Subgroups — #30619429

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461916	C₂	1	2	yes	maximal
2	#21461916	C₂	1	4	yes	maximal
3	#21461872	C₂	1	4	yes	maximal
4	#21461751	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#92420	C₂	1	2	yes	maximal
6	#21461533	C₂	1	1	yes	maximal, characteristic
7	#125×Dic₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#92420	C₂²	1	2	yes	—
9	C₂×C₄⋊#92420	C₂²	1	4	yes	—
10	C₂×C₄⋊#92420	C₂²	1	4	yes	—
11	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92420	C₂²	1	4	yes	—
13	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
14	#16840628	C₂²	1	2	yes	—
15	#16840628	C₂²	1	2	yes	—
16	#16840628	C₂²	1	4	yes	—
17	#16840628	C₂²	1	4	yes	—
18	#16839661	C₄	1	2	yes	—
19	#16839648	C₄	1	4	yes	—
20	#16839628	C₄	1	1	yes	characteristic
21	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
23	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
24	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
25	C₂²×#92420	C₂³	1	1	yes	characteristic
26	C₂²×#92420	C₂³	1	4	yes	—
27	C₂²×#92420	C₂³	1	1	yes	characteristic
28	C₂²×#92420	C₂³	1	2	yes	—
29	C₄⋊#92420	—	2	4	no	—
30	C₄⋊#92420	—	4	4	no	—
31	C₄⋊#92420	—	4	4	no	—
32	C₄×#92420	—	2	2	no	—
33	C₄×#92420	—	4	4	no	—
34	C₄×#92420	Q₈	1	4	yes	—
35	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
40	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
41	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
42	#14519684	C₂×C₄	1	4	yes	—
43	#14519684	C₂×C₄	1	4	yes	—
44	#14519684	C₂×C₄	1	4	yes	—
45	#125×C₆₁	C₈	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
53	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
54	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×#92420	—	2	2	no	—
57	C₂×#92420	—	2	2	no	—
58	C₂×#92420	—	2	1	no	—
59	C₂×#92420	—	2	1	no	—
60	C₂×#92420	—	2	4	no	—
61	C₂×#92420	—	2	4	no	—
62	C₂×#92420	—	2	2	no	—
63	C₂×#92420	—	2	4	no	—
64	C₂×#92420	—	2	2	no	—
65	C₂×#92420	C₂×Q₈	1	4	yes	—
66	C₄⋊C₄×C₆₁	C₂×C₈	1	16	yes	—
67	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
68	C₄²×C₆₁	C₂×C₈	1	4	yes	—
69	C₄⋊Dic₆₁	—	2	4	no	—
70	C₄⋊Dic₆₁	—	4	4	no	—
71	C₄×Dic₆₁	—	2	2	no	—
72	C₄×Dic₆₁	—	2	4	no	—
73	C₄×Dic₆₁	—	2	2	no	—
74	C₄×Dic₆₁	—	2	4	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	—	4	2	no	—
77	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
78	C₄×Dic₆₁	#41	1	2	yes	—
79	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	2	no	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	2	4	no	—
84	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	—
87	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₆₁	#131	1	2	yes	—
89	C₂×Dic₆₁	—	2	2	no	—
90	C₂×Dic₆₁	—	2	4	no	—
91	C₂×Dic₆₁	—	2	2	no	—
92	C₂×Dic₆₁	—	2	2	no	—
93	C₂×Dic₆₁	—	2	4	no	—
94	C₂×Dic₆₁	—	2	2	no	—
95	C₂×Dic₆₁	—	2	4	no	—
96	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
97	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
98	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
99	C₂×Dic₆₁	H(5,𝔽₂)	1	2	yes	—
100	C₂×Dic₆₁	2₋¹⁺⁴	1	2	yes	—
101	#92420	—	2	4	no	—
102	#92420	—	4	1	no	—
103	#92420	—	4	4	no	—
104	#92420	—	4	1	no	—
105	#22299	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂²×C₆₁	#520	1	2	yes	—
108	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
109	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
110	C₂²×C₆₁	#568	1	2	yes	—
111	C₄×C₆₁	—	2	4	no	—
112	C₄×C₆₁	—	2	8	no	—
113	C₄×C₆₁	C₈×Q₈	1	4	yes	—
114	Dic₆₁	—	2	2	no	—
115	Dic₆₁	—	4	2	no	—
116	Dic₆₁	—	4	4	no	—
117	Dic₆₁	#540	1	1	yes	characteristic
118	Dic₆₁	#541	1	1	yes	characteristic
119	Dic₆₁	#544	1	2	yes	—
120	C₄×#125	—	61	1	no	—
121	#1939	—	61	2	no	—
122	#1939	—	61	4	no	—
123	#1916	—	61	4	no	—
124	#1842	—	61	1	no	—
125	C₈×C₄⋊C₄	—	61	2	no	—
126	#1744	—	61	1	no	—
