Subgroups — #36351390

Order 22144.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13063436	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13063436	C₄	1	4	yes	—
4	C₄×#13063436	—	2	2	no	—
5	C₄×#13063436	—	2	2	no	—
6	C₂²×C₈×C₁₇₃	C₄	1	2	yes	—
7	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₇₃	—	2	2	no	—
9	C₂×#13063436	—	2	4	no	—
10	C₂×#13063436	—	2	4	no	—
11	C₂×#13063436	C₈	1	8	yes	complemented
12	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇₃	—	2	2	no	—
15	C₂×C₈×C₁₇₃	—	2	2	no	—
16	C₂×C₈×C₁₇₃	C₈	1	4	yes	complemented
17	C₄²×C₁₇₃	D₄	1	2	yes	—
18	C₄²×C₁₇₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	Q₈	1	1	yes	characteristic
20	#13063436	—	2	8	no	—
21	C₂³×C₁₇₃	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₇₃	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₇₃	M₁₆	1	4	yes	—
29	C₂×C₄×C₁₇₃	M₁₆	1	2	yes	—
30	C₈×C₁₇₃	—	2	4	no	—
31	C₂²×C₁₇₃	#95	1	1	yes	characteristic
32	C₂²×C₁₇₃	C₄×C₈	1	2	yes	—
33	C₂²×C₁₇₃	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₁₇₃	#97	1	2	yes	—
35	C₂²×C₁₇₃	#97	1	1	yes	characteristic
36	C₄×C₁₇₃	C₂²⋊C₈	1	4	yes	—
37	C₄×C₁₇₃	C₂²⋊C₈	1	2	yes	—
38	C₄×C₁₇₃	C₄⋊C₈	1	4	yes	—
39	C₄×C₁₇₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₁₇₃	C₈²	1	1	yes	characteristic
41	C₂×C₁₇₃	#322	1	1	yes	characteristic
42	C₂×C₁₇₃	#322	1	2	yes	—
43	C₂×C₁₇₃	#336	1	2	yes	—
44	C₂×C₁₇₃	#336	1	1	yes	characteristic
45	C₁₇₃	#1273	1	1	yes	characteristic, complemented, Sylow
46	#1273	—	173	1	no	maximal, Sylow
47	C₂×C₄×C₈	—	173	2	no	—
48	C₂×C₄×C₈	D₁₇₃	1	1	yes	characteristic
49	C₂²×C₈	—	173	4	no	—
50	C₂²×C₈	Dic₁₇₃	1	2	yes	—
51	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	346	2	no	—
54	C₄×C₈	—	346	2	no	—
55	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
56	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	346	4	no	—
60	C₂×C₈	—	346	4	no	—
61	C₂×C₈	—	173	8	no	—
62	C₂×C₈	#13063436	1	4	yes	complemented
63	C₄²	C₂²⋊D₁₇₃	1	2	yes	—
64	C₄²	D₆₉₂	1	1	yes	characteristic
65	C₄²	Dic₃₄₆	1	1	yes	characteristic
66	C₂³	C₄×Dic₁₇₃	1	1	yes	characteristic
67	C₂×C₄	#14583803	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
69	C₂×C₄	#14583805	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₁₇₃	1	1	yes	characteristic
71	C₂×C₄	C₈×D₁₇₃	1	4	yes	—
72	C₂×C₄	C₂×#13063436	1	2	yes	—
73	C₂×C₄	#14583809	1	4	yes	—
74	C₂×C₄	#14583810	1	2	yes	—
75	C₈	—	2	4	no	—
76	C₈	—	346	8	no	—
77	C₂²	#19149586	1	1	yes	characteristic
78	C₂²	C₈×Dic₁₇₃	1	2	yes	—
79	C₂²	C₄×#13063436	1	1	yes	characteristic
80	C₂²	#19149589	1	2	yes	—
81	C₂²	#19149590	1	1	yes	characteristic
82	C₄	#19149591	1	4	yes	—
83	C₄	C₂²⋊#13063436	1	2	yes	—
84	C₄	#19149608	1	4	yes	—
85	C₄	C₄⋊#13063436	1	2	yes	—
86	C₂	C₈×#13063436	1	1	yes	characteristic
87	C₂	#26011212	1	1	yes	characteristic
88	C₂	#26011213	1	2	yes	—
89	C₂	#26011245	1	2	yes	—
90	C₂	#26011246	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×D₁₇₃
commutator	C₂×C₁₇₃	C₈²
Fitting	C₂×C₄×C₈×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₄²
cosocle	C₂×C₄²×C₁₇₃	C₂²
hypercenter	C₂×C₄×C₈	D₁₇₃
hypocenter	C₁₇₃	#1273
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×#13063436	C₈	semidirect
16. C₂×C₈×C₁₇₃	C₈	semidirect
45. C₁₇₃	#1273	semidirect
62. C₂×C₈	#13063436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13063436	2
C₂×C₄×C₈×C₁₇₃	2
#1273	173