Subgroups — #12017519

Order 1280.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#64784	C₂	1	4	yes	maximal
3	#10600558	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×Q₁₆	—	2	1	no	—
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₅×#494	—	2	2	no	—
9	C₅×#494	C₄	1	4	yes	complemented
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3950	—	2	1	no	—
12	C₂×#3950	C₂²	1	2	yes	—
13	C₂²×#672	C₂²	1	2	yes	—
14	#65224	—	2	1	no	—
15	#64784	—	2	8	no	—
16	C₄×#672	—	2	2	no	—
17	#13203	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	4	no	—
20	C₂×C₅×Q₁₆	—	2	4	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	4	2	no	—
24	C₂×C₅×Q₁₆	—	4	2	no	—
25	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	—	2	4	no	—
29	C₅×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₅×C₄⋊C₄	—	4	2	no	—
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅	—	2	2	no	—
33	C₂×C₄⋊Dic₅	—	4	2	no	—
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	#3950	—	2	4	no	—
37	#3950	—	4	2	no	—
38	C₂×#672	—	2	2	no	—
39	C₂×#672	—	2	2	no	—
40	C₂×#672	D₄	1	4	yes	—
41	C₂×#494	D₅	1	1	yes	characteristic
42	C₂×#358	—	5	4	no	—
43	C₂×C₄×C₁₆	—	5	1	no	—
44	#1845	—	5	1	no	—
45	C₂×C₅×Q₈	—	2	4	no	—
46	C₂×C₅×Q₈	—	4	2	no	—
47	C₂×C₅×Q₈	—	4	2	no	—
48	C₂×C₅×Q₈	—	4	2	no	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	D₈	1	2	yes	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅	—	4	2	no	—
55	C₅×Q₁₆	—	2	8	no	—
56	C₅×Q₁₆	—	4	8	no	—
57	C₅×Q₁₆	—	4	4	no	—
58	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	D₈	1	2	yes	—
60	C₂×C₅×C₈	SD₁₆	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	2	yes	—
65	C₅×C₄⋊C₄	—	4	8	no	—
66	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅	SD₁₆	1	2	yes	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₄⋊Dic₅	—	4	2	no	—
70	C₄⋊Dic₅	—	8	4	no	—
71	#672	—	2	4	no	—
72	C₂²×Q₁₆	—	2	1	no	—
73	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
74	C₂×C₄⋊Q₈	—	2	1	no	—
75	C₂²×C₁₆	—	5	2	no	—
76	#494	—	2	2	no	—
77	#494	Dic₅	1	4	yes	complemented
78	C₂×#107	—	5	2	no	—
79	C₂×#107	—	10	1	no	—
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	#384	—	10	1	no	—
82	#358	—	10	8	no	—
83	C₄×C₁₆	—	10	2	no	—
84	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
85	C₅×Q₈	—	4	4	no	—
86	C₅×Q₈	—	4	4	no	—
87	C₅×Q₈	—	8	2	no	—
88	C₅×Q₈	—	8	2	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	8	2	no	—
91	C₂×C₄×C₅	—	8	2	no	—
92	C₂×C₄×C₅	#102	1	4	yes	—
93	C₂×C₄×C₅	SD₃₂	1	4	yes	—
94	C₂×C₄×C₅	Q₃₂	1	4	yes	—
95	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	#124	1	1	yes	characteristic
97	C₂×C₄×C₅	#127	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×Dic₅	—	8	2	no	—
102	C₂×Dic₅	—	8	4	no	—
103	C₅×C₈	#102	1	4	yes	—
104	C₅×C₈	C₄×D₄	1	2	yes	—
105	C₅×C₈	#121	1	2	yes	—
106	C₂²×Q₈	—	2	2	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	2	4	no	—
109	C₂×Q₁₆	—	2	4	no	—
110	C₂×Q₁₆	—	2	2	no	—
111	C₂×Q₁₆	—	4	2	no	—
112	C₂×Q₁₆	—	4	2	no	—
113	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
114	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	—	10	2	no	—
120	C₂×C₄⋊C₄	—	20	2	no	—
