Subgroups — #29595212

Order 15616.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447504	C₂	1	1	yes	maximal, characteristic
2	#21447484	C₂	1	1	yes	maximal, characteristic
3	#21445619	C₂	1	2	yes	maximal, complemented
4	#21443948	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2932	C₂	1	1	yes	maximal, characteristic
6	#16839746	C₂²	1	2	yes	—
7	#16839736	C₂²	1	2	yes	—
8	#16839717	—	2	2	no	—
9	#16839561	C₄	1	2	yes	complemented
10	#16839559	C₄	1	2	yes	complemented
11	#16839557	C₄	1	2	yes	—
12	#16839555	C₄	1	2	yes	—
13	#16839533	—	2	2	no	—
14	C₂×#131×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#532	C₂²	1	1	yes	characteristic
17	C₆₁×#435	—	2	2	no	—
18	C₆₁×#435	—	2	2	no	—
19	C₆₁×#423	C₄	1	1	yes	characteristic, complemented
20	C₆₁×#408	C₄	1	1	yes	characteristic, complemented
21	C₆₁×#408	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₁	C₄	1	1	yes	characteristic, complemented
23	C₂×#11758194	—	2	2	no	—
24	#14519736	—	2	2	no	—
25	#14519696	C₂×C₄	1	4	yes	—
26	#14519694	—	2	2	no	—
27	#14519692	—	2	2	no	—
28	#14519692	C₂×C₄	1	4	yes	—
29	#14519687	—	4	2	no	—
30	C₈×Dic₆₁	—	4	2	no	—
31	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
33	#131×C₆₁	—	4	2	no	—
34	#131×C₆₁	—	4	2	no	—
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
39	C₂²×C₈×C₆₁	—	2	1	no	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	D₄	1	2	yes	—
43	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	#127×C₆₁	C₂×C₄	1	1	yes	characteristic
45	#124×C₆₁	C₂×C₄	1	1	yes	characteristic
46	#124×C₆₁	C₂×C₄	1	1	yes	characteristic
47	#121×C₆₁	—	2	2	no	—
48	C₄×D₄×C₆₁	—	2	2	no	—
49	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₆₁	—	2	2	no	—
51	C₂²⋊C₈×C₆₁	—	2	2	no	—
52	C₂²⋊C₈×C₆₁	—	2	2	no	—
53	#97×C₆₁	—	2	1	no	—
54	#97×C₆₁	—	2	1	no	—
55	#41×C₆₁	—	4	1	no	—
56	#41×C₆₁	—	4	1	no	—
57	#41×C₆₁	—	4	1	no	—
58	#41×C₆₁	—	4	1	no	—
59	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
60	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	—	2	2	no	—
63	C₂×D₄×C₆₁	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×D₄×C₆₁	#41	1	2	yes	—
66	C₂²×C₄×C₆₁	—	2	1	no	—
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
70	M₁₆×C₆₁	—	4	1	no	—
71	M₁₆×C₆₁	—	4	2	no	—
72	M₁₆×C₆₁	—	4	2	no	—
73	M₁₆×C₆₁	—	4	1	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	2	1	no	—
79	C₂×C₈×C₆₁	—	4	2	no	—
80	C₂×C₈×C₆₁	—	4	1	no	—
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₂²⋊C₄×C₆₁	—	2	2	no	—
86	C₂²⋊C₄×C₆₁	—	2	2	no	—
87	C₄²×C₆₁	—	2	1	no	—
88	C₄²×C₆₁	C₄²	1	1	yes	characteristic
89	C₄²×C₆₁	C₄²	1	1	yes	characteristic
90	C₂²⋊Dic₆₁	—	2	4	no	—
91	C₄⋊Dic₆₁	—	4	2	no	—
92	C₄×Dic₆₁	—	4	2	no	—
93	#11758194	—	2	4	no	—
94	#11758194	—	4	2	no	—
95	C₂×#92418	—	4	2	no	—
96	C₂³×C₆₁	—	2	1	no	—
97	C₂³×C₆₁	#117	1	1	yes	characteristic
98	C₂³×C₆₁	C₄×D₄	1	2	yes	—
99	Q₈×C₆₁	—	4	1	no	—
100	Q₈×C₆₁	—	4	1	no	—
101	D₄×C₆₁	—	4	1	no	—
102	D₄×C₆₁	—	4	2	no	—
103	D₄×C₆₁	—	4	1	no	—
104	D₄×C₆₁	—	4	2	no	—
105	D₄×C₆₁	—	8	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	8	1	no	—
116	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
119	C₂×Dic₆₁	—	4	4	no	—
120	C₈×C₆₁	—	2	1	no	—
121	C₈×C₆₁	—	2	2	no	—
122	C₈×C₆₁	—	4	1	no	—
123	C₈×C₆₁	—	4	1	no	—
124	C₈×C₆₁	#99	1	2	yes	—
125	#92418	—	4	4	no	—
126	#12634	—	61	1	no	maximal, Sylow
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	1	no	—
129	C₂²×C₆₁	—	4	1	no	—
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	1	no	—
133	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₆₁	—	2	2	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	4	1	no	—
137	C₄×C₆₁	—	4	1	no	—
138	C₄×C₆₁	—	4	1	no	—
139	C₄×C₆₁	C₂×#99	1	1	yes	characteristic
140	C₄×C₆₁	#410	1	1	yes	characteristic
141	Dic₆₁	—	8	4	no	—
142	#2932	D₆₁	1	1	yes	characteristic
143	#2132	—	61	1	no	—
144	#2126	—	61	1	no	—
145	#1757	—	61	2	no	—
