Subgroups — #29571330

Order 15616.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444109	C₂	1	1	yes	maximal, characteristic
3	#21444100	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839484	C₂	1	1	yes	maximal, characteristic
5	#21443655	C₂	1	1	yes	maximal, characteristic
6	#21443655	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519713	C₄	1	2	yes	—
9	C₄×#11758194	C₂²	1	1	yes	characteristic
10	C₄×#11758194	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839527	C₄	1	2	yes	—
14	#16839484	—	2	1	no	—
15	#16839484	—	2	1	no	—
16	#16839484	C₄	1	2	yes	—
17	#16839484	C₄	1	2	yes	—
18	#16839483	—	2	1	no	—
19	C₈×#92418	—	2	1	no	—
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₆₁×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#335	C₄	1	2	yes	—
24	C₆₁×#334	C₄	1	2	yes	—
25	C₂×#11758194	C₂×C₄	1	2	yes	—
26	C₂×#11758194	C₂×C₄	1	2	yes	—
27	C₂²×#92418	—	2	1	no	—
28	C₂²×#92418	Q₈	1	2	yes	—
29	#14519713	—	2	4	no	—
30	#14519688	—	2	1	no	—
31	#14519688	—	2	1	no	—
32	C₄×#92418	—	2	1	no	—
33	C₄×#92418	—	2	1	no	—
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	C₂×C₄	1	2	yes	—
36	C₄×#92418	C₂×C₄	1	2	yes	—
37	C₄×#92418	D₄	1	1	yes	characteristic
38	C₄×#92418	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈×C₆₁	—	2	1	no	—
45	C₂²⋊C₈×C₆₁	—	2	1	no	—
46	C₄×C₈×C₆₁	—	2	1	no	—
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
63	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₁	#41	1	1	yes	characteristic
67	#11758194	—	2	2	no	—
68	#11758194	—	2	2	no	—
69	#11758194	—	2	2	no	—
70	#11758194	M₁₆	1	4	yes	—
71	C₂×#92418	—	2	4	no	—
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	—	2	2	no	—
74	C₂×#92418	—	2	2	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	C₄⋊C₄	1	2	yes	—
77	C₂×#92418	C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₁	#131	1	2	yes	—
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	4	2	no	—
94	C₈×C₆₁	—	4	2	no	—
95	#92418	—	2	4	no	—
96	#92418	—	2	2	no	—
97	#92418	—	2	2	no	—
98	#92418	—	2	4	no	—
99	#92418	—	4	2	no	—
100	#8296	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₆₁	#446	1	2	yes	—
105	C₄×C₆₁	—	2	1	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
108	C₄×C₆₁	#405	1	1	yes	characteristic
109	C₄×C₆₁	#430	1	2	yes	—
110	C₄×C₆₁	#435	1	2	yes	—
111	C₄×C₆₁	#443	1	2	yes	—
112	C₄×#108	—	61	1	no	—
113	#1564	D₆₁	1	1	yes	characteristic
114	#1451	—	61	1	no	—
115	C₈×M₁₆	—	61	1	no	—
116	C₂×#322	—	61	1	no	—
117	#1278	—	61	1	no	—
118	#1278	—	61	1	no	—
119	C₂×C₆₁	—	2	2	no	—
120	C₂×C₆₁	#1771	1	1	yes	characteristic
121	C₂×C₆₁	#1772	1	1	yes	characteristic
122	C₂×C₆₁	#1778	1	1	yes	characteristic
123	#431	C₂×D₆₁	1	1	yes	characteristic
124	#431	C₂×D₆₁	1	1	yes	characteristic
125	C₂×#108	—	61	2	no	—
126	C₄×M₁₆	—	61	1	no	—
127	C₄×M₁₆	—	61	1	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	#341	—	61	2	no	—
132	#335	Dic₆₁	1	2	yes	—
133	#334	Dic₆₁	1	2	yes	—
134	#322	—	61	2	no	—
135	#322	—	61	2	no	—
136	#322	—	122	1	no	—
137	#322	—	122	1	no	—
138	#322	—	122	1	no	—
139	C₈²	—	122	1	no	—
140	C₆₁	#8296	1	1	yes	characteristic, complemented, Sylow
141	C₂×M₁₆	—	61	2	no	—
142	C₂×M₁₆	—	61	2	no	—
143	C₂²×C₈	—	61	2	no	—
144	C₂²×C₈	—	122	1	no	—
145	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
146	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
148	#108	—	122	4	no	—
149	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
150	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₂²⋊C₈	—	2	1	no	—
153	#97	—	122	1	no	—
154	#97	—	122	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	61	2	no	—
158	C₄×C₈	—	61	2	no	—
159	C₄×C₈	—	61	1	no	—
160	C₄×C₈	—	61	1	no	—
161	C₄×C₈	—	122	1	no	—
162	C₄×C₈	—	122	1	no	—
163	C₄×C₈	—	122	1	no	—
164	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
165	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
166	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
169	C₂²×C₄	#11758224	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
171	M₁₆	—	61	4	no	—
172	M₁₆	—	122	2	no	—
173	M₁₆	—	122	2	no	—
174	M₁₆	—	122	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	61	2	no	—
180	C₂×C₈	—	61	2	no	—
181	C₂×C₈	—	122	4	no	—
182	C₂×C₈	—	122	2	no	—
183	C₂×C₈	—	122	2	no	—
184	C₂×C₈	—	122	2	no	—
185	C₂×C₈	—	244	2	no	—
186	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
187	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
188	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
189	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
190	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
191	C₄²	D₄×D₆₁	1	1	yes	characteristic
192	C₄²	#11758226	1	1	yes	characteristic
193	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
194	C₂³	#14519742	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
198	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
199	C₂×C₄	#14519735	1	1	yes	characteristic
200	C₂×C₄	#14519739	1	1	yes	characteristic
201	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
202	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
203	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
204	C₂×C₄	#14519783	1	2	yes	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	122	4	no	—
212	C₈	—	122	2	no	—
213	C₈	—	122	2	no	—
214	C₈	—	122	4	no	—
215	C₈	—	244	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
219	C₂²	#16839891	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	M₁₆×Dic₆₁	1	1	yes	characteristic
223	C₄	#16839717	1	1	yes	characteristic
224	C₄	#16839809	1	2	yes	—
225	C₄	#16839832	1	2	yes	—
226	C₄	#16839879	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#21445696	1	1	yes	characteristic
229	C₂	#21445700	1	1	yes	characteristic
230	C₂	#21445727	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#8296
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
140. C₆₁	#8296	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519713	2
#21444109	2
#21444100	2
C₂×#16839484	2
#21443655	2
C₆₁×#1564	2
#8296	61