Finite Group Databasebeta

Subgroups — #13023007

Order 1280.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₄×#3998	C₂	1	1	yes	maximal, characteristic
2	C₂×#64859	C₂	1	1	yes	maximal, characteristic
3	#10601533	C₂	1	1	yes	maximal, characteristic
4	#10598515	C₂	1	4	yes	maximal, complemented
5	C₂×#3998	C₄	1	2	yes	—
6	C₂²×#674	C₄	1	2	yes	—
7	C₅×#431	C₄	1	1	yes	characteristic
8	#65214	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	#64859	C₄	1	2	yes	—
11	#64859	C₂²	1	2	yes	—
12	#64859	C₂²	1	2	yes	—
13	C₄×#674	C₄	1	2	yes	—
14	#64674	C₂²	1	2	yes	—
15	C₈×#168	C₄	1	2	yes	—
16	#11406	—	5	1	no	maximal, Sylow
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×#168	C₂×C₄	1	1	yes	characteristic
19	C₂²×#168	C₂×C₄	1	1	yes	characteristic
20	#3998	C₈	1	8	yes	complemented
21	C₂×#674	—	2	4	no	—
22	C₂×#674	C₂×C₄	1	4	yes	—
23	C₂×#674	C₂×C₄	1	4	yes	—
24	C₅×C₄⋊C₈	—	2	1	no	—
25	C₅×C₂²⋊C₈	—	2	1	no	—
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²⋊#168	—	2	1	no	—
28	C₄⋊#168	—	2	1	no	—
29	C₄×#168	C₂×C₄	1	2	yes	—
30	C₄×#168	C₂×C₄	1	1	yes	characteristic
31	C₄×#168	C₂×C₄	1	1	yes	characteristic
32	C₄×#168	C₂×C₄	1	1	yes	characteristic
33	C₄×#168	C₂³	1	1	yes	characteristic
34	C₄×M₃₂	—	5	1	no	—
35	C₂×#346	—	5	1	no	—
36	#1450	—	5	1	no	—
37	#1313	—	5	4	no	—
38	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	—	2	2	no	—
42	C₄²×C₅	C₄²	1	1	yes	characteristic
43	C₄²×C₅	M₁₆	1	1	yes	characteristic
44	C₄²×C₅	M₁₆	1	1	yes	characteristic
45	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×#168	—	2	2	no	—
47	C₂×#168	C₄²	1	1	yes	characteristic
48	C₂×#168	C₄²	1	1	yes	characteristic
49	C₂×#168	C₂×C₈	1	4	yes	—
50	C₂×#168	M₁₆	1	4	yes	—
51	C₂×#168	C₂²×C₄	1	1	yes	characteristic
52	C₂×#168	C₂²×C₄	1	1	yes	characteristic
53	#674	—	2	4	no	—
54	#674	C₂×C₈	1	8	yes	—
55	C₂×M₃₂	—	5	2	no	—
56	C₂²×C₁₆	—	5	2	no	—
57	#431	—	5	1	no	—
58	#431	AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₄×C₈	—	5	1	no	—
60	#346	—	5	2	no	—
61	#346	—	5	2	no	—
62	#346	—	5	2	no	—
63	C₄×C₁₆	—	5	2	no	—
64	#322	—	5	2	no	—
65	C₈²	—	5	2	no	—
66	C₂³×C₅	#97	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	1	no	—
68	C₂×C₄×C₅	—	4	1	no	—
69	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₅	#97	1	2	yes	—
71	C₂×C₄×C₅	#97	1	1	yes	characteristic
72	C₂×C₄×C₅	#97	1	1	yes	characteristic
73	C₂×C₄×C₅	#97	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
77	C₅×C₈	—	4	2	no	—
78	#168	—	4	2	no	—
79	#168	C₄×C₈	1	2	yes	—
80	#168	#97	1	2	yes	—
81	#168	C₂²×C₈	1	2	yes	—
82	#168	C₂×M₁₆	1	2	yes	—
83	C₂²×C₈	—	5	1	no	—
84	C₂²×C₈	—	5	1	no	—
85	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	M₃₂	—	5	8	no	—
87	C₂×C₁₆	—	5	4	no	—
88	C₂×C₁₆	—	5	4	no	—
89	C₂×C₁₆	—	10	4	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	10	1	no	—
92	C₂²⋊C₈	—	2	1	no	—
93	C₂²⋊C₈	—	10	1	no	—
94	C₄×C₈	—	5	1	no	—
95	C₄×C₈	—	5	1	no	—
96	C₄×C₈	—	5	1	no	—
97	C₄×C₈	—	5	1	no	—
98	C₄×C₈	—	5	2	no	—
99	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
100	C₂²×C₅	—	2	1	no	—
101	C₂²×C₅	—	2	1	no	—
102	C₂²×C₅	#322	1	2	yes	—
103	C₂²×C₅	C₂×#97	1	1	yes	characteristic
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	—	2	1	no	—
108	C₄×C₅	#322	1	2	yes	—
109	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₅	C₂×#97	1	1	yes	characteristic
111	C₂²×C₄	#700	1	1	yes	characteristic
112	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂²×C₄	#704	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	5	1	no	—
116	C₂×C₈	—	5	1	no	—
117	C₂×C₈	—	5	1	no	—
118	C₂×C₈	—	5	1	no	—
119	C₂×C₈	—	5	4	no	—
120	C₂×C₈	—	5	4	no	—
121	C₂×C₈	—	10	2	no	—
122	C₄²	#700	1	1	yes	characteristic
123	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₄²	#704	1	1	yes	characteristic
125	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₁₆	—	5	8	no	—
127	C₁₆	—	10	4	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	C₂×#322	1	1	yes	characteristic
130	C₂×C₅	#2110	1	1	yes	characteristic
131	C₂×C₅	#2110	1	1	yes	characteristic
132	C₂³	#4002	1	1	yes	characteristic
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
136	C₂×C₄	#3992	1	2	yes	—
137	C₂×C₄	#4002	1	1	yes	characteristic
138	C₂×C₄	#4002	1	1	yes	characteristic
139	C₂×C₄	#4002	1	1	yes	characteristic
140	C₂×C₄	C₂×#700	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₄	C₂×#704	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	5	2	no	—
145	C₈	—	5	2	no	—
146	C₈	—	5	2	no	—
147	C₈	—	5	2	no	—
148	C₈	—	20	2	no	—
149	C₅	#11406	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#170⋊C₈	1	2	yes	—
153	C₂²	C₂×#4002	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#170⋊C₈	1	2	yes	—
159	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₄	C₂×#3992	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#10604369	1	1	yes	characteristic
163	C₂	C₂×#170⋊C₈	1	1	yes	characteristic
164	C₂	#10604748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
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	#11406
LCS	G ▷ 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
20. #3998	C₈	semidirect
149. C₅	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3998	2
C₂×#64859	2
#10601533	2
#10598515	2
#11406	5