Subgroups — C₂×#26012379 #36369763

Order 22144.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14583829	C₂	1	2	yes	maximal
2	#26012379	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#13063436	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583829	C₄	1	8	yes	complemented
5	C₂×#14583829	C₂²	1	4	yes	—
6	C₂²×#13063436	C₄	1	1	yes	characteristic
7	C₂²×#13063436	C₂²	1	1	yes	characteristic
8	C₄×#13063436	C₄	1	2	yes	—
9	C₄×#13063436	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇₃	C₄	1	1	yes	characteristic
11	#14583829	C₂×C₄	1	8	yes	complemented
12	C₂×#13063436	C₈	1	4	yes	—
13	C₂×#13063436	C₂×C₄	1	4	yes	—
14	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
15	C₂×#13063436	C₂×C₄	1	2	yes	—
16	C₂×#13063436	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₇₃	C₈	1	2	yes	—
18	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	C₈	1	2	yes	—
20	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
21	#13063436	C₄²	1	2	yes	—
22	#13063436	C₂×C₈	1	4	yes	—
23	#13063436	C₂²×C₄	1	2	yes	—
24	C₂³×C₁₇₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₇₃	—	2	4	no	—
26	C₂×C₄×C₁₇₃	C₄²	1	2	yes	—
27	C₂×C₄×C₁₇₃	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₇₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₇₃	C₄×C₈	1	2	yes	—
32	C₂²×C₁₇₃	M₃₂	1	4	yes	—
33	C₂²×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₁₇₃	—	2	4	no	—
35	C₄×C₁₇₃	C₄×C₈	1	2	yes	—
36	C₄×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₁₇₃	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₁₇₃	#346	1	4	yes	complemented
39	C₂×C₁₇₃	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₇₃	C₂×M₃₂	1	2	yes	—
41	C₁₇₃	C₂×#346	1	1	yes	characteristic, complemented, Sylow
42	C₂×#346	—	173	1	no	maximal, Sylow
43	C₂²×C₁₆	—	173	2	no	—
44	C₂×C₄×C₈	—	173	1	no	—
45	#346	—	173	4	no	—
46	C₂²×C₈	—	173	1	no	—
47	C₂²×C₈	—	173	1	no	—
48	C₂×C₄²	#10621924	1	1	yes	characteristic
49	C₂×C₁₆	—	173	4	no	—
50	C₂×C₁₆	—	173	8	no	—
51	C₄×C₈	—	173	2	no	—
52	C₄×C₈	—	173	2	no	—
53	C₂²×C₄	#13063443	1	2	yes	—
54	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
55	C₂×C₈	—	173	4	no	—
56	C₂×C₈	—	173	4	no	—
57	C₂×C₈	—	173	1	no	—
58	C₂×C₈	—	173	2	no	—
59	C₂×C₈	—	173	1	no	—
60	C₄²	#13063443	1	2	yes	—
61	C₄²	C₂×#10621924	1	2	yes	—
62	C₁₆	—	173	8	no	—
63	C₂³	C₂×#13063443	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄×#10621924	1	2	yes	—
66	C₂×C₄	C₂×#13063443	1	4	yes	—
67	C₂×C₄	C₂×#13063443	1	1	yes	characteristic
68	C₂×C₄	D₁₇₃⋊C₈	1	2	yes	—
69	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
70	C₈	—	173	2	no	—
71	C₈	—	173	2	no	—
72	C₈	—	173	4	no	—
73	C₂²	C₄×#13063443	1	2	yes	—
74	C₂²	#19149753	1	4	yes	—
75	C₂²	C₂²×#13063443	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×#13063443	1	2	yes	—
78	C₄	C₂×C₄×#10621924	1	1	yes	characteristic
79	C₄	C₂×D₁₇₃⋊C₈	1	1	yes	characteristic
80	C₂	#26012379	1	4	yes	direct product factor
81	C₂	C₂×C₄×#13063443	1	1	yes	characteristic
82	C₂	C₂×#19149753	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×#10621924
commutator	C₂×C₁₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10621924
Fitting	C₂×C₄²×C₁₇₃	C₄
socle, prime radical	C₂³×C₁₇₃	C₂×C₈
cosocle	C₂×#13063436	C₂³
hypercenter	C₂×C₄²	#10621924
hypocenter	C₁₇₃	C₂×#346
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #26012379	C₂	direct (N × H)
4. C₂×#14583829	C₄	semidirect
11. #14583829	C₂×C₄	semidirect
38. C₂×C₁₇₃	#346	semidirect
41. C₁₇₃	C₂×#346	semidirect
80. C₂	#26012379	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14583829	2
#26012379	2
C₂×C₄×#13063436	2
C₂×#346	173

Subgroups — C2×#26012379 #36369763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26012379 #36369763