Subgroups — #29699936

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446760	C₂	1	2	yes	maximal
2	#21446385	C₂	1	2	yes	maximal
3	#21446378	C₂	1	2	yes	maximal
4	#21445971	C₂	1	1	yes	maximal, characteristic
5	#21445939	C₂	1	2	yes	maximal
6	#21445694	C₂	1	2	yes	maximal
7	#21445676	C₂	1	2	yes	maximal
8	#21445672	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#2295	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₂²	1	2	yes	—
13	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
16	C₂×#14519688	C₂²	1	2	yes	—
17	C₂×#14519688	C₂²	1	2	yes	—
18	C₂×C₄×#92418	C₂²	1	2	yes	—
19	#16839517	C₂²	1	2	yes	—
20	#16839517	C₂²	1	2	yes	—
21	#16839517	C₂²	1	2	yes	—
22	#16839517	C₂²	1	2	yes	—
23	#16839517	C₂²	1	2	yes	—
24	#16839517	C₂²	1	2	yes	—
25	#16839517	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#391	C₄	1	1	yes	characteristic
28	C₆₁×#391	C₄	1	1	yes	characteristic
29	C₆₁×#389	C₄	1	1	yes	characteristic
30	C₆₁×#389	C₄	1	1	yes	characteristic
31	C₆₁×#387	C₄	1	2	yes	—
32	C₆₁×#387	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
35	C₆₁×#376	C₂²	1	2	yes	—
36	C₄³×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²×#92418	C₂³	1	2	yes	—
38	C₂²×#92418	C₂³	1	2	yes	—
39	C₂²×#92418	C₂³	1	2	yes	—
40	C₂²×#92418	C₂³	1	2	yes	—
41	C₄⋊#92418	—	2	4	no	—
42	C₄⋊#92418	—	2	2	no	—
43	C₄⋊#92418	—	2	4	no	—
44	C₄⋊#92418	—	4	1	no	—
45	C₄⋊#92418	—	4	1	no	—
46	C₄⋊#92418	Q₈	1	4	yes	complemented
47	#14519688	—	2	4	no	—
48	#14519688	—	4	2	no	—
49	C₄×#92418	—	4	2	no	—
50	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₆₁	—	2	2	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
68	#95×C₆₁	C₂×C₄	1	2	yes	—
69	#95×C₆₁	C₂×C₄	1	2	yes	—
70	#95×C₆₁	C₂×C₄	1	2	yes	—
71	#95×C₆₁	C₂×C₄	1	2	yes	—
72	#95×C₆₁	C₂×C₄	1	2	yes	—
73	#95×C₆₁	C₂×C₄	1	2	yes	—
74	#95×C₆₁	C₂×C₄	1	2	yes	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	4	1	no	—
77	C₂×Q₈×C₆₁	—	4	1	no	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₆₁	—	2	2	no	—
90	C₄⋊C₄×C₆₁	—	2	4	no	—
91	C₄⋊C₄×C₆₁	—	2	4	no	—
92	C₄⋊C₄×C₆₁	—	4	1	no	—
93	C₄⋊C₄×C₆₁	—	4	1	no	—
94	C₄⋊C₄×C₆₁	—	4	2	no	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	2	no	—
97	C₄²×C₆₁	—	2	2	no	—
98	C₄²×C₆₁	—	2	2	no	—
99	C₄²×C₆₁	—	2	1	no	—
100	C₄²×C₆₁	—	2	1	no	—
101	C₄²×C₆₁	—	2	2	no	—
102	C₄²×C₆₁	—	4	1	no	—
103	C₄²×C₆₁	—	4	1	no	—
104	C₄²×C₆₁	—	4	1	no	—
105	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
106	C₄²×C₆₁	#41	1	2	yes	—
107	C₄²×C₆₁	#41	1	2	yes	—
108	C₄²×C₆₁	#41	1	2	yes	—
109	C₂×#92418	—	2	2	no	—
110	C₂×#92418	—	2	2	no	—
111	C₂×#92418	—	2	2	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	—	2	2	no	—
114	C₂×#92418	—	2	2	no	—
115	C₂×#92418	—	2	2	no	—
116	C₂×#92418	—	2	2	no	—
117	C₂×#92418	—	2	2	no	—
118	C₂×#92418	—	2	2	no	—
119	C₂×#92418	—	2	2	no	—
120	C₂×#92418	C₂×Q₈	1	4	yes	—
121	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
122	Q₈×C₆₁	—	4	4	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	1	no	—
125	C₂×C₄×C₆₁	—	2	1	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	1	no	—
141	C₂×C₄×C₆₁	—	2	1	no	—
142	C₂×C₄×C₆₁	—	2	2	no	—
143	C₂×C₄×C₆₁	—	2	2	no	—
144	C₂×C₄×C₆₁	—	2	2	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	—	2	2	no	—
