Subgroups — #29834972

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#21448019	C₂	1	2	yes	maximal
3	#21447968	C₂	1	2	yes	maximal, complemented
4	#21447866	C₂	1	2	yes	maximal, complemented
5	#21447752	C₂	1	2	yes	maximal
6	C₄×#14519723	C₂	1	4	yes	maximal, complemented
7	#21447618	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₂₂	—	2	1	no	—
11	C₂×#14519723	C₂²	1	4	yes	—
12	C₄×Dic₂₄₄	—	2	1	no	—
13	C₄×Dic₂₄₄	—	2	1	no	—
14	C₄×Dic₂₄₄	C₂²	1	2	yes	—
15	C₄×Dic₂₄₄	C₂²	1	2	yes	—
16	C₂×#14519711	C₂²	1	1	yes	characteristic
17	C₂×#14519702	—	2	1	no	—
18	#16839587	C₂²	1	4	yes	—
19	C₄⋊#11758185	C₂²	1	2	yes	—
20	#16839566	C₂²	1	4	yes	—
21	#16839565	C₂²	1	4	yes	—
22	C₂²⋊#11758185	C₂²	1	2	yes	—
23	C₄×#11758185	C₂²	1	2	yes	—
24	C₆₁×#533	—	2	1	no	—
25	C₆₁×#500	C₂²	1	1	yes	characteristic
26	C₆₁×#494	C₂²	1	1	yes	characteristic
27	C₆₁×#487	—	2	1	no	—
28	C₆₁×#467	—	2	1	no	—
29	C₆₁×#467	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₆₁	—	2	1	no	—
31	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₂₂	—	2	1	no	—
34	C₂×Dic₂₄₄	—	2	1	no	—
35	C₂×Dic₂₄₄	—	2	4	no	—
36	C₂×Dic₂₄₄	C₂³	1	2	yes	—
37	C₄×Dic₁₂₂	—	2	2	no	—
38	C₄×Dic₁₂₂	—	2	2	no	—
39	C₄×Dic₁₂₂	—	4	1	no	—
40	C₄×Dic₁₂₂	—	4	1	no	—
41	C₂×C₄⋊Dic₆₁	—	2	1	no	—
42	C₂×#11758196	—	4	1	no	—
43	C₂×C₄×Dic₆₁	—	4	1	no	—
44	#14519723	—	2	8	no	—
45	C₂×#11758185	C₂³	1	4	yes	—
46	#14519711	—	2	1	no	—
47	#14519711	C₂³	1	2	yes	—
48	#14519702	—	2	2	no	—
49	#14519702	—	2	2	no	—
50	C₂×Q₁₆×C₆₁	—	2	1	no	—
51	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₁	—	2	1	no	—
55	C₄⋊Q₈×C₆₁	—	2	1	no	—
56	#125×C₆₁	—	4	1	no	—
57	C₂²⋊Q₈×C₆₁	—	2	2	no	—
58	C₂²⋊Q₈×C₆₁	—	4	1	no	—
59	C₄×Q₈×C₆₁	—	2	2	no	—
60	C₄×Q₈×C₆₁	—	4	1	no	—
61	#117×C₆₁	—	4	1	no	—
62	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
63	#107×C₆₁	—	2	1	no	—
64	#107×C₆₁	C₂³	1	2	yes	—
65	#103×C₆₁	—	2	2	no	—
66	#103×C₆₁	—	2	2	no	—
67	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₆₁	—	2	2	no	—
72	C₂×Q₈×C₆₁	—	4	1	no	—
73	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₆₁	—	4	1	no	—
77	C₂²×Dic₆₁	—	4	1	no	—
78	C₂×Dic₁₂₂	—	2	4	no	—
79	C₂×Dic₁₂₂	—	2	2	no	—
80	C₂×Dic₁₂₂	—	4	2	no	—
81	C₂×Dic₁₂₂	—	4	1	no	—
82	Q₁₆×C₆₁	—	2	4	no	—
83	Q₁₆×C₆₁	—	4	2	no	—
84	C₂×C₈×C₆₁	—	2	2	no	—
85	C₂×C₈×C₆₁	—	4	1	no	—
86	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₁	—	2	2	no	—
91	C₄⋊C₄×C₆₁	—	4	1	no	—
92	C₄⋊C₄×C₆₁	—	4	1	no	—
93	C₄⋊C₄×C₆₁	—	4	2	no	—
94	C₄⋊C₄×C₆₁	—	4	2	no	—
95	C₄⋊C₄×C₆₁	—	4	1	no	—
96	C₂²⋊C₄×C₆₁	—	4	2	no	—
97	C₄²×C₆₁	—	4	2	no	—
98	C₄²×C₆₁	D₈	1	1	yes	characteristic
99	C₄²×C₆₁	D₈	1	1	yes	characteristic
100	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₄⋊Dic₆₁	—	2	2	no	—
103	C₄⋊Dic₆₁	—	4	1	no	—
104	#11758196	—	4	2	no	—
