Finite Group Databasebeta

Subgroups — #13023063

Order 1280.

248 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 108 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	#10604152	C₂	1	1	yes	maximal, characteristic
2	C₄×#3998	C₂	1	1	yes	maximal, characteristic
3	#10601534	C₂	1	1	yes	maximal, characteristic
4	#10598515	C₂	1	2	yes	maximal, complemented
5	#10598515	C₂	1	2	yes	maximal, complemented
6	C₂×#3998	C₄	1	2	yes	—
7	C₅×#431	C₄	1	1	yes	characteristic
8	C₄×#681	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	C₂²⋊#674	C₄	1	2	yes	—
11	#64859	C₂²	1	1	yes	characteristic
12	#64859	C₂²	1	1	yes	characteristic
13	#64859	C₂²	1	2	yes	—
14	C₄⋊#674	C₄	1	2	yes	—
15	C₄×#674	C₄	1	2	yes	—
16	#64675	C₄	1	2	yes	—
17	#64674	C₂²	1	2	yes	—
18	#11414	—	5	1	no	maximal, Sylow
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×#681	—	2	1	no	—
21	C₂²×#168	C₂×C₄	1	1	yes	characteristic
22	C₂²×#168	C₂×C₄	1	1	yes	characteristic
23	#3998	—	2	4	no	—
24	C₂×#674	C₂×C₄	1	2	yes	—
25	C₂×#674	C₂×C₄	1	4	yes	—
26	C₂×#674	C₂×C₄	1	2	yes	—
27	C₅×C₄⋊C₈	—	2	1	no	—
28	C₅×C₂²⋊C₈	—	2	1	no	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	#3935	—	2	1	no	—
31	C₄×#168	C₂×C₄	1	2	yes	—
32	C₄×#168	C₂×C₄	1	1	yes	characteristic
33	C₄×#168	C₂×C₄	1	1	yes	characteristic
34	C₄×#168	C₂×C₄	1	1	yes	characteristic
35	C₄×#168	C₂³	1	1	yes	characteristic
36	#2163	—	5	1	no	—
37	C₄×M₃₂	—	5	1	no	—
38	#1451	—	5	1	no	—
39	#1313	—	5	2	no	—
40	#1313	—	5	2	no	—
41	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
44	C₂×C₅×C₈	—	2	2	no	—
45	C₄²×C₅	C₄²	1	1	yes	characteristic
46	C₄²×C₅	M₁₆	1	1	yes	characteristic
47	C₄²×C₅	M₁₆	1	1	yes	characteristic
48	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
49	#681	—	4	2	no	—
50	C₂×#168	—	2	2	no	—
51	C₂×#168	—	2	2	no	—
52	C₂×#168	—	2	2	no	—
53	C₂×#168	C₄²	1	1	yes	characteristic
54	C₂×#168	C₄²	1	1	yes	characteristic
55	C₂×#168	C₂²×C₄	1	1	yes	characteristic
56	C₂×#168	C₂²×C₄	1	1	yes	characteristic
57	#674	—	2	2	no	—
58	#674	—	2	4	no	—
59	#674	M₁₆	1	4	yes	complemented
60	C₂×M₃₂	—	5	2	no	—
61	#431	AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₄×M₁₆	—	5	1	no	—
63	C₂×C₄×C₈	—	5	1	no	—
64	C₄⋊C₁₆	—	5	2	no	—
65	C₂²⋊C₁₆	—	5	2	no	—
66	#346	—	5	1	no	—
67	#346	—	5	1	no	—
68	#346	—	5	2	no	—
69	C₄×C₁₆	—	5	2	no	—
70	#322	—	5	2	no	—
71	#322	—	5	2	no	—
72	C₂³×C₅	#97	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	4	1	no	—
77	C₂×C₄×C₅	#97	1	1	yes	characteristic
78	C₂×C₄×C₅	#97	1	1	yes	characteristic
79	C₂×C₄×C₅	#97	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
83	C₅×C₈	—	4	2	no	—
84	#168	—	2	1	no	—
85	#168	—	2	1	no	—
86	#168	—	4	2	no	—
87	#168	C₂×M₁₆	1	2	yes	—
88	#168	#131	1	2	yes	—
89	C₂×M₁₆	—	10	1	no	—
90	C₂²×C₈	—	5	1	no	—
91	C₂²×C₈	—	5	1	no	—
92	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	M₃₂	—	10	4	no	—
94	C₂×C₁₆	—	5	2	no	—
95	C₂×C₁₆	—	5	2	no	—
96	C₂×C₁₆	—	5	4	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	#97	—	10	1	no	—
100	C₄×C₈	—	5	1	no	—
101	C₄×C₈	—	5	1	no	—
102	C₄×C₈	—	5	1	no	—
103	C₄×C₈	—	5	1	no	—
104	C₄×C₈	—	5	2	no	—
105	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	C₂²×C₅	—	2	1	no	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	C₂×#97	1	1	yes	characteristic
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	—	2	1	no	—
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	#347	1	2	yes	—
115	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₅	#405	1	1	yes	characteristic
117	C₂²×C₄	#700	1	1	yes	characteristic
118	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂²×C₄	#704	1	1	yes	characteristic
120	M₁₆	—	20	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	5	1	no	—
123	C₂×C₈	—	5	1	no	—
124	C₂×C₈	—	5	1	no	—
125	C₂×C₈	—	5	1	no	—
126	C₂×C₈	—	10	2	no	—
127	C₂×C₈	—	10	2	no	—
128	C₂×C₈	—	10	2	no	—
129	C₄²	#700	1	1	yes	characteristic
130	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₄²	#704	1	1	yes	characteristic
132	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₁₆	—	5	4	no	—
134	C₁₆	—	10	2	no	—
135	C₁₆	—	10	4	no	—
136	C₂×C₅	—	4	1	no	—
137	C₂×C₅	#1451	1	1	yes	characteristic
138	C₂×C₅	C₂×#347	1	1	yes	characteristic
139	C₂×C₅	#2110	1	1	yes	characteristic
140	C₂³	#4002	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	#4002	1	1	yes	characteristic
146	C₂×C₄	#4002	1	1	yes	characteristic
147	C₂×C₄	#4002	1	1	yes	characteristic
148	C₂×C₄	C₂×#700	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₄	C₂×#704	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₈	—	5	2	no	—
153	C₈	—	5	2	no	—
154	C₈	—	10	1	no	—
155	C₈	—	10	1	no	—
156	C₈	—	20	2	no	—
157	C₅	#11414	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂×#4002	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	#64890	1	2	yes	—
167	C₄	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₄	#65728	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#10604369	1	1	yes	characteristic
171	C₂	C₂×#64890	1	1	yes	characteristic
172	C₂	#10604735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#431	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	#97
cosocle	C₄×#168	C₂³
hypercenter	#431	AGL(1,𝔽₅)
hypocenter	C₅	#11414
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
4. #10598515	C₂	semidirect
5. #10598515	C₂	semidirect
59. #674	M₁₆	semidirect
157. C₅	#11414	semidirect

Maximal subgroups

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

subgroup	index
#10604152	2
C₄×#3998	2
#10601534	2
#10598515	2
#11414	5