Subgroups — #29833677

Order 15616.

444 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454751	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	2	yes	maximal, complemented
3	#21447726	C₂	1	2	yes	maximal, complemented
4	#21447725	C₂	1	2	yes	maximal, complemented
5	#21447724	C₂	1	2	yes	maximal, complemented
6	#21447626	C₂	1	1	yes	maximal, characteristic
7	#21447623	C₂	1	2	yes	maximal
8	#21447619	C₂	1	2	yes	maximal
9	C₆₁×#2164	C₂	1	1	yes	maximal, characteristic
10	#16840315	C₄	1	1	yes	characteristic
11	#16839890	C₄	1	2	yes	—
12	#16839890	C₄	1	2	yes	—
13	C₈×Dic₁₂₂	C₂²	1	2	yes	—
14	C₈×Dic₁₂₂	C₂²	1	2	yes	—
15	#16839790	C₄	1	1	yes	characteristic
16	#16839789	C₄	1	1	yes	characteristic
17	#16839785	C₄	1	1	yes	characteristic
18	#16839717	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊#11758185	C₂²	1	2	yes	—
21	#16839580	C₂²	1	2	yes	—
22	#16839580	C₂²	1	2	yes	—
23	#16839580	C₂²	1	2	yes	—
24	#16839580	C₂²	1	2	yes	—
25	C₂²⋊#11758185	C₂²	1	2	yes	—
26	#16839538	C₂²	1	1	yes	characteristic
27	#16839537	C₂²	1	2	yes	—
28	#16839537	C₂²	1	2	yes	—
29	#16839537	C₂²	1	2	yes	—
30	C₄×#11758185	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₆₁	C₂²	1	2	yes	—
32	C₆₁×#405	C₂²	1	1	yes	characteristic
33	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₆₁×#346	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
38	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
39	#14519776	C₈	1	2	yes	—
40	#14519768	C₈	1	2	yes	—
41	#14519754	C₈	1	2	yes	—
42	#14519754	C₈	1	2	yes	—
43	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
45	#14519736	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×#11758185	C₂³	1	2	yes	—
48	C₂×#11758185	C₂³	1	2	yes	—
49	C₄⋊#92418	C₂×C₄	1	2	yes	—
50	C₄⋊#92418	C₂×C₄	1	2	yes	—
51	#14519706	C₂×C₄	1	2	yes	—
52	#14519706	C₂×C₄	1	2	yes	—
53	#14519706	C₂×C₄	1	2	yes	—
54	#14519706	C₂×C₄	1	2	yes	—
55	#14519687	C₂×C₄	1	2	yes	—
56	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
57	C₈×Dic₆₁	C₂³	1	2	yes	—
58	C₈×Dic₆₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
64	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
65	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
70	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
72	M₁₆×C₆₁	—	4	1	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
80	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
87	#11758196	C₂×C₈	1	2	yes	—
88	#11758196	C₂×C₈	1	2	yes	—
89	#11758196	C₂×C₈	1	2	yes	—
90	#11758196	C₂×C₈	1	2	yes	—
91	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
94	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
95	#11758194	—	4	1	no	—
96	C₂×#92418	—	4	1	no	—
97	C₂×#92418	C₂²×C₄	1	2	yes	—
98	C₂×#92418	C₂²×C₄	1	2	yes	—
99	C₁₆×C₆₁	—	2	2	no	—
100	C₁₆×C₆₁	—	2	2	no	—
101	#11758185	—	2	2	no	—
102	#11758185	—	2	2	no	—
103	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₆₁	—	4	2	no	—
111	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
112	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
113	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
114	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
115	Dic₁₂₂	—	4	2	no	—
116	Dic₁₂₂	—	4	2	no	—
117	C₈×C₆₁	—	2	1	no	—
118	C₈×C₆₁	—	2	1	no	—
119	C₈×C₆₁	—	2	1	no	—
120	C₈×C₆₁	2₋¹⁺⁴	1	2	yes	—
121	#92418	—	2	2	no	—
122	#92418	—	2	2	no	—
123	#33501	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	—	2	1	no	—
126	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₆₁	—	2	2	no	—
128	C₄×C₆₁	—	2	2	no	—
129	C₄×C₆₁	—	2	1	no	—
130	C₄×C₆₁	#519	1	1	yes	characteristic
131	C₄×C₆₁	#568	1	1	yes	characteristic
132	Dic₆₁	—	2	2	no	—
133	Dic₆₁	—	2	2	no	—
134	Dic₆₁	—	2	2	no	—
135	Dic₆₁	—	2	2	no	—
136	#2960	—	61	1	no	—
137	#2183	—	61	2	no	—
138	#2183	—	61	2	no	—
139	#2183	—	61	2	no	—
140	#2183	—	61	2	no	—
141	#2164	—	61	1	no	—
142	#2164	—	61	2	no	—
143	#2164	D₆₁	1	1	yes	characteristic
