Subgroups — #36374482

Order 22336.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013598	C₂	1	1	yes	maximal, characteristic
2	#26013572	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013570	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013563	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013563	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#14584095	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊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	C₂²⋊Dic₃₄₉	—	2	1	no	—
13	C₄×Dic₃₄₉	—	2	1	no	—
14	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₄₉	—	2	1	no	—
17	C₄⋊C₄×C₃₄₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₃₄₉	—	2	1	no	—
19	C₂×Dic₃₄₉	—	2	1	no	—
20	C₂×Dic₃₄₉	—	2	1	no	—
21	#14584095	—	2	1	no	—
22	#14584095	—	2	1	no	—
23	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
24	Q₈×C₃₄₉	—	2	1	no	—
25	Q₈×C₃₄₉	—	2	1	no	—
26	C₂×C₄×C₃₄₉	—	2	1	no	—
27	C₂×C₄×C₃₄₉	—	2	1	no	—
28	C₂×C₄×C₃₄₉	—	2	1	no	—
29	C₂×C₄×C₃₄₉	—	2	1	no	—
30	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
32	C₂²×C₃₄₉	—	2	1	no	—
33	C₂²×C₃₄₉	—	2	1	no	—
34	C₂²×C₃₄₉	—	2	1	no	—
35	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₄₉	—	2	1	no	—
37	C₄×C₃₄₉	—	4	1	no	—
38	C₄×C₃₄₉	—	4	1	no	—
39	C₄×C₃₄₉	#41	1	1	yes	characteristic
40	C₄×C₃₄₉	#41	1	1	yes	characteristic
41	Dic₃₄₉	—	4	1	no	—
42	Dic₃₄₉	—	4	1	no	—
43	C₂×C₃₄₉	—	4	1	no	—
44	C₂×C₃₄₉	#123	1	1	yes	characteristic
45	C₂×C₃₄₉	#135	1	1	yes	characteristic
46	C₂×C₃₄₉	#137	1	1	yes	characteristic
47	C₃₄₉	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	349	1	no	maximal, Sylow
49	C₂²⋊Q₈	D₃₄₉	1	1	yes	characteristic
50	#117	—	349	1	no	—
51	#107	—	349	1	no	—
52	#106	—	349	1	no	—
53	#103	—	349	1	no	—
54	#103	—	349	1	no	—
55	C₂²⋊C₈	—	349	1	no	—
56	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
58	C₂×C₈	—	349	1	no	—
59	C₂×C₈	—	349	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	349	1	no	—
62	C₄⋊C₄	—	349	1	no	—
63	C₄⋊C₄	C₂×D₃₄₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	698	1	no	—
66	C₄²	—	698	1	no	—
67	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	698	1	no	—
75	C₂×C₄	—	698	1	no	—
76	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
78	C₈	—	698	1	no	—
79	C₈	—	698	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	1396	1	no	—
88	C₄	—	1396	1	no	—
89	C₄	#19150232	1	1	yes	characteristic
90	C₄	#19150232	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#26013620	1	1	yes	characteristic
93	C₂	#26013659	1	1	yes	characteristic
94	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	C₂²⋊Q₈×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂³×C₃₄₉	D₄
hypercenter	C₂²⋊Q₈	D₃₄₉
hypocenter	C₃₄₉	#485
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #26013572	C₂	semidirect
3. #26013570	C₂	semidirect
4. #26013563	C₂	semidirect
5. #26013563	C₂	semidirect
47. C₃₄₉	#485	semidirect

Maximal subgroups

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

subgroup	index
#26013598	2
#26013572	2
#26013570	2
#26013563	2
C₂²⋊#14584095	2
C₂²⋊Q₈×C₃₄₉	2
#485	349