Subgroups — #36374451

Order 22336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013631	C₂	1	1	yes	maximal, characteristic
2	#26013572	C₂	1	1	yes	maximal, characteristic
3	#26013570	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#14584095	C₂	1	1	yes	maximal, characteristic
5	#26013563	C₂	1	1	yes	maximal, characteristic
6	#26013563	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14584095	C₂²	1	1	yes	characteristic
9	C₂×#14584095	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
12	#19150212	—	2	1	no	—
13	#19150212	—	2	1	no	—
14	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₄₉	—	2	1	no	—
16	C₄⋊C₄×C₃₄₉	C₂²	1	1	yes	characteristic
17	C₄²×C₃₄₉	—	2	1	no	—
18	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₄₉	—	2	1	no	—
20	C₂×Dic₃₄₉	—	2	1	no	—
21	#14584095	—	2	1	no	—
22	#14584095	—	2	1	no	—
23	Q₈×C₃₄₉	—	2	1	no	—
24	Q₈×C₃₄₉	Q₈	1	2	yes	—
25	C₂×C₄×C₃₄₉	—	2	1	no	—
26	C₂×C₄×C₃₄₉	—	2	1	no	—
27	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₃₄₉	—	2	2	no	—
32	C₄×C₃₄₉	—	2	1	no	—
33	C₄×C₃₄₉	—	2	1	no	—
34	C₄×C₃₄₉	—	4	1	no	—
35	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₃₄₉	#41	1	1	yes	characteristic
37	Dic₃₄₉	—	4	1	no	—
38	Dic₃₄₉	—	4	1	no	—
39	C₂×C₃₄₉	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₃₄₉	#135	1	1	yes	characteristic
41	C₂×C₃₄₉	#137	1	1	yes	characteristic
42	C₃₄₉	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	349	1	no	maximal, Sylow
44	#125	—	349	1	no	—
45	C₄×Q₈	D₃₄₉	1	1	yes	characteristic
46	#107	—	349	1	no	—
47	#106	—	349	1	no	—
48	C₄⋊C₈	—	349	1	no	—
49	#103	—	349	1	no	—
50	#103	—	349	1	no	—
51	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
52	C₂×C₈	—	349	1	no	—
53	C₂×C₈	—	349	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	349	1	no	—
56	C₄⋊C₄	—	349	1	no	—
57	C₄⋊C₄	—	698	1	no	—
58	C₄⋊C₄	—	698	1	no	—
59	C₄⋊C₄	C₂×D₃₄₉	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
62	Q₈	—	2	1	no	—
63	Q₈	Dic₆₉₈	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	698	1	no	—
67	C₂×C₄	—	698	1	no	—
68	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
71	C₈	—	698	1	no	—
72	C₈	—	698	1	no	—
73	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1396	1	no	—
79	C₄	—	1396	1	no	—
80	C₄	C₂×Dic₆₉₈	1	1	yes	characteristic
81	C₄	#19150233	1	1	yes	characteristic
82	C₂	#26013618	1	1	yes	characteristic
83	C₂	#26013659	1	1	yes	characteristic
84	C₂	#26013670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₄₉
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₄×Q₈×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×D₄
hypercenter	C₄×Q₈	D₃₄₉
hypocenter	C₃₄₉	#479
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
42. C₃₄₉	#479	semidirect

Maximal subgroups

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

subgroup	index
#26013631	2
#26013572	2
#26013570	2
C₄⋊#14584095	2
#26013563	2
C₄×Q₈×C₃₄₉	2
#479	349