147	C₂×C₄×C₆₁	—	2	2	no	—
148	C₂×C₄×C₆₁	#117	1	2	yes	—
149	C₂×C₄×C₆₁	#117	1	2	yes	—
150	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
151	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
152	C₂×C₄×C₆₁	#131	1	2	yes	—
153	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
159	#92418	—	2	4	no	—
160	#92418	—	4	2	no	—
161	#92418	—	4	2	no	—
162	#92418	—	4	2	no	—
163	#22824	—	61	1	no	maximal, Sylow
164	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
165	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₆₁	#520	1	1	yes	characteristic
167	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
168	C₂²×C₆₁	#568	1	1	yes	characteristic
169	C₂²×C₆₁	#568	1	1	yes	characteristic
170	C₂²×C₆₁	#568	1	1	yes	characteristic
171	C₄×C₆₁	—	2	2	no	—
172	C₄×C₆₁	—	2	2	no	—
173	C₄×C₆₁	—	2	2	no	—
174	C₄×C₆₁	—	2	2	no	—
175	C₄×C₆₁	—	2	2	no	—
176	C₄×C₆₁	—	4	2	no	—
177	C₄×C₆₁	—	4	2	no	—
178	C₄×C₆₁	—	4	2	no	—
179	C₄×C₆₁	—	4	1	no	—
180	C₄×C₆₁	—	4	2	no	—
181	C₄×C₆₁	#554	1	1	yes	characteristic
182	C₄×C₆₁	#556	1	1	yes	characteristic
183	C₄×C₆₁	#557	1	1	yes	characteristic
184	C₄×C₆₁	#559	1	1	yes	characteristic
185	#2295	D₆₁	1	1	yes	characteristic
186	#1985	—	61	2	no	—
187	#1917	—	61	2	no	—
188	#1916	—	61	2	no	—
189	#1842	—	61	1	no	—
190	#1832	—	61	2	no	—
191	#1771	—	61	2	no	—
192	#1768	—	61	2	no	—
193	#1767	—	61	1	no	—
194	C₂×C₆₁	#2355	1	1	yes	characteristic
195	C₂×C₆₁	#2921	1	1	yes	characteristic
196	C₂×C₆₁	#2925	1	1	yes	characteristic
197	C₂×C₆₁	#2961	1	1	yes	characteristic
198	C₂×C₆₁	#2962	1	1	yes	characteristic
199	C₂×C₆₁	#2992	1	1	yes	characteristic
200	C₂×C₆₁	#2995	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
202	C₂×C₄⋊C₈	—	61	1	no	—
203	C₂×C₄⋊C₈	—	61	1	no	—
204	C₂×C₄⋊C₈	—	61	2	no	—
205	C₂×C₄⋊C₈	—	61	2	no	—
206	C₂×C₄⋊C₈	—	61	1	no	—
207	C₂×C₄⋊C₈	—	61	1	no	—
208	C₂×#97	—	61	2	no	—
209	C₂×#97	—	61	2	no	—
210	C₂×C₄×C₈	—	61	2	no	—
211	#391	Dic₆₁	1	1	yes	characteristic
212	#391	Dic₆₁	1	1	yes	characteristic
213	#389	Dic₆₁	1	1	yes	characteristic
214	#389	Dic₆₁	1	1	yes	characteristic
215	#387	Dic₆₁	1	2	yes	—
216	#387	Dic₆₁	1	2	yes	—
217	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
218	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
219	#376	C₂×D₆₁	1	2	yes	—
220	C₄³	C₂×D₆₁	1	1	yes	characteristic
221	#336	—	61	2	no	—
222	#336	—	61	2	no	—
223	#336	—	61	2	no	—
224	#336	—	61	2	no	—
225	#336	—	61	2	no	—
226	#336	—	61	2	no	—
227	#336	—	61	2	no	—
228	C₆₁	#22824	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
230	C₂²×C₈	—	61	2	no	—
231	C₂²×C₈	—	61	2	no	—
232	C₂²×C₈	—	61	2	no	—
233	C₂²×C₈	—	61	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
239	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
242	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
243	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
244	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
245	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
247	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
249	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
251	C₄⋊C₈	—	61	4	no	—
252	C₄⋊C₈	—	122	4	no	—
253	C₄⋊C₈	—	122	2	no	—
254	C₄⋊C₈	—	122	4	no	—
