Finite Group Databasebeta

Subgroups — #28406707

Order 14464.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20299602	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16813687	C₂	1	4	yes	maximal, complemented
3	C₄²×#90673	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299149	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268657	C₂²	1	2	yes	—
6	C₂×C₄×#90673	—	2	2	no	—
7	C₂×C₄×#90673	C₂²	1	2	yes	—
8	C₄⋊#11744753	—	2	2	no	—
9	#16813687	—	2	8	no	—
10	C₂×#14268648	C₂²	1	2	yes	—
11	#16813619	C₄	1	1	yes	characteristic, complemented
12	#16813611	C₄	1	2	yes	complemented
13	#16813597	—	2	2	no	—
14	#16813584	—	2	2	no	—
15	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#90673	—	2	2	no	—
18	#14268657	—	2	4	no	—
19	C₂×#11744753	—	2	2	no	—
20	C₄×#90673	—	2	8	no	—
21	C₄×#90673	—	2	2	no	—
22	C₄×#90673	D₄	1	4	yes	complemented
23	#14268648	—	2	4	no	—
24	#14268648	—	2	4	no	—
25	C₂×D₄₅₂	—	2	2	no	—
26	C₂×D₄₅₂	—	2	2	no	—
27	C₂×D₄₅₂	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₁₃	—	2	2	no	—
29	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
30	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
31	#14268624	—	2	4	no	—
32	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₁₃	D₄	1	2	yes	—
34	C₂×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
35	C₄⋊C₄×C₁₁₃	—	2	2	no	—
36	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₁₁₃	—	2	2	no	—
38	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#90673	—	2	8	no	—
40	C₂×#90673	—	2	4	no	—
41	Q₈×C₁₁₃	—	2	4	no	—
42	C₂×C₄×C₁₁₃	—	2	2	no	—
43	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
47	D₄₅₂	—	2	4	no	—
48	D₄₅₂	—	4	4	no	—
49	D₄₅₂	—	4	4	no	—
50	C₄×D₁₁₃	—	2	4	no	—
51	C₄×D₁₁₃	—	4	2	no	—
52	C₄×D₁₁₃	—	4	2	no	—
53	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
54	C₄×D₁₁₃	#41	1	2	yes	—
55	#11744753	—	4	2	no	—
56	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₁₁₃	—	2	2	no	—
58	C₂×D₁₁₃	—	4	2	no	—
59	C₂×D₁₁₃	—	4	4	no	—
60	C₂×D₁₁₃	#121	1	2	yes	—
61	#90673	—	2	8	no	—
62	C₄×C₁₁₃	—	4	2	no	—
63	C₄×C₁₁₃	#104	1	4	yes	complemented
64	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
65	Dic₁₁₃	—	2	2	no	—
66	Dic₁₁₃	#124	1	1	yes	characteristic
67	Dic₁₁₃	#127	1	1	yes	characteristic
68	C₂×C₁₁₃	#390	1	1	yes	characteristic
69	C₂×C₁₁₃	C₂×#104	1	2	yes	—
70	D₁₁₃	—	2	2	no	—
71	D₁₁₃	—	8	2	no	—
72	#1962	—	113	1	no	maximal, Sylow
73	C₁₁₃	#1962	1	1	yes	characteristic, complemented, Sylow
74	#532	—	113	1	no	—
75	#423	—	113	1	no	—
76	C₂×#104	—	113	4	no	—
77	C₄³	—	113	1	no	—
78	C₂×#41	—	113	2	no	—
79	C₂×M₁₆	—	113	2	no	—
80	C₄⋊Q₈	#90673	1	1	yes	characteristic, complemented
81	#127	—	113	1	no	—
82	#124	—	113	2	no	—
83	#121	—	226	2	no	—
84	C₄×D₄	—	226	2	no	—
85	C₂×C₄²	—	113	1	no	—
86	C₂×C₄²	—	113	2	no	—
87	C₂×C₄²	—	226	2	no	—
88	C₄⋊C₈	—	226	2	no	—
89	#104	—	226	8	no	—
90	#41	—	226	4	no	—
91	#41	—	226	4	no	—
92	C₂×Q₈	C₂×#90673	1	2	yes	—
93	C₂×D₄	—	113	2	no	—
94	C₂×D₄	—	226	2	no	—
95	C₂×D₄	—	226	2	no	—
96	C₂²×C₄	—	113	1	no	—
97	C₂²×C₄	—	113	2	no	—
98	C₂²×C₄	—	226	2	no	—
99	C₂²×C₄	—	226	2	no	—
100	M₁₆	—	226	4	no	—
101	C₂×C₈	—	226	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	226	4	no	—
104	C₄²	—	113	2	no	—
105	C₄²	—	113	1	no	—
106	C₄²	—	113	4	no	—
107	C₄²	—	226	2	no	—
108	C₄²	—	226	8	no	—
109	C₄²	C₂×#90673	1	1	yes	characteristic
110	C₂³	—	113	1	no	—
111	C₂³	—	226	2	no	—
112	Q₈	—	2	4	no	—
113	D₄	—	452	4	no	—
114	D₄	—	452	4	no	—
115	D₄	—	226	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	113	1	no	—
118	C₂×C₄	—	113	2	no	—
119	C₂×C₄	—	113	2	no	—
120	C₂×C₄	—	113	2	no	—
121	C₂×C₄	—	452	2	no	—
122	C₂×C₄	—	452	2	no	—
123	C₂×C₄	—	226	4	no	—
124	C₂×C₄	—	226	8	no	—
125	C₂×C₄	—	226	4	no	—
126	C₂×C₄	C₂²⋊#90673	1	2	yes	—
127	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
128	C₈	—	452	2	no	—
129	C₂²	—	113	2	no	—
130	C₂²	—	452	2	no	—
131	C₂²	—	452	4	no	—
132	C₂²	—	226	2	no	—
133	C₂²	C₂×C₂²⋊#90673	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	113	1	no	—
136	C₄	—	113	1	no	—
137	C₄	—	226	8	no	—
138	C₄	—	226	2	no	—
139	C₄	#16813687	1	4	yes	complemented
140	C₄	D₄×#90673	1	2	yes	—
141	C₂	—	904	2	no	—
142	C₂	—	226	2	no	—
143	C₂	#20299550	1	1	yes	characteristic
144	C₂	C₂×#16813687	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#90673
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#90673
Fitting	C₄⋊Q₈×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	#20299149	C₂
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	C₄⋊Q₈	#90673
hypocenter	C₁₁₃	#1962
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299602	C₂	semidirect
2. C₂×#16813687	C₂	semidirect
3. C₄²×#90673	C₂	semidirect
11. #16813619	C₄	semidirect
12. #16813611	C₄	semidirect
16. C₄⋊Q₈×C₁₁₃	C₄	semidirect
22. C₄×#90673	D₄	semidirect
63. C₄×C₁₁₃	#104	semidirect
73. C₁₁₃	#1962	semidirect
80. C₄⋊Q₈	#90673	semidirect
139. C₄	#16813687	semidirect

Maximal subgroups

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

subgroup	index
#20299602	2
C₂×#16813687	2
C₄²×#90673	2
#20299149	2
#1962	113