Subgroups — #36355322

Order 22144.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149619	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14583801	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14583801	C₂	1	4	yes	maximal, complemented
4	#405×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	#19149619	—	2	2	no	—
6	C₂×#14583801	—	2	2	no	—
7	C₂×#14583801	C₂²	1	2	yes	—
8	C₂×#14583801	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₇₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	#117×C₁₇₃	C₄	1	1	yes	characteristic
12	#97×C₁₇₃	C₄	1	2	yes	—
13	C₄×C₈×C₁₇₃	C₂²	1	2	yes	—
14	#14583801	—	2	2	no	—
15	#14583801	—	2	2	no	—
16	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₁₇₃	—	2	2	no	—
18	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
21	C₂×C₈×C₁₇₃	Q₈	1	2	yes	—
22	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₇₃	C₈	1	2	yes	—
24	C₂²⋊C₄×C₁₇₃	C₈	1	2	yes	—
25	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
26	C₂³×C₁₇₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₇₃	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₇₃	—	2	2	no	—
33	C₈×C₁₇₃	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₇₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₇₃	—	2	2	no	—
37	C₂²×C₁₇₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₇₃	—	2	4	no	—
40	C₄×C₁₇₃	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₇₃	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₁₇₃	—	2	2	no	—
44	C₂×C₁₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₇₃	#504	1	2	yes	—
46	C₁₇₃	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	173	1	no	maximal, Sylow
48	C₂×M₃₂	—	173	1	no	—
49	C₂²×C₁₆	—	173	1	no	—
50	#405	D₁₇₃	1	1	yes	characteristic
51	C₄⋊C₁₆	—	173	4	no	—
52	C₂×M₁₆	Dic₁₇₃	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
54	#117	Dic₁₇₃	1	1	yes	characteristic
55	M₃₂	—	346	2	no	—
56	C₂×C₁₆	—	346	2	no	—
57	C₂×C₁₆	—	173	2	no	—
58	C₂×C₁₆	—	173	2	no	—
59	#97	Dic₁₇₃	1	2	yes	—
60	C₄×C₈	C₂×D₁₇₃	1	2	yes	—
61	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	C₂×Dic₁₇₃	1	2	yes	—
64	C₂×C₈	C₂×Dic₁₇₃	1	1	yes	characteristic
65	C₂×C₈	D₆₉₂	1	2	yes	—
66	C₂×C₈	Dic₃₄₆	1	2	yes	—
67	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
68	C₄⋊C₄	#13063436	1	2	yes	—
69	C₂²⋊C₄	#13063436	1	2	yes	—
70	C₄²	C₂×Dic₁₇₃	1	2	yes	—
71	C₁₆	—	346	2	no	—
72	C₁₆	—	346	2	no	—
73	C₂³	C₂×#13063436	1	1	yes	characteristic
74	C₂×C₄	C₄⋊Dic₁₇₃	1	2	yes	—
75	C₂×C₄	C₂×#13063436	1	4	yes	—
76	C₂×C₄	C₂×#13063436	1	1	yes	characteristic
77	C₂×C₄	#14583810	1	2	yes	—
78	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊Dic₁₇₃	1	2	yes	—
81	C₈	C₂×D₆₉₂	1	1	yes	characteristic
82	C₈	C₂×Dic₃₄₆	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#13063436	1	2	yes	—
85	C₂²	C₂²×#13063436	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#13063436	1	2	yes	—
88	C₄	C₂×C₄⋊Dic₁₇₃	1	1	yes	characteristic
89	C₄	C₂×#14583810	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#13063436	1	1	yes	characteristic
92	C₂	#26011908	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂²×C₈
Fitting	#405×C₁₇₃	C₂
prime radical	C₂³×C₁₇₃	C₂×C₈
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#405	D₁₇₃
hypocenter	C₁₇₃	#2151
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊#14583801	C₂	semidirect
46. C₁₇₃	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149619	2
C₂²×#14583801	2
C₄⋊#14583801	2
#405×C₁₇₃	2
#2151	173