Subgroups — #29584216

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446747	C₂	1	2	yes	maximal, complemented
2	#21446537	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16839506	C₂	1	2	yes	maximal, complemented
4	#21443654	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
8	C₂×#14519706	—	2	1	no	—
9	C₂×#14519698	—	2	2	no	—
10	C₂×#14519698	—	2	2	no	—
11	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	—	2	1	no	—
13	#16839506	—	2	4	no	—
14	C₂²×D₈×C₆₁	—	2	1	no	—
15	C₂²×D₈×C₆₁	—	2	1	no	—
16	C₂×#127×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#493	—	2	4	no	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	—	2	1	no	—
21	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	—	2	1	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	#14519706	—	4	2	no	—
28	#14519698	—	4	4	no	—
29	#14519698	—	4	4	no	—
30	#14519684	—	2	1	no	—
31	C₂²×D₄×C₆₁	—	2	2	no	—
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	—	2	2	no	—
34	C₂×D₈×C₆₁	—	2	4	no	—
35	C₂×D₈×C₆₁	—	2	4	no	—
36	C₂×D₈×C₆₁	—	4	2	no	—
37	C₂×D₈×C₆₁	—	4	2	no	—
38	C₂×D₈×C₆₁	—	4	2	no	—
39	C₂×D₈×C₆₁	—	4	2	no	—
40	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
42	#127×C₆₁	—	2	2	no	—
43	#127×C₆₁	—	2	2	no	—
44	#127×C₆₁	—	2	2	no	—
45	#127×C₆₁	D₄	1	4	yes	—
46	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₁	—	2	2	no	—
48	C₂⁴×C₆₁	—	4	2	no	—
49	C₂×D₄×C₆₁	—	2	4	no	—
50	C₂×D₄×C₆₁	—	2	4	no	—
51	C₂×D₄×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	4	2	no	—
53	C₂×D₄×C₆₁	—	4	2	no	—
54	C₂×D₄×C₆₁	—	4	2	no	—
55	C₂×D₄×C₆₁	—	4	2	no	—
56	C₂×D₄×C₆₁	—	4	2	no	—
57	C₂×D₄×C₆₁	—	4	2	no	—
58	C₂×D₄×C₆₁	—	4	2	no	—
59	C₂×D₄×C₆₁	—	4	2	no	—
60	C₂×D₄×C₆₁	—	4	2	no	—
61	C₂×D₄×C₆₁	—	4	2	no	—
62	C₂×D₄×C₆₁	—	4	2	no	—
63	C₂×D₄×C₆₁	—	8	2	no	—
64	C₂×D₄×C₆₁	—	8	2	no	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
68	C₂²×Dic₆₁	—	2	1	no	—
69	C₂²×Dic₆₁	—	2	1	no	—
70	D₈×C₆₁	—	4	8	no	—
71	D₈×C₆₁	—	4	8	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	1	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	D₈	1	2	yes	—
77	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
78	C₄²×C₆₁	C₂×D₄	1	2	yes	—
79	C₄²×C₆₁	#41	1	2	yes	—
80	C₄⋊Dic₆₁	—	2	1	no	—
81	C₄⋊Dic₆₁	—	2	2	no	—
82	C₄⋊Dic₆₁	D₈	1	2	yes	complemented
83	C₄×Dic₆₁	—	4	1	no	—
84	C₄×Dic₆₁	—	4	1	no	—
85	C₂×#92418	—	4	1	no	—
86	C₂×#92418	—	4	2	no	—
87	C₂×#92418	—	4	2	no	—
88	C₂×#92418	—	4	1	no	—
89	C₂×#92418	—	4	2	no	—
90	C₂×#92418	—	4	1	no	—
91	C₂³×C₆₁	—	4	4	no	—
92	C₂³×C₆₁	—	8	2	no	—
93	C₂³×C₆₁	—	8	2	no	—
94	C₂³×C₆₁	—	8	2	no	—
95	C₂³×C₆₁	—	8	2	no	—
96	C₂³×C₆₁	—	8	2	no	—
97	C₂³×C₆₁	—	8	2	no	—
98	C₂³×C₆₁	#121	1	1	yes	characteristic
99	D₄×C₆₁	—	4	4	no	—
100	D₄×C₆₁	—	4	4	no	—
101	D₄×C₆₁	—	4	4	no	—
102	D₄×C₆₁	—	4	4	no	—
103	D₄×C₆₁	—	8	4	no	—
104	D₄×C₆₁	—	8	4	no	—
105	D₄×C₆₁	—	8	2	no	—
106	D₄×C₆₁	—	8	2	no	—
107	D₄×C₆₁	—	8	2	no	—
108	D₄×C₆₁	—	8	2	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#121	1	2	yes	—
113	C₂×C₄×C₆₁	#123	1	2	yes	—
114	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
119	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×Dic₆₁	—	2	2	no	—
123	C₂×Dic₆₁	—	4	1	no	—
124	C₂×Dic₆₁	—	4	1	no	—
125	C₂×Dic₆₁	—	4	1	no	—
126	C₂×Dic₆₁	—	4	1	no	—
127	C₂×Dic₆₁	—	4	1	no	—
128	C₈×C₆₁	—	2	2	no	—
129	C₈×C₆₁	—	2	2	no	—
130	#92418	—	8	1	no	—
