Subgroups — #36354668

Order 22144.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012291	C₂	1	3	yes	maximal, complemented
2	#26011459	C₂	1	3	yes	maximal, complemented
3	#338×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583826	C₂²	1	3	yes	—
5	#19149712	—	2	6	no	—
6	#19149707	—	2	6	no	—
7	#19149701	—	2	6	no	—
8	C₂²×#13063436	C₂²	1	1	yes	characteristic
9	#19149591	—	2	6	no	—
10	C₂×M₁₆×C₁₇₃	C₂²	1	3	yes	—
11	#14583826	—	2	6	no	—
12	#14583826	—	2	6	no	—
13	C₂×Dic₃₄₆	D₄	1	3	yes	—
14	C₂×D₆₉₂	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₇₃	—	2	3	no	—
16	C₂×C₄×D₁₇₃	—	2	3	no	—
17	Dic₆₉₂	—	2	6	no	—
18	#14583815	—	2	12	no	—
19	D₁₃₈₄	—	2	6	no	—
20	C₂×#13063436	—	2	3	no	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₃	—	2	6	no	—
23	C₂×C₈×C₁₇₃	D₄	1	6	yes	—
24	C₂²×D₁₇₃	—	2	3	no	—
25	Dic₃₄₆	—	2	6	no	—
26	D₆₉₂	—	2	6	no	—
27	C₂²⋊D₁₇₃	—	4	6	no	—
28	C₂×Dic₁₇₃	—	2	3	no	—
29	C₄×D₁₇₃	—	4	3	no	—
30	C₄×D₁₇₃	—	4	3	no	—
31	#13063436	—	4	1	no	—
32	C₂³×C₁₇₃	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
35	C₈×C₁₇₃	—	2	6	no	—
36	C₂²×C₁₇₃	—	4	1	no	—
37	C₂²×C₁₇₃	#121	1	3	yes	—
38	C₂×D₁₇₃	—	4	6	no	—
39	C₂×D₁₇₃	—	4	3	no	—
40	C₄×C₁₇₃	#120	1	3	yes	—
41	C₄×C₁₇₃	#127	1	1	yes	characteristic
42	Dic₁₇₃	—	4	3	no	—
43	C₂×C₁₇₃	—	2	3	no	—
44	C₂×C₁₇₃	#392	1	1	yes	characteristic
45	D₁₇₃	—	8	3	no	—
46	C₁₇₃	#2017	1	1	yes	characteristic, complemented, Sylow
47	#2017	—	173	1	no	maximal, Sylow
48	#575	—	173	3	no	—
49	#408	—	173	3	no	—
50	#338	D₁₇₃	1	1	yes	characteristic, complemented
51	C₂×#41	—	173	3	no	—
52	#137	—	346	6	no	—
53	AGL(1,ℤ/8)	—	346	6	no	—
54	#135	—	346	6	no	—
55	C₂×M₁₆	C₂×D₁₇₃	1	3	yes	—
56	C₂²×C₈	—	173	1	no	—
57	C₂²⋊C₈	—	346	6	no	—
58	#41	—	346	6	no	—
59	#41	—	346	6	no	—
60	C₂×Q₈	—	173	3	no	—
61	C₂×D₄	—	346	3	no	—
62	C₂×D₄	—	173	3	no	—
63	C₂²×C₄	—	346	3	no	—
64	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
65	Q₁₆	—	346	6	no	—
66	SD₁₆	—	346	12	no	—
67	D₈	—	346	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	346	3	no	—
70	C₂×C₈	D₆₉₂	1	6	yes	—
71	C₂³	—	346	3	no	—
72	C₂³	#14583827	1	1	yes	characteristic
73	Q₈	—	346	6	no	—
74	D₄	—	346	6	no	—
75	D₄	—	692	6	no	—
76	C₂×C₄	—	346	3	no	—
77	C₂×C₄	—	692	3	no	—
78	C₂×C₄	—	692	3	no	—
79	C₂×C₄	D₄×D₁₇₃	1	3	yes	—
80	C₂×C₄	C₂×D₆₉₂	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	692	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	692	6	no	—
85	C₂²	—	692	3	no	—
86	C₂²	#19149654	1	3	yes	—
87	C₄	—	692	3	no	—
88	C₄	#19149649	1	3	yes	—
89	C₄	#19149676	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	1384	3	no	—
92	C₂	#26011401	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149676
commutator, cosocle	C₂²×C₄×C₁₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#338×C₁₇₃	C₂
socle	C₂×C₁₇₃	#392
hypercenter	#338	D₁₇₃
hypocenter	C₁₇₃	#2017
LCS	G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #26012291	C₂	semidirect
2. #26011459	C₂	semidirect
3. #338×C₁₇₃	C₂	semidirect
46. C₁₇₃	#2017	semidirect
50. #338	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26012291	2
#26011459	2
#338×C₁₇₃	2
#2017	173