Subgroups — #29706610

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 370 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	#21446756	C₂	1	2	yes	maximal
3	#21446385	C₂	1	2	yes	maximal
4	#21446385	C₂	1	2	yes	maximal
5	#21446378	C₂	1	2	yes	maximal
6	#21446378	C₂	1	2	yes	maximal
7	#21445962	C₂	1	1	yes	maximal, characteristic
8	#21445962	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×#14519688	C₂²	1	2	yes	—
13	C₂×C₄×#92418	C₂²	1	2	yes	—
14	#16839517	C₂²	1	1	yes	characteristic
15	#16839517	C₂²	1	2	yes	—
16	#16839517	C₂²	1	1	yes	characteristic
17	#16839517	C₂²	1	1	yes	characteristic
18	#16839517	C₂²	1	2	yes	—
19	#16839517	C₂²	1	2	yes	—
20	#16839517	C₂²	1	2	yes	—
21	#16839517	C₂²	1	1	yes	characteristic
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	C₄	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#391	C₄	1	2	yes	—
29	C₆₁×#389	C₄	1	2	yes	—
30	C₆₁×#389	C₄	1	1	yes	characteristic
31	C₆₁×#389	C₄	1	1	yes	characteristic
32	C₆₁×#384	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
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	—	4	2	no	—
42	C₄⋊#92418	—	4	2	no	—
43	#14519688	—	4	2	no	—
44	C₄×#92418	—	4	2	no	—
45	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₆₁	—	2	4	no	—
47	C₄×Q₈×C₆₁	—	4	2	no	—
48	C₄×Q₈×C₆₁	—	4	2	no	—
49	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×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	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
67	#95×C₆₁	C₂×C₄	1	2	yes	—
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	C₂×Q₈×C₆₁	—	2	4	no	—
72	C₂×Q₈×C₆₁	—	2	4	no	—
73	C₂×Q₈×C₆₁	—	4	2	no	—
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	4	1	no	—
89	C₄⋊C₄×C₆₁	—	4	1	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₄⋊C₄×C₆₁	—	4	2	no	—
92	C₄⋊C₄×C₆₁	—	4	2	no	—
93	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
94	C₄²×C₆₁	—	2	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	2	no	—
100	C₄²×C₆₁	—	2	2	no	—
101	C₄²×C₆₁	—	2	2	no	—
102	C₄²×C₆₁	—	2	1	no	—
103	C₄²×C₆₁	—	2	1	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	4	1	no	—
106	C₂×#92418	—	2	2	no	—
107	C₂×#92418	—	2	2	no	—
108	C₂×#92418	—	2	2	no	—
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	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	Q₈×C₆₁	—	4	8	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	1	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	2	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	1	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	2	2	no	—
142	C₂×C₄×C₆₁	—	2	1	no	—
143	C₂×C₄×C₆₁	—	2	1	no	—
144	C₂×C₄×C₆₁	—	2	2	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₆₁	#131	1	2	yes	—
149	C₂×C₄×C₆₁	#131	1	2	yes	—
150	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
151	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
154	#92418	—	4	2	no	—
155	#92418	—	4	2	no	—
156	#92418	—	4	2	no	—
157	#92418	—	4	2	no	—
158	#23364	—	61	1	no	maximal, Sylow
159	C₂²×C₆₁	#518	1	1	yes	characteristic
160	C₂²×C₆₁	#519	1	1	yes	characteristic
161	C₂²×C₆₁	#519	1	1	yes	characteristic
162	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
164	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
165	C₂²×C₆₁	#568	1	1	yes	characteristic
166	C₄×C₆₁	—	2	4	no	—
167	C₄×C₆₁	—	2	2	no	—
168	C₄×C₆₁	—	2	2	no	—
169	C₄×C₆₁	—	2	2	no	—
170	C₄×C₆₁	—	4	2	no	—
171	C₄×C₆₁	—	4	1	no	—
172	C₄×C₆₁	—	4	2	no	—
173	C₄×C₆₁	—	4	2	no	—
174	C₄×C₆₁	—	4	1	no	—
175	C₄×C₆₁	—	4	1	no	—
176	C₄×C₆₁	#563	1	2	yes	—
177	C₄×C₆₁	#563	1	2	yes	—
178	C₄×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
179	#1985	—	61	2	no	—
180	#1984	—	61	2	no	—
181	#1917	—	61	2	no	—
182	#1917	—	61	2	no	—
183	#1916	—	61	2	no	—
184	#1916	—	61	2	no	—
185	#1839	—	61	1	no	—
186	#1839	—	61	1	no	—
187	C₂×C₆₁	#2381	1	1	yes	characteristic
188	C₂×C₆₁	#2974	1	1	yes	characteristic
