Subgroups — #29588131

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454409	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839534	C₂	1	2	yes	maximal
3	C₂×#16839520	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839518	C₂	1	2	yes	maximal
5	C₆₁×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	2	yes	—
7	C₂×#14519736	C₂²	1	2	yes	—
8	#16839804	—	2	4	no	—
9	#16839800	—	2	2	no	—
10	#16839534	—	2	4	no	—
11	#16839520	—	2	2	no	—
12	#16839518	—	2	4	no	—
13	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#424	—	2	2	no	—
17	C₆₁×#424	—	2	2	no	—
18	C₆₁×#423	C₄	1	4	yes	complemented
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	—	2	2	no	—
25	C₂²×#92418	Q₈	1	4	yes	—
26	#14519736	—	2	2	no	—
27	#14519736	—	2	2	no	—
28	#14519736	—	2	2	no	—
29	#14519736	D₄	1	4	yes	—
30	C₂×C₄⋊Dic₆₁	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
32	C₂×C₄×Dic₆₁	—	2	2	no	—
33	#14519711	—	2	4	no	—
34	#14519709	—	2	4	no	—
35	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	—	2	2	no	—
38	C₂×M₁₆×C₆₁	—	2	1	no	—
39	C₂×M₁₆×C₆₁	—	2	2	no	—
40	C₂×M₁₆×C₆₁	—	2	2	no	—
41	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
44	C₂²×C₈×C₆₁	—	2	1	no	—
45	C₂²×C₈×C₆₁	D₄	1	2	yes	—
46	#117×C₆₁	—	2	1	no	—
47	#117×C₆₁	—	2	1	no	—
48	#117×C₆₁	D₄	1	2	yes	—
49	#117×C₆₁	Q₈	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
52	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
53	C₄⋊C₈×C₆₁	—	2	8	no	—
54	C₂⁴×C₆₁	#41	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₆₁	#41	1	2	yes	—
64	C₂²×C₄×C₆₁	#41	1	2	yes	—
65	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
66	C₂²×Dic₆₁	—	4	2	no	—
67	M₁₆×C₆₁	—	2	4	no	—
68	M₁₆×C₆₁	—	2	4	no	—
69	M₁₆×C₆₁	—	2	4	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	M₁₆×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	4	2	no	—
76	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₂²⋊C₄×C₆₁	—	2	2	no	—
80	C₂²⋊C₄×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	2	2	no	—
82	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
83	C₂²⋊Dic₆₁	—	2	4	no	—
84	C₂²⋊Dic₆₁	—	4	2	no	—
85	C₄⋊Dic₆₁	—	2	4	no	—
86	C₄⋊Dic₆₁	—	4	2	no	—
87	C₄×Dic₆₁	—	2	4	no	—
88	C₄×Dic₆₁	—	4	2	no	—
89	#11758194	—	4	2	no	—
90	#11758194	—	4	2	no	—
91	#11758194	—	4	2	no	—
92	#11758194	—	4	2	no	—
93	C₂×#92418	—	2	4	no	—
94	C₂×#92418	—	2	4	no	—
95	C₂×#92418	—	2	4	no	—
96	C₂³×C₆₁	—	8	1	no	—
97	C₂³×C₆₁	#117	1	1	yes	characteristic
98	C₂³×C₆₁	#121	1	1	yes	characteristic
99	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₆₁	#123	1	2	yes	—
101	C₂³×C₆₁	#125	1	2	yes	—
102	C₂×C₄×C₆₁	—	2	4	no	—
103	C₂×C₄×C₆₁	—	8	2	no	—
104	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#121	1	2	yes	—
110	C₂×C₄×C₆₁	#121	1	2	yes	—
111	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
117	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#125	1	2	yes	—
119	C₂×C₄×C₆₁	#126	1	2	yes	—
120	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×Dic₆₁	—	4	4	no	—
123	C₂×Dic₆₁	—	8	4	no	—
124	C₈×C₆₁	—	2	4	no	—
125	C₈×C₆₁	—	4	2	no	—
126	#92418	—	4	4	no	—
127	#11245	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	—	4	2	no	—
129	C₂²×C₆₁	—	4	1	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	1	no	—
132	C₂²×C₆₁	—	4	2	no	—
133	C₂²×C₆₁	#380	1	1	yes	characteristic
134	C₂²×C₆₁	#387	1	2	yes	—
135	C₂²×C₆₁	#394	1	1	yes	characteristic
136	C₂²×C₆₁	#398	1	1	yes	characteristic
137	C₂²×C₆₁	#398	1	2	yes	—
138	C₄×C₆₁	—	4	4	no	—
139	C₄×C₆₁	#386	1	1	yes	characteristic
140	C₄×C₆₁	#389	1	2	yes	—
141	C₄×C₆₁	#391	1	1	yes	characteristic
142	C₄×C₆₁	#395	1	1	yes	characteristic
143	C₄×C₆₁	#396	1	2	yes	—
144	C₄×C₆₁	#400	1	1	yes	characteristic
145	Dic₆₁	—	8	4	no	—
146	#2911	—	61	1	no	—
147	#2905	D₆₁	1	1	yes	characteristic
148	C₂×#344	—	61	2	no	—
149	C₂×#338	—	61	1	no	—
150	C₂×H(3,ℤ/4)	—	61	2	no	—
151	C₂×C₆₁	—	2	1	no	—
152	C₂×C₆₁	—	2	1	no	—
153	C₂×C₆₁	—	2	2	no	—
154	C₂×C₆₁	—	2	1	no	—
155	C₂×C₆₁	—	2	2	no	—
156	C₂×C₆₁	#1441	1	1	yes	characteristic
157	C₂²×M₁₆	—	61	2	no	—
158	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
159	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
