Finite Group Databasebeta

Subgroups — #28349609

Order 13696.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293444	C₂	1	1	yes	maximal, characteristic, complemented
2	#20293432	C₂	1	1	yes	maximal, characteristic, complemented
3	#20293070	C₂	1	3	yes	maximal, complemented
4	#20293065	C₂	1	3	yes	maximal, complemented
5	#20293059	C₂	1	3	yes	maximal, complemented
6	C₂×#16812054	C₂	1	3	yes	maximal, complemented
7	#504×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16812156	—	2	1	no	—
9	#16812155	—	2	1	no	—
10	#16812142	—	2	3	no	—
11	#16812137	—	2	3	no	—
12	#16812131	C₂²	1	3	yes	—
13	#16812131	C₂²	1	3	yes	complemented
14	C₂×Dic₄₂₈	C₂²	1	3	yes	complemented
15	C₂×D₈₅₆	C₂²	1	3	yes	—
16	Dic₈₅₆	C₂²	1	3	yes	complemented
17	#16812054	C₂²	1	3	yes	complemented
18	#16812054	C₂²	1	6	yes	complemented
19	#16812054	C₂²	1	1	yes	characteristic, complemented
20	D₁₇₁₂	C₂²	1	3	yes	complemented
21	#131×C₁₀₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₀₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₀₇	C₂²	1	3	yes	complemented
24	#13133819	—	4	1	no	—
25	#13133818	—	2	3	no	—
26	#13133818	—	2	3	no	—
27	#13133817	—	4	3	no	—
28	C₂×Dic₂₁₄	—	2	3	no	—
29	Q₈×D₁₀₇	—	4	1	no	—
30	C₂×D₄₂₈	—	2	3	no	—
31	D₄×D₁₀₇	—	4	3	no	—
32	Dic₄₂₈	—	2	3	no	—
33	Dic₄₂₈	C₂³	1	3	yes	—
34	Dic₄₂₈	C₂³	1	1	yes	characteristic
35	#13133807	—	2	3	no	—
36	#13133807	—	2	3	no	—
37	D₈₅₆	—	2	3	no	—
38	D₈₅₆	C₂³	1	1	yes	characteristic
39	D₈₅₆	C₂³	1	3	yes	—
40	#41×C₁₀₇	D₄	1	1	yes	characteristic
41	M₁₆×C₁₀₇	D₄	1	3	yes	—
42	C₂×C₈×C₁₀₇	C₂³	1	3	yes	—
43	C₁₆×C₁₀₇	C₂³	1	3	yes	—
44	C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
45	C₂²×D₁₀₇	—	4	3	no	—
46	Q₈×C₁₀₇	D₈	1	1	yes	characteristic
47	D₄×C₁₀₇	D₈	1	3	yes	—
48	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
49	C₂²⋊D₁₀₇	—	4	3	no	—
50	C₂²⋊D₁₀₇	—	4	3	no	—
51	C₂×Dic₁₀₇	—	4	3	no	—
52	D₄₂₈	—	2	3	no	—
53	D₄₂₈	—	2	1	no	—
54	D₄₂₈	—	4	3	no	—
55	C₄×D₁₀₇	—	4	3	no	—
56	C₄×D₁₀₇	—	4	3	no	—
57	Dic₂₁₄	—	2	1	no	—
58	Dic₂₁₄	—	2	3	no	—
59	Dic₂₁₄	—	4	3	no	—
60	C₈×C₁₀₇	C₂×D₄	1	3	yes	—
61	C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₀₇	C₂×D₈	1	3	yes	—
63	C₂×D₁₀₇	—	4	1	no	—
64	C₂×D₁₀₇	—	4	3	no	—
65	C₂×D₁₀₇	—	16	3	no	—
66	C₄×C₁₀₇	C₂×D₈	1	3	yes	—
67	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₀₇	—	4	3	no	—
69	Dic₁₀₇	—	4	1	no	—
70	C₂×C₁₀₇	—	2	3	no	—
71	C₂×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
72	D₁₀₇	—	8	1	no	—
73	D₁₀₇	—	8	3	no	—
74	#3416	—	107	1	no	maximal, Sylow
75	C₁₀₇	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	107	1	no	—
77	#576	—	107	1	no	—
78	#510	—	107	3	no	—
79	#509	—	107	3	no	—
80	#508	—	107	3	no	—
81	C₂×SD₃₂	—	107	3	no	—
