Finite Group Databasebeta

Subgroups — #13024821

Order 1280.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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	#10615068	C₂	1	1	yes	maximal, characteristic
2	#10604155	C₂	1	2	yes	maximal
3	#10598737	C₂	1	2	yes	maximal, complemented
4	#10598736	C₂	1	2	yes	maximal, complemented
5	C₅×#533	C₄	1	1	yes	characteristic
6	C₂×#4015	C₄	1	1	yes	characteristic
7	#65382	—	2	2	no	—
8	#65358	C₄	1	1	yes	characteristic
9	#65358	C₄	1	1	yes	characteristic
10	#65358	C₂²	1	1	yes	characteristic
11	#65358	C₂²	1	1	yes	characteristic
12	#65293	—	2	2	no	—
13	C₄×#681	C₂²	1	1	yes	characteristic
14	#64912	—	2	2	no	—
15	#64861	—	2	2	no	—
16	#64860	C₂²	1	4	yes	—
17	#11916	—	5	1	no	maximal, Sylow
18	C₂²×Dic₁₀	—	2	1	no	—
19	C₅×C₄⋊Q₈	—	2	1	no	—
20	C₅×#125	C₂×C₄	1	1	yes	characteristic
21	C₅×#125	C₂×C₄	1	1	yes	characteristic
22	C₅×C₂²⋊Q₈	—	4	1	no	—
23	C₄×C₅×Q₈	—	4	1	no	—
24	C₅×#117	—	2	1	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅	—	4	1	no	—
27	C₂×#681	D₄	1	2	yes	—
28	#4015	—	2	1	no	—
29	#4015	—	4	1	no	—
30	#4015	C₂×C₄	1	1	yes	characteristic
31	#4015	C₂×C₄	1	1	yes	characteristic
32	#3998	—	2	4	no	—
33	#3942	—	2	2	no	—
34	#3942	—	2	2	no	—
35	#3942	—	2	2	no	—
36	#3942	—	2	2	no	—
37	#3935	C₂×C₄	1	1	yes	characteristic
38	#3935	C₂³	1	1	yes	characteristic
39	C₄×#168	D₄	1	2	yes	—
40	#3139	—	5	1	no	—
41	#2165	—	5	2	no	—
42	#1356	—	5	2	no	—
43	#1356	—	5	2	no	—
44	C₂×C₅×Q₈	—	4	1	no	—
45	C₂²×C₄×C₅	—	2	1	no	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	4	1	no	—
48	C₂×Dic₁₀	—	2	1	no	—
49	C₂×Dic₁₀	—	2	1	no	—
50	C₂×Dic₁₀	—	4	4	no	—
51	C₂×Dic₁₀	—	4	1	no	—
52	C₅×C₄⋊C₄	—	2	1	no	—
53	C₅×C₄⋊C₄	—	2	1	no	—
54	C₅×C₄⋊C₄	—	4	1	no	—
55	C₅×C₄⋊C₄	—	4	1	no	—
56	C₅×C₄⋊C₄	—	4	1	no	—
57	C₅×C₂²⋊C₄	—	4	1	no	—
58	C₄²×C₅	—	2	1	no	—
59	C₄²×C₅	—	4	1	no	—
60	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₄⋊Dic₅	—	4	2	no	—
63	C₄⋊Dic₅	—	4	1	no	—
64	C₄⋊Dic₅	—	4	1	no	—
65	#681	—	4	2	no	—
66	C₂×#168	C₂²⋊C₄	1	2	yes	—
67	C₂×#168	C₂×D₄	1	2	yes	—
68	#674	—	4	4	no	—
69	#533	AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	5	1	no	—
71	#491	—	5	1	no	—
72	#491	—	5	1	no	—
73	#491	—	5	1	no	—
74	#491	—	5	1	no	—
75	#487	—	10	2	no	—
76	#419	—	10	2	no	—
77	C₄×M₁₆	—	5	1	no	—
78	#364	—	10	2	no	—
79	#349	—	10	2	no	—
80	#347	—	5	4	no	—
81	C₂³×C₅	#100	1	1	yes	characteristic
82	C₅×Q₈	—	8	2	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	1	no	—
88	C₂×C₄×C₅	—	4	1	no	—
89	C₂×C₄×C₅	#100	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₅	—	4	1	no	—
92	C₂×Dic₅	—	4	1	no	—
93	C₂×Dic₅	—	8	2	no	—
94	Dic₁₀	—	4	2	no	—
95	Dic₁₀	—	4	2	no	—
96	Dic₁₀	—	8	2	no	—
97	#168	—	2	2	no	—
98	#168	—	2	2	no	—
