Subgroups — #29684428

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454470	C₂	1	2	yes	maximal, complemented
2	#21446713	C₂	1	2	yes	maximal
3	#21446573	C₂	1	2	yes	maximal, complemented
4	#21446525	C₂	1	4	yes	maximal, complemented
5	C₄×#14519713	C₂	1	1	yes	maximal, characteristic, complemented
6	#21445629	C₂	1	2	yes	maximal, complemented
7	#21445505	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
9	#16839823	—	2	4	no	—
10	#16839819	—	2	4	no	—
11	C₂×#14519713	C₂²	1	2	yes	—
12	#16839789	C₂²	1	4	yes	—
13	C₂×#14519704	C₂²	1	4	yes	—
14	#16839729	C₂²	1	4	yes	—
15	C₄×#11758194	C₂²	1	2	yes	—
16	C₂×#14519688	C₂²	1	1	yes	characteristic
17	C₂×#14519687	C₂²	1	2	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
19	#16839608	C₂²	1	1	yes	characteristic
20	#16839527	C₂²	1	2	yes	—
21	#16839521	C₂²	1	4	yes	—
22	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₁	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₁	C₄	1	2	yes	complemented
25	C₄×SD₁₆×C₆₁	C₄	1	4	yes	complemented
26	C₄×D₈×C₆₁	C₄	1	2	yes	complemented
27	C₆₁×#427	C₂²	1	1	yes	characteristic
28	C₆₁×#416	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#11758194	C₂³	1	4	yes	—
31	C₂²×#92418	C₂³	1	2	yes	—
32	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
33	#14519713	—	2	4	no	—
34	C₄⋊#92418	—	2	8	no	—
35	#14519704	—	2	8	no	—
36	C₂²⋊#92418	—	2	8	no	—
37	#14519688	—	2	2	no	—
38	#14519688	D₄	1	2	yes	—
39	#14519688	D₄	1	2	yes	—
40	#14519687	—	2	4	no	—
41	C₄×#92418	—	2	2	no	—
42	C₈×Dic₆₁	—	2	4	no	—
43	#14519684	—	2	2	no	—
44	C₂×#41×C₆₁	C₂³	1	2	yes	—
45	#135×C₆₁	—	2	4	no	—
46	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₆₁	—	2	4	no	—
53	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
54	#117×C₆₁	—	2	2	no	—
55	#117×C₆₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₆₁	—	2	2	no	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	#107×C₆₁	C₂×C₄	1	2	yes	—
59	#106×C₆₁	C₂×C₄	1	2	yes	—
60	#103×C₆₁	C₂×C₄	1	4	yes	—
61	#102×C₆₁	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
64	#41×C₆₁	—	2	4	no	—
65	#41×C₆₁	—	2	4	no	—
66	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₆₁	—	2	2	no	—
68	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₁	—	2	2	no	—
70	C₂²×C₄×C₆₁	—	2	4	no	—
71	C₂²×C₄×C₆₁	#41	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₆₁	—	2	2	no	—
74	Q₁₆×C₆₁	—	2	2	no	—
75	SD₁₆×C₆₁	—	2	4	no	—
76	D₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	2	no	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	#41	1	4	yes	—
82	C₄⋊C₄×C₆₁	—	2	4	no	—
83	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
84	C₂²⋊C₄×C₆₁	—	2	2	no	—
85	C₂²⋊C₄×C₆₁	—	2	4	no	—
86	C₄²×C₆₁	—	2	4	no	—
87	C₄²×C₆₁	—	2	2	no	—
88	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₆₁	—	2	2	no	—
92	C₄×Dic₆₁	#41	1	4	yes	—
93	#11758194	—	2	4	no	—
94	#11758194	—	2	4	no	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	2	2	no	—
97	C₂×#92418	—	2	4	no	—
98	C₂×#92418	C₂×D₄	1	4	yes	—
99	C₂³×C₆₁	—	2	2	no	—
100	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
101	Q₈×C₆₁	—	2	2	no	—
102	Q₈×C₆₁	—	2	2	no	—
103	D₄×C₆₁	—	2	2	no	—
104	D₄×C₆₁	—	2	2	no	—
105	D₄×C₆₁	—	4	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	4	no	—
110	C₂×C₄×C₆₁	—	8	4	no	—
111	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
116	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
117	C₂×Dic₆₁	—	2	4	no	—
118	C₂×Dic₆₁	—	4	4	no	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	#117	1	2	yes	—
121	C₈×C₆₁	#117	1	2	yes	—
122	#92418	—	2	4	no	—
123	#92418	—	4	4	no	—
124	#21295	—	61	1	no	maximal, Sylow
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	4	4	no	—
127	C₂²×C₆₁	—	4	2	no	—
128	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₆₁	#529	1	2	yes	—
130	C₄×C₆₁	—	2	2	no	—
131	C₄×C₆₁	—	2	2	no	—
132	C₄×C₆₁	—	4	4	no	—
133	C₄×C₆₁	—	4	2	no	—
134	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
135	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
136	C₄×C₆₁	#532	1	2	yes	—
137	Dic₆₁	—	4	4	no	—
138	C₄×#135	D₆₁	1	1	yes	characteristic
139	#2923	—	61	2	no	—
140	#1975	—	61	2	no	—
141	#1946	—	61	2	no	—
142	#1937	—	61	4	no	—
143	C₄×#108	—	61	1	no	—
144	#1759	—	61	2	no	—
145	#1726	—	61	1	no	—
146	C₂×C₆₁	—	2	2	no	—
147	C₂×C₆₁	—	8	2	no	—