144	#2163	—	61	2	no	—
145	C₂×C₆₁	—	4	1	no	—
146	C₂×C₆₁	#2982	1	1	yes	characteristic
147	C₂×C₆₁	#3407	1	2	yes	—
148	#533	—	61	1	no	—
149	#446	—	61	2	no	—
150	#446	—	61	2	no	—
151	C₈×Q₈	—	61	2	no	—
152	C₈×Q₈	—	61	2	no	—
153	#424	—	61	1	no	—
154	#424	—	61	1	no	—
155	#423	—	61	1	no	—
156	#405	—	61	1	no	—
157	#405	C₂×D₆₁	1	1	yes	characteristic
158	C₂×C₄×C₈	—	61	1	no	—
159	C₄⋊C₁₆	—	61	2	no	—
160	C₄⋊C₁₆	—	61	2	no	—
161	C₄⋊C₁₆	—	61	2	no	—
162	C₄⋊C₁₆	—	61	2	no	—
163	C₄⋊C₁₆	—	61	2	no	—
164	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
165	C₂²⋊C₁₆	—	61	2	no	—
166	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
167	#346	—	61	2	no	—
168	#346	—	61	2	no	—
169	#346	—	61	2	no	—
170	#346	—	61	1	no	—
171	#346	C₂×D₆₁	1	1	yes	characteristic
172	C₄×C₁₆	—	61	2	no	—
173	C₄×C₁₆	—	61	1	no	—
174	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
175	C₆₁	#33501	1	1	yes	characteristic, complemented, Sylow
176	C₂×M₁₆	—	61	1	no	—
177	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
178	C₂²×C₈	—	61	1	no	—
179	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₄⋊Q₈	—	61	2	no	—
181	#125	—	61	2	no	—
182	C₂²⋊Q₈	—	61	2	no	—
183	C₂²⋊Q₈	—	61	2	no	—
184	C₄×Q₈	—	61	2	no	—
185	C₄×Q₈	—	61	2	no	—
186	#117	—	61	1	no	—
187	#117	C₄×D₆₁	1	1	yes	characteristic
188	C₂×C₄²	—	61	1	no	—
189	C₂×C₁₆	—	61	2	no	—
190	C₂×C₁₆	—	61	2	no	—
191	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
192	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
193	C₄⋊C₈	—	61	2	no	—
194	C₄⋊C₈	—	61	2	no	—
195	C₄⋊C₈	—	61	2	no	—
196	C₄⋊C₈	—	61	2	no	—
197	C₄⋊C₈	—	61	2	no	—
198	C₄⋊C₈	—	61	2	no	—
199	#97	—	61	2	no	—
200	#97	C₄×D₆₁	1	1	yes	characteristic
201	#97	C₄×D₆₁	1	1	yes	characteristic
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	—	61	2	no	—
205	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
207	C₂×Q₈	—	61	2	no	—
208	C₂×Q₈	—	61	2	no	—
209	C₂²×C₄	—	61	2	no	—
210	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
211	M₁₆	—	4	1	no	—
212	M₁₆	—	244	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	61	2	no	—
215	C₂×C₈	—	61	2	no	—
216	C₂×C₈	—	244	1	no	—
217	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
221	C₄⋊C₄	—	61	2	no	—
222	C₄⋊C₄	—	61	2	no	—
223	C₄⋊C₄	—	61	2	no	—
224	C₄⋊C₄	—	61	2	no	—
225	C₄⋊C₄	—	61	2	no	—
226	C₄⋊C₄	—	61	2	no	—
227	C₄⋊C₄	—	61	2	no	—
228	C₄⋊C₄	C₈×D₆₁	1	2	yes	—
229	C₂²⋊C₄	—	61	1	no	—
230	C₂²⋊C₄	—	61	1	no	—
231	C₂²⋊C₄	C₈×D₆₁	1	2	yes	—
232	C₄²	—	61	2	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	61	1	no	—
235	C₄²	—	61	1	no	—
236	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
238	C₁₆	—	2	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	122	2	no	—
241	C₁₆	—	122	2	no	—
242	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
243	Q₈	—	244	2	no	—
244	Q₈	—	244	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	61	2	no	—
248	C₂×C₄	—	61	2	no	—
249	C₂×C₄	—	61	2	no	—
250	C₂×C₄	—	61	2	no	—
251	C₂×C₄	—	244	2	no	—
252	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	122	2	no	—
260	C₈	—	122	2	no	—
261	C₈	#14519824	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	122	2	no	—
269	C₄	—	122	2	no	—
270	C₄	—	122	2	no	—
271	C₄	—	122	2	no	—
272	C₄	#16840239	1	1	yes	characteristic
273	C₄	#16840528	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	#21454899	1	1	yes	characteristic
276	C₂	#21458920	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2164	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2164	D₆₁
hypocenter	C₆₁	#33501
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447726	C₂	semidirect
4. #21447725	C₂	semidirect
5. #21447724	C₂	semidirect
175. C₆₁	#33501	semidirect

Maximal subgroups

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

subgroup	index
#21454751	2
#21447727	2
#21447726	2
#21447725	2
#21447724	2
#21447626	2
#21447623	2
#21447619	2
C₆₁×#2164	2
#33501	61