Subgroups — #36373908

Order 22336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150217	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	#19150217	—	2	1	no	—
5	#19150217	C₄	1	2	yes	—
6	C₂×#14584095	—	2	1	no	—
7	C₂×#14584095	—	2	1	no	—
8	C₂×#14584095	—	2	1	no	—
9	C₂×#14584095	C₄	1	2	yes	—
10	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₃₄₉	—	2	1	no	—
12	M₁₆×C₃₄₉	C₄	1	2	yes	—
13	C₂×C₈×C₃₄₉	—	2	1	no	—
14	#14584095	—	2	2	no	—
15	#14584095	—	2	2	no	—
16	C₂³×C₃₄₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
23	C₈×C₃₄₉	—	2	2	no	—
24	C₂²×C₃₄₉	—	2	2	no	—
25	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
29	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃₄₉	—	2	1	no	—
33	C₂×C₃₄₉	—	2	1	no	—
34	C₂×C₃₄₉	#95	1	1	yes	characteristic
35	C₂×C₃₄₉	#108	1	2	yes	—
36	C₃₄₉	#341	1	1	yes	characteristic, complemented, Sylow
37	#341	—	349	1	no	maximal, Sylow
38	C₂×M₁₆	—	349	1	no	—
39	C₂×M₁₆	D₃₄₉	1	1	yes	characteristic
40	C₂²×C₈	—	349	1	no	—
41	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
42	M₁₆	—	2	1	no	—
43	M₁₆	—	349	2	no	—
44	M₁₆	—	698	1	no	—
45	M₁₆	Dic₃₄₉	1	2	yes	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	349	2	no	—
48	C₂×C₈	—	698	1	no	—
49	C₂×C₈	—	698	1	no	—
50	C₂×C₈	—	698	1	no	—
51	C₂³	Dic₆₉₈	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
53	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
57	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	698	2	no	—
60	C₈	—	698	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	#19150212	1	1	yes	characteristic
63	C₂²	#19150212	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₃₄₉	1	1	yes	characteristic
65	C₄	C₄×Dic₃₄₉	1	1	yes	characteristic
66	C₄	#19150210	1	1	yes	characteristic
67	C₄	#19150210	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#26013546	1	1	yes	characteristic
72	C₂	#26013574	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₄₉
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂²×C₄	C₂×D₃₄₉
Fitting	C₂×M₁₆×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₄⋊C₄
prime radical	C₂³×C₃₄₉	Q₈
cosocle	C₂²×C₄×C₃₄₉	C₂²
hypercenter	C₂×M₁₆	D₃₄₉
hypocenter	C₃₄₉	#341
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
36. C₃₄₉	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150217	2
C₂²×#14584095	2
C₂×M₁₆×C₃₄₉	2
#341	349