Subgroups — #36355435

Order 22144.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₈₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₆₉₂	C₂	1	2	yes	maximal
3	#26011319	C₂	1	1	yes	maximal, characteristic
4	#26011296	C₂	1	2	yes	maximal
5	#346×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₉₂	C₂²	1	2	yes	—
7	C₄×Dic₃₄₆	—	2	2	no	—
8	Dic₁₃₈₄	C₄	1	4	yes	complemented
9	#19149613	C₂²	1	2	yes	—
10	#19149604	—	2	2	no	—
11	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₄₆	—	2	2	no	—
15	Dic₆₉₂	—	2	2	no	—
16	Dic₆₉₂	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₇₃	—	2	2	no	—
18	#14583805	—	4	2	no	—
19	C₄×Dic₁₇₃	—	4	2	no	—
20	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
22	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
23	C₁₆×C₁₇₃	—	2	1	no	—
24	C₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
25	Dic₃₄₆	—	2	4	no	—
26	Dic₃₄₆	—	4	2	no	—
27	C₂×Dic₁₇₃	—	4	2	no	—
28	C₂×Dic₁₇₃	—	4	2	no	—
29	C₂×C₄×C₁₇₃	D₈	1	2	yes	—
30	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₇₃	—	2	1	no	—
32	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇₃	#41	1	1	yes	characteristic
34	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₇₃	—	2	2	no	—
36	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₇₃	#135	1	1	yes	characteristic
38	Dic₁₇₃	—	4	4	no	—
39	Dic₁₇₃	—	8	2	no	—
40	C₂×C₁₇₃	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₇₃	#510	1	2	yes	—
42	C₁₇₃	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	173	1	no	maximal, Sylow
44	C₂×Q₃₂	—	173	1	no	—
45	C₄×Q₁₆	—	173	2	no	—
46	#367	—	173	1	no	—
47	#358	—	173	2	no	—
48	#346	D₁₇₃	1	1	yes	characteristic
49	C₂×Q₁₆	—	173	2	no	—
50	C₄×Q₈	—	346	2	no	—
51	Q₃₂	—	173	4	no	—
52	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
54	#107	—	173	2	no	—
55	#103	—	346	2	no	—
56	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
57	C₂×Q₈	—	346	2	no	—
58	Q₁₆	—	346	2	no	—
59	Q₁₆	—	173	4	no	—
60	C₂×C₈	D₆₉₂	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
62	C₄⋊C₄	—	346	2	no	—
63	C₄⋊C₄	—	692	2	no	—
64	C₄²	—	692	2	no	—
65	C₄²	D₆₉₂	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₇₃	1	2	yes	—
68	Q₈	—	346	4	no	—
69	Q₈	—	692	2	no	—
70	C₂×C₄	—	692	2	no	—
71	C₂×C₄	—	692	2	no	—
72	C₂×C₄	D₁₃₈₄	1	2	yes	—
73	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₇₃	1	1	yes	characteristic
76	C₈	#14583826	1	1	yes	characteristic
77	C₂²	C₂×D₁₃₈₄	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	1384	2	no	—
80	C₄	—	692	4	no	—
81	C₄	C₄×D₆₉₂	1	1	yes	characteristic
82	C₄	#19149701	1	1	yes	characteristic
83	C₂	C₄×D₁₃₈₄	1	1	yes	characteristic
84	C₂	#26011934	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₈₄
commutator	C₈×C₁₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#346×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₈
cosocle	C₂×C₈×C₁₇₃	C₂³
hypercenter	#346	D₁₇₃
hypocenter	C₁₇₃	#2176
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₁₃₈₄	C₄	semidirect
42. C₁₇₃	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₈₄	2
C₄×Dic₆₉₂	2
#26011319	2
#26011296	2
#346×C₁₇₃	2
#2176	173