Subgroups — #29699907

Order 15616.

784 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	C₄²×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21446754	C₂	1	2	yes	maximal
3	#21446373	C₂	1	4	yes	maximal
4	#21445935	C₂	1	2	yes	maximal
5	#21445684	C₂	1	2	yes	maximal
6	C₄⋊C₈×Dic₆₁	C₂	1	2	yes	maximal
7	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄×#92418	C₂²	1	2	yes	—
15	C₂×C₈×Dic₆₁	C₂²	1	4	yes	—
16	C₄×#11758196	C₄	1	2	yes	—
17	C₄×#11758196	C₄	1	2	yes	—
18	C₄×#11758196	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
21	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
22	C₄²×Dic₆₁	C₂²	1	2	yes	—
23	#16839517	C₂²	1	2	yes	—
24	#16839516	C₂²	1	4	yes	—
25	#16839516	C₂²	1	4	yes	—
26	#16839516	C₂²	1	4	yes	—
27	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
30	C₄³×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
32	C₂²×#92418	C₂³	1	2	yes	—
33	C₂²×#92418	C₂³	1	2	yes	—
34	C₄×Dic₁₂₂	—	2	2	no	—
35	C₄×Dic₁₂₂	—	2	2	no	—
36	C₄×Dic₁₂₂	—	2	4	no	—
37	C₄×Dic₁₂₂	C₈	1	8	yes	complemented
38	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758196	C₂×C₄	1	2	yes	—
42	C₂×#11758196	C₂×C₄	1	2	yes	—
43	C₂×#11758196	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
49	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
50	C₄⋊#92418	—	2	2	no	—
51	C₄⋊#92418	Q₈	1	4	yes	complemented
52	C₄×#92418	—	2	4	no	—
53	C₈×Dic₆₁	—	4	4	no	—
54	#14519684	C₂×C₄	1	2	yes	—
55	#14519684	C₂×C₄	1	2	yes	—
56	#14519684	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
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	1	yes	characteristic
63	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₁	—	2	4	no	—
66	C₄⋊C₈×C₆₁	—	2	4	no	—
67	C₄⋊C₈×C₆₁	—	4	2	no	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₂₂	—	2	2	no	—
80	C₂×Dic₁₂₂	—	2	2	no	—
81	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
82	C₂×C₈×C₆₁	—	2	4	no	—
83	C₂×C₈×C₆₁	—	2	4	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₄²×C₆₁	—	2	2	no	—
86	C₄²×C₆₁	—	2	2	no	—
87	C₄²×C₆₁	—	2	1	no	—
88	C₄²×C₆₁	—	2	1	no	—
89	C₄²×C₆₁	—	4	1	no	—
90	C₄²×C₆₁	C₂×C₈	1	4	yes	—
91	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
92	C₄²×C₆₁	#41	1	2	yes	—
93	C₄²×C₆₁	#41	1	2	yes	—
94	C₄²×C₆₁	#41	1	2	yes	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
97	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
98	#11758196	—	2	4	no	—
99	#11758196	—	2	4	no	—
100	#11758196	C₂×C₈	1	8	yes	—
101	C₄×Dic₆₁	—	2	4	no	—
102	C₄×Dic₆₁	—	2	2	no	—
103	C₄×Dic₆₁	—	2	2	no	—
104	C₄×Dic₆₁	—	2	2	no	—
105	C₄×Dic₆₁	—	2	2	no	—
106	C₄×Dic₆₁	—	2	4	no	—
107	C₄×Dic₆₁	—	4	2	no	—
108	C₄×Dic₆₁	C₂×C₈	1	8	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	C₂×Q₈	1	4	yes	—
115	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	—	2	1	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	1	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	#117	1	2	yes	—
126	C₂×C₄×C₆₁	#117	1	2	yes	—
127	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₆₁	#131	1	2	yes	—
134	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×Dic₆₁	—	2	2	no	—
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	—	2	2	no	—
143	C₂×Dic₆₁	—	2	2	no	—
144	C₂×Dic₆₁	—	2	2	no	—
145	C₂×Dic₆₁	—	2	2	no	—
