Subgroups — #29688897

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456342	C₂	1	1	yes	maximal, characteristic
2	#21454751	C₂	1	2	yes	maximal
3	#21446798	C₂	1	2	yes	maximal
4	#21446568	C₂	1	1	yes	maximal, characteristic
5	#21446566	C₂	1	2	yes	maximal
6	#21446565	C₂	1	1	yes	maximal, characteristic
7	#21445992	C₂	1	1	yes	maximal, characteristic
8	#21445704	C₂	1	2	yes	maximal
9	#21445703	C₂	1	1	yes	maximal, characteristic
10	#21445507	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1773	C₂	1	1	yes	maximal, characteristic
12	#16840315	C₂²	1	2	yes	—
13	#16840170	C₄	1	4	yes	complemented
14	#16840169	C₄	1	4	yes	complemented
15	#16839890	—	2	4	no	—
16	C₈×Dic₁₂₂	—	2	4	no	—
17	C₂×#14519713	C₂²	1	1	yes	characteristic
18	C₂×#14519712	C₂²	1	2	yes	—
19	#16839800	C₂²	1	1	yes	characteristic
20	#16839798	C₂²	1	2	yes	—
21	#16839790	C₂²	1	2	yes	—
22	#16839789	C₂²	1	2	yes	—
23	#16839785	C₂²	1	2	yes	—
24	#16839717	C₂²	1	2	yes	—
25	C₂×#14519687	C₂²	1	1	yes	characteristic
26	C₂×#14519687	C₂²	1	1	yes	characteristic
27	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
30	#16839608	C₂²	1	1	yes	characteristic
31	#16839523	C₂²	1	2	yes	—
32	#16839522	C₂²	1	2	yes	—
33	#16839521	C₂²	1	2	yes	—
34	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#427	C₂²	1	1	yes	characteristic
36	C₆₁×#405	C₂²	1	2	yes	—
37	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#339	C₂²	1	2	yes	—
39	C₂×#11758194	C₂³	1	2	yes	—
40	C₂²×#92418	C₂³	1	1	yes	characteristic
41	C₂²×#92418	C₂³	1	1	yes	characteristic
42	#14519776	C₂×C₄	1	4	yes	—
43	#14519768	C₂×C₄	1	4	yes	—
44	#14519754	—	2	4	no	—
45	C₄×Dic₁₂₂	—	2	4	no	—
46	#14519736	C₂³	1	2	yes	—
47	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
49	#14519713	—	2	2	no	—
50	#14519712	—	2	4	no	—
51	#14519711	C₂×C₄	1	2	yes	—
52	#14519710	C₂×C₄	1	4	yes	—
53	#14519709	C₂×C₄	1	2	yes	—
54	#14519708	C₂×C₄	1	4	yes	—
55	C₄⋊#92418	—	2	4	no	—
56	#14519706	—	2	4	no	—
57	#14519706	—	2	4	no	—
58	#14519687	—	2	2	no	—
59	#14519687	C₂×C₄	1	2	yes	—
60	#14519687	C₂×C₄	1	2	yes	—
61	#14519687	Q₈	1	4	yes	complemented
62	C₄×#92418	—	2	2	no	—
63	C₈×Dic₆₁	—	2	2	no	—
64	C₈×Dic₆₁	—	2	2	no	—
65	C₈×Dic₆₁	D₄	1	4	yes	—
66	#14519684	—	2	2	no	—
67	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	#117×C₆₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	#108×C₆₁	—	2	2	no	—
73	#107×C₆₁	C₂×C₄	1	2	yes	—
74	#106×C₆₁	C₂×C₄	1	2	yes	—
75	#97×C₆₁	—	2	2	no	—
76	#97×C₆₁	Q₈	1	4	yes	complemented
77	C₄×C₈×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	1	no	—
79	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₆₁	—	2	1	no	—
81	C₂²×Dic₆₁	#41	1	2	yes	—
82	C₂×Dic₁₂₂	—	2	4	no	—
83	M₁₆×C₆₁	—	2	2	no	—
84	M₁₆×C₆₁	—	2	2	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₆₁	#41	1	2	yes	—
93	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₆₁	—	2	2	no	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	1	no	—
97	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
98	C₂²⋊Dic₆₁	—	2	2	no	—
99	C₄⋊Dic₆₁	—	2	4	no	—
100	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
