Subgroups — #29579257

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840139	C₂	1	1	yes	maximal, characteristic
2	#21446745	C₂	1	2	yes	maximal, complemented
3	C₈×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16839505	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₈₈	—	2	1	no	—
7	C₂²×D₄₈₈	—	2	1	no	—
8	C₂×#14519774	C₂²	1	2	yes	—
9	#16840139	—	2	2	no	—
10	#16840139	C₄	1	4	yes	complemented
11	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₂×#14519699	—	2	4	no	—
13	C₂×C₈×Dic₆₁	—	2	1	no	—
14	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
15	#16839516	—	2	1	no	—
16	#16839505	—	2	4	no	—
17	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#335	—	2	2	no	—
20	C₂²×D₂₄₄	—	2	2	no	—
21	C₂²×D₂₄₄	—	2	2	no	—
22	C₂²×D₂₄₄	—	2	2	no	—
23	C₂×D₄₈₈	—	2	4	no	—
24	C₂×D₄₈₈	—	2	4	no	—
25	C₂×D₄₈₈	—	4	2	no	—
26	C₂×D₄₈₈	—	4	2	no	—
27	C₂×D₄₈₈	—	4	2	no	—
28	C₂×D₄₈₈	—	4	2	no	—
29	C₂²×#92418	—	2	1	no	—
30	#14519774	—	2	2	no	—
31	#14519774	—	2	2	no	—
32	#14519774	—	2	2	no	—
33	#14519774	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
35	C₂×C₄×Dic₆₁	—	2	1	no	—
36	C₄⋊#92418	—	2	1	no	—
37	C₄⋊#92418	D₄	1	2	yes	—
38	#14519699	—	4	8	no	—
39	C₈×Dic₆₁	—	4	2	no	—
40	#14519684	—	2	1	no	—
41	C₂²×C₈×C₆₁	—	2	2	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₁	—	2	2	no	—
46	C₄⋊C₈×C₆₁	—	2	2	no	—
47	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₆₁	D₄	1	2	yes	—
49	C₂³×D₆₁	—	4	2	no	—
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	2	2	no	—
54	C₂×D₂₄₄	—	2	4	no	—
55	C₂×D₂₄₄	—	2	4	no	—
56	C₂×D₂₄₄	—	2	4	no	—
57	C₂×D₂₄₄	—	4	4	no	—
58	C₂×D₂₄₄	—	4	2	no	—
59	C₂×D₂₄₄	—	4	2	no	—
60	C₂×D₂₄₄	—	4	4	no	—
61	C₂×D₂₄₄	—	4	2	no	—
62	C₂×D₂₄₄	—	4	2	no	—
63	C₂×D₂₄₄	—	4	2	no	—
64	C₂×D₂₄₄	—	4	2	no	—
65	C₂×D₂₄₄	—	4	2	no	—
66	C₂×D₂₄₄	—	8	4	no	—
67	C₂×C₈×C₆₁	—	2	1	no	—
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	—	4	4	no	—
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₁	D₈	1	2	yes	—
74	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
75	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₁	#41	1	1	yes	characteristic
79	C₄⋊Dic₆₁	—	2	2	no	—
80	C₄⋊Dic₆₁	D₈	1	4	yes	complemented
81	C₄×Dic₆₁	—	4	1	no	—
82	C₄×Dic₆₁	—	4	1	no	—
83	C₂×#92418	—	2	2	no	—
84	C₂×#92418	—	4	1	no	—
85	C₂×#92418	—	4	1	no	—
86	D₄₈₈	—	4	8	no	—
87	D₄₈₈	—	4	8	no	—
88	C₂³×C₆₁	#121	1	1	yes	characteristic
89	C₂²×D₆₁	—	4	4	no	—
90	C₂²×D₆₁	—	8	2	no	—
91	C₂²×D₆₁	—	8	2	no	—
92	C₂²×D₆₁	—	8	4	no	—
93	C₂²×D₆₁	—	8	4	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#135	1	2	yes	—
105	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×Dic₆₁	—	2	4	no	—
108	C₂×Dic₆₁	—	4	2	no	—
109	C₂×Dic₆₁	—	4	2	no	—
110	D₂₄₄	—	4	4	no	—
111	D₂₄₄	—	4	4	no	—
112	D₂₄₄	—	4	8	no	—
113	D₂₄₄	—	8	4	no	—
114	D₂₄₄	—	8	4	no	—
115	D₂₄₄	—	8	2	no	—
116	D₂₄₄	—	8	2	no	—
117	D₂₄₄	—	8	2	no	—
118	D₂₄₄	—	8	2	no	—
119	C₈×C₆₁	—	2	2	no	—
120	C₈×C₆₁	—	8	2	no	—
