Subgroups — #125×#168 #12107604

Order 1280.

742 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#10600277	C₂	1	2	yes	maximal
3	#10600275	C₂	1	4	yes	maximal
4	C₄⋊C₄×#168	C₂	1	4	yes	maximal
5	C₄⋊C₄×#168	C₂	1	2	yes	maximal
6	#10600185	C₂	1	1	yes	maximal, characteristic
7	C₄²×#168	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#125	C₄	1	1	yes	characteristic
9	C₅×#389	C₄	1	2	yes	—
10	C₅×#387	C₄	1	4	yes	—
11	C₅×#382	C₄	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#168	C₂²	1	4	yes	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	C₂×C₄×#168	C₂²	1	4	yes	—
18	C₂×C₄×#168	C₂²	1	2	yes	—
19	C₂×C₄×#168	C₂²	1	2	yes	—
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₂×C₄×#168	C₂²	1	1	yes	characteristic
22	#64744	C₂²	1	4	yes	—
23	#64744	C₂²	1	4	yes	—
24	#64744	C₂²	1	4	yes	—
25	C₈×#125	—	5	1	no	maximal, Sylow
26	C₅×#125	C₈	1	8	yes	complemented
27	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	4	yes	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂²×#168	C₂³	1	4	yes	—
38	C₂²×#168	C₂³	1	1	yes	characteristic
39	C₂²×#168	C₂³	1	2	yes	—
40	C₂²×#168	C₂³	1	1	yes	characteristic
41	C₅×#95	C₂×C₄	1	4	yes	—
42	C₅×#95	C₂×C₄	1	4	yes	—
43	C₅×#95	C₂×C₄	1	4	yes	—
44	C₄⋊#168	—	2	4	no	—
45	C₄⋊#168	—	2	8	no	—
46	C₄×#168	—	2	2	no	—
47	C₄×#168	—	2	4	no	—
48	C₄×#168	—	2	4	no	—
49	C₄×#168	—	2	8	no	—
50	C₄×#168	Q₈	1	4	yes	—
51	C₄×#125	D₅	1	1	yes	characteristic
52	#1939	—	5	2	no	—
53	#1916	—	5	4	no	—
54	#1831	—	5	1	no	—
55	C₈×C₄⋊C₄	—	5	4	no	—
56	C₈×C₄⋊C₄	—	5	2	no	—
57	C₄²×C₈	—	5	1	no	—
58	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	8	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	C₂×C₈	1	16	yes	—
68	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
69	C₄²×C₅	—	2	4	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	4	no	—
74	C₄²×C₅	—	2	4	no	—
75	C₄²×C₅	C₂×C₈	1	4	yes	—
76	C₄²×C₅	C₂×Q₈	1	2	yes	—
77	C₄²×C₅	#41	1	2	yes	—
78	C₂×#168	—	2	2	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	—	2	2	no	—
81	C₂×#168	—	2	4	no	—
82	C₂×#168	—	2	4	no	—
83	C₂×#168	—	2	4	no	—
84	C₂×#168	C₂×Q₈	1	2	yes	—
85	C₂×#168	C₂×Q₈	1	2	yes	—
86	C₂×#168	#41	1	4	yes	—
87	C₂×#168	#41	1	4	yes	—
88	C₂×#125	Dic₅	1	1	yes	characteristic
89	C₂×C₄⋊C₈	—	5	2	no	—
90	C₂×C₄⋊C₈	—	5	4	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	C₂×C₄×C₈	—	5	2	no	—
93	C₂×C₄×C₈	—	5	1	no	—
94	C₂×C₄×C₈	—	5	2	no	—
95	C₂×C₄×C₈	—	5	4	no	—
96	#389	Dic₅	1	2	yes	—
97	#387	Dic₅	1	4	yes	—
98	#382	Dic₅	1	1	yes	characteristic
99	C₄×C₄⋊C₄	C₂×D₅	1	4	yes	—
100	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
101	C₄³	C₂×D₅	1	1	yes	characteristic
102	#336	—	5	4	no	—
103	#336	—	5	4	no	—
104	#336	—	5	4	no	—
105	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
119	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
120	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₅	#131	1	4	yes	—
122	C₂×C₄×C₅	#131	1	2	yes	—
123	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×#41	1	2	yes	—
126	C₂×C₄×C₅	C₂×#41	1	2	yes	—
127	#168	—	2	4	no	—
128	#168	—	2	2	no	—
129	#168	—	4	4	no	—
130	#168	#125	1	4	yes	direct product factor
131	C₂²×C₈	—	5	1	no	—
132	C₂²×C₈	—	5	1	no	—
133	C₂²×C₈	—	5	2	no	—
134	C₂²×C₈	—	5	4	no	—
135	#125	#168	1	8	yes	direct product factor
136	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
