Subgroups — #29569545

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519736	C₂	1	1	yes	maximal, characteristic
3	#21443659	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂³×#92418	—	2	1	no	—
6	C₂×#14519736	C₄	1	2	yes	—
7	C₂×C₄⋊#92418	—	2	1	no	—
8	C₂×C₄⋊#92418	—	2	2	no	—
9	C₂×C₄⋊#92418	C₄	1	4	yes	complemented
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₆₁	C₄	1	2	yes	—
12	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₄×C₂²⋊Dic₆₁	—	2	1	no	—
14	C₄×C₂²⋊Dic₆₁	—	2	1	no	—
15	#16839608	—	2	1	no	—
16	C₄²×Dic₆₁	—	2	1	no	—
17	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#431	—	2	2	no	—
19	C₆₁×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
23	C₂²×#92418	—	2	4	no	—
24	C₂²×#92418	—	2	2	no	—
25	C₂²×#92418	—	4	4	no	—
26	#14519736	—	2	4	no	—
27	#14519736	C₈	1	8	yes	complemented
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	C₂×C₄×Dic₆₁	—	2	2	no	—
33	C₄⋊#92418	—	2	4	no	—
34	C₄⋊#92418	—	2	2	no	—
35	C₄⋊#92418	C₂×C₄	1	4	yes	—
36	#14519684	—	2	2	no	—
37	#14519684	—	2	2	no	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	—	2	2	no	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	D₄	1	2	yes	—
48	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
49	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₆₁	—	2	4	no	—
51	C₂²⋊C₈×C₆₁	—	2	4	no	—
52	C₄×C₈×C₆₁	—	2	2	no	—
53	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
54	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₆₁	—	2	4	no	—
73	C₂×C₈×C₆₁	—	2	4	no	—
74	C₂×C₈×C₆₁	—	4	4	no	—
75	C₄²×C₆₁	C₄²	1	2	yes	—
76	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
80	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
85	C₂²⋊Dic₆₁	—	2	8	no	—
86	C₄⋊Dic₆₁	—	2	4	no	—
87	C₄⋊Dic₆₁	C₂×C₈	1	8	yes	complemented
88	C₄×Dic₆₁	—	2	8	no	—
89	C₄×Dic₆₁	—	4	4	no	—
90	C₄×Dic₆₁	—	4	2	no	—
91	C₄×Dic₆₁	—	4	2	no	—
92	C₂×#92418	—	2	4	no	—
93	C₂×#92418	—	4	8	no	—
94	C₂×#92418	—	4	4	no	—
95	C₂³×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	—	2	2	no	—
97	C₂³×C₆₁	#95	1	2	yes	—
98	C₂³×C₆₁	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	#95	1	2	yes	—
109	C₂×C₄×C₆₁	#95	1	2	yes	—
110	C₂×C₄×C₆₁	#95	1	2	yes	—
111	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
112	C₂×C₄×C₆₁	#97	1	4	yes	—
113	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
115	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₆₁	#137	1	2	yes	—
125	C₂×Dic₆₁	—	2	8	no	—
126	C₂×Dic₆₁	—	4	4	no	—
127	C₂×Dic₆₁	—	4	4	no	—
128	C₈×C₆₁	—	4	4	no	—
129	#92418	—	4	4	no	—
130	#7891	—	61	1	no	maximal, Sylow
131	C₂²×C₆₁	—	2	8	no	—
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	#336	1	4	yes	—
135	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₆₁	#417	1	2	yes	—
138	C₂²×C₆₁	#421	1	2	yes	—
139	C₄×C₆₁	—	2	1	no	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	1	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	#336	1	4	yes	—
144	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
145	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
146	C₄×C₆₁	#418	1	1	yes	characteristic
147	C₄×C₆₁	#419	1	1	yes	characteristic
148	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
149	C₄×C₆₁	#428	1	2	yes	—
150	Dic₆₁	—	4	8	no	—
151	C₂×#431	D₆₁	1	1	yes	characteristic
152	C₂²×C₄⋊C₈	—	61	1	no	—
153	C₄×#117	—	61	1	no	—
154	#1279	—	61	4	no	—
155	C₂×C₆₁	—	2	4	no	—
156	C₂×C₆₁	#1477	1	4	yes	—
157	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
158	C₂×C₆₁	#1733	1	1	yes	characteristic
159	C₂×C₆₁	#1736	1	1	yes	characteristic
160	C₂³×C₈	—	122	1	no	—
161	C₂×#117	—	61	2	no	—
162	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
163	#431	—	2	2	no	—
164	#431	Dic₆₁	1	4	yes	complemented
165	C₂×C₄⋊C₈	—	2	1	no	—
166	C₂×C₄⋊C₈	—	61	4	no	—
167	C₂×C₄⋊C₈	—	61	2	no	—
168	C₂×C₄⋊C₈	—	122	1	no	—
169	C₂×C₄⋊C₈	—	122	2	no	—
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