121	C₈×C₆₁	#102	1	4	yes	complemented
122	#92418	—	4	2	no	—
123	#9734	—	61	1	no	maximal, Sylow
124	C₂²×C₆₁	#390	1	1	yes	characteristic
125	C₂²×C₆₁	#459	1	1	yes	characteristic
126	C₂²×C₆₁	#460	1	1	yes	characteristic
127	C₂²×C₆₁	#466	1	2	yes	—
128	C₂²×C₆₁	#472	1	2	yes	—
129	C₂×D₆₁	—	8	4	no	—
130	C₂×D₆₁	—	8	4	no	—
131	C₂×D₆₁	—	8	4	no	—
132	C₂×D₆₁	—	16	4	no	—
133	C₂×D₆₁	—	16	4	no	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	C₂×#102	1	1	yes	characteristic
137	C₄×C₆₁	#418	1	1	yes	characteristic
138	C₄×C₆₁	C₄×D₈	1	2	yes	—
139	C₄×C₆₁	#459	1	1	yes	characteristic
140	C₄×C₆₁	#460	1	1	yes	characteristic
141	C₄×C₆₁	#493	1	1	yes	characteristic
142	C₄×C₆₁	#495	1	1	yes	characteristic
143	Dic₆₁	—	4	4	no	—
144	C₂×#493	—	61	1	no	—
145	#1982	—	61	2	no	—
146	C₈×C₄⋊C₄	—	61	1	no	—
147	C₂×#335	D₆₁	1	1	yes	characteristic, complemented
148	C₂×#331	—	61	2	no	—
149	C₂×C₆₁	#1667	1	2	yes	—
150	C₂×C₆₁	#1668	1	2	yes	—
151	C₂×C₆₁	#1970	1	1	yes	characteristic
152	C₂×C₆₁	#1978	1	1	yes	characteristic
153	C₂×C₆₁	#1982	1	1	yes	characteristic
154	D₆₁	—	16	4	no	—
155	C₂²×D₈	—	122	1	no	—
156	C₂²×D₈	—	122	1	no	—
157	C₂×#127	—	61	2	no	—
158	#493	—	61	4	no	—
159	#493	—	122	2	no	—
160	C₂×C₄⋊C₈	—	61	1	no	—
161	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
162	C₂×#102	—	122	4	no	—
163	C₂×C₄×C₈	—	122	1	no	—
164	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	61	1	no	—
166	#336	—	122	1	no	—
167	#335	—	2	2	no	—
168	#331	—	122	4	no	—
169	C₆₁	#9734	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	122	2	no	—
171	C₂²×D₄	—	122	2	no	—
172	C₂²×D₄	—	122	2	no	—
173	C₂×D₈	—	122	4	no	—
174	C₂×D₈	—	122	4	no	—
175	C₂×D₈	—	244	2	no	—
176	C₂×D₈	—	244	2	no	—
177	C₂×D₈	—	244	2	no	—
178	C₂×D₈	—	244	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	122	1	no	—
181	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
182	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
183	#127	—	61	4	no	—
184	#127	—	122	2	no	—
185	#127	—	122	2	no	—
186	#127	—	122	2	no	—
187	C₂×C₄⋊C₄	—	61	2	no	—
188	C₂×C₄²	—	122	1	no	—
189	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	61	2	no	—
193	C₄⋊C₈	—	122	1	no	—
194	#102	—	244	8	no	—
195	C₄×C₈	—	244	2	no	—
196	C₄×C₈	C₄×D₆₁	1	2	yes	—
197	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
198	#95	—	122	1	no	—
199	C₂⁴	—	244	2	no	—
200	C₂×D₄	—	122	4	no	—
201	C₂×D₄	—	122	4	no	—
202	C₂×D₄	—	122	4	no	—
203	C₂×D₄	—	488	4	no	—
204	C₂×D₄	—	244	2	no	—
205	C₂×D₄	—	244	2	no	—
206	C₂×D₄	—	244	2	no	—
207	C₂×D₄	—	244	4	no	—
208	C₂×D₄	—	244	2	no	—
209	C₂×D₄	—	244	2	no	—
210	C₂×D₄	—	244	4	no	—
211	C₂×D₄	—	244	2	no	—
212	C₂×D₄	—	244	2	no	—
213	C₂²×C₄	—	122	2	no	—
214	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	#11758226	1	1	yes	characteristic
217	D₈	—	244	8	no	—
218	D₈	—	244	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	122	2	no	—
224	C₂×C₈	—	244	1	no	—
225	C₂×C₈	—	244	1	no	—
226	C₂×C₈	#11758190	1	2	yes	—
