Subgroups — #36375063

Order 22336.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013714	C₂	1	1	yes	maximal, characteristic
2	#26013709	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	#19150243	—	2	1	no	—
5	#19150236	—	2	1	no	—
6	Q₈×D₃₄₉	—	2	1	no	—
7	Dic₁₃₉₆	C₄	1	1	yes	characteristic
8	#19150223	—	2	1	no	—
9	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₄₉	C₄	1	1	yes	characteristic
11	#14584102	—	4	1	no	—
12	Dic₆₉₈	—	2	1	no	—
13	Dic₆₉₈	—	4	1	no	—
14	C₄×D₃₄₉	—	4	1	no	—
15	C₄×D₃₄₉	D₄	1	1	yes	characteristic
16	#14584095	D₄	1	1	yes	characteristic
17	Q₈×C₃₄₉	—	2	1	no	—
18	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₄₉	SD₁₆	1	1	yes	characteristic
20	C₄×C₃₄₉	—	4	1	no	—
21	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₃₄₉	—	4	1	no	—
23	Dic₃₄₉	D₈	1	1	yes	characteristic
24	C₂×C₃₄₉	#102	1	1	yes	characteristic
25	D₃₄₉	—	2	1	no	—
26	C₃₄₉	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	349	1	no	maximal, Sylow
28	C₂×Q₁₆	—	349	1	no	—
29	M₃₂	—	349	1	no	—
30	#108	—	349	1	no	—
31	C₂×Q₈	—	698	1	no	—
32	Q₁₆	—	349	1	no	—
33	Q₁₆	—	698	1	no	—
34	Q₁₆	#13063703	1	1	yes	characteristic
35	M₁₆	—	698	1	no	—
36	C₂×C₈	—	349	1	no	—
37	C₁₆	—	698	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	1396	1	no	—
40	Q₈	—	698	1	no	—
41	C₂×C₄	—	349	1	no	—
42	C₂×C₄	—	1396	1	no	—
43	C₈	—	349	1	no	—
44	C₈	—	1396	1	no	—
45	C₈	C₂×#13063703	1	1	yes	characteristic
46	C₂²	—	349	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	349	1	no	—
49	C₄	—	1396	1	no	—
50	C₄	C₂²⋊#13063703	1	1	yes	characteristic
51	C₂	—	698	1	no	—
52	C₂	#26013701	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013701
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₈	C₂×#13063703
Fitting	Q₁₆×C₃₄₉	C₄
socle	C₂×C₃₄₉	#102
prime radical	C₂×D₃₄₉	SD₁₆
cosocle	C₈×D₃₄₉	C₂²
hypercenter	Q₁₆	#13063703
hypocenter	C₃₄₉	#362
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₃₄₉	#362	semidirect

Maximal subgroups

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

subgroup	index
#26013714	2
#26013709	2
Q₁₆×D₃₄₉	2
#362	349