Subgroups — #29700021

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446756	C₂	1	2	yes	maximal
2	#21446385	C₂	1	4	yes	maximal
3	#21445962	C₂	1	2	yes	maximal
4	#21445936	C₂	1	1	yes	maximal, characteristic
5	#21445688	C₂	1	2	yes	maximal
6	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
7	#21445568	C₂	1	2	yes	maximal
8	C₄²×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#92418	C₂²	1	2	yes	—
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	C₂×#14519688	C₂²	1	4	yes	—
12	C₂×C₄×#92418	C₂²	1	2	yes	—
13	#16839517	C₂²	1	1	yes	characteristic
14	#16839517	C₂²	1	1	yes	characteristic
15	#16839517	C₂²	1	2	yes	—
16	#16839517	C₂²	1	2	yes	—
17	#16839517	C₂²	1	4	yes	—
18	#16839517	C₂²	1	4	yes	—
19	#16839517	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
29	C₄³×C₆₁	C₂²	1	1	yes	characteristic
30	C₄³×C₆₁	C₂²	1	2	yes	—
31	C₂²×#92418	C₂³	1	2	yes	—
32	C₂²×#92418	C₂³	1	4	yes	—
33	C₂²×#92418	C₂³	1	2	yes	—
34	C₄⋊#92418	—	2	4	no	—
35	C₄⋊#92418	—	2	4	no	—
36	#14519688	—	2	8	no	—
37	C₄×#92418	—	2	4	no	—
38	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₆₁	—	2	4	no	—
40	C₄×Q₈×C₆₁	—	2	4	no	—
41	C₄×Q₈×C₆₁	—	2	2	no	—
42	C₄×Q₈×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	#95×C₆₁	C₂×C₄	1	2	yes	—
63	#95×C₆₁	C₂×C₄	1	2	yes	—
64	#95×C₆₁	C₂×C₄	1	2	yes	—
65	C₂×Q₈×C₆₁	—	2	2	no	—
66	C₂×Q₈×C₆₁	—	2	2	no	—
67	C₂×Q₈×C₆₁	M₁₆	1	4	yes	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₄⋊C₄×C₆₁	—	2	4	no	—
82	C₄⋊C₄×C₆₁	—	2	4	no	—
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄²×C₆₁	—	2	4	no	—
86	C₄²×C₆₁	—	2	2	no	—
87	C₄²×C₆₁	—	2	4	no	—
88	C₄²×C₆₁	—	2	2	no	—
89	C₄²×C₆₁	—	2	4	no	—
90	C₄²×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	2	4	no	—
93	C₄²×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	2	no	—
95	C₄²×C₆₁	—	4	1	no	—
96	C₄²×C₆₁	—	4	2	no	—
97	C₄²×C₆₁	M₁₆	1	4	yes	—
98	C₄²×C₆₁	#41	1	2	yes	—
99	C₄²×C₆₁	#41	1	2	yes	—
100	C₂×#92418	—	2	2	no	—
101	C₂×#92418	—	2	2	no	—
102	C₂×#92418	—	2	2	no	—
103	C₂×#92418	—	2	4	no	—
104	C₂×#92418	—	2	2	no	—
105	C₂×#92418	—	2	4	no	—
106	C₂×#92418	—	2	2	no	—
107	C₂×#92418	—	2	4	no	—
108	C₂×#92418	#41	1	4	yes	—
109	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₆₁	—	2	8	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	1	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	1	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	#117	1	2	yes	—
134	C₂×C₄×C₆₁	#117	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₆₁	#131	1	2	yes	—
141	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
143	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
145	#92418	—	2	4	no	—
146	#92418	—	4	2	no	—
147	#92418	—	4	4	no	—
148	#22832	—	61	1	no	maximal, Sylow
149	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
150	C₂²×C₆₁	C₄×#41	1	1	yes	characteristic
151	C₂²×C₆₁	#519	1	1	yes	characteristic
152	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
153	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
155	C₂²×C₆₁	#568	1	1	yes	characteristic
156	C₄×C₆₁	—	2	4	no	—
157	C₄×C₆₁	—	2	2	no	—
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	—	2	2	no	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	—	4	2	no	—
162	C₄×C₆₁	—	4	2	no	—
163	C₄×C₆₁	—	4	2	no	—
164	C₄×C₆₁	—	4	1	no	—
165	C₄×C₆₁	#432	1	4	yes	—
166	C₄×C₆₁	#552	1	2	yes	—
167	C₄×C₆₁	#556	1	2	yes	—
168	C₄²×Q₈	D₆₁	1	1	yes	characteristic
169	#1984	—	61	2	no	—
170	#1917	—	61	4	no	—
171	#1839	—	61	2	no	—
172	#1831	—	61	1	no	—
173	#1770	—	61	2	no	—
174	C₄×C₄⋊C₈	—	61	1	no	—
175	#1745	—	61	2	no	—
176	C₂×C₆₁	#2356	1	1	yes	characteristic
177	C₂×C₆₁	C₂×#432	1	1	yes	characteristic
178	C₂×C₆₁	#2925	1	1	yes	characteristic
179	C₂×C₆₁	#2965	1	1	yes	characteristic
180	C₂×C₆₁	#2971	1	1	yes	characteristic
181	C₂×C₆₁	#2983	1	1	yes	characteristic