105	#11758196	—	4	2	no	—
106	C₄×Dic₆₁	—	4	2	no	—
107	C₄×Dic₆₁	—	4	2	no	—
108	Dic₂₄₄	—	2	4	no	—
109	Dic₂₄₄	—	2	2	no	—
110	Dic₂₄₄	#41	1	4	yes	—
111	#11758185	—	2	4	no	—
112	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₆₁	—	4	2	no	—
114	Q₈×C₆₁	—	4	2	no	—
115	Q₈×C₆₁	—	8	2	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	2	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	Q₃₂	1	4	yes	complemented
122	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₆₁	—	4	4	no	—
129	C₂×Dic₆₁	—	4	2	no	—
130	C₂×Dic₆₁	—	4	2	no	—
131	C₂×Dic₆₁	—	8	2	no	—
132	Dic₁₂₂	—	2	4	no	—
133	Dic₁₂₂	—	4	4	no	—
134	Dic₁₂₂	—	4	2	no	—
135	C₈×C₆₁	—	2	1	no	—
136	C₈×C₆₁	—	2	1	no	—
137	C₈×C₆₁	—	2	1	no	—
138	C₈×C₆₁	C₂×#41	1	2	yes	—
139	#33664	—	61	1	no	maximal, Sylow
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	2	1	no	—
142	C₂²×C₆₁	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₆₁	—	2	1	no	—
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	8	2	no	—
148	C₄×C₆₁	—	8	2	no	—
149	C₄×C₆₁	C₂×Q₃₂	1	2	yes	—
150	C₄×C₆₁	C₂×Q₃₂	1	2	yes	—
151	C₄×C₆₁	#525	1	1	yes	characteristic
152	C₄×C₆₁	#575	1	1	yes	characteristic
153	Dic₆₁	—	4	4	no	—
154	Dic₆₁	—	8	2	no	—
155	#3170	D₆₁	1	1	yes	characteristic
156	C₂×C₄×Q₁₆	—	61	1	no	—
157	#2236	—	61	2	no	—
158	#2227	—	61	2	no	—
159	#2209	—	61	2	no	—
160	#2188	—	61	2	no	—
161	C₄×Q₃₂	—	61	4	no	—
162	#2162	—	61	1	no	—
163	C₂×C₆₁	—	2	2	no	—
164	C₂×C₆₁	#3100	1	1	yes	characteristic
165	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
166	C₂×C₆₁	#3414	1	1	yes	characteristic
167	C₂²×Q₁₆	—	61	1	no	—
168	#533	—	2	1	no	—
169	C₂×C₄×Q₈	—	122	1	no	—
170	C₂×Q₃₂	—	61	4	no	—
171	#500	C₂×D₆₁	1	1	yes	characteristic
172	#494	C₂×D₆₁	1	1	yes	characteristic
173	#487	—	2	1	no	—
174	#467	—	2	1	no	—
175	#467	C₂×D₆₁	1	2	yes	—
176	C₄×Q₁₆	—	2	1	no	—
177	C₄×Q₁₆	—	61	2	no	—
178	C₄×Q₁₆	—	61	2	no	—
179	C₄×Q₁₆	—	122	1	no	—
180	C₄×Q₁₆	—	122	1	no	—
181	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
182	C₂×#107	—	61	1	no	—
183	C₂×#103	—	122	1	no	—
184	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
185	#366	—	61	4	no	—
186	C₄⋊C₁₆	—	61	2	no	—
187	#358	—	61	4	no	—
188	#358	—	61	4	no	—
189	C₂²⋊C₁₆	—	61	2	no	—
190	C₄×C₁₆	—	61	2	no	—
191	C₆₁	#33664	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	122	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	61	2	no	—
195	C₂×Q₁₆	—	122	4	no	—
196	C₂×Q₁₆	—	122	1	no	—
197	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	#125	—	4	1	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	4	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	1	no	—
207	C₄×Q₈	—	122	2	no	—
208	C₄×Q₈	—	122	2	no	—
209	C₄×Q₈	—	244	1	no	—
210	C₄×Q₈	—	244	1	no	—
211	#117	—	4	1	no	—
212	C₂×C₄⋊C₄	—	122	1	no	—
