Subgroups — #12251950

Order 1280.

818 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#65429	C₂	1	2	yes	maximal
3	#10600674	C₂	1	6	yes	maximal, complemented
4	C₄×#3944	C₂	1	3	yes	maximal, complemented
5	C₂×#64681	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₄×C₅×#41	C₄	1	1	yes	characteristic
9	C₅×#405	C₄	1	3	yes	—
10	C₅×#405	C₂²	1	3	yes	—
11	C₄×C₅×M₁₆	C₄	1	3	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
13	#65429	C₄	1	16	yes	complemented
14	C₂×#3944	C₂²	1	6	yes	—
15	C₂²×#672	C₂²	1	6	yes	—
16	#64681	C₂²	1	1	yes	characteristic
17	#64681	C₂²	1	3	yes	—
18	#64681	C₂²	1	6	yes	—
19	C₄×#672	C₂²	1	6	yes	—
20	#33326	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
22	C₅×#131	C₈	1	8	yes	—
23	C₅×#131	C₂×C₄	1	8	yes	—
24	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
25	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₅×C₈	C₂³	1	3	yes	—
28	C₄×C₅×Q₈	C₈	1	2	yes	—
29	C₄×C₅×D₄	C₈	1	6	yes	—
30	C₅×#117	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
32	C₅×#97	C₂×C₄	1	6	yes	—
33	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
34	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂³	1	3	yes	—
37	#3944	C₂×C₄	1	24	yes	complemented
38	C₂×#672	C₂×C₄	1	24	yes	complemented
39	C₂×#672	C₂³	1	2	yes	—
40	C₂×#672	C₂³	1	6	yes	—
41	C₂×#504	—	5	2	no	—
42	C₄×#131	D₅	1	1	yes	characteristic
43	#2108	—	5	6	no	—
44	C₄×M₃₂	—	5	3	no	—
45	C₂×#346	—	5	3	no	—
46	C₅×#41	C₄²	1	4	yes	—
47	C₅×#41	C₂×C₈	1	4	yes	—
48	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
51	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
52	C₅×M₁₆	C₄²	1	12	yes	—
53	C₅×M₁₆	C₂×C₈	1	12	yes	—
54	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	12	yes	—
56	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₂×C₈	1	6	yes	—
61	C₅×C₂²⋊C₄	C₂×C₈	1	6	yes	—
62	C₄²×C₅	C₂×C₈	1	6	yes	—
63	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅	C₂²×C₄	1	3	yes	—
65	#672	C₂²×C₄	1	4	yes	—
66	#672	C₂²×C₄	1	12	yes	—
67	C₂×#131	Dic₅	1	1	yes	characteristic
68	C₂×#131	C₂×D₅	1	1	yes	characteristic
69	C₄×#41	Dic₅	1	1	yes	characteristic
70	#504	—	5	16	no	—
71	C₂×M₃₂	—	5	6	no	—
72	C₂²×C₁₆	—	5	6	no	—
73	#405	Dic₅	1	3	yes	—
74	#405	C₂×D₅	1	3	yes	—
75	C₄×M₁₆	Dic₅	1	3	yes	—
76	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
77	#346	—	5	1	no	—
78	#346	—	5	3	no	—
79	#346	—	5	6	no	—
80	C₄×C₁₆	—	5	6	no	—
81	C₂³×C₅	C₂²×C₈	1	3	yes	—
82	C₅×Q₈	C₄×C₈	1	4	yes	—
83	C₅×D₄	C₄×C₈	1	12	yes	—
84	C₂×C₄×C₅	—	4	6	no	—
85	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
86	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
89	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
90	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
91	C₅×C₈	C₂×C₄²	1	6	yes	—
92	C₅×C₈	C₂²×C₈	1	2	yes	—
93	C₅×C₈	C₂²×C₈	1	6	yes	—
94	C₅×C₈	C₂³×C₄	1	2	yes	—
95	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