101	#11758196	—	2	4	no	—
102	#11758196	—	2	4	no	—
103	C₄×Dic₆₁	—	2	2	no	—
104	C₄×Dic₆₁	—	2	1	no	—
105	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
106	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
107	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
108	#11758194	—	2	2	no	—
109	#11758194	—	2	2	no	—
110	C₂×#92418	—	2	1	no	—
111	C₂×#92418	—	2	1	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	C₂²×C₄	1	2	yes	—
114	C₂×#92418	C₂×D₄	1	2	yes	—
115	C₂×#92418	C₂×Q₈	1	2	yes	—
116	C₂×#92418	#41	1	2	yes	—
117	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₆₁	—	2	4	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₆₁	—	2	4	no	—
128	C₂×Dic₆₁	—	2	2	no	—
129	C₂×Dic₆₁	—	4	2	no	—
130	C₂×Dic₆₁	—	4	2	no	—
131	C₂×Dic₆₁	C₄×D₄	1	4	yes	—
132	Dic₁₂₂	—	4	8	no	—
133	C₈×C₆₁	—	4	2	no	—
134	C₈×C₆₁	C₄×Q₈	1	2	yes	—
135	C₈×C₆₁	C₄×Q₈	1	2	yes	—
136	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
137	#92418	—	2	2	no	—
138	#92418	—	4	2	no	—
139	#92418	C₄×Q₈	1	4	yes	—
140	#21737	—	61	1	no	maximal, Sylow
141	C₂²×C₆₁	—	2	2	no	—
142	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₆₁	#533	1	1	yes	characteristic
144	C₂²×C₆₁	#533	1	1	yes	characteristic
145	C₄×C₆₁	—	4	2	no	—
146	C₄×C₆₁	—	4	4	no	—
147	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₄×C₆₁	#532	1	1	yes	characteristic
151	Dic₆₁	—	2	4	no	—
152	Dic₆₁	—	4	2	no	—
153	Dic₆₁	—	4	4	no	—
154	#3162	—	61	1	no	—
155	#2960	—	61	2	no	—
156	#1995	—	61	2	no	—
157	#1945	—	61	1	no	—
158	#1945	—	61	1	no	—
159	#1945	—	61	2	no	—
160	#1849	—	61	1	no	—
161	#1773	—	61	2	no	—
162	#1773	—	61	1	no	—
163	#1773	D₆₁	1	1	yes	characteristic
164	#1726	—	61	1	no	—
165	C₂×C₆₁	—	2	1	no	—
166	C₂×C₆₁	—	2	1	no	—
167	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₆₁	#2955	1	2	yes	—
169	#533	—	61	2	no	—
170	#501	—	61	4	no	—
171	#501	—	61	4	no	—
172	#446	—	122	4	no	—
173	C₈×Q₈	—	122	4	no	—
174	C₂×#108	—	61	2	no	—
175	C₂×#108	—	61	1	no	—
176	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
177	#427	—	61	1	no	—
178	#427	—	61	2	no	—
179	#427	C₂×D₆₁	1	1	yes	characteristic
180	#424	—	61	2	no	—
181	#424	—	61	2	no	—
182	#423	—	61	2	no	—
183	#405	—	61	2	no	—
184	#405	C₂×D₆₁	1	2	yes	—
185	C₂×#97	—	61	1	no	—
186	C₂×#97	—	61	1	no	—
187	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
188	C₂×C₄×C₈	—	61	1	no	—
189	C₂×C₄×C₈	—	61	1	no	—
190	C₂×C₄×C₈	—	61	1	no	—
191	#376	—	61	1	no	—
192	#339	—	61	2	no	—
193	#339	—	61	2	no	—
194	#339	—	61	2	no	—
195	#339	C₂×D₆₁	1	2	yes	—
196	C₆₁	#21737	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	61	2	no	—
198	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
199	C₂²×C₈	—	61	1	no	—
200	C₂²×C₈	—	61	1	no	—
201	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
203	C₄⋊Q₈	—	61	4	no	—
204	#125	—	61	4	no	—
205	C₂²⋊Q₈	—	122	4	no	—
206	C₄×Q₈	—	122	4	no	—
207	#117	—	61	2	no	—
208	#117	C₂²×D₆₁	1	2	yes	—
209	C₂×C₄²	—	61	1	no	—
210	C₂×C₄²	—	61	1	no	—
211	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
212	#108	—	2	2	no	—
213	#108	—	122	4	no	—
214	#108	—	122	2	no	—
