Subgroups — #12014667

Order 1280.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic
2	#10602212	C₂	1	2	yes	maximal
3	#10600572	C₂	1	1	yes	maximal, characteristic
4	C₂×#64681	C₂	1	1	yes	maximal, characteristic
5	#10598306	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3951	C₄	1	1	yes	characteristic
9	#65395	C₄	1	1	yes	characteristic
10	#65390	C₂²	1	2	yes	—
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₅×#364	C₄	1	4	yes	complemented
14	#64743	C₄	1	2	yes	—
15	#64722	—	2	2	no	—
16	C₁₆×Dic₅	—	2	2	no	—
17	#64681	C₄	1	4	yes	complemented
18	#12665	—	5	1	no	maximal, Sylow
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	#4072	C₂×C₄	1	2	yes	—
23	C₂×#681	C₂×C₄	1	2	yes	—
24	C₅×M₃₂	—	2	2	no	—
25	C₅×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₅×C₁₆	—	2	2	no	—
27	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	#3951	C₈	1	4	yes	—
30	#3950	C₈	1	2	yes	—
31	#3949	C₈	1	2	yes	—
32	#3947	—	2	2	no	—
33	C₈×Dic₅	—	2	2	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	—	2	1	no	—
36	C₂×#672	C₂×C₄	1	2	yes	—
37	C₂×#672	C₂×C₄	1	2	yes	—
38	C₂×#672	D₄	1	2	yes	—
39	C₂×#672	Q₈	1	2	yes	—
40	C₂×#364	D₅	1	1	yes	characteristic
41	#2108	—	5	2	no	—
42	C₂×#346	—	5	1	no	—
43	#1772	—	5	1	no	—
44	#1379	—	5	2	no	—
45	C₂²×C₄×C₅	—	2	1	no	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₄²	1	2	yes	—
48	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	1	yes	characteristic
56	C₂×C₅×C₈	#41	1	1	yes	characteristic
57	C₄²×C₅	—	2	1	no	—
58	C₄²×C₅	C₄²	1	2	yes	—
59	C₂²⋊Dic₅	—	2	2	no	—
60	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
61	C₄×Dic₅	—	2	2	no	—
62	#681	—	2	2	no	—
63	#681	C₂×C₈	1	4	yes	—
64	C₂×#168	—	2	2	no	—
65	C₅×C₁₆	—	2	4	no	—
66	#672	—	2	2	no	—
67	#672	C₄⋊C₄	1	4	yes	—
68	C₂×M₃₂	C₂×D₅	1	2	yes	—
69	C₂²×C₁₆	—	5	1	no	—
70	C₂²×C₁₆	—	5	1	no	—
71	C₂×#108	—	5	1	no	—
72	#427	—	5	1	no	—
73	#405	—	5	2	no	—
74	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
75	#364	Dic₅	1	4	yes	complemented
76	#346	—	5	4	no	—
77	#346	—	10	2	no	—
78	C₄×C₁₆	—	10	2	no	—
79	#339	—	5	2	no	—
80	C₂³×C₅	#131	1	1	yes	characteristic
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅	#131	1	1	yes	characteristic
89	C₂×Dic₅	—	2	4	no	—
90	C₅×C₈	C₄×C₈	1	2	yes	—
91	C₅×C₈	C₄×C₈	1	2	yes	—
92	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₅×C₈	#117	1	1	yes	characteristic
94	C₅×C₈	C₄×D₄	1	1	yes	characteristic
95	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
96	#168	—	2	4	no	—
97	C₂×M₁₆	—	5	2	no	—
98	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
100	#117	—	5	2	no	—
101	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	C₂×Dic₅	1	4	yes	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	5	2	no	—
106	C₂×C₁₆	—	5	2	no	—
107	C₂×C₁₆	—	5	2	no	—
108	C₂×C₁₆	—	5	2	no	—
109	C₂×C₁₆	—	10	1	no	—
110	C₂×C₁₆	—	10	1	no	—
111	#108	—	5	4	no	—
112	#107	—	5	2	no	—
113	#106	—	5	2	no	—
114	#97	—	10	2	no	—
115	C₄×C₈	—	10	2	no	—
116	C₄×C₈	C₄×D₅	1	2	yes	—
117	C₄×C₈	C₂×Dic₅	1	2	yes	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	#405	1	1	yes	characteristic
120	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
122	C₄×C₅	—	4	2	no	—
123	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₅	C₈×D₄	1	1	yes	characteristic
126	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
127	Dic₅	—	4	4	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
130	M₁₆	—	5	4	no	—
131	M₁₆	—	10	2	no	—
132	C₂×C₈	—	10	2	no	—
133	C₂×C₈	C₈×D₅	1	1	yes	characteristic
134	C₂×C₈	C₈×D₅	1	1	yes	characteristic
135	C₂×C₈	C₈×D₅	1	2	yes	—
136	C₂×C₈	C₄×Dic₅	1	2	yes	—
137	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₈	D₄×D₅	1	1	yes	characteristic
139	C₂×C₈	#711	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
141	C₂×C₈	#713	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
143	C₄⋊C₄	—	5	4	no	—
144	C₂²⋊C₄	—	10	2	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	10	2	no	—
147	C₄²	C₄×Dic₅	1	2	yes	—
148	C₁₆	—	2	4	no	—
149	C₁₆	—	5	4	no	—
150	C₁₆	—	10	2	no	—
151	C₂×C₅	—	2	1	no	—
152	C₂×C₅	—	2	1	no	—
153	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₅	#2171	1	2	yes	—
155	C₂³	#4053	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	10	4	no	—
159	C₂×C₄	#4033	1	1	yes	characteristic
160	C₂×C₄	#4039	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
162	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
163	C₂×C₄	#4053	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
165	C₈	—	10	4	no	—
166	C₈	C₈×Dic₅	1	2	yes	—
167	C₈	C₈×Dic₅	1	2	yes	—
168	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
169	C₈	#4038	1	1	yes	characteristic
170	C₈	D₄×Dic₅	1	1	yes	characteristic
171	C₈	Q₈×Dic₅	1	1	yes	characteristic
172	C₅	#12665	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	2	no	—
174	C₂²	#65119	1	1	yes	characteristic
175	C₂²	#65123	1	1	yes	characteristic
176	C₂²	#65411	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	20	4	no	—
179	C₄	#65119	1	1	yes	characteristic
180	C₄	#65123	1	1	yes	characteristic
181	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
182	C₄	C₂×C₈×Dic₅	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#10600094	1	2	yes	—
186	C₂	#10601258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂²×C₅	C₈×D₄
prime radical	C₂³×C₅	#131
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#364	D₅
hypocenter	C₅	#12665
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. C₅×#364	C₄	semidirect
17. #64681	C₄	semidirect
75. #364	Dic₅	semidirect
172. C₅	#12665	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#364	2
#10602212	2
#10600572	2
C₂×#64681	2
#10598306	2
#12665	5