160	C₂×#117	—	61	2	no	—
161	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
162	#428	—	122	4	no	—
163	#427	—	122	2	no	—
164	#424	—	2	2	no	—
165	#424	—	2	2	no	—
166	#423	Dic₆₁	1	4	yes	complemented
167	#344	—	122	4	no	—
168	#338	—	122	2	no	—
169	H(3,ℤ/4)	—	122	4	no	—
170	C₆₁	#11245	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	122	2	no	—
178	C₂×M₁₆	—	122	2	no	—
179	C₂×M₁₆	—	122	2	no	—
180	C₂×M₁₆	—	122	2	no	—
181	C₂×M₁₆	—	122	2	no	—
182	C₂×M₁₆	—	122	2	no	—
183	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
184	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	61	4	no	—
188	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
189	#117	—	2	1	no	—
190	#117	—	2	1	no	—
191	#117	—	61	4	no	—
192	#117	—	122	2	no	—
193	#117	—	122	2	no	—
194	#117	—	122	2	no	—
195	#117	Dic₁₂₂	1	2	yes	—
196	#117	D₂₄₄	1	2	yes	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	122	2	no	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	122	2	no	—
201	C₂×C₄²	—	122	2	no	—
202	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
203	#107	—	122	4	no	—
204	#106	—	122	4	no	—
205	C₄⋊C₈	—	2	8	no	—
206	C₂⁴	#11758222	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	244	2	no	—
209	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
211	C₂²×C₄	#11758222	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₆₁	1	2	yes	—
213	C₂²×C₄	#11758224	1	2	yes	—
214	C₂²×C₄	Q₈×D₆₁	1	2	yes	—
215	C₂²×C₄	#11758226	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	244	2	no	—
225	M₁₆	—	244	2	no	—
226	M₁₆	—	244	2	no	—
227	M₁₆	—	244	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	122	4	no	—
233	C₂×C₈	—	122	4	no	—
234	C₂×C₈	—	122	4	no	—
235	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	122	4	no	—
239	C₄⋊C₄	—	244	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	122	4	no	—
243	C₂²⋊C₄	—	244	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	122	4	no	—
246	C₄²	—	244	2	no	—
247	C₄²	C₄⋊Dic₆₁	1	4	yes	—
248	C₂³	—	8	1	no	—
249	C₂³	#14519736	1	1	yes	characteristic
250	C₂³	#14519751	1	1	yes	characteristic
251	C₂³	#14519756	1	1	yes	characteristic
252	C₂³	#14519762	1	2	yes	—
253	C₂³	#14519767	1	2	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	488	4	no	—
257	C₂×C₄	—	244	4	no	—
258	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
261	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
262	C₂×C₄	#14519745	1	1	yes	characteristic
263	C₂×C₄	#14519749	1	2	yes	—
264	C₂×C₄	#14519751	1	1	yes	characteristic
265	C₂×C₄	#14519752	1	2	yes	—
266	C₂×C₄	#14519753	1	2	yes	—
267	C₂×C₄	#14519755	1	2	yes	—
268	C₂×C₄	#14519756	1	1	yes	characteristic
269	C₂×C₄	#14519757	1	2	yes	—
270	C₂×C₄	#14519758	1	1	yes	characteristic
271	C₂×C₄	#14519766	1	1	yes	characteristic
272	C₂×C₄	#14519768	1	2	yes	—
273	C₂×C₄	#14519772	1	2	yes	—
274	C₂×C₄	#14519775	1	1	yes	characteristic
275	C₂×C₄	#14519776	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	244	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	#16839622	1	1	yes	characteristic
285	C₂²	#16839648	1	2	yes	—
286	C₂²	#16839679	1	1	yes	characteristic
287	C₂²	#16839692	1	1	yes	characteristic
288	C₂²	#16839695	1	2	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	488	4	no	—
291	C₄	#16839642	1	1	yes	characteristic
292	C₄	#16839661	1	2	yes	—
293	C₄	#16839669	1	1	yes	characteristic
294	C₄	#16839682	1	1	yes	characteristic
295	C₄	#16839684	1	2	yes	—
296	C₄	#16839701	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#21444072	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839669
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#2905	C₂
socle	C₂×C₆₁	#1441
prime radical	C₂⁴×C₆₁	#41
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#2905	D₆₁
hypocenter	C₆₁	#11245
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
18. C₆₁×#423	C₄	semidirect
166. #423	Dic₆₁	semidirect
170. C₆₁	#11245	semidirect

Maximal subgroups

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

subgroup	index
#21454409	2
C₂×#16839534	2
C₂×#16839520	2
C₂×#16839518	2
C₆₁×#2905	2
#11245	61