Subgroups — #36351541

Order 22144.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₃₄₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#14583801	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₃₄₆	C₄	1	1	yes	characteristic
5	C₂×#14583801	—	2	1	no	—
6	C₄⋊#13063436	C₄	1	1	yes	characteristic
7	#19149608	—	2	1	no	—
8	C₈×Dic₁₇₃	—	2	1	no	—
9	C₂×C₁₆×C₁₇₃	—	2	1	no	—
10	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₆	C₈	1	1	yes	characteristic
12	C₂×#13063436	—	2	1	no	—
13	C₄⋊Dic₁₇₃	C₈	1	1	yes	characteristic
14	#14583805	—	2	1	no	—
15	C₄×Dic₁₇₃	—	2	1	no	—
16	#14583801	—	2	2	no	—
17	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₇₃	—	4	1	no	—
21	Dic₃₄₆	—	2	1	no	—
22	Dic₃₄₆	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₇₃	—	2	1	no	—
24	C₂×Dic₁₇₃	—	2	1	no	—
25	#13063436	—	4	1	no	—
26	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
29	C₈×C₁₇₃	—	2	1	no	—
30	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₇₃	—	2	1	no	—
34	C₄×C₁₇₃	#103	1	1	yes	characteristic
35	C₄×C₁₇₃	#104	1	1	yes	characteristic
36	C₄×C₁₇₃	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₇₃	M₃₂	1	1	yes	characteristic
38	Dic₁₇₃	—	2	2	no	—
39	Dic₁₇₃	—	4	1	no	—
40	C₂×C₁₇₃	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₇₃	#350	1	1	yes	characteristic
43	C₁₇₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	173	1	no	maximal, Sylow
45	C₈×Q₈	—	173	1	no	—
46	C₄⋊C₁₆	D₁₇₃	1	1	yes	characteristic
47	C₄×C₁₆	—	173	1	no	—
48	C₄×Q₈	—	173	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	346	1	no	—
51	C₄⋊C₈	—	346	1	no	—
52	C₄⋊C₈	—	173	1	no	—
53	C₄×C₈	—	346	1	no	—
54	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
55	C₂×Q₈	—	173	1	no	—
56	C₂×C₈	—	346	1	no	—
57	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
58	C₂×C₈	D₆₉₂	1	1	yes	characteristic
59	C₄⋊C₄	—	346	1	no	—
60	C₄⋊C₄	—	173	1	no	—
61	C₄²	—	346	1	no	—
62	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
63	C₁₆	—	4	1	no	—
64	C₁₆	—	346	2	no	—
65	Q₈	—	346	1	no	—
66	Q₈	—	173	2	no	—
67	C₂×C₄	—	346	1	no	—
68	C₂×C₄	—	346	1	no	—
69	C₂×C₄	#14583803	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₇₃	1	1	yes	characteristic
71	C₂×C₄	#14583809	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	692	1	no	—
74	C₈	#14583814	1	1	yes	characteristic
75	C₈	#14583816	1	1	yes	characteristic
76	C₂²	#19149591	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	346	2	no	—
79	C₄	—	692	1	no	—
80	C₄	#19149602	1	1	yes	characteristic
81	C₄	#19149605	1	1	yes	characteristic
82	C₄	C₁₆×D₁₇₃	1	1	yes	characteristic
83	C₄	#19149618	1	1	yes	characteristic
84	C₂	#26011221	1	1	yes	characteristic
85	C₂	#26011270	1	1	yes	characteristic
86	C₂	#26011276	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₇₃
commutator	C₄×C₁₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	C₄⋊C₁₆×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	C₄⋊C₁₆	D₁₇₃
hypocenter	C₁₇₃	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₃₄₆	C₁₆	semidirect
43. C₁₇₃	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₃₄₆	2
C₄×#14583801	2
C₄⋊C₁₆×C₁₇₃	2
Q₈⋊C₁₆	173