215	#107	—	61	2	no	—
216	#107	—	61	4	no	—
217	#107	C₄×D₆₁	1	2	yes	—
218	#106	—	61	2	no	—
219	#106	—	61	4	no	—
220	#106	C₄×D₆₁	1	2	yes	—
221	C₄⋊C₈	—	122	4	no	—
222	C₄⋊C₈	—	122	4	no	—
223	C₄⋊C₈	—	122	4	no	—
224	#97	—	2	2	no	—
225	#97	—	61	2	no	—
226	#97	—	61	2	no	—
227	#97	—	61	4	no	—
228	#97	—	122	2	no	—
229	#97	Dic₁₂₂	1	4	yes	complemented
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	61	4	no	—
232	C₄×C₈	—	122	2	no	—
233	C₄×C₈	—	122	2	no	—
234	C₄×C₈	—	122	2	no	—
235	#95	—	122	2	no	—
236	C₂×Q₈	—	122	4	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	61	2	no	—
239	C₂²×C₄	—	122	1	no	—
240	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	122	2	no	—
244	M₁₆	—	122	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	61	2	no	—
247	C₂×C₈	—	61	2	no	—
248	C₂×C₈	—	61	2	no	—
249	C₂×C₈	—	61	2	no	—
250	C₂×C₈	—	122	1	no	—
251	C₂×C₈	—	122	1	no	—
252	C₂×C₈	—	122	2	no	—
253	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
254	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
257	C₂×C₈	#11758222	1	2	yes	—
258	C₂×C₈	D₄×D₆₁	1	2	yes	—
259	C₂×C₈	Q₈×D₆₁	1	2	yes	—
260	C₄⋊C₄	—	61	4	no	—
261	C₄⋊C₄	—	122	4	no	—
262	C₄⋊C₄	—	122	4	no	—
263	C₄⋊C₄	—	122	4	no	—
264	C₄⋊C₄	C₂×C₄×D₆₁	1	4	yes	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	122	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	61	2	no	—
270	C₄²	—	61	2	no	—
271	C₄²	—	61	2	no	—
272	C₄²	—	122	2	no	—
273	C₄²	—	122	1	no	—
274	C₄²	C₂×Dic₁₂₂	1	2	yes	—
275	C₂³	#41×D₆₁	1	1	yes	characteristic
276	Q₈	—	244	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	61	4	no	—
281	C₂×C₄	—	122	4	no	—
282	C₂×C₄	—	122	2	no	—
283	C₂×C₄	—	244	2	no	—
284	C₂×C₄	—	244	2	no	—
285	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
289	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
290	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	61	4	no	—
293	C₈	—	122	2	no	—
294	C₈	—	244	2	no	—
295	C₈	C₄×Dic₁₂₂	1	2	yes	—
296	C₈	C₄×Dic₁₂₂	1	2	yes	—
297	C₈	#14519776	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
300	C₂²	#16840315	1	1	yes	characteristic
301	C₂²	#16840316	1	1	yes	characteristic
302	C₄	—	4	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	122	4	no	—
305	C₄	—	244	2	no	—
306	C₄	—	244	4	no	—
307	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
308	C₄	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
309	C₄	C₂×#14519776	1	1	yes	characteristic
310	C₄	#16840309	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	C₄×#14519776	1	1	yes	characteristic
314	C₂	#21454719	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1773	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1773	D₆₁
hypocenter	C₆₁	#21737
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
13. #16840170	C₄	semidirect
14. #16840169	C₄	semidirect
61. #14519687	Q₈	semidirect
76. #97×C₆₁	Q₈	semidirect
196. C₆₁	#21737	semidirect
229. #97	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456342	2
#21454751	2
#21446798	2
#21446568	2
#21446566	2
#21446565	2
#21445992	2
#21445704	2
#21445703	2
#21445507	2
C₆₁×#1773	2
#21737	61