Subgroups — #36374892

Order 22336.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013627	C₂	1	2	yes	maximal, complemented
2	#26013624	C₂	1	4	yes	maximal, complemented
3	#26013617	C₂	1	4	yes	maximal, complemented
4	#26013614	C₂	1	4	yes	maximal, complemented
5	#125×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₉₈	C₂²	1	2	yes	—
7	C₂×D₁₃₉₆	C₂²	1	4	yes	—
8	C₂×D₁₃₉₆	C₂²	1	2	yes	—
9	C₂×C₄×D₃₄₉	C₂²	1	4	yes	—
10	#19150212	C₂²	1	4	yes	—
11	#19150210	C₂²	1	4	yes	—
12	#19150210	C₂²	1	4	yes	—
13	#19150210	C₂²	1	4	yes	—
14	C₄⋊C₄×C₃₄₉	C₂²	1	4	yes	—
15	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
16	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
17	C₂²×D₃₄₉	C₂³	1	4	yes	—
18	Dic₆₉₈	—	4	2	no	—
19	D₁₃₉₆	—	4	4	no	—
20	D₁₃₉₆	—	4	2	no	—
21	C₂×Dic₃₄₉	C₂³	1	4	yes	—
22	C₄×D₃₄₉	—	4	4	no	—
23	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
25	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
26	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₃₄₉	—	2	4	no	—
28	C₂×D₃₄₉	—	2	4	no	—
29	C₂×D₃₄₉	—	2	4	no	—
30	C₄×C₃₄₉	—	2	4	no	—
31	C₄×C₃₄₉	—	2	2	no	—
32	C₄×C₃₄₉	—	2	1	no	—
33	Dic₃₄₉	—	2	4	no	—
34	C₂×C₃₄₉	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₃₄₉	2₋¹⁺⁴	1	1	yes	characteristic
36	D₃₄₉	—	4	4	no	—
37	C₃₄₉	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	349	1	no	maximal, Sylow
39	#125	D₃₄₉	1	1	yes	characteristic, complemented
40	#124	—	349	2	no	—
41	#123	—	349	4	no	—
42	C₂²⋊Q₈	—	349	4	no	—
43	#121	—	349	4	no	—
44	C₂×Q₈	—	349	2	no	—
45	C₂×D₄	—	349	4	no	—
46	C₂×D₄	—	349	2	no	—
47	C₂²×C₄	—	349	4	no	—
48	C₄⋊C₄	—	349	4	no	—
49	C₄⋊C₄	C₂×D₃₄₉	1	4	yes	—
50	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
51	C₂²⋊C₄	—	349	4	no	—
52	C₂²⋊C₄	—	349	4	no	—
53	C₂²⋊C₄	—	349	4	no	—
54	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
55	C₂³	—	349	4	no	—
56	Q₈	—	1396	2	no	—
57	D₄	—	1396	4	no	—
58	D₄	—	1396	2	no	—
59	C₂×C₄	—	349	4	no	—
60	C₂×C₄	—	1396	4	no	—
61	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
63	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
64	C₂²	—	698	4	no	—
65	C₂²	—	698	4	no	—
66	C₂²	—	698	4	no	—
67	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	698	4	no	—
72	C₂	—	1396	4	no	—
73	C₂	#26013685	1	2	yes	—
74	C₂	#26013687	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₄₉
commutator, socle, cosocle	C₂²×C₃₄₉	C₂⁴
Fitting	#125×C₃₄₉	C₂
hypercenter	#125	D₃₄₉
hypocenter	C₃₄₉	#562
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #26013627	C₂	semidirect
2. #26013624	C₂	semidirect
3. #26013617	C₂	semidirect
4. #26013614	C₂	semidirect
5. #125×C₃₄₉	C₂	semidirect
37. C₃₄₉	#562	semidirect
39. #125	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013627	2
#26013624	2
#26013617	2
#26013614	2
#125×C₃₄₉	2
#562	349