146	#1395	—	61	2	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	—	4	2	no	—
149	C₂×C₆₁	—	4	1	no	—
150	C₂×C₆₁	—	8	1	no	—
151	C₂×C₆₁	C₄×#99	1	1	yes	characteristic
152	C₂×#131	Dic₆₁	1	1	yes	characteristic, complemented
153	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
154	#532	C₂×D₆₁	1	1	yes	characteristic
155	#435	—	2	2	no	—
156	#435	—	2	2	no	—
157	#423	Dic₆₁	1	1	yes	characteristic, complemented
158	#413	—	61	2	no	—
159	#410	—	61	2	no	—
160	#408	Dic₆₁	1	1	yes	characteristic, complemented
161	#408	C₂×D₆₁	1	1	yes	characteristic
162	#405	—	122	2	no	—
163	C₂×#97	Dic₆₁	1	1	yes	characteristic, complemented
164	#356	—	61	2	no	—
165	#355	—	61	2	no	—
166	#354	—	61	2	no	—
167	#353	—	61	2	no	—
168	#344	—	122	2	no	—
169	C₆₁	#12634	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	C₂×Dic₆₁	1	1	yes	characteristic
171	C₂×#41	C₂²×D₆₁	1	1	yes	characteristic
172	#131	—	4	2	no	—
173	#131	—	4	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	122	2	no	—
176	C₂×M₁₆	C₂×Dic₆₁	1	1	yes	characteristic
177	C₂×M₁₆	C₂×Dic₆₁	1	1	yes	characteristic
178	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
183	C₄⋊Q₈	C₄×D₆₁	1	1	yes	characteristic
184	#127	C₄×D₆₁	1	1	yes	characteristic
185	#124	C₄×D₆₁	1	1	yes	characteristic
186	#124	C₄×D₆₁	1	1	yes	characteristic
187	#121	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	#117	—	122	2	no	—
190	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	#101	—	61	4	no	—
193	#100	—	122	2	no	—
194	#99	—	61	4	no	—
195	#99	—	122	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	244	2	no	—
201	C₄×C₈	—	244	2	no	—
202	#41	—	4	1	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	C₄×Dic₆₁	1	1	yes	characteristic
207	C₂×Q₈	C₂×C₄×D₆₁	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₄×Dic₆₁	1	1	yes	characteristic
211	C₂×D₄	C₂×C₄×D₆₁	1	1	yes	characteristic
212	C₂×D₄	#11758222	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	122	4	no	—
222	M₁₆	—	244	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	244	2	no	—
231	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	244	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	122	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	244	2	no	—
241	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
242	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
243	C₂³	—	2	1	no	—
244	C₂³	#14519736	1	1	yes	characteristic
245	C₂³	C₄×C₂²⋊D₆₁	1	2	yes	—
246	Q₈	—	4	1	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	4	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	244	4	no	—
264	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
267	C₈	—	2	1	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	244	4	no	—
272	C₈	#14519692	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	488	4	no	—
286	C₄	C₂×#14519692	1	1	yes	characteristic
287	C₄	#16839736	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×#14519692	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519692
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×#41	C₂²×D₆₁
Fitting	C₆₁×#2932	C₂
socle	C₂×C₆₁	C₄×#99
prime radical	C₆₁×#532	C₂²
cosocle	C₂×#41×C₆₁	C₂³
hypercenter	#2932	D₆₁
hypocenter	C₆₁	#12634
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2932

Complements

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

N	H ≅ G/N	type
3. #21445619	C₂	semidirect
4. #21443948	C₂	semidirect
9. #16839561	C₄	semidirect
10. #16839559	C₄	semidirect
14. C₂×#131×C₆₁	C₄	semidirect
19. C₆₁×#423	C₄	semidirect
20. C₆₁×#408	C₄	semidirect
22. C₂×#97×C₆₁	C₄	semidirect
152. C₂×#131	Dic₆₁	semidirect
157. #423	Dic₆₁	semidirect
160. #408	Dic₆₁	semidirect
163. C₂×#97	Dic₆₁	semidirect
169. C₆₁	#12634	semidirect

Maximal subgroups

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

subgroup	index
#21447504	2
#21447484	2
#21445619	2
#21443948	2
C₆₁×#2932	2
#12634	61