96	#131	#168	1	8	yes	—
97	#131	C₄×D₅	1	8	yes	—
98	C₂×M₁₆	C₂×Dic₅	1	3	yes	—
99	C₂×M₁₆	C₂×Dic₅	1	3	yes	—
100	C₂²×C₈	C₂×Dic₅	1	3	yes	—
101	C₂²×C₈	C₂²×D₅	1	3	yes	—
102	C₄×Q₈	#168	1	2	yes	—
103	C₄×D₄	#168	1	6	yes	—
104	#117	C₂×Dic₅	1	3	yes	—
105	C₂×C₄²	C₂×Dic₅	1	3	yes	—
106	M₃₂	—	5	24	no	—
107	C₂×C₁₆	—	5	24	no	—
108	C₂×C₁₆	—	5	6	no	—
109	C₂×C₁₆	—	5	2	no	—
110	#97	C₂×Dic₅	1	6	yes	—
111	C₄×C₈	C₂×Dic₅	1	3	yes	—
112	C₄×C₈	C₂×Dic₅	1	3	yes	—
113	C₄×C₈	C₂²×D₅	1	3	yes	—
114	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
115	C₂²×C₅	—	2	6	no	—
116	C₂²×C₅	C₂×C₄×C₈	1	6	yes	—
117	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	—	2	6	no	—
120	C₄×C₅	C₂×C₄×C₈	1	6	yes	—
121	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
122	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
123	#41	C₂×#168	1	4	yes	—
124	#41	C₄×Dic₅	1	4	yes	—
125	C₂×Q₈	C₂×#168	1	1	yes	characteristic
126	C₂×D₄	C₂×#168	1	3	yes	—
127	C₂²×C₄	C₂×#168	1	6	yes	—
128	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
129	M₁₆	C₂×#168	1	12	yes	—
130	M₁₆	C₄×Dic₅	1	12	yes	—
131	C₂×C₈	C₂×#168	1	12	yes	—
132	C₂×C₈	C₂×C₄×D₅	1	12	yes	—
133	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
134	C₂×C₈	C₂²×Dic₅	1	3	yes	—
135	C₂×C₈	C₂²×Dic₅	1	3	yes	—
136	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
137	C₄⋊C₄	C₂×#168	1	6	yes	—
138	C₂²⋊C₄	C₂×#168	1	6	yes	—
139	C₄²	C₂×#168	1	6	yes	—
140	C₄²	C₂²×Dic₅	1	1	yes	characteristic
141	C₄²	C₂²×Dic₅	1	3	yes	—
142	C₁₆	—	5	4	no	—
143	C₁₆	—	5	12	no	—
144	C₂×C₅	—	2	6	no	—
145	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
146	C₂×C₅	#3407	1	2	yes	—
147	C₂³	C₂²×#168	1	3	yes	—
148	Q₈	C₄×#168	1	4	yes	—
149	D₄	C₄×#168	1	12	yes	—
150	C₂×C₄	—	4	6	no	—
151	C₂×C₄	C₂²×#168	1	2	yes	—
152	C₂×C₄	C₂²×#168	1	6	yes	—
153	C₂×C₄	C₂²×#168	1	6	yes	—
154	C₂×C₄	C₂²×#168	1	3	yes	—
155	C₂×C₄	C₂×C₄×Dic₅	1	6	yes	—
156	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
157	C₈	C₂²×#168	1	2	yes	—
158	C₈	C₂²×#168	1	6	yes	—
159	C₈	C₂×C₄×Dic₅	1	6	yes	—
160	C₈	C₂²×C₄×D₅	1	2	yes	—
161	C₅	#33326	1	1	yes	characteristic, complemented, Sylow
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#168	1	6	yes	—
164	C₂²	C₂³×#168	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×#168	1	6	yes	—
168	C₄	C₂³×#168	1	1	yes	characteristic
169	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#168	1	1	yes	characteristic
172	C₂	#10615511	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Fitting	C₄×C₅×#131	C₂
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#33326
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600674	C₂	semidirect
4. C₄×#3944	C₂	semidirect
5. C₂×#64681	C₂	semidirect
13. #65429	C₄	semidirect
37. #3944	C₂×C₄	semidirect
38. C₂×#672	C₂×C₄	semidirect
161. C₅	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
C₂×#65429	2
#10600674	2
C₄×#3944	2
C₂×#64681	2
#33326	5