Finite Group Databasebeta

Subgroups — #12010993

Order 1280.

214 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₅×#335	C₂	1	1	yes	maximal, characteristic
2	#10600537	C₂	1	1	yes	maximal, characteristic
3	#10600525	C₂	1	1	yes	maximal, characteristic
4	#10597961	C₂	1	2	yes	maximal, complemented
5	#10597960	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	#65215	—	2	1	no	—
9	C₅×#335	—	2	1	no	—
10	C₅×#335	C₂²	1	2	yes	—
11	C₂²⋊#672	C₄	1	2	yes	—
12	C₄⋊#672	C₄	1	2	yes	—
13	C₄×#672	C₂²	1	2	yes	—
14	#64678	C₂²	1	1	yes	characteristic
15	#64678	C₂²	1	1	yes	characteristic
16	#64677	—	2	1	no	—
17	#12035	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	—	4	1	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₅×C₄⋊C₈	—	2	2	no	—
23	C₅×C₄⋊C₈	—	4	1	no	—
24	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×C₈	D₄	1	1	yes	characteristic
26	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²⋊#168	—	4	1	no	—
29	C₂×#672	C₂×C₄	1	4	yes	—
30	C₄⋊#168	—	2	1	no	—
31	C₄⋊#168	—	2	1	no	—
32	#3935	—	4	1	no	—
33	#2162	—	5	1	no	—
34	#1568	—	5	1	no	—
35	C₂×#335	D₅	1	1	yes	characteristic
36	#1372	—	5	2	no	—
37	#1371	—	5	2	no	—
38	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	4	1	no	—
43	C₂×C₅×C₈	—	4	2	no	—
44	C₂×C₅×C₈	—	8	2	no	—
45	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅	D₈	1	1	yes	characteristic
51	C₄²×C₅	Q₁₆	1	1	yes	characteristic
52	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×#168	—	4	1	no	—
54	C₂×#168	—	4	1	no	—
55	#672	—	2	4	no	—
56	#432	—	10	1	no	—
57	C₂×C₄⋊C₈	—	2	1	no	—
58	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
59	C₄⋊C₁₆	—	5	2	no	—
60	C₂²⋊C₁₆	—	5	2	no	—
61	C₄×C₁₆	—	5	2	no	—
62	#335	—	2	1	no	—
63	#335	—	5	1	no	—
64	#335	—	5	1	no	—
65	#335	C₂×D₅	1	2	yes	—
66	#334	—	10	1	no	—
67	C₂³×C₅	#107	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	1	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	—	2	1	no	—
72	C₂×C₄×C₅	—	2	1	no	—
73	C₂×C₄×C₅	#107	1	1	yes	characteristic
74	C₂×C₄×C₅	#107	1	1	yes	characteristic
75	C₂×C₄×C₅	#107	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
79	C₅×C₈	—	2	1	no	—
80	C₅×C₈	—	2	1	no	—
81	C₅×C₈	—	2	1	no	—
82	C₅×C₈	—	8	2	no	—
83	C₅×C₈	#131	1	2	yes	—
84	#168	—	8	1	no	—
85	#168	—	8	1	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
88	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
89	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
90	C₂×C₁₆	—	5	4	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	10	1	no	—
94	C₄⋊C₈	—	10	1	no	—
95	C₂²⋊C₈	—	20	1	no	—
96	#97	—	20	1	no	—
97	C₄×C₈	Dic₁₀	1	1	yes	characteristic
98	C₄×C₈	C₄×D₅	1	1	yes	characteristic
99	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	—	2	1	no	—
103	C₂²×C₅	#368	1	2	yes	—
104	C₂²×C₅	C₂×#107	1	1	yes	characteristic
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	—	2	1	no	—
109	C₄×C₅	#424	1	1	yes	characteristic
110	C₄×C₅	#430	1	1	yes	characteristic
111	C₄×C₅	#509	1	1	yes	characteristic
112	C₄×C₅	#510	1	1	yes	characteristic
113	C₂²×C₄	#683	1	1	yes	characteristic
114	C₂²×C₄	#686	1	1	yes	characteristic
115	C₂²×C₄	#689	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	—	20	1	no	—
121	C₂×C₈	—	20	1	no	—
122	C₂×C₈	#683	1	1	yes	characteristic
123	C₂×C₈	#683	1	1	yes	characteristic
124	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
126	C₄²	#683	1	1	yes	characteristic
127	C₄²	#686	1	1	yes	characteristic
128	C₄²	#689	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
130	C₁₆	—	10	4	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	#1571	1	1	yes	characteristic
133	C₂×C₅	#2159	1	1	yes	characteristic
134	C₂×C₅	C₂×#368	1	1	yes	characteristic
135	C₂³	#3939	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	#3939	1	1	yes	characteristic
142	C₂×C₄	#3939	1	1	yes	characteristic
143	C₂×C₄	#3939	1	1	yes	characteristic
144	C₂×C₄	C₂×#683	1	1	yes	characteristic
145	C₂×C₄	C₂×#686	1	1	yes	characteristic
146	C₂×C₄	C₂×#689	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	2	no	—
151	C₈	—	40	1	no	—
152	C₈	—	40	1	no	—
153	C₈	#4047	1	2	yes	—
154	C₅	#12035	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#64713	1	2	yes	—
158	C₂²	C₂×#3939	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#65391	1	1	yes	characteristic
164	C₄	#65413	1	1	yes	characteristic
165	C₄	#65436	1	1	yes	characteristic
166	C₄	#65468	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#10601215	1	1	yes	characteristic
169	C₂	#10601389	1	1	yes	characteristic
170	C₂	C₂×#64713	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×#335	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	#107
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×#335	D₅
hypocenter	C₅	#12035
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #10597961	C₂	semidirect
5. #10597960	C₂	semidirect
154. C₅	#12035	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#335	2
#10600537	2
#10600525	2
#10597961	2
#10597960	2
#12035	5