146	C₂×Dic₆₁	—	2	2	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
150	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
151	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
152	Dic₁₂₂	—	2	8	no	—
153	C₈×C₆₁	—	4	4	no	—
154	#92418	—	2	4	no	—
155	#92418	—	4	2	no	—
156	#22821	—	61	1	no	maximal, Sylow
157	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
158	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₆₁	#520	1	1	yes	characteristic
160	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
161	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
162	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
163	C₂²×C₆₁	#568	1	1	yes	characteristic
164	C₄×C₆₁	—	2	2	no	—
165	C₄×C₆₁	—	2	2	no	—
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	—	4	1	no	—
168	C₄×C₆₁	#431	1	4	yes	—
169	C₄×C₆₁	C₈×Q₈	1	4	yes	—
170	C₄×C₆₁	#554	1	1	yes	characteristic
171	C₄×C₆₁	#556	1	1	yes	characteristic
172	C₄×C₆₁	#557	1	1	yes	characteristic
173	C₄×C₆₁	#559	1	1	yes	characteristic
174	Dic₆₁	—	2	4	no	—
175	Dic₆₁	—	2	4	no	—
176	Dic₆₁	—	2	4	no	—
177	Dic₆₁	—	4	2	no	—
178	C₄²×Q₈	—	61	1	no	—
179	#1984	—	61	2	no	—
180	#1916	—	61	4	no	—
181	#1842	D₆₁	1	1	yes	characteristic
182	#1831	—	61	2	no	—
183	C₈×C₄⋊C₄	—	61	2	no	—
184	C₄×C₄⋊C₈	—	61	2	no	—
185	C₄×C₄⋊C₈	—	61	1	no	—
186	C₂×C₆₁	#2355	1	1	yes	characteristic
187	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
188	C₂×C₆₁	#2925	1	1	yes	characteristic
189	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
190	C₂×C₆₁	Q₈×M₁₆	1	1	yes	characteristic
191	C₂×C₆₁	#2988	1	1	yes	characteristic
192	C₂×C₆₁	#2990	1	1	yes	characteristic
193	C₂×C₄×Q₈	—	61	1	no	—
194	C₂×C₄×Q₈	—	61	1	no	—
195	C₂×C₄×Q₈	—	61	1	no	—
196	C₂×C₄⋊C₈	—	61	1	no	—
197	C₂×C₄⋊C₈	—	61	1	no	—
198	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
199	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
200	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
201	C₂×C₄×C₈	—	61	2	no	—
202	C₂×C₄×C₈	—	61	4	no	—
203	C₄×C₄⋊C₄	—	61	1	no	—
204	C₄×C₄⋊C₄	—	61	1	no	—
205	C₄×C₄⋊C₄	—	61	2	no	—
206	C₄×C₄⋊C₄	—	61	2	no	—
207	C₄×C₄⋊C₄	—	61	2	no	—
208	C₄×C₄⋊C₄	—	61	1	no	—
209	C₄³	—	61	2	no	—
210	C₄³	C₂×D₆₁	1	1	yes	characteristic
211	#336	—	61	4	no	—
212	#336	—	61	4	no	—
213	#336	—	61	4	no	—
214	#336	—	61	2	no	—
215	C₆₁	#22821	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	61	1	no	—
217	C₂²×C₈	—	61	2	no	—
218	C₂²×C₈	—	61	2	no	—
219	C₂²×C₈	C₂²×D₆₁	1	4	yes	—
220	C₄×Q₈	—	61	8	no	—
221	C₄×Q₈	—	122	4	no	—
222	C₄×Q₈	—	122	2	no	—
223	C₄×Q₈	—	122	2	no	—
224	C₂×C₄⋊C₄	—	61	2	no	—
225	C₂×C₄⋊C₄	—	61	1	no	—
226	C₂×C₄⋊C₄	—	61	1	no	—
227	C₂×C₄⋊C₄	—	61	2	no	—
228	C₂×C₄⋊C₄	—	61	2	no	—
229	C₂×C₄⋊C₄	—	61	1	no	—
230	C₂×C₄²	—	61	2	no	—
231	C₂×C₄²	—	61	2	no	—
232	C₂×C₄²	—	61	2	no	—
233	C₂×C₄²	—	61	2	no	—
234	C₂×C₄²	—	61	2	no	—
235	C₂×C₄²	—	61	2	no	—
236	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
237	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
238	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
239	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	4	2	no	—
246	C₄⋊C₈	—	61	4	no	—
247	C₄⋊C₈	—	122	2	no	—
248	C₄×C₈	—	122	4	no	—
249	C₄×C₈	—	244	4	no	—
250	#95	—	61	2	no	—
251	#95	—	61	2	no	—
252	#95	—	61	2	no	—
253	C₂×Q₈	—	61	4	no	—
254	C₂×Q₈	—	122	2	no	—
