Subgroups — #12015715

Order 1280.

382 conjugacy classes of subgroups.

The subgroup lattice has 220 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	C₂×#64731	C₂	1	2	yes	maximal
3	#10602213	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64683	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	C₄²×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₅×#364	—	2	2	no	—
16	#64731	—	2	4	no	—
17	#64721	—	2	4	no	—
18	#64683	—	2	2	no	—
19	#12900	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	—	2	2	no	—
24	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×#168	C₂×C₄	1	1	yes	characteristic
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	C₅×M₃₂	—	2	2	no	—
29	C₅×M₃₂	—	2	2	no	—
30	C₂×C₅×C₁₆	—	2	2	no	—
31	C₄×C₅×C₈	D₄	1	2	yes	—
32	C₄×C₅×C₈	Q₈	1	2	yes	—
33	C₈×Dic₅	—	2	2	no	—
34	C₈×Dic₅	C₈	1	2	yes	—
35	C₈×Dic₅	C₈	1	2	yes	—
36	C₈×Dic₅	D₄	1	2	yes	—
37	C₈×Dic₅	D₄	1	2	yes	—
38	C₈×Dic₅	Q₈	1	2	yes	—
39	C₈×Dic₅	Q₈	1	2	yes	—
40	#3944	—	2	2	no	—
41	#3944	—	2	2	no	—
42	C₂×#672	—	2	2	no	—
43	C₄×#168	—	2	2	no	—
44	C₂×#364	—	5	2	no	—
45	C₂×#364	—	5	1	no	—
46	C₂×#364	D₅	1	1	yes	characteristic
47	#2150	—	5	2	no	—
48	C₄²×C₈	—	5	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅	—	2	1	no	—
52	C₂²×Dic₅	M₁₆	1	2	yes	—
53	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₅×C₈	M₁₆	1	2	yes	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅	—	2	1	no	—
63	C₄²×C₅	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₅	—	2	8	no	—
65	C₄×Dic₅	—	2	1	no	—
66	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
67	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
68	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
69	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
70	C₂×#168	—	2	1	no	—
71	C₂×#168	—	2	1	no	—
72	C₂×#168	C₄⋊C₄	1	2	yes	—
73	C₂×#168	C₄⋊C₄	1	2	yes	—
74	C₂×#168	C₂×C₈	1	2	yes	—
75	C₂×#168	M₁₆	1	2	yes	—
76	C₅×C₁₆	—	4	2	no	—
77	#672	—	4	2	no	—
78	C₂×M₃₂	—	5	2	no	—
79	C₂×M₃₂	C₂×D₅	1	2	yes	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	—	5	1	no	—
83	C₂×C₄×C₈	—	5	1	no	—
84	C₂×C₄×C₈	—	5	1	no	—
85	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
86	C₄³	—	5	1	no	—
87	#364	—	2	2	no	—
88	#364	—	10	4	no	—
89	#364	—	10	2	no	—
90	C₄⋊C₁₆	—	10	4	no	—
91	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅	—	2	4	no	—
101	C₂×Dic₅	—	2	2	no	—
102	C₂×Dic₅	—	2	4	no	—
103	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
104	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
105	C₅×C₈	C₄⋊C₈	1	4	yes	—
106	C₅×C₈	#127	1	1	yes	characteristic
107	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
108	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
109	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
110	#168	—	2	2	no	—
111	#168	C₄⋊C₈	1	4	yes	—
112	C₂²×C₈	—	5	1	no	—
113	C₂²×C₈	—	5	1	no	—
114	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
116	C₂×C₄²	—	5	1	no	—
117	C₂×C₄²	—	5	1	no	—
118	C₂×C₄²	—	10	2	no	—
119	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	10	2	no	—
123	M₃₂	—	10	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	10	2	no	—
126	C₄×C₈	—	5	2	no	—
127	C₄×C₈	—	5	2	no	—
128	C₄×C₈	—	5	2	no	—
129	C₄×C₈	—	5	2	no	—
130	C₄×C₈	—	5	2	no	—
131	C₄×C₈	—	5	2	no	—
132	C₄×C₈	—	10	2	no	—
133	C₄×C₈	—	10	2	no	—
134	C₄×C₈	Dic₁₀	1	2	yes	—
135	C₄×C₈	C₂²⋊D₅	1	2	yes	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₅	#423	1	1	yes	characteristic
139	C₂²×C₅	#423	1	1	yes	characteristic
140	C₄×C₅	—	2	4	no	—
141	C₄×C₅	#384	1	1	yes	characteristic
142	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₄×C₅	#423	1	1	yes	characteristic
145	Dic₅	—	2	4	no	—
146	Dic₅	—	2	2	no	—
147	Dic₅	#364	1	4	yes	complemented
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	5	2	no	—
150	C₂²×C₄	—	10	1	no	—
151	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	5	2	no	—
154	C₂×C₈	—	5	2	no	—
155	C₂×C₈	—	5	2	no	—
156	C₂×C₈	—	10	1	no	—
157	C₂×C₈	—	10	1	no	—
158	C₂×C₈	C₈×D₅	1	2	yes	—
159	C₂×C₈	#676	1	2	yes	—
160	C₂×C₈	#683	1	2	yes	—
161	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
162	C₂×C₈	D₄×D₅	1	1	yes	characteristic
163	C₂×C₈	D₄×D₅	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₄²	—	2	1	no	—
168	C₄²	—	5	2	no	—
169	C₄²	—	5	1	no	—
170	C₄²	—	5	1	no	—
171	C₄²	—	5	2	no	—
172	C₄²	—	10	8	no	—
173	C₄²	—	10	1	no	—
174	C₄²	#683	1	2	yes	—
175	C₁₆	—	4	2	no	—
176	C₁₆	—	20	2	no	—
177	C₂×C₅	—	2	1	no	—
178	C₂×C₅	—	2	1	no	—
179	C₂×C₅	#1842	1	1	yes	characteristic
180	C₂×C₅	C₂×#364	1	2	yes	—
181	C₂³	D₅×M₁₆	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	5	2	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	10	4	no	—
187	C₂×C₄	—	10	4	no	—
188	C₂×C₄	—	10	2	no	—
189	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
192	C₂×C₄	C₂×#676	1	1	yes	characteristic
193	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
194	C₂×C₄	C₂×#683	1	1	yes	characteristic
195	C₈	—	5	4	no	—
196	C₈	—	10	2	no	—
197	C₈	#3946	1	4	yes	—
198	C₈	#4034	1	1	yes	characteristic
199	C₈	#4034	1	1	yes	characteristic
200	C₈	#4086	1	1	yes	characteristic
201	C₈	#4090	1	1	yes	characteristic
202	C₅	#12900	1	1	yes	characteristic, complemented, Sylow
203	C₂²	—	2	2	no	—
204	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
205	C₂²	#65126	1	1	yes	characteristic
206	C₂²	#65407	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	5	4	no	—
209	C₄	—	10	4	no	—
210	C₄	—	10	2	no	—
211	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
212	C₄	#65126	1	1	yes	characteristic
213	C₄	#65265	1	1	yes	characteristic
214	C₄	C₂×#3946	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	D₅×#364	1	2	yes	—
218	C₂	#10601238	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂²×C₅	C₂×C₄⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#364	D₅
hypocenter	C₅	#12900
LCS	G ▷ C₂×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
147. Dic₅	#364	semidirect
202. C₅	#12900	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
C₂×#64731	2
#10602213	2
C₄×C₈×Dic₅	2
C₂×#64683	2
#12900	5