Subgroups — #36351735

Order 22144.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011859	C₂	1	1	yes	maximal, characteristic
2	#26011290	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149671	C₄	1	1	yes	characteristic
5	#19149603	—	2	1	no	—
6	#19149603	—	2	1	no	—
7	#19149598	—	2	1	no	—
8	#19149590	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	#101×C₁₇₃	—	2	1	no	—
11	#14583810	—	4	1	no	—
12	C₂×#13063436	—	2	1	no	—
13	C₄⋊Dic₁₇₃	—	2	1	no	—
14	#14583805	—	4	1	no	—
15	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
17	M₁₆×C₁₇₃	—	4	1	no	—
18	C₄⋊C₄×C₁₇₃	—	4	1	no	—
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₇₃	—	4	1	no	—
21	#13063436	—	4	1	no	—
22	#13063436	—	8	1	no	—
23	Q₈×C₁₇₃	—	4	1	no	—
24	Q₈×C₁₇₃	—	4	1	no	—
25	C₂×C₄×C₁₇₃	—	2	1	no	—
26	C₂×C₄×C₁₇₃	—	2	1	no	—
27	C₂×C₄×C₁₇₃	—	2	1	no	—
28	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₁₇₃	—	8	1	no	—
30	C₂²×C₁₇₃	#99	1	1	yes	characteristic
31	C₄×C₁₇₃	—	2	1	no	—
32	C₄×C₁₇₃	—	4	1	no	—
33	C₄×C₁₇₃	—	4	1	no	—
34	C₄×C₁₇₃	—	4	1	no	—
35	Dic₁₇₃	—	8	1	no	—
36	C₂×C₁₇₃	—	2	1	no	—
37	C₂×C₁₇₃	#354	1	1	yes	characteristic
38	C₁₇₃	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	173	1	no	maximal, Sylow
40	#491	—	173	1	no	—
41	#356	—	173	1	no	—
42	#356	D₁₇₃	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
44	#125	—	173	1	no	—
45	#103	—	346	1	no	—
46	#103	—	346	1	no	—
47	#101	—	2	1	no	—
48	#101	—	346	1	no	—
49	#97	—	173	1	no	—
50	C₂×Q₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
51	C₂×Q₈	D₆₉₂	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	692	1	no	—
54	C₂×C₈	—	346	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	346	1	no	—
57	C₄⋊C₄	—	692	1	no	—
58	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	692	1	no	—
65	C₂×C₄	#14583803	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	1384	1	no	—
68	C₈	—	692	1	no	—
69	C₂²	#19149593	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1384	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#26011285	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011285
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₇₃
Fitting	#356×C₁₇₃	C₂
socle	C₂×C₁₇₃	#354
prime radical	C₂²×C₁₇₃	#99
cosocle	C₄⋊Q₈×C₁₇₃	C₂²
hypercenter	#356	D₁₇₃
hypocenter	C₁₇₃	#1413
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₁₇₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#26011859	2
#26011290	2
#356×C₁₇₃	2
#1413	173