Subgroups — #36374311

Order 22336.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013684	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013665	C₂	1	2	yes	maximal, complemented
3	#26013567	C₂	1	2	yes	maximal, complemented
4	#26013556	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150233	C₂²	1	2	yes	—
7	#19150232	—	2	2	no	—
8	C₂×C₂²⋊D₃₄₉	—	2	2	no	—
9	D₄×D₃₄₉	—	2	2	no	—
10	#19150221	—	2	2	no	—
11	#19150218	—	2	2	no	—
12	#19150217	C₂²	1	2	yes	complemented
13	C₂×D₄×C₃₄₉	—	2	2	no	—
14	C₂×D₄×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₃₄₉	C₂²	1	1	yes	characteristic, complemented
16	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₃₄₉	—	2	2	no	—
18	Dic₆₉₈	D₄	1	2	yes	complemented
19	D₁₃₉₆	D₄	1	2	yes	complemented
20	C₂²⋊D₃₄₉	—	2	4	no	—
21	C₂²⋊D₃₄₉	—	2	2	no	—
22	C₂×Dic₃₄₉	—	2	2	no	—
23	C₄×D₃₄₉	—	2	2	no	—
24	#14584095	—	2	2	no	—
25	C₂³×C₃₄₉	—	2	1	no	—
26	C₂³×C₃₄₉	D₄	1	2	yes	—
27	D₄×C₃₄₉	—	2	2	no	—
28	D₄×C₃₄₉	—	2	2	no	—
29	D₄×C₃₄₉	—	4	4	no	—
30	C₂×C₄×C₃₄₉	—	2	1	no	—
31	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
32	C₂²×C₃₄₉	—	2	2	no	—
33	C₂²×C₃₄₉	—	4	2	no	—
34	C₂²×C₃₄₉	—	4	2	no	—
35	C₂²×C₃₄₉	—	4	1	no	—
36	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
37	C₂×D₃₄₉	—	2	2	no	—
38	C₂×D₃₄₉	—	8	2	no	—
39	C₄×C₃₄₉	—	2	2	no	—
40	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
41	Dic₃₄₉	—	2	2	no	—
42	C₂×C₃₄₉	—	2	1	no	—
43	C₂×C₃₄₉	—	4	2	no	—
44	C₂×C₃₄₉	—	8	1	no	—
45	C₂×C₃₄₉	#120	1	1	yes	characteristic
46	D₃₄₉	—	4	2	no	—
47	C₃₄₉	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	349	1	no	maximal, Sylow
49	H(5,𝔽₂)	—	349	1	no	—
50	AGL(1,ℤ/8)	—	349	2	no	—
51	#127	D₃₄₉	1	1	yes	characteristic, complemented
52	#104	—	349	2	no	—
53	#100	—	349	1	no	—
54	#41	—	349	2	no	—
55	#41	—	698	2	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	698	2	no	—
58	C₂×D₄	—	698	2	no	—
59	C₂×D₄	C₂×D₃₄₉	1	1	yes	characteristic
60	C₂×D₄	C₂×D₃₄₉	1	1	yes	characteristic, complemented
61	SD₁₆	—	698	2	no	—
62	D₈	—	698	2	no	—
63	M₁₆	—	349	2	no	—
64	C₄²	C₂×D₃₄₉	1	1	yes	characteristic, complemented
65	C₂³	—	2	1	no	—
66	C₂³	—	698	2	no	—
67	C₂³	C₂²⋊D₃₄₉	1	2	yes	—
68	Q₈	—	349	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	2	2	no	—
71	D₄	—	4	4	no	—
72	D₄	—	349	2	no	—
73	D₄	—	698	4	no	—
74	D₄	—	698	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	698	2	no	—
77	C₂×C₄	—	698	2	no	—
78	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
79	C₈	—	698	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	698	2	no	—
85	C₂²	—	2792	2	no	—
86	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	698	2	no	—
89	C₄	D₄×D₃₄₉	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	8	1	no	—
93	C₂	—	1396	2	no	—
94	C₂	#26013607	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013607
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#127×C₃₄₉	C₂
socle	C₂×C₃₄₉	#120
hypercenter	#127	D₃₄₉
hypocenter	C₃₄₉	#453
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #26013684	C₂	semidirect
2. #26013665	C₂	semidirect
3. #26013567	C₂	semidirect
4. #26013556	C₂	semidirect
5. #127×C₃₄₉	C₂	semidirect
12. #19150217	C₂²	semidirect
15. C₂×D₄×C₃₄₉	C₂²	semidirect
16. C₄²×C₃₄₉	C₂²	semidirect
18. Dic₆₉₈	D₄	semidirect
19. D₁₃₉₆	D₄	semidirect
47. C₃₄₉	#453	semidirect
51. #127	D₃₄₉	semidirect
60. C₂×D₄	C₂×D₃₄₉	semidirect
64. C₄²	C₂×D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013684	2
#26013665	2
#26013567	2
#26013556	2
#127×C₃₄₉	2
#453	349