Group #34000852 — subgroups

Subgroups — #34000852

Order 18688.

152 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 36 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	#22624877	C₂	1	2	yes	maximal
2	#22610102	C₂	1	1	yes	maximal, characteristic
3	#17989483	—	2	2	no	—
4	C₂×#14556526	—	2	2	no	—
5	#17988611	C₂²	1	1	yes	characteristic
6	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
7	#423×C₇₃	C₄	1	1	yes	characteristic
8	#14556665	—	4	2	no	—
9	C₂×#11924012	D₄	1	2	yes	—
10	C₂²×#10596831	—	2	1	no	—
11	#14556526	—	4	4	no	—
12	C₄⋊#10596831	—	4	1	no	—
13	C₄×#10596831	—	2	1	no	—
14	C₄×#10596831	C₈	1	2	yes	—
15	C₂×M₁₆×C₇₃	—	2	1	no	—
16	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	—	4	1	no	—
18	#11924031	—	8	2	no	—
19	#11924012	—	4	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₂²×C₄×C₇₃	—	2	1	no	—
25	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	M₁₆×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	—	4	1	no	—
28	C₄²×C₇₃	—	2	1	no	—
29	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
31	C₂³×C₇₃	#101	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	—	4	2	no	—
37	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
39	C₈×C₇₃	—	8	1	no	—
40	#10596831	—	2	4	no	—
41	#10596831	—	4	2	no	—
42	C₂²×C₇₃	—	2	1	no	—
43	C₂²×C₇₃	—	4	1	no	—
44	C₂²×C₇₃	#324	1	1	yes	characteristic
45	C₄×C₇₃	—	2	2	no	—
46	C₄×C₇₃	—	2	1	no	—
47	C₄×C₇₃	—	4	1	no	—
48	C₄×C₇₃	#353	1	1	yes	characteristic
49	C₄×C₇₃	#356	1	1	yes	characteristic
50	#7373	—	73	1	no	maximal, Sylow
51	C₂×C₇₃	—	2	1	no	—
52	C₂×C₇₃	—	4	1	no	—
53	C₂×C₇₃	#1325	1	1	yes	characteristic
54	#1848	—	73	1	no	—
55	#1327	—	73	2	no	—
56	C₇₃	#7373	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	146	2	no	—
58	#423	—	73	1	no	—
59	#423	#64408	1	1	yes	characteristic
60	C₂×C₄×C₈	—	73	1	no	—
61	#349	—	146	2	no	—
62	C₂×M₁₆	—	2	1	no	—
63	C₂×M₁₆	—	73	2	no	—
64	C₂²×C₈	—	146	1	no	—
65	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
66	M₃₂	—	292	2	no	—
67	#108	—	292	4	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	292	1	no	—
70	C₄×C₈	—	73	2	no	—
71	C₄×C₈	—	146	1	no	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
74	M₁₆	—	4	2	no	—
75	M₁₆	—	292	2	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	146	2	no	—
78	C₂×C₈	—	146	2	no	—
79	C₂×C₈	—	146	2	no	—
80	C₂×C₈	—	146	2	no	—
81	C₄²	—	2	1	no	—
82	C₄²	#11924036	1	1	yes	characteristic
83	C₄²	#11924038	1	1	yes	characteristic
84	C₁₆	—	584	2	no	—
85	C₂³	#14556652	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	#14556648	1	1	yes	characteristic
92	C₂×C₄	#14556649	1	1	yes	characteristic
93	C₈	—	8	1	no	—
94	C₈	—	292	2	no	—
95	C₈	—	146	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	#17989433	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	#17989491	1	1	yes	characteristic
103	C₄	#17989497	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#22624874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624877

C₂

yes

maximal

#22610102

C₂

yes

maximal, characteristic

#17989483

—

C₂×#14556526

—

#17988611

C₂²

yes

characteristic

C₂×C₄×#10596831

C₄

yes

characteristic

#423×C₇₃

C₄

yes

characteristic

#14556665

—

C₂×#11924012

D₄

yes

—

—

—

—

—

C₈

yes

—

C₂×M₁₆×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

—

—

—

—

—

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

#101

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#99

yes

characteristic

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#324

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

C₇₃

#7373

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

—

yes

C₂×C₄×C₈

—

#349

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

146

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

292

—

#108

—

292

—

C₄⋊C₈

—

C₄⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#64408

yes

characteristic

M₁₆

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

584

—

C₂³

#14556652

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

292

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

#17989433

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#17989491

yes

characteristic

103

C₄

#17989497

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#22624874

yes

characteristic

	subgroup	G/H
center	C₄	#17989491
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	#423×C₇₃	C₄
socle	C₂×C₇₃	#1325
prime radical	C₂³×C₇₃	#101
cosocle	#17988611	C₂²
hypercenter	#423	#64408
hypocenter	C₇₃	#7373
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#17989491

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

#423×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

C₂²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
56. C₇₃	#7373	semidirect

H ≅ G/N

type

56. C₇₃

#7373

semidirect

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

subgroup	index
#22624877	2
#22610102	2
#7373	73

subgroup

index

#22624877

#22610102

#7373

Subgroups — #34000852

Conjugacy classes of subgroups

Complements

Maximal subgroups