255	C₂×Q₈	—	122	2	no	—
256	C₂²×C₄	—	61	2	no	—
257	C₂²×C₄	—	61	2	no	—
258	C₂²×C₄	—	61	2	no	—
259	C₂²×C₄	—	61	2	no	—
260	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	61	4	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	C₄⋊C₄	—	61	8	no	—
277	C₄⋊C₄	—	61	4	no	—
278	C₄⋊C₄	—	61	4	no	—
279	C₄⋊C₄	—	122	4	no	—
280	C₄⋊C₄	—	122	4	no	—
281	C₄⋊C₄	—	122	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	61	8	no	—
288	C₄²	—	122	4	no	—
289	C₄²	—	122	2	no	—
290	C₄²	—	122	2	no	—
291	C₄²	—	122	2	no	—
292	C₄²	—	122	2	no	—
293	C₄²	—	122	4	no	—
294	C₄²	—	244	2	no	—
295	C₄²	C₈×D₆₁	1	4	yes	—
296	C₄²	#11758224	1	2	yes	—
297	C₄²	#11758224	1	2	yes	—
298	C₄²	Q₈×D₆₁	1	2	yes	—
299	C₄²	#11758226	1	2	yes	—
300	C₂³	C₂²×C₄×D₆₁	1	1	yes	characteristic
301	Q₈	—	122	8	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	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	—	61	4	no	—
312	C₂×C₄	—	61	4	no	—
313	C₂×C₄	—	61	4	no	—
314	C₂×C₄	—	122	2	no	—
315	C₂×C₄	—	122	2	no	—
316	C₂×C₄	—	122	2	no	—
317	C₂×C₄	—	122	2	no	—
318	C₂×C₄	—	122	2	no	—
319	C₂×C₄	—	122	2	no	—
320	C₂×C₄	—	122	2	no	—
321	C₂×C₄	—	122	2	no	—
322	C₂×C₄	—	122	2	no	—
323	C₂×C₄	#14519735	1	2	yes	—
324	C₂×C₄	#14519735	1	2	yes	—
325	C₂×C₄	#14519742	1	2	yes	—
326	C₂×C₄	#14519742	1	2	yes	—
327	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
328	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
330	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
331	C₂×C₄	#14519785	1	2	yes	—
332	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
333	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
334	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
335	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
336	C₂×C₄	#14519822	1	1	yes	characteristic
337	C₂×C₄	#14519825	1	1	yes	characteristic
338	C₈	—	4	4	no	—
339	C₈	—	122	4	no	—
340	C₈	—	244	2	no	—
341	C₂²	C₂×#14519735	1	1	yes	characteristic
342	C₂²	C₂×#14519742	1	1	yes	characteristic
343	C₂²	#16840243	1	1	yes	characteristic
344	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
345	C₂²	C₂×M₁₆×D₆₁	1	1	yes	characteristic
346	C₂²	C₂×#14519785	1	1	yes	characteristic
347	C₂²	#16840528	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	122	4	no	—
353	C₄	—	122	4	no	—
354	C₄	—	122	4	no	—
355	C₄	—	244	2	no	—
356	C₄	#16839814	1	4	yes	—
357	C₄	#16839886	1	4	yes	—
358	C₄	#16840473	1	1	yes	characteristic
359	C₄	#16840486	1	1	yes	characteristic
360	C₄	#16840487	1	1	yes	characteristic
361	C₄	#16840498	1	1	yes	characteristic
362	C₂	#21448778	1	1	yes	characteristic
363	C₂	C₂×#16839814	1	1	yes	characteristic
364	C₂	#21454487	1	1	yes	characteristic
365	C₂	C₂×#16839886	1	1	yes	characteristic
366	C₂	#21454770	1	1	yes	characteristic
367	C₂	#21454928	1	1	yes	characteristic
368	C₂	#21454945	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#1842	C₂
socle, prime radical	C₂³×C₆₁	C₂³×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1842	D₆₁
hypocenter	C₆₁	#22821
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
37. C₄×Dic₁₂₂	C₈	semidirect
51. C₄⋊#92418	Q₈	semidirect
215. C₆₁	#22821	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₂₂	2
#21446754	2
#21446373	2
#21445935	2
#21445684	2
C₄⋊C₈×Dic₆₁	2
C₄×C₄⋊#92418	2
C₆₁×#1842	2
#22821	61