Subgroups — #36355144

Order 22144.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011540	C₂	1	1	yes	maximal, characteristic
2	#26011519	C₂	1	1	yes	maximal, characteristic
3	#26011263	C₂	1	2	yes	maximal
4	#26011255	C₂	1	2	yes	maximal
5	#338×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583810	C₂²	1	2	yes	—
7	C₂²×#13063436	C₂²	1	1	yes	characteristic
8	#19149638	C₂²	1	1	yes	characteristic
9	#19149615	—	2	2	no	—
10	#19149608	—	2	2	no	—
11	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
13	#14583810	—	2	2	no	—
14	#14583810	—	2	2	no	—
15	C₂×#13063436	—	2	1	no	—
16	C₂×#13063436	—	2	2	no	—
17	C₂×#13063436	—	2	2	no	—
18	C₄⋊Dic₁₇₃	—	2	1	no	—
19	C₂²⋊Dic₁₇₃	Q₈	1	2	yes	—
20	C₄×Dic₁₇₃	—	2	1	no	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₃	—	2	2	no	—
23	M₁₆×C₁₇₃	—	2	2	no	—
24	M₁₆×C₁₇₃	—	2	2	no	—
25	C₂×C₈×C₁₇₃	—	2	2	no	—
26	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
27	C₂×Dic₁₇₃	—	2	2	no	—
28	#13063436	—	4	1	no	—
29	#13063436	—	4	2	no	—
30	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	#41	1	2	yes	—
33	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₇₃	#41	1	2	yes	—
35	C₈×C₁₇₃	—	2	2	no	—
36	C₈×C₁₇₃	—	4	2	no	—
37	C₂²×C₁₇₃	—	4	1	no	—
38	C₂²×C₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₁₇₃	#125	1	2	yes	—
40	C₄×C₁₇₃	#123	1	1	yes	characteristic
41	C₄×C₁₇₃	#124	1	1	yes	characteristic
42	C₄×C₁₇₃	#126	1	2	yes	—
43	Dic₁₇₃	—	4	2	no	—
44	C₂×C₁₇₃	—	2	1	no	—
45	C₂×C₁₇₃	—	2	2	no	—
46	C₂×C₁₇₃	#400	1	1	yes	characteristic
47	C₁₇₃	#2103	1	1	yes	characteristic, complemented, Sylow
48	#2103	—	173	1	no	maximal, Sylow
49	#430	—	173	1	no	—
50	#424	—	173	1	no	—
51	#344	—	173	2	no	—
52	#341	—	173	2	no	—
53	#338	D₁₇₃	1	1	yes	characteristic
54	C₂×M₁₆	—	173	2	no	—
55	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	—
56	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
57	C₂²×C₈	—	173	1	no	—
58	#117	—	173	1	no	—
59	#108	—	346	2	no	—
60	C₄⋊C₈	—	346	2	no	—
61	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	2	2	no	—
64	M₁₆	—	2	2	no	—
65	M₁₆	—	346	2	no	—
66	M₁₆	—	346	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	346	1	no	—
69	C₂×C₈	—	346	2	no	—
70	C₂×C₈	—	346	2	no	—
71	C₂×C₈	D₆₉₂	1	2	yes	—
72	C₄⋊C₄	—	346	1	no	—
73	C₂²⋊C₄	—	173	2	no	—
74	C₄²	—	346	1	no	—
75	C₂³	Q₈×D₁₇₃	1	1	yes	characteristic
76	C₂×C₄	—	346	2	no	—
77	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
78	C₂×C₄	#14583825	1	2	yes	—
79	C₂×C₄	#14583825	1	1	yes	characteristic
80	C₂×C₄	#14583826	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	692	1	no	—
84	C₈	—	692	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#19149659	1	1	yes	characteristic
87	C₂²	#19149669	1	2	yes	—
88	C₄	—	692	2	no	—
89	C₄	#19149663	1	1	yes	characteristic
90	C₄	#19149667	1	1	yes	characteristic
91	C₄	#19149672	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#26011431	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149667
commutator, cosocle	C₂²×C₄×C₁₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#338×C₁₇₃	C₂
socle	C₂×C₁₇₃	#400
prime radical	C₂³×C₁₇₃	C₂×Q₈
hypercenter	#338	D₁₇₃
hypocenter	C₁₇₃	#2103
LCS	G ▷ C₂²×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
47. C₁₇₃	#2103	semidirect

Maximal subgroups

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

subgroup	index
#26011540	2
#26011519	2
#26011263	2
#26011255	2
#338×C₁₇₃	2
#2103	173