Subgroups — #36353634

Order 22144.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#13063436	C₂	1	2	yes	maximal
2	#26011246	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063436	C₂²	1	4	yes	—
5	C₂²⋊#13063436	—	2	4	no	—
6	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
7	#117×C₁₇₃	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₇₃	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₇₃	C₄	1	2	yes	—
10	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
11	C₂×#13063436	—	2	4	no	—
12	C₂×#13063436	—	2	8	no	—
13	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₇₃	—	2	4	no	—
15	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇₃	D₄	1	4	yes	—
18	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₇₃	—	2	4	no	—
20	C₂²⋊C₄×C₁₇₃	—	2	4	no	—
21	C₄²×C₁₇₃	—	2	4	no	—
22	#13063436	—	4	4	no	—
23	C₂³×C₁₇₃	—	2	2	no	—
24	C₂³×C₁₇₃	—	2	4	no	—
25	C₂³×C₁₇₃	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	—	2	2	no	—
28	C₂×C₄×C₁₇₃	—	2	8	no	—
29	C₂×C₄×C₁₇₃	—	2	4	no	—
30	C₂×C₄×C₁₇₃	—	2	8	no	—
31	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₇₃	#41	1	4	yes	—
34	C₂²×C₁₇₃	—	2	2	no	—
35	C₂²×C₁₇₃	—	2	4	no	—
36	C₂²×C₁₇₃	—	4	4	no	—
37	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₇₃	#117	1	2	yes	—
39	C₂²×C₁₇₃	#131	1	4	yes	—
40	C₄×C₁₇₃	—	2	2	no	—
41	C₄×C₁₇₃	—	4	4	no	—
42	C₄×C₁₇₃	#123	1	4	yes	—
43	C₂×C₁₇₃	—	4	2	no	—
44	C₂×C₁₇₃	#381	1	1	yes	characteristic
45	C₂×C₁₇₃	#408	1	2	yes	—
46	C₂×C₁₇₃	#433	1	4	yes	—
47	C₁₇₃	#1818	1	1	yes	characteristic, complemented, Sylow
48	#1818	—	173	1	no	maximal, Sylow
49	C₂×#117	D₁₇₃	1	1	yes	characteristic
50	C₂×C₂²⋊C₈	—	173	2	no	—
51	#336	—	173	4	no	—
52	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
53	C₂²×C₈	—	173	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	Dic₁₇₃	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₁₇₃	1	2	yes	—
57	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
58	C₂²⋊C₈	—	346	4	no	—
59	C₂⁴	C₂×Dic₁₇₃	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₁₇₃	1	4	yes	—
62	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₁₇₃	1	4	yes	—
64	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
65	C₂×C₈	—	346	4	no	—
66	C₂×C₈	—	346	8	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	4	no	—
72	C₂³	C₂²⋊Dic₁₇₃	1	2	yes	—
73	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	C₂²⋊Dic₁₇₃	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
80	C₂×C₄	#14583826	1	4	yes	—
81	C₈	—	692	4	no	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
86	C₂²	#19149638	1	2	yes	—
87	C₂²	#19149686	1	4	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#19149662	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#26011354	1	1	yes	characteristic
93	C₂	#26011458	1	2	yes	—
94	C₂	#26011552	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇₃
commutator, socle	C₂³×C₁₇₃	C₂²×C₄
Fitting	C₂×#117×C₁₇₃	C₂
prime radical	C₂⁴×C₁₇₃	C₂×C₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×#117	D₁₇₃
hypocenter	C₁₇₃	#1818
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #26011246	C₂	semidirect
47. C₁₇₃	#1818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#13063436	2
#26011246	2
C₂×#117×C₁₇₃	2
#1818	173