Subgroups — #125×#92418 #29688317

Order 15616.

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	#21446538	C₂	1	2	yes	maximal
2	#21446378	C₂	1	4	yes	maximal
3	#21445936	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#92418	C₂	1	4	yes	maximal
5	C₄⋊C₄×#92418	C₂	1	2	yes	maximal
6	C₄²×#92418	C₂	1	1	yes	maximal, characteristic
7	C₄×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	C₂×C₄⋊#92418	C₂²	1	4	yes	—
10	C₂×C₄×#92418	C₂²	1	4	yes	—
11	C₂×C₄×#92418	C₂²	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92418	C₂²	1	2	yes	—
14	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
15	#16839517	C₂²	1	4	yes	—
16	#16839517	C₂²	1	4	yes	—
17	#16839517	C₂²	1	4	yes	—
18	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
19	C₆₁×#389	C₄	1	2	yes	—
20	C₆₁×#387	C₄	1	4	yes	—
21	C₆₁×#382	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
24	C₄³×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×#92418	C₂³	1	4	yes	—
26	C₂²×#92418	C₂³	1	1	yes	characteristic
27	C₂²×#92418	C₂³	1	2	yes	—
28	C₂²×#92418	C₂³	1	1	yes	characteristic
29	C₄⋊#92418	—	2	4	no	—
30	C₄⋊#92418	—	2	8	no	—
31	C₄×#92418	—	2	2	no	—
32	C₄×#92418	—	2	4	no	—
33	C₄×#92418	—	2	4	no	—
34	C₄×#92418	—	2	8	no	—
35	C₄×#92418	Q₈	1	4	yes	—
36	#125×C₆₁	C₈	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	#95×C₆₁	C₂×C₄	1	4	yes	—
48	#95×C₆₁	C₂×C₄	1	4	yes	—
49	#95×C₆₁	C₂×C₄	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	—	2	8	no	—
58	C₄⋊C₄×C₆₁	—	2	4	no	—
59	C₄⋊C₄×C₆₁	C₂×C₈	1	16	yes	—
60	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
61	C₄²×C₆₁	—	2	4	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	—	2	4	no	—
66	C₄²×C₆₁	—	2	4	no	—
67	C₄²×C₆₁	C₂×C₈	1	4	yes	—
68	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
69	C₄²×C₆₁	#41	1	2	yes	—
70	C₂×#92418	—	2	2	no	—
71	C₂×#92418	—	2	2	no	—
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	—	2	4	no	—
74	C₂×#92418	—	2	4	no	—
75	C₂×#92418	—	2	4	no	—
76	C₂×#92418	C₂×Q₈	1	2	yes	—
77	C₂×#92418	C₂×Q₈	1	2	yes	—
78	C₂×#92418	#41	1	4	yes	—
79	C₂×#92418	#41	1	4	yes	—
80	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	—	2	2	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	4	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
95	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₆₁	#131	1	4	yes	—
97	C₂×C₄×C₆₁	#131	1	2	yes	—
98	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
102	#92418	—	2	4	no	—
103	#92418	—	2	2	no	—
104	#92418	—	4	4	no	—
105	#92418	#125	1	4	yes	direct product factor
106	C₈×#125	—	61	1	no	maximal, Sylow
107	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₂²×C₆₁	C₄×#41	1	2	yes	—
109	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
110	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
111	C₂²×C₆₁	C₂×#131	1	2	yes	—
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	—	2	4	no	—
114	C₄×C₆₁	—	2	8	no	—
115	C₄×C₆₁	—	2	4	no	—
116	C₄×C₆₁	—	4	4	no	—
117	C₄×C₆₁	C₈×Q₈	1	4	yes	—
118	C₄×C₆₁	C₂×#125	1	1	yes	characteristic
119	C₄×C₆₁	#529	1	2	yes	—
120	C₄×C₆₁	#533	1	1	yes	characteristic
121	C₄×#125	D₆₁	1	1	yes	characteristic
122	#1939	—	61	2	no	—
123	#1916	—	61	4	no	—
124	#1831	—	61	1	no	—
125	C₈×C₄⋊C₄	—	61	4	no	—
126	C₈×C₄⋊C₄	—	61	2	no	—
127	C₄²×C₈	—	61	1	no	—
128	C₂×C₆₁	C₄×#125	1	1	yes	characteristic
129	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₆₁	#2960	1	1	yes	characteristic
131	C₂×C₆₁	C₈×#41	1	2	yes	—
132	C₂×C₆₁	#2966	1	2	yes	—
133	C₂×#125	Dic₆₁	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	61	2	no	—
135	C₂×C₄⋊C₈	—	61	4	no	—
136	C₂×C₄×C₈	—	61	1	no	—
