Subgroups — #36353674

Order 22144.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149596	C₂	1	2	yes	maximal
2	#26011261	C₂	1	4	yes	maximal, complemented
3	#518×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583810	C₂²	1	4	yes	—
5	#19149596	—	2	4	no	—
6	C₂²×D₄×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₄×D₄×C₁₇₃	—	2	4	no	—
8	#117×C₁₇₃	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₇₃	C₄	1	4	yes	—
10	#14583810	—	2	4	no	—
11	#14583810	—	2	4	no	—
12	C₂×#13063436	—	2	4	no	—
13	C₂⁴×C₁₇₃	C₂×C₄	1	2	yes	—
14	C₂×D₄×C₁₇₃	—	2	4	no	—
15	C₂×D₄×C₁₇₃	D₄	1	4	yes	—
16	C₂²×C₄×C₁₇₃	—	2	4	no	—
17	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇₃	—	2	2	no	—
19	C₂²⋊C₄×C₁₇₃	—	2	4	no	—
20	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
21	C₄²×C₁₇₃	—	2	2	no	—
22	#13063436	—	4	4	no	—
23	C₂³×C₁₇₃	—	2	4	no	—
24	C₂³×C₁₇₃	—	8	2	no	—
25	C₂³×C₁₇₃	C₂²⋊C₄	1	4	yes	—
26	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
27	D₄×C₁₇₃	—	2	8	no	—
28	C₂×C₄×C₁₇₃	—	2	4	no	—
29	C₂×C₄×C₁₇₃	—	8	4	no	—
30	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₇₃	#41	1	4	yes	—
32	C₂²×C₁₇₃	—	2	4	no	—
33	C₂²×C₁₇₃	—	4	1	no	—
34	C₂²×C₁₇₃	—	4	4	no	—
35	C₂²×C₁₇₃	—	8	4	no	—
36	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₁₇₃	#117	1	2	yes	—
38	C₄×C₁₇₃	—	4	4	no	—
39	C₄×C₁₇₃	#123	1	4	yes	—
40	C₂×C₁₇₃	—	2	1	no	—
41	C₂×C₁₇₃	—	2	2	no	—
42	C₂×C₁₇₃	—	4	4	no	—
43	C₂×C₁₇₃	#381	1	1	yes	characteristic
44	C₁₇₃	#1829	1	1	yes	characteristic, complemented, Sylow
45	#1829	—	173	1	no	maximal, Sylow
46	#518	D₁₇₃	1	1	yes	characteristic
47	C₂×#100	—	173	2	no	—
48	#344	—	173	4	no	—
49	C₂²×D₄	C₂×D₁₇₃	1	1	yes	characteristic
50	C₂×M₁₆	—	173	4	no	—
51	C₄×D₄	—	2	4	no	—
52	#117	C₂×D₁₇₃	1	2	yes	—
53	C₂×C₂²⋊C₄	Dic₁₇₃	1	4	yes	—
54	#100	—	346	4	no	—
55	C₂⁴	C₂×Dic₁₇₃	1	2	yes	—
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	C₂²⋊D₁₇₃	1	4	yes	—
58	C₂²×C₄	—	2	4	no	—
59	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
60	M₁₆	—	346	4	no	—
61	M₁₆	—	346	4	no	—
62	C₂×C₈	—	346	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	2	4	no	—
65	C₂²⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
66	C₄²	—	2	2	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	8	2	no	—
69	C₂³	C₂²⋊Dic₁₇₃	1	4	yes	—
70	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	8	4	no	—
74	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
75	C₂×C₄	#14583826	1	4	yes	—
76	C₈	—	692	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	4	no	—
80	C₂²	—	8	4	no	—
81	C₂²	C₂×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
82	C₂²	#19149638	1	2	yes	—
83	C₄	—	4	4	no	—
84	C₄	#19149662	1	4	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	C₂	#26011354	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011354
commutator	C₂³×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#518×C₁₇₃	C₂
socle	C₂×C₁₇₃	#381
prime radical	C₂²×D₄×C₁₇₃	C₂²
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#518	D₁₇₃
hypocenter	C₁₇₃	#1829
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #518

Complements

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

N	H ≅ G/N	type
2. #26011261	C₂	semidirect
44. C₁₇₃	#1829	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149596	2
#26011261	2
#518×C₁₇₃	2
#1829	173