47	#12898	—	73	1	no	maximal, Sylow
48	C₂×C₇₃	#1842	1	1	yes	characteristic
49	C₂×C₇₃	C₂×C₄⋊C₁₆	1	3	yes	—
50	C₂×C₇₃	#2150	1	3	yes	—
51	C₂×C₄⋊C₁₆	—	73	6	no	—
52	C₄²×C₈	—	73	1	no	—
53	C₇₃	#12898	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₁₆	—	73	4	no	—
55	C₂×C₄×C₈	—	73	3	no	—
56	C₂×C₄×C₈	—	73	3	no	—
57	C₄³	#64408	1	1	yes	characteristic
58	C₄⋊C₁₆	—	146	12	no	—
59	C₂²×C₈	—	73	1	no	—
60	C₂²×C₈	—	73	3	no	—
61	C₂×C₄²	#10596833	1	3	yes	—
62	C₂×C₄²	#10596833	1	1	yes	characteristic
63	C₂×C₄²	C₂×#64408	1	3	yes	—
64	C₂×C₁₆	—	146	12	no	—
65	C₄×C₈	—	73	6	no	—
66	C₄×C₈	—	73	6	no	—
67	C₄×C₈	—	146	6	no	—
68	C₂²×C₄	C₂×#10596833	1	3	yes	—
69	C₂²×C₄	C₂×#10596833	1	3	yes	—
70	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
71	C₂×C₈	—	73	3	no	—
72	C₂×C₈	—	73	3	no	—
73	C₂×C₈	—	73	6	no	—
74	C₂×C₈	—	146	6	no	—
75	C₄²	—	2	6	no	—
76	C₄²	#11924031	1	4	yes	complemented
77	C₄²	C₄⋊#64408	1	6	yes	—
78	C₄²	#11924038	1	6	yes	—
79	C₁₆	—	292	4	no	—
80	C₂³	C₂²×#10596833	1	1	yes	characteristic
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₄⋊#10596833	1	6	yes	—
83	C₂×C₄	C₄⋊#10596833	1	6	yes	—
84	C₂×C₄	C₂×#11924031	1	6	yes	—
85	C₂×C₄	#14556665	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
87	C₂×C₄	C₂×#11924038	1	3	yes	—
88	C₈	—	73	1	no	—
89	C₈	—	73	3	no	—
90	C₈	—	146	6	no	—
91	C₂²	C₂×C₄⋊#10596833	1	3	yes	—
92	C₂²	C₂²×#11924031	1	1	yes	characteristic
93	C₂²	C₂×#14556665	1	3	yes	—
94	C₄	—	2	6	no	—
95	C₄	C₄⋊#11924031	1	12	yes	complemented
96	C₄	#17989533	1	1	yes	characteristic
97	C₄	#17989591	1	3	yes	—
98	C₂	#22625863	1	1	yes	characteristic
99	C₂	C₂×C₄⋊#11924031	1	3	yes	—
100	C₂	#22626263	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11924031

C₂

yes

maximal

C₄²×#10596831

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

C₄⋊#11924031

—

C₂×C₄×#10596831

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

—

C₄³×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

C₄×#10596831

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂×C₄²×C₇₃

C₈

yes

—

C₂×C₄²×C₇₃

C₈

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

—

#11924031

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₁₆

yes

complemented

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

—

yes

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₃₂

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

—

C₂×C₇₃

#2150

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

—

C₇₃

#12898

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#64408

yes

characteristic

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#10596833

yes

—

C₂×C₄²

#10596833

yes

characteristic

C₂×C₄²

C₂×#64408

yes

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

C₂×#10596833

yes

—

C₂²×C₄

C₂×#10596833

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄²

—

C₄²

#11924031

yes

complemented

C₄²

C₄⋊#64408

yes

—

C₄²

#11924038

yes

—

C₁₆

—

292

—

C₂³

C₂²×#10596833

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊#10596833

yes

—

C₂×C₄

C₄⋊#10596833

yes

—

C₂×C₄

C₂×#11924031

yes

—

C₂×C₄

#14556665

yes

—

C₂×C₄

C₂×C₄⋊#64408

yes

—

C₂×C₄

C₂×#11924038

yes

—

C₈

—

C₈

—

C₈

—

146

—

C₂²

C₂×C₄⋊#10596833

yes

—

C₂²

C₂²×#11924031

yes

characteristic

C₂²

C₂×#14556665

yes

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

#22625863

yes

characteristic

C₂

C₂×C₄⋊#11924031

yes

—

100

C₂

#22626263

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄³×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12898
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₄³×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₄³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
26. C₄²×C₇₃	C₁₆	semidirect
44. C₄×C₇₃	C₄⋊C₁₆	semidirect
53. C₇₃	#12898	semidirect
76. C₄²	#11924031	semidirect
95. C₄	C₄⋊#11924031	semidirect

H ≅ G/N

type

26. C₄²×C₇₃

C₁₆

semidirect

44. C₄×C₇₃

C₄⋊C₁₆

semidirect

53. C₇₃

#12898

semidirect

76. C₄²

#11924031

semidirect

95. C₄

C₄⋊#11924031

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊#11924031	2
C₄²×#10596831	2
#12898	73

subgroup

index

C₂×C₄⋊#11924031

C₄²×#10596831

#12898

Subgroups — #34013314

Conjugacy classes of subgroups

Complements

Maximal subgroups