148	C₂×C₆₁	#2293	1	1	yes	characteristic
149	C₂×C₆₁	#2957	1	2	yes	—
150	C₂×#135	C₂×D₆₁	1	1	yes	characteristic
151	C₄×#41	C₂×D₆₁	1	2	yes	—
152	C₄×Q₁₆	Dic₆₁	1	2	yes	complemented
153	C₄×SD₁₆	Dic₆₁	1	4	yes	complemented
154	C₄×D₈	Dic₆₁	1	2	yes	complemented
155	#433	—	122	4	no	—
156	#432	—	122	4	no	—
157	C₂×#108	—	61	2	no	—
158	#427	C₂×D₆₁	1	1	yes	characteristic
159	#424	—	61	4	no	—
160	C₂×#104	—	61	4	no	—
161	#416	C₂×D₆₁	1	2	yes	—
162	#408	—	61	4	no	—
163	C₄×M₁₆	—	61	2	no	—
164	C₂×#97	—	61	1	no	—
165	C₂×#97	—	61	2	no	—
166	C₂×C₄×C₈	—	61	2	no	—
167	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
168	#376	—	61	1	no	—
169	#341	—	61	2	no	—
170	#339	—	61	4	no	—
171	C₆₁	#21295	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂²×D₆₁	1	2	yes	—
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂×Dic₆₁	1	1	yes	characteristic
175	C₂×SD₁₆	C₂×Dic₆₁	1	2	yes	—
176	C₂×D₈	C₂×Dic₆₁	1	1	yes	characteristic
177	C₂×M₁₆	—	61	4	no	—
178	C₂²×C₈	—	61	2	no	—
179	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₄×Q₈	C₂×Dic₆₁	1	4	yes	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×Dic₆₁	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	C₂²×D₆₁	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	61	2	no	—
188	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
189	#108	—	122	4	no	—
190	#107	C₂×Dic₆₁	1	2	yes	—
191	#106	C₂×Dic₆₁	1	2	yes	—
192	C₄⋊C₈	—	122	8	no	—
193	#104	—	122	8	no	—
194	#103	C₂×Dic₆₁	1	4	yes	—
195	#102	C₂×Dic₆₁	1	4	yes	—
196	C₂²⋊C₈	—	122	8	no	—
197	#97	—	61	2	no	—
198	#97	—	61	2	no	—
199	#97	—	122	4	no	—
200	#97	—	122	2	no	—
201	C₄×C₈	—	122	4	no	—
202	C₄×C₈	—	122	2	no	—
203	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
204	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
205	#95	—	122	2	no	—
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×Dic₆₁	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×Dic₆₁	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	122	2	no	—
214	C₂²×C₄	#11758224	1	2	yes	—
215	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	122	4	no	—
220	M₁₆	—	122	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	61	4	no	—
223	C₂×C₈	—	122	2	no	—
224	C₂×C₈	—	122	2	no	—
225	C₂×C₈	—	122	4	no	—
226	C₂×C₈	#11758222	1	4	yes	—
227	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
228	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₆₁	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	61	4	no	—
237	C₄²	—	122	2	no	—
238	C₄²	D₄×D₆₁	1	1	yes	characteristic
239	C₄²	D₄×D₆₁	1	1	yes	characteristic
240	C₄²	C₂²×Dic₆₁	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×#11758224	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	122	4	no	—
254	C₂×C₄	—	244	4	no	—
255	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
256	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
259	C₂×C₄	#41×D₆₁	1	2	yes	—
260	C₂×C₄	C₂×#11758222	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	122	4	no	—
263	C₈	—	244	4	no	—
264	C₈	#14519736	1	2	yes	—
265	C₈	#14519736	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
270	C₂²	#16840294	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	244	4	no	—
276	C₄	C₂×#14519736	1	1	yes	characteristic
277	C₄	#41×Dic₆₁	1	1	yes	characteristic
278	C₄	#16840309	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	2	no	—
281	C₂	#21448284	1	1	yes	characteristic
282	C₂	#21454736	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₄×#135×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#135×C₆₁	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×#135	D₆₁
hypocenter	C₆₁	#21295
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #21454470	C₂	semidirect
3. #21446573	C₂	semidirect
4. #21446525	C₂	semidirect
5. C₄×#14519713	C₂	semidirect
6. #21445629	C₂	semidirect
7. #21445505	C₂	semidirect
24. C₄×Q₁₆×C₆₁	C₄	semidirect
25. C₄×SD₁₆×C₆₁	C₄	semidirect
26. C₄×D₈×C₆₁	C₄	semidirect
152. C₄×Q₁₆	Dic₆₁	semidirect
153. C₄×SD₁₆	Dic₆₁	semidirect
154. C₄×D₈	Dic₆₁	semidirect
171. C₆₁	#21295	semidirect

Maximal subgroups

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

subgroup	index
#21454470	2
#21446713	2
#21446573	2
#21446525	2
C₄×#14519713	2
#21445629	2
#21445505	2
C₄×#135×C₆₁	2
#21295	61