Subgroups — #29835537

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456333	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
3	#21448105	C₂	1	4	yes	maximal, complemented
4	#21448101	C₂	1	2	yes	maximal, complemented
5	#21448094	C₂	1	2	yes	maximal, complemented
6	#21447607	C₂	1	4	yes	maximal
7	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
8	#16840319	—	2	1	no	—
9	C₂×#14519777	—	2	1	no	—
10	#16840167	—	2	1	no	—
11	#16840167	—	2	1	no	—
12	#16840167	C₂²	1	1	yes	characteristic
13	#16840167	C₂²	1	2	yes	—
14	#16840167	C₂²	1	1	yes	characteristic
15	#16840167	C₂²	1	2	yes	—
16	#16840164	—	2	1	no	—
17	#16840164	C₂²	1	2	yes	—
18	#16840160	—	2	1	no	—
19	#16839800	—	2	1	no	—
20	#16839800	C₂²	1	2	yes	—
21	C₂×#14519711	—	2	1	no	—
22	C₂×#14519711	C₂²	1	2	yes	—
23	#16839590	C₂²	1	8	yes	—
24	#16839588	C₂²	1	8	yes	—
25	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
29	C₂²×Dic₁₂₂	—	4	1	no	—
30	#14519777	—	2	2	no	—
31	#14519777	—	2	2	no	—
32	#14519777	—	2	1	no	—
33	#14519777	—	2	1	no	—
34	#14519777	—	4	1	no	—
35	#14519777	—	4	1	no	—
36	#14519769	—	2	2	no	—
37	#14519756	—	4	1	no	—
38	#14519756	—	4	1	no	—
39	C₄×Dic₁₂₂	—	4	1	no	—
40	C₄×Dic₁₂₂	—	4	1	no	—
41	#14519736	—	2	2	no	—
42	C₂×C₄⋊Dic₆₁	—	2	2	no	—
43	C₂×C₄⋊Dic₆₁	—	4	1	no	—
44	#14519711	—	2	2	no	—
45	#14519711	—	2	2	no	—
46	#14519711	—	2	2	no	—
47	#14519711	—	2	1	no	—
48	#14519711	—	2	1	no	—
49	#14519711	C₂³	1	4	yes	—
50	#14519711	C₂³	1	2	yes	—
51	#14519711	C₂³	1	2	yes	—
52	#14519709	—	2	2	no	—
53	#14519709	—	2	2	no	—
54	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
57	M₃₂×C₆₁	Q₈	1	8	yes	complemented
58	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
59	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	—	4	2	no	—
67	C₂×Dic₁₂₂	—	4	2	no	—
68	C₂×Dic₁₂₂	—	4	2	no	—
69	C₂×Dic₁₂₂	—	4	1	no	—
70	C₂×Dic₁₂₂	—	4	1	no	—
71	C₂×Dic₁₂₂	—	8	4	no	—
72	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄²×C₆₁	D₈	1	1	yes	characteristic
79	C₄²×C₆₁	D₈	1	1	yes	characteristic
80	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₆₁	—	4	2	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	—	2	4	no	—
86	C₄⋊Dic₆₁	—	4	2	no	—
87	C₄⋊Dic₆₁	—	4	2	no	—
88	C₄⋊Dic₆₁	—	4	2	no	—
89	C₄⋊Dic₆₁	—	4	1	no	—
90	C₄⋊Dic₆₁	—	4	1	no	—
91	#11758196	—	4	2	no	—
92	#11758196	—	4	2	no	—
93	C₄×Dic₆₁	—	4	2	no	—
94	C₁₆×C₆₁	C₂×Q₈	1	8	yes	complemented
95	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	—	2	2	no	—
98	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₆₁	—	4	2	no	—
107	C₂×Dic₆₁	—	4	2	no	—
108	C₂×Dic₆₁	—	4	4	no	—
109	C₂×Dic₆₁	—	8	4	no	—
110	Dic₁₂₂	—	8	4	no	—
111	Dic₁₂₂	—	8	2	no	—
112	Dic₁₂₂	—	8	4	no	—
113	Dic₁₂₂	—	8	2	no	—
114	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
115	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
116	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
117	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
118	#33731	—	61	1	no	maximal, Sylow
119	C₂²×C₆₁	—	2	1	no	—
120	C₂²×C₆₁	—	2	1	no	—
121	C₂²×C₆₁	#500	1	2	yes	—
122	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	2	1	no	—
125	C₄×C₆₁	#500	1	2	yes	—
126	C₄×C₆₁	#509	1	2	yes	—
127	C₄×C₆₁	#510	1	2	yes	—
128	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
130	Dic₆₁	—	8	4	no	—
131	Dic₆₁	—	8	2	no	—
132	Dic₆₁	—	8	2	no	—
133	#3160	—	61	1	no	—
134	C₂×#500	—	61	1	no	—
135	#2255	—	61	4	no	—
136	#2254	—	61	2	no	—
137	#2252	—	61	2	no	—
138	#2159	—	61	4	no	—
139	C₄×M₃₂	D₆₁	1	1	yes	characteristic
140	C₂×C₆₁	—	2	2	no	—
141	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
142	C₂×C₆₁	C₂×#509	1	1	yes	characteristic
143	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
144	#533	—	122	1	no	—
145	C₂×C₄⋊Q₈	—	122	1	no	—
146	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
147	#500	—	61	2	no	—
148	#500	—	61	2	no	—
