Subgroups — #28365318

Order 13952.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295431	C₂	1	1	yes	maximal, characteristic
2	#20295428	C₂	1	1	yes	maximal, characteristic
3	#20294854	C₂	1	2	yes	maximal
4	#20294852	C₂	1	2	yes	maximal
5	#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812699	—	2	1	no	—
7	#16812691	—	2	1	no	—
8	#16812633	—	2	1	no	—
9	#16812633	C₂²	1	1	yes	characteristic
10	#16812633	C₂²	1	1	yes	characteristic
11	#16812633	C₂²	1	2	yes	—
12	#16812631	—	2	1	no	—
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	4	1	no	—
16	C₄⋊Dic₁₀₉	—	2	1	no	—
17	C₄⋊Dic₁₀₉	—	2	1	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	4	1	no	—
20	#13182008	—	4	1	no	—
21	#13182008	—	4	1	no	—
22	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₁₆×C₁₀₉	Q₈	1	4	yes	complemented
26	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	4	1	no	—
30	C₂×Dic₁₀₉	—	4	1	no	—
31	C₂×Dic₁₀₉	—	4	2	no	—
32	Dic₂₁₈	—	8	2	no	—
33	C₈×C₁₀₉	Q₁₆	1	2	yes	—
34	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₀₉	—	2	1	no	—
37	C₄×C₁₀₉	—	2	1	no	—
38	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
40	Dic₁₀₉	—	8	2	no	—
41	Dic₁₀₉	—	8	1	no	—
42	Dic₁₀₉	—	8	1	no	—
43	C₂×C₁₀₉	#500	1	1	yes	characteristic
44	C₂×C₁₀₉	#509	1	1	yes	characteristic
45	C₂×C₁₀₉	#510	1	1	yes	characteristic
46	#2255	—	109	1	no	maximal, Sylow
47	C₁₀₉	#2255	1	1	yes	characteristic, complemented, Sylow
48	#500	—	109	1	no	—
49	#499	—	109	1	no	—
50	#367	—	109	2	no	—
51	#366	—	109	2	no	—
52	#346	D₁₀₉	1	1	yes	characteristic
53	C₄⋊Q₈	—	218	1	no	—
54	#125	—	218	1	no	—
55	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
56	#107	—	109	1	no	—
57	#107	—	109	2	no	—
58	#107	—	109	1	no	—
59	#107	—	218	1	no	—
60	#106	—	218	1	no	—
61	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂×Q₈	—	436	1	no	—
63	C₂×C₈	D₄₃₆	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₄	—	436	1	no	—
66	C₄⋊C₄	—	436	1	no	—
67	C₄⋊C₄	—	436	1	no	—
68	C₄⋊C₄	—	218	1	no	—
69	C₄⋊C₄	—	218	1	no	—
70	C₄⋊C₄	—	218	2	no	—
71	C₄²	D₄₃₆	1	1	yes	characteristic
72	C₁₆	Dic₂₁₈	1	4	yes	complemented
73	Q₈	—	872	2	no	—
74	C₂×C₄	—	436	1	no	—
75	C₂×C₄	—	436	1	no	—
76	C₂×C₄	—	436	2	no	—
77	C₂×C₄	D₈₇₂	1	1	yes	characteristic
78	C₂×C₄	D₈₇₂	1	1	yes	characteristic
79	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
80	C₈	Dic₄₃₆	1	2	yes	—
81	C₈	C₂×Dic₂₁₈	1	2	yes	—
82	C₂²	C₂×D₈₇₂	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	872	2	no	—
86	C₄	—	872	1	no	—
87	C₄	—	872	1	no	—
88	C₄	#16812699	1	1	yes	characteristic
89	C₄	C₂×Dic₄₃₆	1	1	yes	characteristic
90	C₂	#20295431	1	1	yes	characteristic
91	C₂	#20295464	1	1	yes	characteristic
92	C₂	#20295469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₇₂
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#346×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#346	D₁₀₉
hypocenter	C₁₀₉	#2255
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
25. C₁₆×C₁₀₉	Q₈	semidirect
47. C₁₀₉	#2255	semidirect
72. C₁₆	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295431	2
#20295428	2
#20294854	2
#20294852	2
#346×C₁₀₉	2
#2255	109