Finite Group Databasebeta

Subgroups — #28338327

Order 13696.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂×#16812054	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈₅₆	C₂	1	1	yes	maximal, characteristic
4	#20292453	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292432	C₂	1	1	yes	maximal, characteristic, complemented
6	#20292429	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic
9	C₂×D₈₅₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₁₄	—	2	1	no	—
11	C₄×D₄₂₈	—	2	1	no	—
12	#16812054	C₄	1	4	yes	complemented
13	#16812043	C₂²	1	1	yes	characteristic
14	#16812043	C₂²	1	1	yes	characteristic
15	#16812034	—	2	1	no	—
16	#16812031	—	2	1	no	—
17	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₄	—	2	1	no	—
21	C₂×D₄₂₈	—	2	1	no	—
22	C₂×C₄×D₁₀₇	—	4	1	no	—
23	Dic₄₂₈	—	2	1	no	—
24	Dic₄₂₈	C₂×C₄	1	2	yes	complemented
25	D₈₅₆	—	2	1	no	—
26	D₈₅₆	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₁₀₇	—	2	1	no	—
28	C₄⋊Dic₁₀₇	—	2	1	no	—
29	Dic₁₀₇⋊C₄	—	4	1	no	—
30	(C₂×D₁₀₇)⋊C₄	—	4	1	no	—
31	C₄×Dic₁₀₇	—	4	1	no	—
32	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
35	C₁₆×C₁₀₇	—	2	1	no	—
36	C₁₆×C₁₀₇	C₂×C₄	1	2	yes	complemented
37	C₂²×D₁₀₇	—	4	1	no	—
38	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₇	—	4	1	no	—
42	C₂×Dic₁₀₇	—	4	1	no	—
43	C₂×Dic₁₀₇	—	4	1	no	—
44	D₄₂₈	—	2	2	no	—
45	D₄₂₈	—	4	1	no	—
46	C₄×D₁₀₇	—	8	2	no	—
47	Dic₂₁₄	—	2	2	no	—
48	Dic₂₁₄	—	4	1	no	—
49	C₈×C₁₀₇	—	2	1	no	—
50	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₀₇	#41	1	1	yes	characteristic
52	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₀₇	—	4	2	no	—
54	C₂×D₁₀₇	—	8	1	no	—
55	C₂×D₁₀₇	—	8	1	no	—
56	C₄×C₁₀₇	—	2	1	no	—
57	C₄×C₁₀₇	—	2	1	no	—
58	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₀₇	#135	1	1	yes	characteristic
60	Dic₁₀₇	—	4	2	no	—
61	Dic₁₀₇	—	8	1	no	—
62	Dic₁₀₇	—	8	1	no	—
63	C₂×C₁₀₇	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₀₇	#509	1	1	yes	characteristic
65	C₂×C₁₀₇	#510	1	1	yes	characteristic
66	D₁₀₇	—	8	2	no	—
67	#2175	—	107	1	no	maximal, Sylow
68	C₁₀₇	#2175	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₃₂	—	107	1	no	—
70	C₄×Q₁₆	—	107	1	no	—
71	C₄×D₈	—	107	1	no	—
72	#366	—	107	1	no	—
73	#358	—	107	1	no	—
74	#357	—	107	1	no	—
75	#346	D₁₀₇	1	1	yes	characteristic, complemented
76	C₂×Q₁₆	—	107	1	no	—
77	C₂×D₈	—	107	1	no	—
78	C₄×Q₈	—	214	1	no	—
79	C₄×D₄	—	214	1	no	—
80	SD₃₂	—	107	4	no	—
81	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
83	#107	—	107	1	no	—
84	#107	—	107	1	no	—
85	#103	—	214	1	no	—
86	#102	—	214	1	no	—
87	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
88	C₂×Q₈	—	214	1	no	—
89	C₂×D₄	—	214	1	no	—
90	C₂²×C₄	—	428	1	no	—
91	Q₁₆	—	107	2	no	—
92	Q₁₆	—	214	1	no	—
93	D₈	—	107	2	no	—
94	D₈	—	214	1	no	—
95	C₂×C₈	D₄₂₈	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
97	C₄⋊C₄	—	428	1	no	—
98	C₄⋊C₄	—	214	1	no	—
99	C₄⋊C₄	—	214	1	no	—
100	C₂²⋊C₄	—	428	1	no	—
101	C₄²	—	428	1	no	—
102	C₄²	D₄₂₈	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	C₄×D₁₀₇	1	2	yes	complemented
105	C₂³	—	428	1	no	—
106	Q₈	—	428	1	no	—
107	Q₈	—	214	2	no	—
108	D₄	—	428	1	no	—
109	D₄	—	214	2	no	—
110	C₂×C₄	—	856	2	no	—
111	C₂×C₄	—	428	1	no	—
112	C₂×C₄	—	428	1	no	—
113	C₂×C₄	—	428	1	no	—
114	C₂×C₄	D₈₅₆	1	1	yes	characteristic
115	C₂×C₄	D₈₅₆	1	1	yes	characteristic
116	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	C₂×C₄×D₁₀₇	1	1	yes	characteristic
119	C₈	#13133818	1	1	yes	characteristic
120	C₂²	—	856	1	no	—
121	C₂²	—	856	1	no	—
122	C₂²	—	428	2	no	—
123	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	856	1	no	—
127	C₄	—	856	1	no	—
128	C₄	—	428	2	no	—
129	C₄	C₄×D₄₂₈	1	1	yes	characteristic
130	C₄	#16812131	1	1	yes	characteristic
131	C₂	—	856	2	no	—
132	C₂	C₄×D₈₅₆	1	1	yes	characteristic
133	C₂	#20293065	1	1	yes	characteristic
134	C₂	#20293070	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#346×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₈
prime radical	C₂×D₈₅₆	C₂²
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	#346	D₁₀₇
hypocenter	C₁₀₇	#2175
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₄₂₈	C₂	semidirect
4. #20292453	C₂	semidirect
5. #20292432	C₂	semidirect
7. #346×C₁₀₇	C₂	semidirect
12. #16812054	C₄	semidirect
24. Dic₄₂₈	C₂×C₄	semidirect
36. C₁₆×C₁₀₇	C₂×C₄	semidirect
68. C₁₀₇	#2175	semidirect
75. #346	D₁₀₇	semidirect
104. C₁₆	C₄×D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812054	2
C₄×Dic₄₂₈	2
C₄×D₈₅₆	2
#20292453	2
#20292432	2
#20292429	2
#346×C₁₀₇	2
#2175	107