Finite Group Databasebeta

Subgroups — #28405781

Order 14464.

208 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 98 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	#20299614	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11744753	C₂	1	1	yes	maximal, characteristic
3	#20299440	C₂	1	2	yes	maximal, complemented
4	#20299440	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11744753	C₄	1	2	yes	—
7	C₄⋊#11744753	C₄	1	2	yes	—
8	C₂²⋊#11744753	C₄	1	2	yes	—
9	C₄×#11744753	C₄	1	2	yes	—
10	C₄×#11744753	C₂²	1	1	yes	characteristic
11	C₄×#11744753	C₂²	1	1	yes	characteristic
12	C₄×#11744753	C₂²	1	2	yes	—
13	C₂×#14268631	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
15	#16813532	C₄	1	2	yes	—
16	C₈×Dic₁₁₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₁₃	C₄	1	1	yes	characteristic
18	C₂×#11744753	—	2	4	no	—
19	C₂×#11744753	C₂×C₄	1	2	yes	—
20	C₂×#11744753	C₂×C₄	1	4	yes	—
21	C₂×#11744753	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
24	#14268631	—	2	2	no	—
25	C₂×#11744751	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
31	M₁₆×C₁₁₃	—	2	2	no	—
32	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
33	C₂³×C₁₁₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₁₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₁₁₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₁₃	—	2	2	no	—
39	C₂×Dic₁₁₃	—	2	2	no	—
40	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
41	C₂×Dic₁₁₃	C₄²	1	1	yes	characteristic
42	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
44	#11744753	—	2	2	no	—
45	#11744753	—	2	4	no	—
46	#11744753	M₁₆	1	4	yes	complemented
47	C₈×C₁₁₃	—	2	2	no	—
48	#11744751	—	2	2	no	—
49	C₂²×C₁₁₃	—	2	1	no	—
50	C₂²×C₁₁₃	—	2	1	no	—
51	C₂²×C₁₁₃	#97	1	2	yes	—
52	C₂²×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₁₁₃	#97	1	2	yes	—
54	C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
56	Dic₁₁₃	—	2	1	no	—
57	Dic₁₁₃	—	2	1	no	—
58	Dic₁₁₃	C₂×M₁₆	1	2	yes	—
59	Dic₁₁₃	#131	1	2	yes	—
60	C₂×C₁₁₃	—	2	2	no	—
61	C₂×C₁₁₃	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₁₃	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₁₃	#405	1	1	yes	characteristic
64	#1451	—	113	1	no	maximal, Sylow
65	C₁₁₃	#1451	1	1	yes	characteristic, complemented, Sylow
66	#431	—	113	1	no	—
67	C₄×M₁₆	—	113	1	no	—
68	C₂×C₄×C₈	—	113	1	no	—
69	#322	—	113	2	no	—
70	#322	—	113	2	no	—
71	C₂×M₁₆	—	113	1	no	—
72	C₂×M₁₆	#90673	1	1	yes	characteristic
73	C₂²×C₈	—	113	2	no	—
74	C₂×C₄²	—	113	1	no	—
75	C₄⋊C₈	—	113	2	no	—
76	C₂²⋊C₈	—	113	2	no	—
77	#97	—	113	2	no	—
78	C₄×C₈	—	113	1	no	—
79	C₄×C₈	—	113	1	no	—
80	C₄×C₈	—	113	2	no	—
81	C₄×C₈	—	113	2	no	—
82	C₄×C₈	—	113	2	no	—
83	C₂²×C₄	—	113	1	no	—
84	C₂²×C₄	—	113	1	no	—
85	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	226	2	no	—
88	C₂×C₈	—	113	2	no	—
89	C₂×C₈	—	113	2	no	—
90	C₂×C₈	—	113	4	no	—
91	C₂×C₈	—	113	2	no	—
92	C₂×C₈	—	226	4	no	—
93	C₂×C₈	C₂×#90673	1	2	yes	—
94	C₄²	—	113	1	no	—
95	C₄²	—	113	1	no	—
96	C₄²	—	113	1	no	—
97	C₄²	—	113	1	no	—
98	C₂³	C₄×#90673	1	1	yes	characteristic
99	C₂×C₄	—	113	1	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	113	1	no	—
102	C₂×C₄	—	113	1	no	—
103	C₂×C₄	—	226	2	no	—
104	C₂×C₄	—	226	2	no	—
105	C₂×C₄	C₄×#90673	1	1	yes	characteristic
106	C₂×C₄	#14268656	1	2	yes	—
107	C₂×C₄	#14268657	1	2	yes	—
108	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	113	4	no	—
111	C₈	—	226	2	no	—
112	C₈	—	226	4	no	—
113	C₈	—	226	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#16813677	1	2	yes	—
117	C₂²	C₂×C₄×#90673	1	1	yes	characteristic
118	C₄	—	113	2	no	—
119	C₄	—	113	2	no	—
120	C₄	—	226	1	no	—
121	C₄	—	226	1	no	—
122	C₄	#16813677	1	2	yes	—
123	C₄	C₂×#14268656	1	1	yes	characteristic
124	C₄	C₂×#14268657	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	C₂×#16813677	1	1	yes	characteristic
127	C₂	M₁₆×#90673	1	1	yes	characteristic
128	C₂	#20299564	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂×C₄²
Fitting	C₂×M₁₆×C₁₁₃	C₄
prime radical	C₂³×C₁₁₃	C₄²
cosocle	C₄×Dic₁₁₃	C₂³
hypercenter	C₂×M₁₆	#90673
hypocenter	C₁₁₃	#1451
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20299440	C₂	semidirect
4. #20299440	C₂	semidirect
46. #11744753	M₁₆	semidirect
65. C₁₁₃	#1451	semidirect

Maximal subgroups

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

subgroup	index
#20299614	2
C₂×C₄×#11744753	2
#20299440	2
M₁₆×Dic₁₁₃	2
#1451	113