255	C₄⋊C₈	—	244	1	no	—
256	C₄⋊C₈	—	244	1	no	—
257	#97	—	122	4	no	—
258	#97	—	244	2	no	—
259	C₄×C₈	—	244	2	no	—
260	#95	C₂×Dic₆₁	1	2	yes	—
261	#95	C₂×Dic₆₁	1	2	yes	—
262	#95	C₂×Dic₆₁	1	2	yes	—
263	#95	C₂×Dic₆₁	1	2	yes	—
264	#95	C₂×Dic₆₁	1	2	yes	—
265	#95	C₂×Dic₆₁	1	2	yes	—
266	#95	C₂×Dic₆₁	1	2	yes	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
271	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
272	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
273	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
278	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
279	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
280	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
281	C₂×C₈	—	61	4	no	—
282	C₂×C₈	—	122	2	no	—
283	C₂×C₈	—	122	2	no	—
284	C₂×C₈	—	122	2	no	—
285	C₂×C₈	—	122	2	no	—
286	C₂×C₈	—	122	2	no	—
287	C₂×C₈	—	122	2	no	—
288	C₂×C₈	—	122	2	no	—
289	C₂×C₈	—	122	2	no	—
290	C₂×C₈	—	122	2	no	—
291	C₂×C₈	—	122	2	no	—
292	C₂×C₈	—	122	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	4	1	no	—
308	C₄²	—	4	1	no	—
309	C₄²	#11758222	1	2	yes	—
310	C₄²	#11758222	1	2	yes	—
311	C₄²	Q₈×D₆₁	1	2	yes	—
312	C₄²	#11758226	1	2	yes	—
313	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
314	Q₈	—	4	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	#14519736	1	2	yes	—
341	C₂×C₄	#14519736	1	2	yes	—
342	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
343	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
344	C₂×C₄	#14519784	1	2	yes	—
345	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
346	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
347	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
348	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
349	C₂×C₄	#14519823	1	1	yes	characteristic
350	C₂×C₄	#14519824	1	1	yes	characteristic
351	C₈	—	122	4	no	—
352	C₈	—	244	2	no	—
353	C₈	—	244	2	no	—
354	C₈	—	244	2	no	—
355	C₂²	C₂×#14519736	1	1	yes	characteristic
356	C₂²	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
357	C₂²	#16840245	1	1	yes	characteristic
358	C₂²	C₂×#14519784	1	1	yes	characteristic
359	C₂²	#16840529	1	1	yes	characteristic
360	C₂²	#16840529	1	1	yes	characteristic
361	C₂²	#16840529	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	#16840472	1	1	yes	characteristic
373	C₄	#16840485	1	1	yes	characteristic
374	C₄	#16840490	1	1	yes	characteristic
375	C₄	#16840497	1	1	yes	characteristic
376	C₂	#21448776	1	1	yes	characteristic
377	C₂	#21454457	1	1	yes	characteristic
378	C₂	#21454485	1	1	yes	characteristic
379	C₂	#21454760	1	1	yes	characteristic
380	C₂	#21454768	1	1	yes	characteristic
381	C₂	#21454958	1	1	yes	characteristic
382	C₂	#21454989	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₆₁×#2295	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#2295	D₆₁
hypocenter	C₆₁	#22824
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
46. C₄⋊#92418	Q₈	semidirect
228. C₆₁	#22824	semidirect

Maximal subgroups

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

subgroup	index
#21446760	2
#21446385	2
#21446378	2
#21445971	2
#21445939	2
#21445694	2
#21445676	2
#21445672	2
C₆₁×#2295	2
#22824	61