Subgroups — #11999312

Order 1280.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#10600185	C₂	1	1	yes	maximal, characteristic
3	C₄²×#168	C₂	1	1	yes	maximal, characteristic
4	#10598110	C₂	1	4	yes	maximal
5	C₂×C₅×#125	C₄	1	1	yes	characteristic
6	C₅×#389	C₄	1	2	yes	—
7	C₅×#387	—	2	2	no	—
8	C₅×#382	C₄	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄×#168	—	2	2	no	—
14	C₂×C₄×#168	C₂²	1	2	yes	—
15	#64744	—	2	2	no	—
16	#9871	—	5	1	no	maximal, Sylow
17	C₅×#125	—	2	2	no	—
18	C₅×#125	C₈	1	4	yes	complemented
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂²×#168	—	2	2	no	—
30	C₂²×#168	—	2	2	no	—
31	C₅×#95	—	2	2	no	—
32	C₅×#95	—	2	2	no	—
33	C₅×#95	—	2	2	no	—
34	C₄⋊#168	—	2	4	no	—
35	C₄×#168	—	2	8	no	—
36	C₄×#168	—	2	2	no	—
37	C₄×#168	Q₈	1	4	yes	—
38	C₄×#125	D₅	1	1	yes	characteristic
39	#1831	—	5	1	no	—
40	C₄²×C₈	—	5	1	no	—
41	#1279	—	5	4	no	—
42	C₂²×C₄×C₅	—	2	2	no	—
43	C₂²×C₄×C₅	—	2	2	no	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₅×C₄⋊C₄	—	2	8	no	—
50	C₅×C₄⋊C₄	—	2	2	no	—
51	C₅×C₄⋊C₄	—	2	4	no	—
52	C₅×C₄⋊C₄	—	4	4	no	—
53	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
54	C₄²×C₅	—	2	4	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	—	2	4	no	—
57	C₄²×C₅	—	4	4	no	—
58	C₄²×C₅	C₂×C₈	1	2	yes	—
59	C₄²×C₅	M₁₆	1	2	yes	—
60	C₄²×C₅	C₂×Q₈	1	2	yes	—
61	C₄²×C₅	#41	1	2	yes	—
62	C₂×#168	—	2	8	no	—
63	C₂×#168	—	2	4	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	4	no	—
66	C₂×#125	Dic₅	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	5	2	no	—
68	C₂×C₄×C₈	—	5	2	no	—
69	C₂×C₄×C₈	—	10	2	no	—
70	#389	Dic₅	1	2	yes	—
71	#387	—	2	2	no	—
72	#382	Dic₅	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	2	2	no	—
74	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
75	C₄³	C₂×D₅	1	1	yes	characteristic
76	#336	—	10	2	no	—
77	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₅	#104	1	4	yes	—
91	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₅	#124	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅	#131	1	2	yes	—
98	C₂×C₄×C₅	#135	1	4	yes	—
99	#168	—	2	8	no	—
100	#168	—	8	2	no	—
101	C₂²×C₈	—	10	2	no	—
102	C₂²×C₈	—	10	2	no	—
103	#125	—	2	2	no	—
104	#125	#168	1	4	yes	complemented
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
114	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
115	C₄⋊C₈	—	10	4	no	—
116	C₄×C₈	—	5	4	no	—
117	C₄×C₈	—	10	8	no	—
118	C₄×C₈	—	10	2	no	—
119	#95	—	2	2	no	—
120	#95	—	2	2	no	—
121	#95	—	2	2	no	—
122	C₂²×C₅	#391	1	1	yes	characteristic
123	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₅	#408	1	1	yes	characteristic
125	C₂²×C₅	#416	1	2	yes	—
126	C₂²×C₅	C₂×#104	1	2	yes	—
127	C₄×C₅	—	2	4	no	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	4	no	—
130	C₄×C₅	—	4	4	no	—
131	C₄×C₅	—	8	2	no	—
132	C₄×C₅	C₈×Q₈	1	2	yes	—
133	C₄×C₅	#446	1	2	yes	—
134	C₄×C₅	#488	1	2	yes	—
135	C₄×C₅	#499	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
139	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
143	C₂×C₈	—	10	8	no	—
144	C₂×C₈	—	10	4	no	—
145	C₂×C₈	—	20	4	no	—
146	C₂×C₈	—	20	2	no	—
147	C₄⋊C₄	—	2	8	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	C₂×#168	1	4	yes	—
152	C₄²	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×#168	1	2	yes	—
157	C₄²	#681	1	2	yes	—
158	C₄²	Q₈×D₅	1	2	yes	—
159	C₄²	#713	1	2	yes	—
160	C₂×C₅	#1476	1	2	yes	—
161	C₂×C₅	#1494	1	2	yes	—
162	C₂×C₅	#1984	1	1	yes	characteristic
163	C₂×C₅	#1991	1	1	yes	characteristic
164	C₂×C₅	#1994	1	1	yes	characteristic
165	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	C₂²⋊#168	1	4	yes	—
178	C₂×C₄	#3969	1	4	yes	—
179	C₂×C₄	C₂²×#168	1	1	yes	characteristic
180	C₂×C₄	C₂×#681	1	1	yes	characteristic
181	C₂×C₄	#4090	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₅	1	2	yes	—
183	C₂×C₄	#4092	1	2	yes	—
184	C₂×C₄	#4093	1	1	yes	characteristic
185	C₂×C₄	#4094	1	2	yes	—
186	C₂×C₄	#4096	1	4	yes	—
187	C₈	—	10	8	no	—
188	C₈	—	40	2	no	—
189	C₅	#9871	1	1	yes	characteristic, complemented, Sylow
190	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
191	C₂²	#65519	1	1	yes	characteristic
192	C₂²	#65522	1	2	yes	—
193	C₂²	#65523	1	1	yes	characteristic
194	C₂²	C₂×#3969	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₄	Q₈×#168	1	2	yes	—
201	C₄	#65313	1	2	yes	—
202	C₄	#65353	1	2	yes	—
203	C₄	#65357	1	2	yes	—
204	C₂	#10602427	1	1	yes	characteristic
205	C₂	#10602431	1	2	yes	—
206	C₂	#10602533	1	1	yes	characteristic
207	C₂	#10602545	1	2	yes	—
208	C₂	#10602632	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#125	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#125	D₅
hypocenter	C₅	#9871
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
18. C₅×#125	C₈	semidirect
104. #125	#168	semidirect
189. C₅	#9871	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#125	2
#10600185	2
C₄²×#168	2
#10598110	2
#9871	5