Subgroups — #36369977

Order 22144.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012412	C₂	1	2	yes	maximal
2	C₄⋊#14583829	C₂	1	1	yes	maximal, characteristic
3	#26012377	C₂	1	2	yes	maximal
4	C₄×#14583829	C₂	1	1	yes	maximal, characteristic
5	#26011615	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583829	C₂²	1	1	yes	characteristic
7	C₂×#14583829	C₂²	1	2	yes	—
8	C₂×#14583829	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₄₆	C₄	1	1	yes	characteristic
10	#19149608	C₄	1	2	yes	—
11	C₄×#13063436	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₇₃	C₂²	1	2	yes	—
13	C₄⋊C₈×C₁₇₃	C₄	1	1	yes	characteristic
14	#14583829	—	2	1	no	—
15	#14583829	—	2	2	no	—
16	#14583829	Q₈	1	2	yes	complemented
17	C₂×Dic₃₄₆	C₈	1	1	yes	characteristic
18	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
19	C₂×#13063436	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₇₃	C₈	1	1	yes	characteristic
21	#14583805	C₈	1	2	yes	—
22	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
24	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
25	Dic₃₄₆	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₇₃	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₇₃	C₂×C₈	1	2	yes	—
28	#13063436	—	2	1	no	—
29	#13063436	C₂×Q₈	1	1	yes	characteristic
30	#13063436	#41	1	1	yes	characteristic
31	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₇₃	—	2	2	no	—
35	C₂²×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₇₃	—	2	1	no	—
37	C₄×C₁₇₃	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₇₃	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₇₃	#131	1	1	yes	characteristic
40	Dic₁₇₃	—	2	2	no	—
41	Dic₁₇₃	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₇₃	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₇₃	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₇₃	#504	1	1	yes	characteristic
45	C₁₇₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	Q₈×C₁₆	—	173	1	no	maximal, Sylow
47	C₈×Q₈	—	173	1	no	—
48	C₄⋊C₁₆	—	173	1	no	—
49	C₄⋊C₁₆	—	173	2	no	—
50	C₄×C₁₆	—	173	1	no	—
51	C₄×C₁₆	—	173	2	no	—
52	C₄×Q₈	—	173	1	no	—
53	C₂×C₁₆	—	173	1	no	—
54	C₂×C₁₆	—	173	2	no	—
55	C₂×C₁₆	—	173	1	no	—
56	C₄⋊C₈	—	173	2	no	—
57	C₄⋊C₈	#10621924	1	1	yes	characteristic
58	C₄×C₈	—	173	2	no	—
59	C₄×C₈	—	173	1	no	—
60	C₂×Q₈	—	173	1	no	—
61	C₂×C₈	—	173	1	no	—
62	C₂×C₈	—	173	1	no	—
63	C₂×C₈	C₂×#10621924	1	2	yes	—
64	C₄⋊C₄	—	173	2	no	—
65	C₄⋊C₄	—	173	1	no	—
66	C₄²	—	173	2	no	—
67	C₄²	C₂×#10621924	1	1	yes	characteristic
68	C₁₆	—	346	1	no	—
69	C₁₆	—	346	2	no	—
70	C₁₆	—	173	2	no	—
71	Q₈	—	173	4	no	—
72	C₂×C₄	—	173	2	no	—
73	C₂×C₄	—	173	2	no	—
74	C₂×C₄	D₁₇₃⋊C₈	1	1	yes	characteristic
75	C₂×C₄	D₁₇₃⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	346	1	no	—
79	C₈	—	173	1	no	—
80	C₈	—	173	1	no	—
81	C₂²	C₂×D₁₇₃⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	346	2	no	—
84	C₄	—	173	4	no	—
85	C₄	D₁₇₃⋊C₁₆	1	2	yes	—
86	C₄	Q₈×#10621924	1	1	yes	characteristic
87	C₄	#19149766	1	1	yes	characteristic
88	C₂	#26012540	1	1	yes	characteristic
89	C₂	C₂×D₁₇₃⋊C₁₆	1	1	yes	characteristic
90	C₂	#26012550	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10621924
commutator	C₂×C₁₇₃	C₂²×C₁₆
Fitting	C₄⋊C₈×C₁₇₃	C₄
socle, prime radical	C₂²×C₁₇₃	C₂²×C₈
cosocle	C₂×#13063436	C₂³
hypercenter	C₄⋊C₈	#10621924
hypocenter	C₁₇₃	Q₈×C₁₆
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #14583829	Q₈	semidirect
25. Dic₃₄₆	C₁₆	semidirect
45. 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
#26012412	2
C₄⋊#14583829	2
#26012377	2
C₄×#14583829	2
#26011615	2
Q₈×C₁₆	173