182	C₂×C₆₁	#2984	1	1	yes	characteristic
183	C₂×C₄×Q₈	Dic₆₁	1	1	yes	characteristic
184	C₂×C₄×Q₈	Dic₆₁	1	1	yes	characteristic
185	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	61	2	no	—
187	C₂×C₄⋊C₈	—	61	2	no	—
188	C₂×#97	—	61	4	no	—
189	C₂×C₄×C₈	—	61	2	no	—
190	C₄×C₄⋊C₄	Dic₆₁	1	1	yes	characteristic
191	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
192	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
193	C₄×C₄⋊C₄	Dic₆₁	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
195	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
196	C₄³	C₂×D₆₁	1	1	yes	characteristic
197	C₄³	C₂×D₆₁	1	2	yes	—
198	#336	—	61	2	no	—
199	#336	—	61	2	no	—
200	#336	—	61	1	no	—
201	#336	—	61	4	no	—
202	#336	—	61	4	no	—
203	#336	—	61	4	no	—
204	#336	—	61	1	no	—
205	C₆₁	#22832	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
207	C₂²×C₈	—	61	2	no	—
208	C₂²×C₈	—	61	2	no	—
209	C₂²×C₈	—	61	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
220	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
221	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
222	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
224	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
226	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
229	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
231	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
233	C₄⋊C₈	—	122	4	no	—
234	C₄⋊C₈	—	122	4	no	—
235	#97	—	122	8	no	—
236	C₄×C₈	—	122	4	no	—
237	#95	C₂×Dic₆₁	1	2	yes	—
238	#95	C₂×Dic₆₁	1	2	yes	—
239	#95	C₂×Dic₆₁	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	#11758194	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
254	C₂×C₈	—	61	4	no	—
255	C₂×C₈	—	122	2	no	—
256	C₂×C₈	—	122	2	no	—
257	C₂×C₈	—	122	4	no	—
258	C₂×C₈	—	122	2	no	—
259	C₂×C₈	—	122	2	no	—
260	C₂×C₈	—	122	4	no	—
261	C₂×C₈	—	122	2	no	—
262	C₂×C₈	—	122	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	2	no	—
281	C₄²	#11758194	1	4	yes	—
282	C₄²	#11758222	1	2	yes	—
283	C₄²	#11758222	1	2	yes	—
284	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
285	Q₈	—	2	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	#14519736	1	2	yes	—
309	C₂×C₄	#14519736	1	2	yes	—
310	C₂×C₄	C₂×#11758194	1	2	yes	—
311	C₂×C₄	C₂×#11758194	1	4	yes	—
312	C₂×C₄	C₂×#11758194	1	2	yes	—
313	C₂×C₄	C₂×#11758194	1	2	yes	—
314	C₂×C₄	C₂×#11758194	1	2	yes	—
315	C₂×C₄	#14519784	1	2	yes	—
316	C₂×C₄	#41×D₆₁	1	2	yes	—
317	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
318	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
319	C₂×C₄	#14519825	1	2	yes	—
320	C₈	—	122	4	no	—
321	C₈	—	244	2	no	—
322	C₈	—	244	4	no	—
323	C₂²	C₂×#14519736	1	1	yes	characteristic
324	C₂²	#41×Dic₆₁	1	1	yes	characteristic
325	C₂²	#16840240	1	1	yes	characteristic
326	C₂²	C₂²×#11758194	1	1	yes	characteristic
327	C₂²	C₂²×#11758194	1	1	yes	characteristic
328	C₂²	C₂×#14519784	1	1	yes	characteristic
329	C₂²	#16840529	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	1	no	—
339	C₄	#16839819	1	4	yes	—
340	C₄	#16840453	1	2	yes	—
341	C₄	#16840484	1	2	yes	—
342	C₂	#21448784	1	1	yes	characteristic
343	C₂	C₂×#16839819	1	1	yes	characteristic
344	C₂	#21454485	1	1	yes	characteristic
345	C₂	#21454786	1	1	yes	characteristic
346	C₂	#21454839	1	1	yes	characteristic
347	C₂	#21454904	1	1	yes	characteristic
348	C₂	#21454908	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₄²×Q₈×C₆₁	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄²×Q₈	D₆₁
hypocenter	C₆₁	#22832
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
205. C₆₁	#22832	semidirect

Maximal subgroups

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

subgroup	index
#21446756	2
#21446385	2
#21445962	2
#21445936	2
#21445688	2
C₄×C₄⋊#92418	2
#21445568	2
C₄²×Q₈×C₆₁	2
#22832	61