Subgroups — #36375025

Order 22336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150242	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150242	C₂	1	1	yes	maximal, characteristic
3	#26013595	C₂	1	1	yes	maximal, characteristic
4	#19150242	—	2	1	no	—
5	#19150242	—	2	1	no	—
6	#19150242	—	2	1	no	—
7	#19150242	C₄	1	2	yes	complemented
8	C₂×#14584102	—	2	1	no	—
9	C₂×#14584102	C₄	1	2	yes	complemented
10	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
11	#19150212	—	2	1	no	—
12	C₂²⋊Dic₃₄₉	C₄	1	1	yes	characteristic
13	C₄×Dic₃₄₉	—	2	1	no	—
14	C₂²⋊C₄×C₃₄₉	C₄	1	1	yes	characteristic
15	#14584102	—	2	2	no	—
16	#14584102	—	2	2	no	—
17	C₂×Dic₃₄₉	—	2	1	no	—
18	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
21	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
22	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
23	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
24	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	—	2	1	no	—
26	C₂²×C₃₄₉	—	4	1	no	—
27	C₂²×C₃₄₉	C₄²	1	1	yes	characteristic
28	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃₄₉	—	4	1	no	—
31	Dic₃₄₉	—	4	1	no	—
32	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₃₄₉	—	2	1	no	—
37	C₂×C₃₄₉	—	2	1	no	—
38	C₂×C₃₄₉	—	2	1	no	—
39	C₂×C₃₄₉	#95	1	1	yes	characteristic
40	C₃₄₉	#344	1	1	yes	characteristic, complemented, Sylow
41	#344	—	349	1	no	maximal, Sylow
42	C₂×M₁₆	—	349	1	no	—
43	C₂×M₁₆	—	349	1	no	—
44	#117	—	349	1	no	—
45	C₂²×C₄	—	349	1	no	—
46	M₁₆	—	349	2	no	—
47	M₁₆	—	698	1	no	—
48	M₁₆	—	698	1	no	—
49	M₁₆	—	698	1	no	—
50	C₂×C₈	—	349	2	no	—
51	C₂×C₈	—	698	1	no	—
52	C₄⋊C₄	—	698	1	no	—
53	C₂²⋊C₄	—	349	1	no	—
54	C₂²⋊C₄	#13063703	1	1	yes	characteristic
55	C₄²	—	698	1	no	—
56	C₂³	C₂×#13063703	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	349	1	no	—
59	C₂×C₄	—	349	1	no	—
60	C₂×C₄	—	349	1	no	—
61	C₂×C₄	—	349	1	no	—
62	C₂×C₄	—	349	1	no	—
63	C₂×C₄	—	349	1	no	—
64	C₂×C₄	—	698	1	no	—
65	C₈	—	698	2	no	—
66	C₈	—	698	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	C₄×#13063703	1	1	yes	characteristic
69	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
70	C₂²	C₄⋊#13063703	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	349	1	no	—
73	C₄	—	349	1	no	—
74	C₄	—	349	1	no	—
75	C₄	—	349	1	no	—
76	C₄	—	1396	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#13063703)⋊C₄
commutator	C₂²×C₃₄₉	C₄²
Frattini	C₂³	C₂×#13063703
Fitting	C₂²⋊C₄×C₃₄₉	C₄
socle	C₂×C₃₄₉	#95
prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂²×Dic₃₄₉	C₂²
hypercenter	C₂²⋊C₄	#13063703
hypocenter	C₃₄₉	#344
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. #19150242	C₄	semidirect
9. C₂×#14584102	C₄	semidirect
40. C₃₄₉	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150242	2
#26013595	2
#344	349