137	C₂×C₄×C₈	—	61	2	no	—
138	C₂×C₄×C₈	—	61	1	no	—
139	C₂×C₄×C₈	—	61	2	no	—
140	C₂×C₄×C₈	—	61	4	no	—
141	#389	Dic₆₁	1	2	yes	—
142	#387	Dic₆₁	1	4	yes	—
143	#382	Dic₆₁	1	1	yes	characteristic
144	C₄×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
145	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
146	C₄³	C₂×D₆₁	1	1	yes	characteristic
147	#336	—	61	4	no	—
148	#336	—	61	4	no	—
149	#336	—	61	4	no	—
150	C₆₁	C₈×#125	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₈	—	61	1	no	—
152	C₂²×C₈	—	61	1	no	—
153	C₂²×C₈	—	61	2	no	—
154	C₂²×C₈	—	61	4	no	—
155	#125	#92418	1	8	yes	direct product factor
156	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
157	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
160	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
161	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
164	C₂×C₄²	C₂²×D₆₁	1	4	yes	—
165	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
166	C₄⋊C₈	—	122	8	no	—
167	C₄⋊C₈	—	122	4	no	—
168	C₄×C₈	—	61	4	no	—
169	C₄×C₈	—	122	4	no	—
170	C₄×C₈	—	122	8	no	—
171	C₄×C₈	—	122	4	no	—
172	C₄×C₈	—	122	2	no	—
173	#95	C₂×Dic₆₁	1	4	yes	—
174	#95	C₂×Dic₆₁	1	4	yes	—
175	#95	C₂×Dic₆₁	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₆₁	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₆₁	1	4	yes	—
180	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
183	C₂×C₈	—	61	2	no	—
184	C₂×C₈	—	61	4	no	—
185	C₂×C₈	—	61	2	no	—
186	C₂×C₈	—	61	4	no	—
187	C₂×C₈	—	122	2	no	—
188	C₂×C₈	—	122	2	no	—
189	C₂×C₈	—	122	2	no	—
190	C₂×C₈	—	122	4	no	—
191	C₂×C₈	—	122	4	no	—
192	C₂×C₈	—	122	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	C₂×#92418	1	16	yes	—
196	C₄⋊C₄	C₂×#92418	1	8	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	2	no	—
203	C₄²	C₂×#92418	1	4	yes	—
204	C₄²	Q₈×D₆₁	1	2	yes	—
205	C₄²	#11758226	1	2	yes	—
206	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
218	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
219	C₂×C₄	C₂²×#92418	1	4	yes	—
220	C₂×C₄	C₂²×#92418	1	8	yes	—
221	C₂×C₄	C₂²×#92418	1	2	yes	—
222	C₂×C₄	#14519784	1	4	yes	—
223	C₂×C₄	#14519784	1	2	yes	—
224	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
225	C₂×C₄	#41×D₆₁	1	2	yes	—
226	C₂×C₄	#41×D₆₁	1	2	yes	—
227	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
228	C₈	—	61	4	no	—
229	C₈	—	122	4	no	—
230	C₈	—	122	2	no	—
231	C₈	—	244	4	no	—
232	C₂²	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
233	C₂²	#41×Dic₆₁	1	2	yes	—
234	C₂²	C₂³×#92418	1	1	yes	characteristic
235	C₂²	C₂×#14519784	1	1	yes	characteristic
236	C₂²	C₂×#14519784	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	8	no	—
240	C₄	—	2	4	no	—
241	C₄	—	4	4	no	—
242	C₄	Q₈×#92418	1	4	yes	—
243	C₄	#125×D₆₁	1	1	yes	characteristic
244	C₄	#16840297	1	2	yes	—
245	C₄	#16840318	1	1	yes	characteristic
246	C₂	#125×Dic₆₁	1	1	yes	characteristic
247	C₂	C₂×Q₈×#92418	1	1	yes	characteristic
248	C₂	#21454754	1	1	yes	characteristic
249	C₂	#41×#92418	1	2	yes	—
250	C₂	#21454795	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₈
Fitting	C₄×#125×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂³×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×#125	D₆₁
hypocenter	C₆₁	C₈×#125
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
36. #125×C₆₁	C₈	semidirect
105. #92418	#125	direct (N × H)
150. C₆₁	C₈×#125	semidirect
155. #125	#92418	direct (N × H)

Maximal subgroups

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

subgroup	index
#21446538	2
#21446378	2
#21445936	2
C₄⋊C₄×#92418	2
C₄²×#92418	2
C₄×#125×C₆₁	2
C₈×#125	61