149	#500	—	61	1	no	—
150	#500	—	61	1	no	—
151	#500	—	122	1	no	—
152	#500	—	122	1	no	—
153	#499	—	61	2	no	—
154	#499	—	122	1	no	—
155	#498	—	122	1	no	—
156	#427	—	61	2	no	—
157	#427	—	122	1	no	—
158	C₂×#107	—	61	2	no	—
159	C₂×#107	—	122	1	no	—
160	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
161	#367	—	61	8	no	—
162	#366	—	61	8	no	—
163	#346	C₂×D₆₁	1	2	yes	—
164	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
165	C₆₁	#33731	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	244	1	no	—
167	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	122	2	no	—
170	C₄⋊Q₈	—	122	2	no	—
171	C₄⋊Q₈	—	122	1	no	—
172	C₄⋊Q₈	—	122	1	no	—
173	C₄⋊Q₈	—	244	1	no	—
174	C₄⋊Q₈	—	244	1	no	—
175	#125	—	122	2	no	—
176	C₂²⋊Q₈	—	244	1	no	—
177	C₂²⋊Q₈	—	244	1	no	—
178	C₄×Q₈	—	244	1	no	—
179	C₄×Q₈	—	244	1	no	—
180	#117	—	122	2	no	—
181	C₂×C₄⋊C₄	—	122	2	no	—
182	C₂×C₄⋊C₄	—	244	1	no	—
183	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
184	M₃₂	Dic₁₂₂	1	8	yes	complemented
185	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
186	#107	—	61	2	no	—
187	#107	—	61	4	no	—
188	#107	—	61	2	no	—
189	#107	—	122	2	no	—
190	#107	—	122	2	no	—
191	#107	—	122	2	no	—
192	#107	—	122	1	no	—
193	#107	—	122	1	no	—
194	#106	—	122	2	no	—
195	#106	—	122	2	no	—
196	C₄×C₈	D₂₄₄	1	1	yes	characteristic
197	C₄×C₈	D₂₄₄	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
200	C₂×Q₈	—	488	4	no	—
201	C₂×Q₈	—	244	1	no	—
202	C₂×Q₈	—	244	1	no	—
203	C₂×Q₈	—	244	2	no	—
204	C₂×Q₈	—	244	2	no	—
205	C₂²×C₄	—	244	2	no	—
206	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
207	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
209	C₂×C₈	Dic₂₄₄	1	4	yes	—
210	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
211	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
212	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
215	C₄⋊C₄	—	122	2	no	—
216	C₄⋊C₄	—	122	2	no	—
217	C₄⋊C₄	—	122	4	no	—
218	C₄⋊C₄	—	244	2	no	—
219	C₄⋊C₄	—	244	2	no	—
220	C₄⋊C₄	—	244	2	no	—
221	C₄⋊C₄	—	244	2	no	—
222	C₄⋊C₄	—	244	2	no	—
223	C₄⋊C₄	—	244	1	no	—
224	C₄⋊C₄	—	244	1	no	—
225	C₂²⋊C₄	—	244	2	no	—
226	C₄²	—	244	2	no	—
227	C₄²	D₄₈₈	1	1	yes	characteristic
228	C₄²	D₄₈₈	1	1	yes	characteristic
229	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
230	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
231	C₁₆	C₂×Dic₁₂₂	1	8	yes	complemented
232	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
233	Q₈	—	488	4	no	—
234	Q₈	—	488	4	no	—
235	Q₈	—	488	2	no	—
236	Q₈	—	488	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	488	4	no	—
240	C₂×C₄	—	244	2	no	—
241	C₂×C₄	—	244	2	no	—
242	C₂×C₄	—	244	4	no	—
243	C₂×C₄	#14519777	1	2	yes	—
244	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
250	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
251	C₈	#14519777	1	2	yes	—
252	C₈	C₂×Dic₂₄₄	1	2	yes	—
253	C₈	C₂×Dic₂₄₄	1	2	yes	—
254	C₈	C₂²×Dic₁₂₂	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#16840167	1	2	yes	—
258	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	488	4	no	—
262	C₄	—	488	2	no	—
263	C₄	—	488	2	no	—
264	C₄	#16840167	1	2	yes	—
265	C₄	#16840200	1	2	yes	—
266	C₄	#16840205	1	2	yes	—
267	C₄	C₂×#14519777	1	1	yes	characteristic
268	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#16840167	1	1	yes	characteristic
271	C₂	C₂×#16840200	1	1	yes	characteristic
272	C₂	C₂×#16840205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	#33731
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21448105	C₂	semidirect
4. #21448101	C₂	semidirect
5. #21448094	C₂	semidirect
57. M₃₂×C₆₁	Q₈	semidirect
94. C₁₆×C₆₁	C₂×Q₈	semidirect
165. C₆₁	#33731	semidirect
184. M₃₂	Dic₁₂₂	semidirect
231. C₁₆	C₂×Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
C₂×#16840167	2
#21448105	2
#21448101	2
#21448094	2
#21447607	2
C₄×M₃₂×C₆₁	2
#33731	61