213	C₂×C₄⋊C₄	—	244	1	no	—
214	C₂×C₄²	—	244	1	no	—
215	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
216	Q₃₂	—	122	8	no	—
217	C₂×C₁₆	—	61	4	no	—
218	#107	—	2	1	no	—
219	#107	—	61	2	no	—
220	#107	—	122	1	no	—
221	#107	C₂²×D₆₁	1	2	yes	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#103	—	122	2	no	—
225	#103	—	122	2	no	—
226	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
227	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	122	4	no	—
233	C₂×Q₈	—	122	2	no	—
234	C₂×Q₈	—	244	2	no	—
235	C₂×Q₈	—	244	1	no	—
236	C₂²×C₄	—	244	1	no	—
237	C₂²×C₄	—	244	1	no	—
238	C₂²×C₄	#11758199	1	1	yes	characteristic
239	C₂²×C₄	#11758199	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	4	2	no	—
243	Q₁₆	—	61	4	no	—
244	Q₁₆	—	122	4	no	—
245	Q₁₆	—	122	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	122	2	no	—
259	C₄⋊C₄	—	244	1	no	—
260	C₄⋊C₄	—	244	2	no	—
261	C₄⋊C₄	—	244	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	244	2	no	—
265	C₄²	—	244	2	no	—
266	C₄²	#11758199	1	1	yes	characteristic
267	C₄²	#11758199	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
270	C₁₆	—	122	4	no	—
271	C₂³	C₂×#11758199	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	8	2	no	—
275	Q₈	—	122	4	no	—
276	Q₈	—	244	4	no	—
277	Q₈	—	244	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	488	2	no	—
284	C₂×C₄	—	244	2	no	—
285	C₂×C₄	—	244	4	no	—
286	C₂×C₄	—	244	2	no	—
287	C₂×C₄	#14519723	1	4	yes	complemented
288	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
289	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
290	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
291	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
292	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#41×D₆₁	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	C₂×#14519723	1	2	yes	—
301	C₂²	C₂²×#11758199	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	488	2	no	—
308	C₄	—	244	4	no	—
309	C₄	C₂×#14519723	1	2	yes	—
310	C₄	C₂×#14519723	1	2	yes	—
311	C₄	#16840269	1	1	yes	characteristic
312	C₄	#16840560	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#21455815	1	1	yes	characteristic
315	C₂	C₂²×#14519723	1	1	yes	characteristic
316	C₂	#21458953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#3170	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#3170	D₆₁
hypocenter	C₆₁	#33664
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #21447968	C₂	semidirect
4. #21447866	C₂	semidirect
6. C₄×#14519723	C₂	semidirect
121. C₂×C₄×C₆₁	Q₃₂	semidirect
191. C₆₁	#33664	semidirect
287. C₂×C₄	#14519723	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
#21448019	2
#21447968	2
#21447866	2
#21447752	2
C₄×#14519723	2
#21447618	2
C₆₁×#3170	2
#33664	61