99	C₂²×Q₈	—	10	1	no	—
100	C₂×M₁₆	—	5	2	no	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	5	1	no	—
103	C₄⋊Q₈	—	5	1	no	—
104	C₄⋊Q₈	—	10	1	no	—
105	C₄⋊Q₈	—	20	1	no	—
106	#125	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
107	#125	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₂²⋊Q₈	—	4	1	no	—
109	C₄×Q₈	—	4	1	no	—
110	#117	—	2	1	no	—
111	C₂×C₄⋊C₄	—	20	1	no	—
112	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
113	M₃₂	—	10	4	no	—
114	#103	—	10	2	no	—
115	#103	—	10	2	no	—
116	#103	—	10	2	no	—
117	#103	—	10	2	no	—
118	#97	—	5	1	no	—
119	#97	—	5	1	no	—
120	C₄×C₈	—	5	2	no	—
121	C₂²×C₅	—	2	1	no	—
122	C₂²×C₅	—	4	1	no	—
123	C₂²×C₅	C₂×#100	1	1	yes	characteristic
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	1	no	—
126	C₄×C₅	—	4	1	no	—
127	C₄×C₅	—	8	1	no	—
128	C₄×C₅	—	8	1	no	—
129	C₄×C₅	#421	1	2	yes	—
130	Dic₅	—	8	1	no	—
131	Dic₅	—	8	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	10	1	no	—
134	C₂×Q₈	—	10	1	no	—
135	C₂×Q₈	—	20	4	no	—
136	C₂×Q₈	—	20	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	20	1	no	—
139	C₂²×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
140	M₁₆	—	20	2	no	—
141	C₂×C₈	—	5	2	no	—
142	C₂×C₈	—	5	2	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	20	1	no	—
149	C₄⋊C₄	—	20	1	no	—
150	C₄⋊C₄	—	20	2	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
155	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₁₆	—	20	4	no	—
157	C₂×C₅	—	2	1	no	—
158	C₂×C₅	—	4	1	no	—
159	C₂×C₅	#1533	1	1	yes	characteristic
160	C₂³	#4013	1	1	yes	characteristic
161	Q₈	—	8	2	no	—
162	Q₈	—	20	2	no	—
163	Q₈	—	20	2	no	—
164	Q₈	—	40	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	20	1	no	—
172	C₂×C₄	—	20	1	no	—
173	C₂×C₄	—	40	2	no	—
174	C₂×C₄	#4013	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₈	—	10	2	no	—
177	C₈	—	10	2	no	—
178	C₅	#11916	1	1	yes	characteristic, complemented, Sylow
179	C₂²	—	2	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	C₂×#4013	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	8	1	no	—
186	C₄	—	8	1	no	—
187	C₄	—	40	1	no	—
188	C₄	—	40	1	no	—
189	C₄	#65784	1	2	yes	—
190	C₂	—	2	1	no	—
191	C₂	—	4	1	no	—
192	C₂	#10604961	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10604961
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#533	C₄
socle	C₂×C₅	#1533
prime radical	C₂³×C₅	#100
cosocle	#3935	C₂³
hypercenter	#533	AGL(1,𝔽₅)
hypocenter	C₅	#11916
LCS	G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
3. #10598737	C₂	semidirect
4. #10598736	C₂	semidirect
178. C₅	#11916	semidirect

Maximal subgroups

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

subgroup	index
#10615068	2
#10604155	2
#10598737	2
#10598736	2
#11916	5