Finite Group Databasebeta

Subgroups — #28379384

Order 13952.

156 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20296036	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812768	C₂	1	1	yes	maximal, characteristic
3	#20295953	C₂	1	2	yes	maximal, complemented
4	#20295952	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812778	—	2	1	no	—
7	C₂×C₄⋊#89137	—	2	1	no	—
8	#16812768	—	2	1	no	—
9	#16812768	C₂²	1	1	yes	characteristic
10	#16812768	C₂²	1	2	yes	—
11	#16812768	C₂²	1	1	yes	characteristic
12	#16812767	—	2	1	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812639	C₄	1	1	yes	characteristic, complemented
15	#16812638	C₄	1	2	yes	complemented
16	C₁₆×D₁₀₉	C₂²	1	2	yes	—
17	M₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×#89137	—	4	1	no	—
19	C₄⋊#89137	—	2	1	no	—
20	C₄⋊#89137	—	2	1	no	—
21	C₄⋊#89137	—	2	2	no	—
22	C₄⋊#89137	—	4	1	no	—
23	C₂²⋊#89137	—	4	1	no	—
24	C₄×#89137	—	4	1	no	—
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×#11724784	Q₈	1	1	yes	characteristic
27	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₀₉	D₄	1	1	yes	characteristic
29	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
30	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
31	#13182004	C₂×C₄	1	2	yes	complemented
32	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
34	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
35	C₂×#89137	—	4	1	no	—
36	C₂×#89137	—	4	1	no	—
37	C₂×#89137	—	4	2	no	—
38	C₂×#89137	—	8	2	no	—
39	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
41	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₁₀₉	D₈	1	1	yes	characteristic
43	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
44	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	#11724784	C₄⋊C₄	1	1	yes	characteristic
48	#11724784	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₉	#107	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	#107	1	1	yes	characteristic
53	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
54	#89137	—	8	1	no	—
55	#89137	—	8	2	no	—
56	#89137	—	8	1	no	—
57	C₄×C₁₀₉	#107	1	1	yes	characteristic
58	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	Dic₁₀₉	#107	1	1	yes	characteristic
60	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
63	D₁₀₉	—	2	1	no	—
64	D₁₀₉	#509	1	1	yes	characteristic
65	D₁₀₉	#510	1	1	yes	characteristic
66	#2159	—	109	1	no	maximal, Sylow
67	C₁₀₉	#2159	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₃₂	—	109	1	no	—
69	#427	—	109	1	no	—
70	C₂×#107	—	109	1	no	—
71	#367	—	109	2	no	—
72	#366	—	109	2	no	—
73	C₂²×C₈	—	109	1	no	—
74	#117	—	218	1	no	—
75	C₂×C₄⋊C₄	—	218	1	no	—
76	M₃₂	—	109	1	no	—
77	M₃₂	—	109	2	no	—
78	M₃₂	#89137	1	1	yes	characteristic, complemented
79	C₂×C₁₆	—	109	2	no	—
80	#107	—	109	1	no	—
81	#107	—	109	2	no	—
82	#107	—	109	1	no	—
83	#107	—	218	1	no	—
84	#106	—	218	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	436	1	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	1	no	—
90	C₂×C₈	—	109	1	no	—
91	C₂×C₈	—	109	1	no	—
92	C₂×C₈	C₂×#89137	1	1	yes	characteristic
93	C₄⋊C₄	—	436	1	no	—
94	C₄⋊C₄	—	218	1	no	—
95	C₄⋊C₄	—	218	1	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₂²⋊C₄	—	436	1	no	—
98	C₄²	—	436	1	no	—
99	C₁₆	—	109	2	no	—
100	C₁₆	C₂×#89137	1	2	yes	complemented
101	C₂³	—	109	1	no	—
102	C₂×C₄	—	872	2	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	109	1	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	436	1	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
112	C₈	—	109	1	no	—
113	C₈	—	109	1	no	—
114	C₈	C₄⋊#89137	1	1	yes	characteristic
115	C₈	C₂²×#89137	1	1	yes	characteristic
116	C₂²	—	109	1	no	—
117	C₂²	—	109	1	no	—
118	C₂²	—	218	1	no	—
119	C₂²	—	218	1	no	—
120	C₂²	#16812768	1	1	yes	characteristic
121	C₄	—	872	2	no	—
122	C₄	—	872	1	no	—
123	C₄	—	872	1	no	—
124	C₄	—	109	1	no	—
125	C₄	—	109	1	no	—
126	C₄	#16812768	1	1	yes	characteristic
127	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	109	1	no	—
130	C₂	—	109	1	no	—
131	C₂	—	218	1	no	—
132	C₂	C₂×#16812768	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16812768
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	M₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#107
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₈×D₁₀₉	C₂³
hypercenter	M₃₂	#89137
hypocenter	C₁₀₉	#2159
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
3. #20295953	C₂	semidirect
4. #20295952	C₂	semidirect
14. #16812639	C₄	semidirect
15. #16812638	C₄	semidirect
17. M₃₂×C₁₀₉	C₄	semidirect
31. #13182004	C₂×C₄	semidirect
33. C₁₆×C₁₀₉	C₂×C₄	semidirect
67. C₁₀₉	#2159	semidirect
78. M₃₂	#89137	semidirect
100. C₁₆	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296036	2
C₂×#16812768	2
#20295953	2
#20295952	2
M₃₂×D₁₀₉	2
#2159	109