Subgroups — #36368509

Order 22144.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012387	C₂	1	2	yes	maximal
2	#26011515	C₂	1	1	yes	maximal, characteristic
3	#19149753	—	2	2	no	—
4	C₂×#14583810	C₄	1	1	yes	characteristic
5	C₂×#14583810	C₂²	1	1	yes	characteristic
6	C₄⋊#13063436	—	2	2	no	—
7	C₂×C₄²×C₁₇₃	C₄	1	1	yes	characteristic
8	#14583829	—	4	2	no	—
9	#14583810	—	4	1	no	—
10	#14583810	—	4	1	no	—
11	C₂×#13063436	—	2	1	no	—
12	C₂×#13063436	D₄	1	2	yes	—
13	C₂²×C₄×C₁₇₃	—	2	1	no	—
14	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
15	C₄²×C₁₇₃	—	2	1	no	—
16	C₄²×C₁₇₃	C₈	1	1	yes	characteristic
17	C₄²×C₁₇₃	C₈	1	1	yes	characteristic
18	#13063436	—	2	2	no	—
19	#13063436	—	4	1	no	—
20	C₂³×C₁₇₃	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₁₇₃	—	2	1	no	—
22	C₂×C₄×C₁₇₃	—	2	1	no	—
23	C₂×C₄×C₁₇₃	—	2	1	no	—
24	C₂×C₄×C₁₇₃	—	2	1	no	—
25	C₂×C₄×C₁₇₃	—	4	2	no	—
26	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₇₃	—	2	1	no	—
29	C₂²×C₁₇₃	—	4	1	no	—
30	C₂²×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₇₃	—	2	2	no	—
32	C₄×C₁₇₃	—	2	1	no	—
33	C₄×C₁₇₃	—	4	1	no	—
34	C₄×C₁₇₃	#99	1	1	yes	characteristic
35	C₄×C₁₇₃	#101	1	1	yes	characteristic
36	C₂×C₁₇₃	—	2	1	no	—
37	C₂×C₁₇₃	—	4	1	no	—
38	C₂×C₁₇₃	#324	1	1	yes	characteristic
39	C₁₇₃	#1327	1	1	yes	characteristic, complemented, Sylow
40	#1327	—	173	1	no	maximal, Sylow
41	#423	—	173	1	no	—
42	#349	—	173	2	no	—
43	C₂×M₁₆	—	173	1	no	—
44	C₂×M₁₆	—	173	1	no	—
45	C₂×C₄²	#10621924	1	1	yes	characteristic
46	M₃₂	—	346	2	no	—
47	C₄⋊C₈	—	346	2	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
50	M₁₆	—	692	1	no	—
51	M₁₆	—	692	1	no	—
52	C₂×C₈	—	346	1	no	—
53	C₂×C₈	—	173	2	no	—
54	C₄²	—	2	1	no	—
55	C₄²	#13063443	1	1	yes	characteristic
56	C₄²	#13063443	1	1	yes	characteristic
57	C₁₆	—	692	2	no	—
58	C₂³	#14583835	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
65	C₂×C₄	C₂×#13063443	1	1	yes	characteristic
66	C₈	—	346	2	no	—
67	C₈	—	692	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂²⋊#13063443	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	#19149738	1	1	yes	characteristic
75	C₄	#19149740	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#26012336	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149738
commutator	C₂×C₄×C₁₇₃	C₂×C₈
Frattini	C₂²×C₄	C₂×#10621924
Fitting	C₂×C₄²×C₁₇₃	C₄
socle	C₂×C₁₇₃	#324
prime radical	C₂³×C₁₇₃	M₁₆
cosocle	C₂×#14583810	C₂²
hypercenter	C₂×C₄²	#10621924
hypocenter	C₁₇₃	#1327
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
39. C₁₇₃	#1327	semidirect

Maximal subgroups

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

subgroup	index
#26012387	2
#26011515	2
#1327	173