Finite Group Databasebeta

Subgroups — #12014032

Order 1280.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₅×#2248	C₂	1	1	yes	maximal, characteristic
2	#10602105	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602091	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602084	C₂	1	1	yes	maximal, characteristic
5	#10602068	C₂	1	1	yes	maximal, characteristic
6	#10599859	C₂	1	1	yes	maximal, characteristic, complemented
7	#10597989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×SD₃₂	—	2	2	no	—
9	C₅×#494	C₂²	1	1	yes	characteristic
10	C₅×#493	C₂²	1	1	yes	characteristic
11	C₂×#3961	—	2	2	no	—
12	C₂×#3959	—	2	2	no	—
13	#65367	—	2	2	no	—
14	#65359	—	2	2	no	—
15	C₂×Dic₄₀	—	2	2	no	—
16	#64988	C₂²	1	1	yes	characteristic
17	#64988	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₄×#672	C₂²	1	1	yes	characteristic
20	C₈×#168	C₂²	1	1	yes	characteristic
21	#12520	—	5	1	no	maximal, Sylow
22	C₂×C₅×Q₁₆	—	2	1	no	—
23	C₂×C₅×Q₁₆	—	2	1	no	—
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×#127	—	2	1	no	—
28	C₂×Dic₂₀	—	2	1	no	—
29	C₂×Dic₂₀	—	2	1	no	—
30	C₂×Dic₂₀	—	2	1	no	—
31	C₂×Dic₂₀	—	2	1	no	—
32	#4015	—	2	1	no	—
33	#4015	—	2	1	no	—
34	C₅×SD₃₂	—	4	4	no	—
35	C₂×C₅×C₁₆	D₄	1	2	yes	—
36	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
37	#3964	—	4	4	no	—
38	#3961	—	4	4	no	—
39	#3959	—	4	4	no	—
40	C₂×#672	D₄	1	2	yes	—
41	#3939	—	4	4	no	—
42	C₄×#168	D₄	1	2	yes	—
43	Dic₄₀	—	4	4	no	—
44	#2248	—	5	1	no	—
45	#2248	D₅	1	1	yes	characteristic
46	#2247	—	5	1	no	—
47	#2247	—	5	1	no	—
48	#1713	—	5	1	no	—
49	#1709	—	5	1	no	—
50	#1311	—	5	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂×C₅×D₄	—	4	1	no	—
53	C₂×C₅×D₄	—	4	2	no	—
54	C₂×Dic₁₀	—	4	1	no	—
55	C₂×Dic₁₀	—	4	1	no	—
56	C₅×Q₁₆	—	4	2	no	—
57	C₅×Q₁₆	—	4	2	no	—
58	C₅×D₈	—	4	2	no	—
59	C₅×D₈	—	4	2	no	—
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	4	2	no	—
63	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅	—	4	2	no	—
65	C₄⋊Dic₅	—	4	2	no	—
66	C₂×#168	D₈	1	4	yes	—
67	Dic₂₀	—	4	2	no	—
68	Dic₂₀	—	4	2	no	—
69	Dic₂₀	—	4	2	no	—
70	Dic₂₀	—	4	2	no	—
71	C₅×C₁₆	D₈	1	4	yes	—
72	#672	D₈	1	4	yes	—
73	C₂×Q₃₂	—	10	2	no	—
74	C₂×Q₃₂	—	10	2	no	—
75	C₂×SD₃₂	—	2	2	no	—
76	C₂×SD₃₂	—	10	2	no	—
77	#500	—	10	2	no	—
78	#494	—	5	1	no	—
79	#494	—	5	1	no	—
80	#494	C₂×D₅	1	1	yes	characteristic
81	#493	C₂×D₅	1	1	yes	characteristic
82	#486	—	10	2	no	—
83	C₄×C₁₆	—	5	1	no	—
84	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
85	C₈²	—	5	1	no	—
86	C₂³×C₅	—	8	1	no	—
87	C₅×Q₈	—	8	1	no	—
88	C₅×Q₈	—	8	1	no	—
89	C₅×D₄	—	8	1	no	—
90	C₅×D₄	—	8	1	no	—
91	C₅×D₄	—	8	4	no	—
92	C₂×C₄×C₅	—	8	1	no	—
93	C₂×C₄×C₅	#127	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
95	C₂×Dic₅	—	8	1	no	—
96	C₂×Dic₅	—	8	1	no	—
97	Dic₁₀	—	8	1	no	—
98	Dic₁₀	—	8	1	no	—
99	Dic₁₀	—	8	1	no	—