82	#504	D₁₀₇	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	214	1	no	—
84	H(5,𝔽₂)	—	214	1	no	—
85	#137	—	214	3	no	—
86	AGL(1,ℤ/8)	—	214	3	no	—
87	#135	—	107	3	no	—
88	#135	—	107	3	no	—
89	C₂×Q₁₆	—	107	3	no	—
90	C₂×D₈	—	107	3	no	—
91	#131	C₂×D₁₀₇	1	1	yes	characteristic
92	Q₃₂	—	107	3	no	—
93	SD₃₂	—	107	3	no	—
94	SD₃₂	—	107	6	no	—
95	SD₃₂	—	107	1	no	—
96	D₁₆	—	107	3	no	—
97	M₃₂	C₂×D₁₀₇	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₀₇	1	3	yes	complemented
99	#41	—	428	1	no	—
100	#41	—	428	3	no	—
101	#41	—	214	3	no	—
102	#41	—	214	3	no	—
103	#41	D₄₂₈	1	1	yes	characteristic
104	C₂×Q₈	—	428	1	no	—
105	C₂×Q₈	—	214	3	no	—
106	C₂×D₄	—	428	3	no	—
107	C₂×D₄	—	214	3	no	—
108	Q₁₆	—	107	3	no	—
109	Q₁₆	—	107	1	no	—
110	Q₁₆	—	214	3	no	—
111	SD₁₆	—	214	3	no	—
112	SD₁₆	—	214	3	no	—
113	D₈	—	107	1	no	—
114	D₈	—	107	3	no	—
115	D₈	—	214	3	no	—
116	M₁₆	D₄₂₈	1	3	yes	—
117	C₂×C₈	C₂²×D₁₀₇	1	3	yes	—
118	C₁₆	C₂²×D₁₀₇	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₀₇	1	3	yes	—
120	C₂³	—	428	3	no	—
121	Q₈	—	428	3	no	—
122	Q₈	—	214	1	no	—
123	Q₈	—	214	3	no	—
124	Q₈	D₈₅₆	1	1	yes	characteristic
125	D₄	—	428	3	no	—
126	D₄	—	428	3	no	—
127	D₄	—	428	3	no	—
128	D₄	—	214	3	no	—
129	D₄	—	214	1	no	—
130	D₄	D₈₅₆	1	3	yes	—
131	C₂×C₄	—	428	3	no	—
132	C₂×C₄	—	428	3	no	—
133	C₂×C₄	—	428	3	no	—
134	C₂×C₄	C₂×D₄₂₈	1	3	yes	—
135	C₈	C₂×D₄₂₈	1	3	yes	—
136	C₈	C₂³×D₁₀₇	1	1	yes	characteristic
137	C₂²	—	428	1	no	—
138	C₂²	—	428	3	no	—
139	C₂²	—	1712	3	no	—
140	C₂²	C₂×D₈₅₆	1	3	yes	—
141	C₄	—	428	1	no	—
142	C₄	—	428	3	no	—
143	C₄	C₂×D₈₅₆	1	3	yes	—
144	C₄	C₂²×D₄₂₈	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	856	3	no	—
147	C₂	—	856	1	no	—
148	C₂	C₂²×D₈₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₈₅₆
commutator, cosocle	C₈×C₁₀₇	C₂⁴
Frattini	C₈	C₂³×D₁₀₇
Fitting	#504×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₂²×D₈
hypercenter	#504	D₁₀₇
hypocenter	C₁₀₇	#3416
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20293444	C₂	semidirect
2. #20293432	C₂	semidirect
3. #20293070	C₂	semidirect
4. #20293065	C₂	semidirect
5. #20293059	C₂	semidirect
6. C₂×#16812054	C₂	semidirect
7. #504×C₁₀₇	C₂	semidirect
13. #16812131	C₂²	semidirect
14. C₂×Dic₄₂₈	C₂²	semidirect
16. Dic₈₅₆	C₂²	semidirect
17. #16812054	C₂²	semidirect
18. #16812054	C₂²	semidirect
19. #16812054	C₂²	semidirect
20. D₁₇₁₂	C₂²	semidirect
22. M₃₂×C₁₀₇	C₂²	semidirect
23. C₂×C₁₆×C₁₀₇	C₂²	semidirect
75. C₁₀₇	#3416	semidirect
82. #504	D₁₀₇	semidirect
97. M₃₂	C₂×D₁₀₇	semidirect
98. C₂×C₁₆	C₂×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293444	2
#20293432	2
#20293070	2
#20293065	2
#20293059	2
C₂×#16812054	2
#504×C₁₀₇	2
#3416	107