Finite Group Databasebeta

Subgroups — #28407787

Order 14464.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	1	yes	maximal, characteristic, complemented
2	C₂×#16813717	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#14268659	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299695	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#14268659	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299692	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299625	C₂	1	1	yes	maximal, characteristic
8	#16813718	—	2	2	no	—
9	#16813717	—	2	2	no	—
10	C₂×#14268659	C₂²	1	1	yes	characteristic
11	C₂×#14268659	C₂²	1	1	yes	characteristic
12	C₂×#14268659	C₂²	1	1	yes	characteristic
13	C₂×#14268659	C₂²	1	1	yes	characteristic
14	C₂×#14268655	C₂²	1	1	yes	characteristic
15	C₂²×#11744753	C₂²	1	1	yes	characteristic
16	#16813689	C₄	1	1	yes	characteristic
17	#16813679	C₄	1	1	yes	characteristic
18	C₂²⋊#11744753	C₄	1	1	yes	characteristic
19	C₄×#11744753	C₂²	1	1	yes	characteristic
20	#16813582	C₄	1	1	yes	characteristic
21	#14268659	—	2	1	no	—
22	#14268659	—	2	1	no	—
23	#14268659	—	2	1	no	—
24	#14268659	D₄	1	2	yes	complemented
25	#14268655	—	2	1	no	—
26	#14268655	—	2	1	no	—
27	C₂×#11744753	—	2	1	no	—
28	C₂×#11744753	—	2	1	no	—
29	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
30	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
31	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
32	C₂×#11744753	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊D₁₁₃	C₈	1	1	yes	characteristic
34	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
36	#14268626	C₈	1	1	yes	characteristic
37	#14268624	C₈	1	1	yes	characteristic
38	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₁₁₃	C₈	1	1	yes	characteristic
40	C₂³×C₁₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₁₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²⋊D₁₁₃	—	2	2	no	—
45	C₂×Dic₁₁₃	—	2	1	no	—
46	C₂×Dic₁₁₃	—	2	1	no	—
47	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₄×D₁₁₃	—	2	1	no	—
51	C₄×D₁₁₃	—	2	1	no	—
52	#11744753	—	2	1	no	—
53	#11744753	—	2	1	no	—
54	#11744753	—	2	1	no	—
55	#11744753	C₂×D₄	1	1	yes	characteristic
56	#11744753	#41	1	1	yes	characteristic
57	C₂²×C₁₁₃	—	2	1	no	—
58	C₂²×C₁₁₃	—	2	1	no	—
59	C₂²×C₁₁₃	—	2	1	no	—
60	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₁₁₃	—	2	1	no	—
62	C₂×D₁₁₃	—	2	1	no	—
63	C₂×D₁₁₃	M₃₂	1	2	yes	complemented
64	C₄×C₁₁₃	—	2	1	no	—
65	C₄×C₁₁₃	—	2	1	no	—
66	Dic₁₁₃	—	2	1	no	—
67	Dic₁₁₃	—	2	1	no	—
68	Dic₁₁₃	C₄×D₄	1	1	yes	characteristic
69	Dic₁₁₃	#131	1	1	yes	characteristic
70	C₂×C₁₁₃	—	4	1	no	—
71	C₂×C₁₁₃	C₈×D₄	1	1	yes	characteristic
72	C₂×C₁₁₃	C₂×M₃₂	1	1	yes	characteristic
73	C₂×C₁₁₃	#504	1	1	yes	characteristic
74	D₁₁₃	—	2	2	no	—
75	#2168	—	113	1	no	maximal, Sylow
76	C₁₁₃	#2168	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	113	1	no	—
78	C₂²×C₁₆	—	113	1	no	—
79	C₈×D₄	—	113	1	no	—
80	C₄⋊C₁₆	—	113	1	no	—
81	C₂²⋊C₁₆	—	113	1	no	—
82	C₂²⋊C₁₆	—	113	1	no	—
83	#346	—	113	1	no	—
84	C₂²×C₈	—	113	1	no	—
85	C₂²×C₈	—	113	1	no	—
86	C₄×D₄	—	113	1	no	—
87	M₃₂	—	226	2	no	—
88	C₂×C₁₆	—	113	1	no	—
89	C₂×C₁₆	—	113	1	no	—
90	C₂×C₁₆	—	113	1	no	—
91	C₂×C₁₆	—	113	1	no	—
92	C₂×C₁₆	—	226	2	no	—
93	C₄⋊C₈	—	113	1	no	—
94	C₂²⋊C₈	—	113	1	no	—
95	C₂²⋊C₈	—	113	1	no	—
96	C₄×C₈	—	113	1	no	—
97	C₂×D₄	—	113	1	no	—
98	C₂²×C₄	—	113	1	no	—
99	C₂²×C₄	—	113	1	no	—
100	C₂×C₈	—	113	1	no	—
101	C₂×C₈	—	113	1	no	—
102	C₂×C₈	—	113	1	no	—
103	C₂×C₈	—	113	1	no	—
104	C₂×C₈	—	226	1	no	—
105	C₂×C₈	—	226	1	no	—
106	C₂×C₈	—	226	1	no	—
107	C₂×C₈	—	226	1	no	—
108	C₄⋊C₄	—	113	1	no	—
109	C₂²⋊C₄	—	113	1	no	—
110	C₂²⋊C₄	#11744762	1	1	yes	characteristic
111	C₄²	—	113	1	no	—
112	C₁₆	—	113	2	no	—
113	C₁₆	—	226	1	no	—
114	C₁₆	—	226	1	no	—
115	C₁₆	—	226	1	no	—
116	C₂³	—	113	1	no	—
117	C₂³	C₂×#11744762	1	1	yes	characteristic
118	D₄	—	226	2	no	—
119	C₂×C₄	—	113	1	no	—
120	C₂×C₄	—	113	1	no	—
121	C₂×C₄	—	113	1	no	—
122	C₂×C₄	—	226	1	no	—
123	C₂×C₄	—	226	1	no	—
124	C₂×C₄	—	226	1	no	—
125	C₂×C₄	—	226	1	no	—
126	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
127	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
128	C₈	—	113	1	no	—
129	C₈	—	113	1	no	—
130	C₈	—	226	1	no	—
131	C₈	—	226	1	no	—
132	C₈	—	226	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	113	2	no	—
137	C₂²	—	226	1	no	—
138	C₂²	—	226	1	no	—
139	C₂²	C₂²×#11744762	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	113	1	no	—
143	C₄	—	113	1	no	—
144	C₄	—	226	1	no	—
145	C₄	—	226	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	226	2	no	—
148	C₂	D₄×#11744762	1	1	yes	characteristic
149	C₂	#20299717	1	1	yes	characteristic
150	C₂	#20299719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11744762
commutator, socle	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂²⋊C₄×C₁₁₃	C₈
prime radical	C₂×C₂²⋊D₁₁₃	C₈
cosocle	C₂×#11744753	C₂³
hypercenter	C₂²⋊C₄	#11744762
hypocenter	C₁₁₃	#2168
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16813718	C₂	semidirect
2. C₂×#16813717	C₂	semidirect
3. C₄⋊#14268659	C₂	semidirect
4. #20299695	C₂	semidirect
5. C₂²⋊#14268659	C₂	semidirect
6. #20299692	C₂	semidirect
24. #14268659	D₄	semidirect
63. C₂×D₁₁₃	M₃₂	semidirect
76. C₁₁₃	#2168	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
C₂×#16813717	2
C₄⋊#14268659	2
#20299695	2
C₂²⋊#14268659	2
#20299692	2
#20299625	2
#2168	113