127	C₂×C₆₁	#2330	1	1	yes	characteristic
128	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
129	C₂×C₆₁	#2961	1	1	yes	characteristic
130	C₂×C₆₁	#2988	1	2	yes	—
131	C₂×C₆₁	#2993	1	2	yes	—
132	C₂×#125	#6923	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	61	2	no	—
134	C₂×C₄⋊C₈	—	61	4	no	—
135	C₂×C₄⋊C₈	—	61	4	no	—
136	C₂×C₄×C₈	—	61	1	no	—
137	C₂×C₄×C₈	—	61	1	no	—
138	C₂×C₄×C₈	—	61	4	no	—
139	#389	—	61	2	no	—
140	#387	—	61	4	no	—
141	#382	—	61	1	no	—
142	C₄×C₄⋊C₄	—	61	4	no	—
143	C₄×C₄⋊C₄	—	61	2	no	—
144	C₄³	—	61	1	no	—
145	#336	—	61	4	no	—
146	#336	—	61	2	no	—
147	#336	—	61	4	no	—
148	#336	—	61	2	no	—
149	C₆₁	#22299	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₈	—	61	2	no	—
151	C₂²×C₈	—	61	1	no	—
152	C₂²×C₈	—	61	1	no	—
153	C₂²×C₈	—	61	4	no	—
154	#125	#92420	1	8	yes	complemented
155	C₂×C₄⋊C₄	—	61	2	no	—
156	C₂×C₄⋊C₄	—	61	4	no	—
157	C₂×C₄⋊C₄	C₂×#6923	1	4	yes	—
158	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
159	C₂×C₄²	—	61	1	no	—
160	C₂×C₄²	—	61	2	no	—
161	C₂×C₄²	—	61	4	no	—
162	C₂×C₄²	—	61	1	no	—
163	C₂×C₄²	—	61	2	no	—
164	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
165	C₄⋊C₈	—	122	4	no	—
166	C₄⋊C₈	—	244	4	no	—
167	C₄⋊C₈	—	244	4	no	—
168	C₄×C₈	—	61	4	no	—
169	C₄×C₈	—	122	2	no	—
170	C₄×C₈	—	244	4	no	—
171	#95	—	61	4	no	—
172	#95	—	61	4	no	—
173	#95	—	61	4	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	61	4	no	—
177	C₂²×C₄	—	61	2	no	—
178	C₂²×C₄	C₂²×#6923	1	2	yes	—
179	C₂²×C₄	C₂²×#6923	1	4	yes	—
180	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
181	C₂×C₈	—	61	4	no	—
182	C₂×C₈	—	122	2	no	—
183	C₂×C₈	—	122	2	no	—
184	C₂×C₈	—	122	1	no	—
185	C₂×C₈	—	122	1	no	—
186	C₂×C₈	—	122	4	no	—
187	C₂×C₈	—	122	4	no	—
188	C₂×C₈	—	122	2	no	—
189	C₂×C₈	—	122	4	no	—
190	C₂×C₈	—	122	2	no	—
191	C₄⋊C₄	—	122	4	no	—
192	C₄⋊C₄	—	244	4	no	—
193	C₄⋊C₄	C₂×#92420	1	16	yes	—
194	C₄⋊C₄	C₂×#92420	1	8	yes	—
195	C₄²	—	61	2	no	—
196	C₄²	—	61	2	no	—
197	C₄²	—	122	4	no	—
198	C₄²	—	122	2	no	—
199	C₄²	—	122	2	no	—
200	C₄²	—	122	4	no	—
201	C₄²	—	122	2	no	—
202	C₄²	—	244	2	no	—
203	C₄²	C₂×#92420	1	4	yes	—
204	C₂³	C₂³×#6923	1	1	yes	characteristic
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	61	1	no	—
210	C₂×C₄	—	61	1	no	—
211	C₂×C₄	—	61	2	no	—
212	C₂×C₄	—	61	2	no	—
213	C₂×C₄	—	61	2	no	—
214	C₂×C₄	—	122	2	no	—
215	C₂×C₄	—	122	4	no	—
216	C₂×C₄	—	122	2	no	—
217	C₂×C₄	—	122	2	no	—
218	C₂×C₄	—	122	4	no	—
219	C₂×C₄	—	122	2	no	—
220	C₂×C₄	—	122	4	no	—
221	C₂×C₄	Q₈×#6923	1	2	yes	—
222	C₂×C₄	C₂²×#92420	1	4	yes	—
223	C₂×C₄	C₂²×#92420	1	8	yes	—
224	C₂×C₄	C₂²×#92420	1	2	yes	—
225	C₂×C₄	#14519865	1	2	yes	—
226	C₈	—	122	4	no	—
227	C₈	—	244	1	no	—
228	C₈	—	244	4	no	—
229	C₈	—	244	1	no	—
230	C₂²	C₂×Q₈×#6923	1	1	yes	characteristic
231	C₂²	#16840831	1	2	yes	—
232	C₂²	C₂³×#92420	1	1	yes	characteristic
233	C₂²	C₂×#14519865	1	1	yes	characteristic
234	C₂²	#16840841	1	2	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	8	no	—
237	C₄	—	61	1	no	—
238	C₄	—	61	1	no	—
239	C₄	—	61	2	no	—
240	C₄	—	122	2	no	—
241	C₄	—	244	2	no	—
242	C₄	—	244	4	no	—
243	C₄	Q₈×#92420	1	4	yes	—
244	C₂	#21462313	1	1	yes	characteristic
245	C₂	C₂×Q₈×#92420	1	1	yes	characteristic
246	C₂	#21462611	1	1	yes	characteristic
247	C₂	#21462734	1	2	yes	—
248	C₂	#21462765	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6923
commutator	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×#125×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂³×C₄
cosocle	C₂²×Dic₆₁	C₂⁴
hypercenter	C₂×#125	#6923
hypocenter	C₆₁	#22299
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
45. #125×C₆₁	C₈	semidirect
149. C₆₁	#22299	semidirect
154. #125	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461916	2
#21461872	2
#21461751	2
C₄⋊C₄×#92420	2
#21461533	2
#125×Dic₆₁	2
#22299	61