Subgroups — #36375147

Order 22336.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150242	C₂	1	2	yes	maximal
2	#26013706	C₂	1	2	yes	maximal, complemented
3	C₄⋊#14584102	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
5	#19150242	—	2	4	no	—
6	C₂×#14584102	C₂²	1	4	yes	—
7	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
8	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic
10	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic
11	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₄₉	C₄	1	1	yes	characteristic
14	#14584102	—	2	4	no	—
15	C₂×Dic₃₄₉	—	2	2	no	—
16	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₃₄₉	D₄	1	2	yes	—
20	C₂×Dic₃₄₉	Q₈	1	2	yes	—
21	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
22	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₄₉	—	2	2	no	—
24	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₄₉	—	2	1	no	—
27	C₂²×C₃₄₉	—	2	1	no	—
28	C₂²×C₃₄₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃₄₉	—	2	1	no	—
31	C₄×C₃₄₉	M₁₆	1	2	yes	complemented
32	Dic₃₄₉	—	2	1	no	—
33	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
34	Dic₃₄₉	M₁₆	1	2	yes	complemented
35	Dic₃₄₉	C₂×D₄	1	1	yes	characteristic
36	Dic₃₄₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₃₄₉	—	2	2	no	—
38	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₃₄₉	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₃₄₉	C₂×M₁₆	1	1	yes	characteristic
41	C₃₄₉	#423	1	1	yes	characteristic, complemented, Sylow
42	#423	—	349	1	no	maximal, Sylow
43	C₂×M₁₆	—	349	2	no	—
44	C₂×C₄²	—	349	1	no	—
45	C₄⋊C₈	—	349	2	no	—
46	C₄⋊C₈	—	349	2	no	—
47	C₂²×C₄	—	349	1	no	—
48	C₂²×C₄	—	349	1	no	—
49	C₂²×C₄	#13063703	1	1	yes	characteristic
50	M₁₆	—	698	4	no	—
51	C₂×C₈	—	349	4	no	—
52	C₄²	—	349	1	no	—
53	C₄²	—	349	1	no	—
54	C₄²	—	349	1	no	—
55	C₄²	—	349	1	no	—
56	C₂³	C₂×#13063703	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	349	2	no	—
59	C₂×C₄	—	349	2	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	2	no	—
65	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
66	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
67	C₈	—	698	4	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	C₄⋊#13063703	1	2	yes	—
71	C₂²	C₂²×#13063703	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	349	1	no	—
74	C₄	—	349	2	no	—
75	C₄	—	349	1	no	—
76	C₄	—	349	2	no	—
77	C₄	—	698	1	no	—
78	C₄	#19150242	1	2	yes	complemented
79	C₂	—	2	2	no	—
80	C₂	C₂×C₄⋊#13063703	1	1	yes	characteristic
81	C₂	C₂×#19150242	1	1	yes	characteristic
82	C₂	D₃₄₉⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13063703
commutator, socle	C₂²×C₃₄₉	C₂²×C₄
Fitting	C₂²×C₄×C₃₄₉	C₄
prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₂²×C₄	#13063703
hypocenter	C₃₄₉	#423
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #26013706	C₂	semidirect
3. C₄⋊#14584102	C₂	semidirect
31. C₄×C₃₄₉	M₁₆	semidirect
34. Dic₃₄₉	M₁₆	semidirect
41. C₃₄₉	#423	semidirect
78. C₄	#19150242	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
#26013706	2
C₄⋊#14584102	2
C₂×C₄×Dic₃₄₉	2
#423	349