189	C₂×C₆₁	#2981	1	1	yes	characteristic
190	C₂×C₆₁	#2983	1	1	yes	characteristic
191	C₂×C₆₁	#2984	1	1	yes	characteristic
192	C₂×C₆₁	#2985	1	1	yes	characteristic
193	C₂×C₆₁	#2986	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
195	C₂×C₄×Q₈	C₂×D₆₁	1	2	yes	—
196	C₂×C₄⋊C₈	—	61	2	no	—
197	C₂×C₄⋊C₈	—	61	2	no	—
198	C₂×#97	—	61	2	no	—
199	C₂×C₄×C₈	—	61	2	no	—
200	#391	Dic₆₁	1	2	yes	—
201	#389	Dic₆₁	1	2	yes	—
202	#389	Dic₆₁	1	1	yes	characteristic
203	#389	Dic₆₁	1	1	yes	characteristic
204	#384	Dic₆₁	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
207	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
208	C₄³	C₂×D₆₁	1	1	yes	characteristic
209	#336	—	61	1	no	—
210	#336	—	61	2	no	—
211	#336	—	61	2	no	—
212	#336	—	61	2	no	—
213	#336	—	61	1	no	—
214	#336	—	61	1	no	—
215	#336	—	61	2	no	—
216	#336	—	61	1	no	—
217	#336	—	61	2	no	—
218	#336	—	61	2	no	—
219	#336	—	61	2	no	—
220	#336	—	61	2	no	—
221	C₆₁	#23364	1	1	yes	characteristic, complemented, Sylow
222	C₂²×Q₈	C₂×Dic₆₁	1	2	yes	—
223	C₂²×C₈	—	61	2	no	—
224	C₂²×C₈	—	61	2	no	—
225	C₂²×C₈	—	61	2	no	—
226	C₂²×C₈	—	61	2	no	—
227	C₄⋊Q₈	—	2	4	no	—
228	C₄×Q₈	—	4	2	no	—
229	C₄×Q₈	—	4	2	no	—
230	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
236	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
242	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
245	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
246	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
247	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
248	C₄⋊C₈	—	244	2	no	—
249	C₄⋊C₈	—	244	2	no	—
250	#97	—	244	2	no	—
251	C₄×C₈	—	244	2	no	—
252	#95	C₂×Dic₆₁	1	2	yes	—
253	#95	C₂×Dic₆₁	1	2	yes	—
254	#95	C₂×Dic₆₁	1	2	yes	—
255	#95	C₂×Dic₆₁	1	2	yes	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
266	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
270	C₂×C₈	—	122	2	no	—
271	C₂×C₈	—	122	2	no	—
272	C₂×C₈	—	122	2	no	—
273	C₂×C₈	—	122	2	no	—
274	C₂×C₈	—	122	2	no	—
275	C₂×C₈	—	122	2	no	—
276	C₂×C₈	—	122	2	no	—
277	C₂×C₈	—	122	2	no	—
278	C₂×C₈	—	122	2	no	—
279	C₂×C₈	—	122	2	no	—
280	C₂×C₈	—	122	2	no	—
281	C₂×C₈	—	122	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	#11758194	1	4	yes	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	1	no	—
303	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
304	Q₈	—	4	8	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	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	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	C₂×#11758194	1	2	yes	—
332	C₂×C₄	C₂×#11758194	1	4	yes	—
333	C₂×C₄	#14519784	1	2	yes	—
334	C₂×C₄	#14519784	1	2	yes	—
335	C₂×C₄	#14519822	1	2	yes	—
336	C₂×C₄	#14519825	1	1	yes	characteristic
337	C₂×C₄	#14519825	1	1	yes	characteristic
338	C₂×C₄	#14519825	1	2	yes	—
339	C₈	—	244	2	no	—
340	C₈	—	244	2	no	—
341	C₈	—	244	2	no	—
342	C₈	—	244	2	no	—
343	C₂²	#16840237	1	1	yes	characteristic
344	C₂²	#16840240	1	1	yes	characteristic
345	C₂²	#16840240	1	1	yes	characteristic
346	C₂²	C₂²×#11758194	1	1	yes	characteristic
347	C₂²	C₂×#14519784	1	1	yes	characteristic
348	C₂²	C₂×#14519784	1	1	yes	characteristic
349	C₂²	#16840529	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	#16840511	1	2	yes	—
361	C₄	#16840511	1	2	yes	—
362	C₂	#21449000	1	1	yes	characteristic
363	C₂	#21454854	1	1	yes	characteristic
364	C₂	#21454895	1	1	yes	characteristic
365	C₂	#21454904	1	1	yes	characteristic
366	C₂	#21454908	1	1	yes	characteristic
367	C₂	#21454914	1	1	yes	characteristic
368	C₂	#21454921	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₆₁	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#23364
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
221. C₆₁	#23364	semidirect

Maximal subgroups

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

subgroup	index
#21446760	2
#21446756	2
#21446385	2
#21446378	2
#21445962	2
C₄×C₄⋊Q₈×C₆₁	2
#23364	61