Finite Group Databasebeta

Subgroups — #28407750

Order 14464.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16813718	C₂	1	2	yes	maximal, complemented
2	#20299697	C₂	1	2	yes	maximal, complemented
3	#20299696	C₂	1	2	yes	maximal, complemented
4	C₂×#16813709	C₂	1	1	yes	maximal, characteristic
5	#16813718	—	2	2	no	—
6	#16813718	C₂²	1	4	yes	—
7	C₂×#14268659	—	2	2	no	—
8	#16813709	—	2	2	no	—
9	#16813709	—	2	2	no	—
10	C₂×#14268657	C₂²	1	1	yes	characteristic
11	C₂×#14268656	C₄	1	1	yes	characteristic
12	C₂×#14268656	C₂²	1	1	yes	characteristic
13	C₂×#14268655	C₄	1	1	yes	characteristic
14	C₂²×#11744753	C₄	1	1	yes	characteristic
15	C₂²×#11744753	C₂²	1	1	yes	characteristic
16	C₂×#14268646	C₄	1	1	yes	characteristic
17	#14268659	—	2	4	no	—
18	#14268657	—	2	2	no	—
19	#14268656	—	2	2	no	—
20	#14268656	—	2	2	no	—
21	#14268655	—	2	1	no	—
22	#14268655	—	2	1	no	—
23	C₂×#11744753	—	2	1	no	—
24	C₂×#11744753	—	2	1	no	—
25	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
26	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
27	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
28	C₂×#11744753	D₄	1	2	yes	—
29	C₂×#11744753	D₄	1	2	yes	—
30	C₂×#11744753	C₂³	1	1	yes	characteristic
31	#14268646	—	2	2	no	—
32	#14268646	—	2	2	no	—
33	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₁₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₁₃	C₈	1	1	yes	characteristic
36	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₁₃	C₈	1	1	yes	characteristic
40	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
43	D₄×C₁₁₃	—	2	2	no	—
44	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²⋊D₁₁₃	—	2	2	no	—
46	C₂²⋊D₁₁₃	—	2	2	no	—
47	C₂×Dic₁₁₃	—	2	1	no	—
48	C₂×Dic₁₁₃	—	2	1	no	—
49	C₂×Dic₁₁₃	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂×Dic₁₁₃	M₁₆	1	2	yes	—
53	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₁₁₃	—	2	1	no	—
55	C₄×D₁₁₃	—	2	1	no	—
56	Dic₂₂₆	—	2	2	no	—
57	#11744753	—	2	1	no	—
58	#11744753	—	2	1	no	—
59	#11744753	C₂²⋊C₄	1	1	yes	characteristic
60	#11744753	C₂²⋊C₄	1	1	yes	characteristic
61	#11744753	C₂×D₄	1	2	yes	—
62	C₂²×C₁₁₃	—	2	1	no	—
63	C₂²×C₁₁₃	—	4	1	no	—
64	C₂²×C₁₁₃	—	4	1	no	—
65	C₂²×C₁₁₃	C₂²⋊C₈	1	2	yes	—
66	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×D₁₁₃	—	2	1	no	—
68	C₂×D₁₁₃	—	4	1	no	—
69	C₄×C₁₁₃	—	2	1	no	—
70	Dic₁₁₃	—	2	1	no	—
71	Dic₁₁₃	C₂²⋊C₈	1	2	yes	—
72	Dic₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	Dic₁₁₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₁₁₃	—	2	1	no	—
75	C₂×C₁₁₃	—	2	2	no	—
76	C₂×C₁₁₃	—	4	1	no	—
77	C₂×C₁₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	D₁₁₃	—	4	1	no	—
79	#2115	—	113	1	no	maximal, Sylow
80	C₁₁₃	#2115	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	113	1	no	—
82	C₂×M₃₂	—	113	2	no	—
83	#349	—	113	2	no	—
84	#349	—	113	2	no	—
85	C₂×#41	—	113	1	no	—
86	#131	—	226	2	no	—
87	#131	—	226	2	no	—
88	C₂×M₁₆	—	113	1	no	—
89	C₂×M₁₆	—	113	1	no	—
90	C₂×M₁₆	—	113	1	no	—
91	C₂²×C₈	—	113	1	no	—
92	C₂²×C₈	—	113	1	no	—
93	C₂²×C₈	—	113	1	no	—
94	M₃₂	—	113	4	no	—
95	M₃₂	—	226	2	no	—
96	C₂×C₁₆	—	226	2	no	—
97	#41	—	226	2	no	—
98	#41	—	226	2	no	—
99	C₂×Q₈	—	113	1	no	—
100	C₂×D₄	—	113	1	no	—
101	C₂×D₄	—	113	1	no	—
102	C₂×D₄	#11744762	1	1	yes	characteristic
103	C₂²×C₄	—	113	1	no	—
104	C₂²×C₄	—	113	1	no	—
105	C₂²×C₄	—	113	1	no	—
106	M₁₆	—	226	2	no	—
107	M₁₆	—	226	2	no	—
108	M₁₆	—	226	2	no	—
109	C₂×C₈	—	113	2	no	—
110	C₂×C₈	—	113	1	no	—
111	C₂×C₈	—	113	2	no	—
112	C₂×C₈	—	113	1	no	—
113	C₂×C₈	—	113	1	no	—
114	C₂×C₈	—	113	1	no	—
115	C₂×C₈	—	226	1	no	—
116	C₂×C₈	—	226	1	no	—
117	C₂×C₈	—	226	1	no	—
118	C₂×C₈	—	226	1	no	—
119	C₁₆	—	226	4	no	—
120	C₂³	—	113	1	no	—
121	C₂³	C₂×#11744762	1	1	yes	characteristic
122	C₂³	C₂×#11744762	1	1	yes	characteristic
123	Q₈	—	226	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	226	2	no	—
126	D₄	—	226	2	no	—
127	C₂×C₄	—	113	2	no	—
128	C₂×C₄	—	113	2	no	—
129	C₂×C₄	—	113	1	no	—
130	C₂×C₄	—	113	1	no	—
131	C₂×C₄	—	113	1	no	—
132	C₂×C₄	—	226	1	no	—
133	C₂×C₄	—	226	1	no	—
134	C₂×C₄	—	226	1	no	—
135	C₂×C₄	—	226	1	no	—
136	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
137	C₈	—	113	2	no	—
138	C₈	—	113	1	no	—
139	C₈	—	113	1	no	—
140	C₈	—	226	1	no	—
141	C₈	—	226	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	452	1	no	—
146	C₂²	—	226	1	no	—
147	C₂²	C₂²⋊#11744762	1	2	yes	—
148	C₂²	C₂²×#11744762	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	113	1	no	—
151	C₄	—	113	1	no	—
152	C₄	—	113	2	no	—
153	C₄	—	226	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	2	no	—
156	C₂	—	4	1	no	—
157	C₂	—	452	1	no	—
158	C₂	C₂×C₂²⋊#11744762	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11744762
commutator	C₂²×C₁₁₃	C₂²×C₈
Frattini	C₂²	C₂²×#11744762
Fitting	C₂×D₄×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₂×C₂²⋊C₈
prime radical	C₂×#14268646	C₄
cosocle	C₂×#11744753	C₂³
hypercenter	C₂×D₄	#11744762
hypocenter	C₁₁₃	#2115
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16813718	C₂	semidirect
2. #20299697	C₂	semidirect
3. #20299696	C₂	semidirect
80. C₁₁₃	#2115	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813718	2
#20299697	2
#20299696	2
C₂×#16813709	2
#2115	113