172	C₄×C₄⋊C₄	—	61	2	no	—
173	C₄×C₄⋊C₄	—	61	2	no	—
174	C₄×C₂²⋊C₄	—	122	1	no	—
175	C₄×C₂²⋊C₄	—	122	1	no	—
176	#376	—	122	1	no	—
177	C₄³	—	122	1	no	—
178	C₆₁	#7891	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
180	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	122	4	no	—
184	C₂²×C₈	—	122	2	no	—
185	C₂²×C₈	—	244	4	no	—
186	#117	—	61	8	no	—
187	#117	—	122	4	no	—
188	C₂×C₄⋊C₄	—	61	4	no	—
189	C₂×C₂²⋊C₄	—	122	2	no	—
190	C₂×C₄²	—	122	2	no	—
191	C₂×C₄²	—	122	2	no	—
192	C₂×C₄²	—	122	2	no	—
193	C₂×C₄²	Dic₁₂₂	1	2	yes	—
194	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₆₁	1	2	yes	—
196	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
200	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	61	4	no	—
203	C₄⋊C₈	—	122	4	no	—
204	C₄⋊C₈	—	122	2	no	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
208	#95	—	122	2	no	—
209	#95	—	122	2	no	—
210	C₂⁴	#11758198	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	122	4	no	—
215	C₂²×C₄	C₈×D₆₁	1	4	yes	—
216	C₂²×C₄	#11758189	1	4	yes	—
217	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
218	C₂²×C₄	#11758196	1	2	yes	—
219	C₂²×C₄	#11758196	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
221	C₂²×C₄	#11758198	1	1	yes	characteristic
222	C₂²×C₄	#11758198	1	1	yes	characteristic
223	C₂²×C₄	#11758198	1	2	yes	—
224	C₂²×C₄	#11758198	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	122	4	no	—
232	C₂×C₈	—	244	8	no	—
233	C₂×C₈	—	244	4	no	—
234	C₄⋊C₄	—	61	8	no	—
235	C₄⋊C₄	—	122	4	no	—
236	C₂²⋊C₄	—	122	8	no	—
237	C₄²	—	122	8	no	—
238	C₄²	—	244	4	no	—
239	C₄²	—	244	2	no	—
240	C₄²	—	244	2	no	—
241	C₄²	C₄×Dic₆₁	1	2	yes	—
242	C₄²	#11758196	1	2	yes	—
243	C₄²	#11758196	1	2	yes	—
244	C₄²	C₄⋊Dic₆₁	1	2	yes	—
245	C₄²	#11758198	1	2	yes	—
246	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
247	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
249	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	#14519684	1	2	yes	—
254	C₂³	#14519689	1	4	yes	—
255	C₂³	C₂×#11758198	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	122	8	no	—
265	C₂×C₄	—	244	4	no	—
266	C₂×C₄	—	244	4	no	—
267	C₂×C₄	#14519684	1	2	yes	—
268	C₂×C₄	#14519684	1	2	yes	—
269	C₂×C₄	#14519684	1	2	yes	—
270	C₂×C₄	C₈×Dic₆₁	1	4	yes	—
271	C₂×C₄	#14519687	1	4	yes	—
272	C₂×C₄	#14519689	1	4	yes	—
273	C₂×C₄	#14519706	1	8	yes	—
274	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
275	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
278	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
281	C₂×C₄	C₂×#11758189	1	2	yes	—
282	C₂×C₄	#14519805	1	2	yes	—
283	C₂×C₄	#14519810	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	244	4	no	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#16839516	1	4	yes	—
290	C₂²	C₂×#14519684	1	1	yes	characteristic
291	C₂²	C₂×#14519689	1	2	yes	—
292	C₂²	#16839764	1	2	yes	—
293	C₂²	#16839780	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	244	8	no	—
299	C₄	#16839516	1	4	yes	—
300	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
301	C₄	C₂×#14519687	1	1	yes	characteristic
302	C₄	#16839768	1	1	yes	characteristic
303	C₄	#16839772	1	1	yes	characteristic
304	C₄	C₂×#14519706	1	2	yes	—
305	C₄	#16839804	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	#21444207	1	4	yes	—
308	C₂	C₂×#16839516	1	1	yes	characteristic
309	C₂	#21445529	1	1	yes	characteristic
310	C₂	#21445539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#431	C₂
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#431	D₆₁
hypocenter	C₆₁	#7891
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #21443659	C₂	semidirect
9. C₂×C₄⋊#92418	C₄	semidirect
19. C₆₁×#431	C₄	semidirect
27. #14519736	C₈	semidirect
87. C₄⋊Dic₆₁	C₂×C₈	semidirect
164. #431	Dic₆₁	semidirect
178. C₆₁	#7891	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#92418	2
C₄×#14519736	2
#21443659	2
C₂×C₆₁×#431	2
#7891	61