140	C₂×C₄²	C₂×Dic₅	1	2	yes	—
141	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
143	C₂×C₄²	C₂²×D₅	1	2	yes	—
144	C₂×C₄²	C₂²×D₅	1	4	yes	—
145	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
146	C₄⋊C₈	—	10	8	no	—
147	C₄⋊C₈	—	10	4	no	—
148	C₄×C₈	—	5	4	no	—
149	C₄×C₈	—	10	4	no	—
150	C₄×C₈	—	10	8	no	—
151	C₄×C₈	—	10	4	no	—
152	C₄×C₈	—	10	2	no	—
153	#95	C₂×Dic₅	1	4	yes	—
154	#95	C₂×Dic₅	1	4	yes	—
155	#95	C₂×Dic₅	1	4	yes	—
156	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₅	C₄×#41	1	2	yes	—
158	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
159	C₂²×C₅	C₂×#131	1	1	yes	characteristic
160	C₂²×C₅	C₂×#131	1	2	yes	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	4	no	—
163	C₄×C₅	—	2	8	no	—
164	C₄×C₅	—	2	4	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	C₈×Q₈	1	4	yes	—
167	C₄×C₅	C₂×#125	1	1	yes	characteristic
168	C₄×C₅	#529	1	2	yes	—
169	C₄×C₅	#533	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
173	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
176	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	5	4	no	—
179	C₂×C₈	—	5	2	no	—
180	C₂×C₈	—	5	4	no	—
181	C₂×C₈	—	10	2	no	—
182	C₂×C₈	—	10	2	no	—
183	C₂×C₈	—	10	2	no	—
184	C₂×C₈	—	10	4	no	—
185	C₂×C₈	—	10	4	no	—
186	C₂×C₈	—	10	4	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	C₂×#168	1	16	yes	—
190	C₄⋊C₄	C₂×#168	1	8	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂×#168	1	4	yes	—
198	C₄²	Q₈×D₅	1	2	yes	—
199	C₄²	#713	1	2	yes	—
200	C₂×C₅	C₄×#125	1	1	yes	characteristic
201	C₂×C₅	C₂×C₈×Q₈	1	1	yes	characteristic
202	C₂×C₅	#2960	1	1	yes	characteristic
203	C₂×C₅	C₈×#41	1	2	yes	—
204	C₂×C₅	#2966	1	2	yes	—
205	C₂³	C₂³×Dic₅	1	1	yes	characteristic
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₂²×#168	1	4	yes	—
217	C₂×C₄	C₂²×#168	1	8	yes	—
218	C₂×C₄	C₂²×#168	1	2	yes	—
219	C₂×C₄	Q₈×Dic₅	1	2	yes	—
220	C₂×C₄	Q₈×Dic₅	1	2	yes	—
221	C₂×C₄	#4094	1	4	yes	—
222	C₂×C₄	#4094	1	2	yes	—
223	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
224	C₂×C₄	C₂×#713	1	1	yes	characteristic
225	C₂×C₄	D₅×#41	1	2	yes	—
226	C₂×C₄	D₅×#41	1	2	yes	—
227	C₈	—	5	4	no	—
228	C₈	—	10	2	no	—
229	C₈	—	10	4	no	—
230	C₈	—	20	4	no	—
231	C₅	C₈×#125	1	1	yes	characteristic, complemented, Sylow
232	C₂²	C₂³×#168	1	1	yes	characteristic
233	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
234	C₂²	C₂×#4094	1	2	yes	—
235	C₂²	C₂×#4094	1	1	yes	characteristic
236	C₂²	#41×Dic₅	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	2	8	no	—
241	C₄	—	4	4	no	—
242	C₄	Q₈×#168	1	4	yes	—
243	C₄	D₅×#125	1	1	yes	characteristic
244	C₄	#66022	1	1	yes	characteristic
245	C₄	#66025	1	2	yes	—
246	C₂	C₂×Q₈×#168	1	1	yes	characteristic
247	C₂	#41×#168	1	2	yes	—
248	C₂	Dic₅×#125	1	1	yes	characteristic
249	C₂	#10614857	1	2	yes	—
250	C₂	#10615059	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₈
Fitting	C₄×C₅×#125	C₂
socle, prime radical	C₂³×C₅	C₂³×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₄×#125	D₅
hypocenter	C₅	C₈×#125
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
26. C₅×#125	C₈	semidirect
130. #168	#125	direct (N × H)
135. #125	#168	direct (N × H)
231. C₅	C₈×#125	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
#10600277	2
#10600275	2
C₄⋊C₄×#168	2
#10600185	2
C₄²×#168	2
C₈×#125	5