Subgroups — #36375010

Order 22336.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14584102	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14584102	—	2	2	no	—
4	C₂×#14584102	—	2	2	no	—
5	C₂×#14584102	C₄	1	4	yes	complemented
6	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₄₉	—	2	2	no	—
9	C₂²×C₄×C₃₄₉	C₄	1	1	yes	characteristic
10	#14584102	—	2	4	no	—
11	C₂×Dic₃₄₉	—	2	1	no	—
12	C₂×Dic₃₄₉	—	2	1	no	—
13	C₂×Dic₃₄₉	C₈	1	2	yes	complemented
14	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
15	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
16	C₂×Dic₃₄₉	D₄	1	2	yes	—
17	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
18	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₄₉	—	2	1	no	—
20	C₂×C₄×C₃₄₉	—	2	1	no	—
21	C₂×C₄×C₃₄₉	C₈	1	2	yes	complemented
22	C₂²×C₃₄₉	C₄²	1	1	yes	characteristic
23	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₃₄₉	M₁₆	1	1	yes	characteristic
28	C₂²×C₃₄₉	M₁₆	1	1	yes	characteristic
29	C₄×C₃₄₉	—	2	2	no	—
30	Dic₃₄₉	—	2	2	no	—
31	Dic₃₄₉	C₂²⋊C₄	1	2	yes	—
32	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
33	C₂×C₃₄₉	#95	1	1	yes	characteristic
34	C₂×C₃₄₉	C₄×C₈	1	1	yes	characteristic
35	C₂×C₃₄₉	#97	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
40	C₃₄₉	#336	1	1	yes	characteristic, complemented, Sylow
41	#336	—	349	1	no	maximal, Sylow
42	C₂²×C₈	—	349	2	no	—
43	C₂×C₄²	—	349	1	no	—
44	C₂²×C₄	—	349	1	no	—
45	C₂²×C₄	—	349	1	no	—
46	C₂²×C₄	#13063703	1	1	yes	characteristic
47	C₂×C₈	—	349	4	no	—
48	C₂×C₈	—	698	2	no	—
49	C₂×C₈	—	698	2	no	—
50	C₄²	—	698	2	no	—
51	C₂³	C₂×#13063703	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	349	2	no	—
55	C₂×C₄	—	349	2	no	—
56	C₂×C₄	—	349	1	no	—
57	C₂×C₄	—	349	1	no	—
58	C₂×C₄	—	349	2	no	—
59	C₂×C₄	—	698	1	no	—
60	C₂×C₄	—	698	1	no	—
61	C₂×C₄	#14584102	1	2	yes	complemented
62	C₈	—	698	4	no	—
63	C₂²	C₄×#13063703	1	1	yes	characteristic
64	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
65	C₂²	C₄⋊#13063703	1	1	yes	characteristic
66	C₂²	C₂×#14584102	1	1	yes	characteristic
67	C₂²	D₃₄₉⋊C₈	1	1	yes	characteristic
68	C₂²	#19150242	1	1	yes	characteristic
69	C₂²	#19150243	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	349	2	no	—
72	C₄	—	349	2	no	—
73	C₄	—	698	2	no	—
74	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic
75	C₂	C₄×#14584102	1	1	yes	characteristic
76	C₂	#26013694	1	1	yes	characteristic
77	C₂	C₂²⋊#14584102	1	1	yes	characteristic
78	C₂	#26013696	1	1	yes	characteristic
79	C₂	C₄⋊#14584102	1	1	yes	characteristic
80	C₂	#26013706	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×#13063703
commutator	C₂×C₃₄₉	C₄×C₈
Fitting	C₂²×C₄×C₃₄₉	C₄
socle, prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂²×Dic₃₄₉	C₂²
hypercenter	C₂²×C₄	#13063703
hypocenter	C₃₄₉	#336
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
5. C₂×#14584102	C₄	semidirect
13. C₂×Dic₃₄₉	C₈	semidirect
21. C₂×C₄×C₃₄₉	C₈	semidirect
40. C₃₄₉	#336	semidirect
61. C₂×C₄	#14584102	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14584102	2
C₂×C₄×Dic₃₄₉	2
#336	349