100	Dic₁₀	—	8	1	no	—
101	C₅×C₈	C₂×D₈	1	2	yes	—
102	C₅×C₈	C₂×D₈	1	2	yes	—
103	#168	—	2	4	no	—
104	C₂×Q₁₆	—	2	1	no	—
105	C₂×Q₁₆	—	2	1	no	—
106	C₂×Q₁₆	—	10	1	no	—
107	C₂×Q₁₆	—	10	1	no	—
108	C₂×Q₁₆	—	10	1	no	—
109	C₂×Q₁₆	—	10	1	no	—
110	C₂×D₈	—	2	1	no	—
111	C₂×D₈	—	2	1	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	10	1	no	—
114	C₄⋊Q₈	—	10	1	no	—
115	#127	—	2	1	no	—
116	Q₃₂	—	20	4	no	—
117	Q₃₂	—	20	4	no	—
118	SD₃₂	—	4	4	no	—
119	SD₃₂	—	20	4	no	—
120	C₂×C₁₆	—	5	2	no	—
121	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
122	#107	—	20	4	no	—
123	#102	—	20	4	no	—
124	C₄×C₈	—	5	2	no	—
125	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
126	C₂²×C₅	—	16	1	no	—
127	C₂²×C₅	—	16	2	no	—
128	C₂²×C₅	#493	1	1	yes	characteristic
129	C₄×C₅	—	16	1	no	—
130	C₄×C₅	#493	1	1	yes	characteristic
131	C₄×C₅	#493	1	1	yes	characteristic
132	C₄×C₅	#510	1	4	yes	—
133	Dic₅	—	16	1	no	—
134	Dic₅	—	16	1	no	—
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	20	1	no	—
137	C₂×Q₈	—	20	1	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂×D₄	—	4	1	no	—
140	Q₁₆	—	4	2	no	—
141	Q₁₆	—	4	2	no	—
142	Q₁₆	—	20	2	no	—
143	Q₁₆	—	20	2	no	—
144	Q₁₆	—	20	2	no	—
145	Q₁₆	—	20	2	no	—
146	D₈	—	4	2	no	—
147	D₈	—	4	2	no	—
148	C₂×C₈	—	5	4	no	—
149	C₂×C₈	D₄×D₅	1	1	yes	characteristic
150	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	20	2	no	—
153	C₄⋊C₄	—	20	2	no	—
154	C₄²	D₄×D₅	1	1	yes	characteristic
155	C₁₆	—	5	4	no	—
156	C₁₆	#686	1	4	yes	—
157	C₂×C₅	—	32	1	no	—
158	C₂×C₅	#1712	1	1	yes	characteristic
159	C₂×C₅	#2251	1	2	yes	—
160	C₂³	—	8	1	no	—
161	Q₈	—	8	1	no	—
162	Q₈	—	8	1	no	—
163	Q₈	—	40	1	no	—
164	Q₈	—	40	1	no	—
165	Q₈	—	40	1	no	—
166	Q₈	—	40	1	no	—
167	D₄	—	8	1	no	—
168	D₄	—	8	1	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	40	1	no	—
172	C₂×C₄	—	40	1	no	—
173	C₂×C₄	D₅×D₈	1	2	yes	—
174	C₂×C₄	#4086	1	1	yes	characteristic
175	C₈	—	10	4	no	—
176	C₈	D₅×D₈	1	2	yes	—
177	C₈	C₂×#686	1	2	yes	—
178	C₅	#12520	1	1	yes	characteristic, complemented, Sylow
179	C₂²	—	16	2	no	—
180	C₂²	—	16	1	no	—
181	C₂²	#65440	1	1	yes	characteristic
182	C₄	—	16	1	no	—
183	C₄	—	80	1	no	—
184	C₄	—	80	1	no	—
185	C₄	#65204	1	4	yes	—
186	C₄	#65362	1	1	yes	characteristic
187	C₄	#65440	1	1	yes	characteristic
188	C₂	—	32	1	no	—
189	C₂	#10602069	1	1	yes	characteristic
190	C₂	#10602275	1	2	yes	—

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2248	C₂
socle	C₂²×C₅	#493
prime radical	C₅×#493	C₂²
hypercenter	#2248	D₅
hypocenter	C₅	#12520
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
2. #10602105	C₂	semidirect
3. #10602091	C₂	semidirect
6. #10599859	C₂	semidirect
7. #10597989	C₂	semidirect
178. C₅	#12520	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2248	2
#10602105	2
#10602091	2
#10602084	2
#10602068	2
#10599859	2
#10597989	2
#12520	5