Finite Group Databasebeta

Subgroups — #28392471

Order 14464.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20299036	C₂	1	1	yes	maximal, characteristic
2	#20298719	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298709	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298410	C₂	1	1	yes	maximal, characteristic, complemented
5	#20298407	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298375	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813644	C₂²	1	1	yes	characteristic
9	C₂×D₉₀₄	C₂²	1	1	yes	characteristic
10	#16813630	—	2	1	no	—
11	#16813584	—	2	1	no	—
12	#16813567	C₄	1	2	yes	complemented
13	D₁₈₀₈	C₄	1	2	yes	complemented
14	#16813562	C₂²	1	1	yes	characteristic
15	#16813548	—	2	1	no	—
16	#16813542	—	2	1	no	—
17	#16813532	C₂²	1	1	yes	characteristic
18	M₃₂×C₁₁₃	C₂²	1	1	yes	characteristic
19	#108×C₁₁₃	C₂²	1	1	yes	characteristic
20	#106×C₁₁₃	C₂²	1	1	yes	characteristic
21	#14268647	—	2	1	no	—
22	C₂×D₄₅₂	—	2	1	no	—
23	C₂×C₄×D₁₁₃	—	4	1	no	—
24	Dic₄₅₂	C₂×C₄	1	1	yes	characteristic, complemented
25	#14268636	—	2	1	no	—
26	D₉₀₄	—	2	1	no	—
27	D₉₀₄	C₂×C₄	1	2	yes	—
28	D₉₀₄	C₂×C₄	1	1	yes	characteristic, complemented
29	#14268630	—	4	1	no	—
30	C₂×#11744751	D₄	1	1	yes	characteristic
31	C₈×D₁₁₃	—	4	1	no	—
32	#14268624	—	4	1	no	—
33	C₄×Dic₁₁₃	D₄	1	1	yes	characteristic
34	#14268622	—	2	1	no	—
35	M₁₆×C₁₁₃	—	2	1	no	—
36	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₁₃	—	2	1	no	—
38	C₁₆×C₁₁₃	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₁₃	—	4	1	no	—
40	C₂×C₄×C₁₁₃	—	4	1	no	—
41	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₁₁₃	—	4	1	no	—
43	C₂×Dic₁₁₃	—	2	1	no	—
44	D₄₅₂	—	2	2	no	—
45	D₄₅₂	—	2	1	no	—
46	D₄₅₂	—	4	1	no	—
47	C₄×D₁₁₃	—	4	1	no	—
48	C₄×D₁₁₃	—	8	2	no	—
49	Dic₂₂₆	—	2	1	no	—
50	C₈×C₁₁₃	—	4	1	no	—
51	C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₁₃	#41	1	1	yes	characteristic
53	#11744751	—	2	1	no	—
54	C₂²×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₁₃	—	4	1	no	—
56	C₂×D₁₁₃	—	4	2	no	—
57	C₂×D₁₁₃	—	8	1	no	—
58	C₂×D₁₁₃	—	8	1	no	—
59	C₄×C₁₁₃	—	8	1	no	—
60	C₄×C₁₁₃	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₁₃	—	4	1	no	—
63	Dic₁₁₃	—	4	1	no	—
64	C₂×C₁₁₃	—	2	1	no	—
65	C₂×C₁₁₃	#442	1	1	yes	characteristic
66	D₁₁₃	—	8	1	no	—
67	D₁₁₃	—	8	2	no	—
68	AGL(1,ℤ/16)	—	113	1	no	maximal, Sylow
69	C₁₁₃	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	—	113	1	no	—
71	#444	—	113	1	no	—
72	#442	—	113	1	no	—
73	#365	D₁₁₃	1	1	yes	characteristic, complemented
74	#361	—	113	1	no	—
75	#360	—	113	1	no	—
76	#347	—	113	1	no	—
77	#135	—	113	1	no	—
78	C₂×D₈	—	113	1	no	—
79	#131	—	226	1	no	—
80	C₄×D₄	—	226	1	no	—
81	SD₃₂	—	113	2	no	—
82	D₁₆	—	113	2	no	—
83	M₃₂	—	113	1	no	—
84	M₃₂	C₂×D₁₁₃	1	1	yes	characteristic
85	#108	C₂×D₁₁₃	1	1	yes	characteristic
86	#106	C₂×D₁₁₃	1	1	yes	characteristic
87	#104	—	226	1	no	—
88	#102	—	226	1	no	—
89	#97	—	113	1	no	—
90	#41	—	226	1	no	—
91	C₂×D₄	—	226	1	no	—
92	C₂²×C₄	—	452	1	no	—
93	Q₁₆	—	113	1	no	—
94	SD₁₆	—	226	1	no	—
95	D₈	—	113	1	no	—
96	D₈	—	113	2	no	—
97	D₈	—	226	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	452	1	no	—
100	C₂×C₈	—	113	1	no	—
101	C₂×C₈	—	452	1	no	—
102	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	452	1	no	—
105	C₄²	—	113	1	no	—
106	C₁₆	—	226	1	no	—
107	C₁₆	C₄×D₁₁₃	1	2	yes	complemented
108	C₂³	—	452	1	no	—
109	Q₈	—	226	1	no	—
110	D₄	—	452	1	no	—
111	D₄	—	452	1	no	—
112	D₄	—	226	2	no	—
113	D₄	—	226	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	904	2	no	—
116	C₂×C₄	—	452	1	no	—
117	C₂×C₄	—	226	1	no	—
118	C₂×C₄	D₄×D₁₁₃	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	226	1	no	—
121	C₈	C₂×C₄×D₁₁₃	1	1	yes	characteristic
122	C₈	#14268648	1	1	yes	characteristic
123	C₂²	—	904	1	no	—
124	C₂²	—	904	1	no	—
125	C₂²	—	452	1	no	—
126	C₂²	—	452	2	no	—
127	C₂²	#16813649	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	452	1	no	—
130	C₄	—	452	1	no	—
131	C₄	#16813584	1	1	yes	characteristic
132	C₄	#16813649	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	904	2	no	—
135	C₂	—	904	1	no	—
136	C₂	#20298709	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20298709
commutator	C₈×C₁₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₁₃
Fitting	#365×C₁₁₃	C₂
socle	C₂×C₁₁₃	#442
prime radical	#20299036	C₂
cosocle	C₂×C₈×C₁₁₃	C₂³
hypercenter	#365	D₁₁₃
hypocenter	C₁₁₃	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

Complements

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

N	H ≅ G/N	type
2. #20298719	C₂	semidirect
3. #20298709	C₂	semidirect
4. #20298410	C₂	semidirect
5. #20298407	C₂	semidirect
6. #20298375	C₂	semidirect
7. #365×C₁₁₃	C₂	semidirect
12. #16813567	C₄	semidirect
13. D₁₈₀₈	C₄	semidirect
24. Dic₄₅₂	C₂×C₄	semidirect
28. D₉₀₄	C₂×C₄	semidirect
38. C₁₆×C₁₁₃	C₂×C₄	semidirect
69. C₁₁₃	AGL(1,ℤ/16)	semidirect
73. #365	D₁₁₃	semidirect
107. C₁₆	C₄×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299036	2
#20298719	2
#20298709	2
#20298410	2
#20298407	2
#20298375	2
#365×C₁₁₃	2
AGL(1,ℤ/16)	113