121	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
122	C₂×C₁₆	—	5	4	no	—
123	C₂×C₁₆	—	10	2	no	—
124	C₂×C₁₆	—	10	2	no	—
125	#107	—	10	4	no	—
126	#107	—	20	2	no	—
127	C₄×C₈	C₂×Dic₅	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅	1	2	yes	—
129	C₂²×C₅	C₂×#102	1	1	yes	characteristic
130	C₂²×C₅	#486	1	1	yes	characteristic
131	C₂²×C₅	#493	1	1	yes	characteristic
132	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
133	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
134	C₄×C₅	—	8	4	no	—
135	C₄×C₅	#358	1	8	yes	complemented
136	C₄×C₅	#390	1	1	yes	characteristic
137	C₄×C₅	C₂×#102	1	1	yes	characteristic
138	C₄×C₅	#486	1	1	yes	characteristic
139	C₄×C₅	#492	1	1	yes	characteristic
140	Dic₅	—	16	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	20	2	no	—
149	C₂²×C₄	#686	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	Q₁₆	—	2	8	no	—
152	Q₁₆	—	4	8	no	—
153	Q₁₆	—	4	4	no	—
154	C₂×C₈	#686	1	2	yes	—
155	C₂×C₈	#687	1	2	yes	—
156	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
157	C₂×C₈	D₄×D₅	1	2	yes	—
158	C₂×C₈	#711	1	2	yes	—
159	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	20	2	no	—
163	C₄⋊C₄	—	20	2	no	—
164	C₄⋊C₄	—	40	4	no	—
165	C₄²	#687	1	2	yes	—
166	C₄²	C₂²⋊Dic₅	1	2	yes	—
167	C₁₆	—	10	4	no	—
168	C₂×C₅	#1957	1	1	yes	characteristic
169	C₂×C₅	C₂×#358	1	2	yes	—
170	C₂×C₅	#2241	1	2	yes	—
171	C₂×C₅	#2247	1	1	yes	characteristic
172	C₂×C₅	#2248	1	1	yes	characteristic
173	C₂³	C₂×#686	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	Q₈	—	4	4	no	—
176	Q₈	—	8	2	no	—
177	Q₈	—	8	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	40	2	no	—
182	C₂×C₄	—	40	4	no	—
183	C₂×C₄	#3960	1	4	yes	—
184	C₂×C₄	#3961	1	4	yes	—
185	C₂×C₄	#3964	1	4	yes	—
186	C₂×C₄	C₂×#686	1	1	yes	characteristic
187	C₂×C₄	C₂×#687	1	1	yes	characteristic
188	C₂×C₄	C₂×#687	1	1	yes	characteristic
189	C₂×C₄	#4082	1	1	yes	characteristic
190	C₂×C₄	#4086	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
192	C₈	#3964	1	4	yes	—
193	C₈	D₄×Dic₅	1	2	yes	—
194	C₈	#4084	1	2	yes	—
195	C₅	#13203	1	1	yes	characteristic, complemented, Sylow
196	C₂²	#65359	1	1	yes	characteristic
197	C₂²	#65362	1	1	yes	characteristic
198	C₂²	C₂×#3960	1	2	yes	—
199	C₂²	C₂×#3961	1	2	yes	—
200	C₂²	C₂×#3964	1	1	yes	characteristic
201	C₄	—	8	4	no	—
202	C₄	—	80	2	no	—
203	C₄	#64784	1	8	yes	complemented
204	C₄	#65359	1	1	yes	characteristic
205	C₄	#65365	1	1	yes	characteristic
206	C₄	C₂×#3964	1	1	yes	characteristic
207	C₄	#65510	1	1	yes	characteristic
208	C₂	#10602103	1	2	yes	—
209	C₂	#10602105	1	1	yes	characteristic
210	C₂	#10602106	1	1	yes	characteristic
211	C₂	#10602557	1	1	yes	characteristic
212	C₂	C₂×#64784	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#494	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#494	D₅
hypocenter	C₅	#13203
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
9. C₅×#494	C₄	semidirect
77. #494	Dic₅	semidirect
135. C₄×C₅	#358	semidirect
195. C₅	#13203	semidirect
203. C₄	#64784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
C₂×#64784	2
#10600558	2
C₂×C₄×#672	2
#13203	5