227	C₂×C₈	D₄₈₈	1	2	yes	—
228	C₂×C₈	#11758198	1	2	yes	—
229	C₂×C₈	#11758198	1	2	yes	—
230	C₄⋊C₄	—	61	4	no	—
231	C₄⋊C₄	—	122	2	no	—
232	C₄²	—	244	1	no	—
233	C₄²	—	244	1	no	—
234	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
235	C₄²	D₄×D₆₁	1	1	yes	characteristic
236	C₄²	#11758226	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
238	C₂³	—	488	2	no	—
239	C₂³	—	488	2	no	—
240	C₂³	—	488	4	no	—
241	C₂³	—	488	4	no	—
242	C₂³	—	244	4	no	—
243	C₂³	#14519752	1	1	yes	characteristic
244	D₄	—	488	4	no	—
245	D₄	—	488	4	no	—
246	D₄	—	488	2	no	—
247	D₄	—	488	2	no	—
248	D₄	—	488	2	no	—
249	D₄	—	488	2	no	—
250	D₄	—	244	4	no	—
251	D₄	—	244	4	no	—
252	D₄	—	244	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	122	4	no	—
255	C₂×C₄	—	244	2	no	—
256	C₂×C₄	—	244	2	no	—
257	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
258	C₂×C₄	#14519739	1	1	yes	characteristic
259	C₂×C₄	#14519739	1	1	yes	characteristic
260	C₂×C₄	#14519752	1	1	yes	characteristic
261	C₂×C₄	#14519766	1	1	yes	characteristic
262	C₂×C₄	#14519773	1	1	yes	characteristic
263	C₂×C₄	D₈×D₆₁	1	2	yes	—
264	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
265	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
266	C₂×C₄	#14519800	1	2	yes	—
267	C₂×C₄	#14519805	1	1	yes	characteristic
268	C₂×C₄	#14519805	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	8	2	no	—
271	C₈	—	244	2	no	—
272	C₈	#14519699	1	4	yes	complemented
273	C₂²	—	976	4	no	—
274	C₂²	—	976	4	no	—
275	C₂²	—	488	4	no	—
276	C₂²	—	488	4	no	—
277	C₂²	—	488	4	no	—
278	C₂²	#16839663	1	1	yes	characteristic
279	C₂²	#16839958	1	1	yes	characteristic
280	C₂²	#16839963	1	1	yes	characteristic
281	C₂²	#16839999	1	2	yes	—
282	C₂²	#16840033	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	244	4	no	—
286	C₄	C₂×#14519699	1	1	yes	characteristic
287	C₄	#16839768	1	1	yes	characteristic
288	C₄	#16839844	1	2	yes	—
289	C₄	#16839958	1	1	yes	characteristic
290	C₄	#16839963	1	1	yes	characteristic
291	C₄	#16840137	1	1	yes	characteristic
292	C₄	#16840147	1	1	yes	characteristic
293	C₂	—	976	4	no	—
294	C₂	#21445175	1	2	yes	—
295	C₂	#21445180	1	2	yes	—
296	C₂	#21446683	1	1	yes	characteristic
297	C₂	#21446727	1	1	yes	characteristic
298	C₂	#21446745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519752
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle	C₂³×C₆₁	#121
prime radical	C₂×#16840139	C₂
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#9734
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #21446745	C₂	semidirect
3. C₈×C₄⋊Dic₆₁	C₂	semidirect
4. C₂×#16839505	C₂	semidirect
5. C₂×C₆₁×#335	C₂	semidirect
10. #16840139	C₄	semidirect
80. C₄⋊Dic₆₁	D₈	semidirect
121. C₈×C₆₁	#102	semidirect
147. C₂×#335	D₆₁	semidirect
169. C₆₁	#9734	semidirect
272. C₈	#14519699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840139	2
#21446745	2
C₈×C₄⋊Dic₆₁	2
C₂×#16839505	2
C₂×C₆₁×#335	2
#9734	61