131	#92418	—	8	2	no	—
132	#10536	—	61	1	no	maximal, Sylow
133	C₂²×C₆₁	—	8	4	no	—
134	C₂²×C₆₁	—	8	4	no	—
135	C₂²×C₆₁	—	8	4	no	—
136	C₂²×C₆₁	—	16	2	no	—
137	C₂²×C₆₁	—	16	2	no	—
138	C₂²×C₆₁	—	16	2	no	—
139	C₂²×C₆₁	—	16	2	no	—
140	C₂²×C₆₁	#394	1	1	yes	characteristic
141	C₂²×C₆₁	#459	1	1	yes	characteristic
142	C₂²×C₆₁	#460	1	1	yes	characteristic
143	C₂²×C₆₁	#466	1	1	yes	characteristic
144	C₂²×C₆₁	#469	1	1	yes	characteristic
145	C₂²×C₆₁	#472	1	1	yes	characteristic
146	C₂²×C₆₁	#472	1	1	yes	characteristic
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	#447	1	1	yes	characteristic
149	C₄×C₆₁	#450	1	1	yes	characteristic
150	C₄×C₆₁	#459	1	2	yes	—
151	C₄×C₆₁	#463	1	2	yes	—
152	C₄×C₆₁	#486	1	1	yes	characteristic
153	C₄×C₆₁	#490	1	1	yes	characteristic
154	Dic₆₁	—	4	2	no	—
155	Dic₆₁	—	8	1	no	—
156	C₂×#493	D₆₁	1	1	yes	characteristic
157	#1982	—	61	2	no	—
158	#1939	—	61	1	no	—
159	C₂×#331	—	61	2	no	—
160	#1277	—	61	1	no	—
161	C₂×C₆₁	—	16	4	no	—
162	C₂×C₆₁	#1667	1	1	yes	characteristic
163	C₂×C₆₁	#1668	1	1	yes	characteristic
164	C₂×C₆₁	#1687	1	1	yes	characteristic
165	C₂×C₆₁	#1688	1	1	yes	characteristic
166	C₂×C₆₁	#2041	1	1	yes	characteristic
167	C₂×C₆₁	#2045	1	1	yes	characteristic
168	C₂×C₆₁	#2054	1	1	yes	characteristic
169	C₂²×D₈	—	2	1	no	—
170	C₂²×D₈	—	2	1	no	—
171	C₂×#127	C₂×D₆₁	1	2	yes	—
172	#493	—	2	4	no	—
173	C₂×C₄⋊C₈	—	61	1	no	—
174	C₂×C₄⋊C₈	—	61	2	no	—
175	C₂×C₄⋊C₈	—	122	1	no	—
176	C₂×#102	—	122	2	no	—
177	C₂×#102	—	122	2	no	—
178	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	61	1	no	—
180	#336	—	122	1	no	—
181	#331	—	122	4	no	—
182	C₆₁	#10536	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂×D₈	—	2	4	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	—	4	2	no	—
189	C₂×D₈	—	4	2	no	—
190	C₂×D₈	—	4	2	no	—
191	C₂×D₈	—	4	2	no	—
192	C₂²×C₈	—	122	2	no	—
193	C₂²×C₈	—	122	1	no	—
194	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
196	#127	—	2	2	no	—
197	#127	—	2	2	no	—
198	#127	—	2	2	no	—
199	#127	C₂²⋊D₆₁	1	4	yes	—
200	C₂×C₄⋊C₄	—	61	1	no	—
201	C₂×C₄⋊C₄	—	61	1	no	—
202	C₂×C₄²	—	122	1	no	—
203	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
204	C₄⋊C₈	—	122	2	no	—
205	C₄⋊C₈	—	122	2	no	—
206	C₄⋊C₈	—	122	2	no	—
207	C₄⋊C₈	—	244	2	no	—
208	#102	—	244	4	no	—
209	#102	—	244	4	no	—
210	C₄×C₈	—	2	2	no	—
211	#95	—	122	1	no	—
212	C₂⁴	—	4	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂×D₄	—	8	2	no	—
229	C₂²×C₄	—	122	1	no	—
230	C₂²×C₄	—	122	1	no	—
231	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
232	C₂²×C₄	#11758224	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
234	D₈	—	4	8	no	—
235	D₈	—	4	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	244	1	no	—
241	C₂×C₈	—	244	2	no	—
242	C₂×C₈	—	244	2	no	—
243	C₂×C₈	—	244	1	no	—
244	C₂×C₈	—	244	2	no	—
245	C₂×C₈	—	244	1	no	—
246	C₂×C₈	#11758199	1	2	yes	—
247	C₂×C₈	#11758200	1	2	yes	—
248	C₄⋊C₄	—	61	2	no	—
249	C₄⋊C₄	—	122	1	no	—
250	C₄⋊C₄	—	122	2	no	—
251	C₄²	—	244	1	no	—
252	C₄²	—	244	1	no	—
253	C₄²	#11758224	1	2	yes	—
254	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
255	C₂³	—	4	4	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	8	2	no	—
262	C₂³	#14519748	1	1	yes	characteristic
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	122	2	no	—
276	C₂×C₄	—	244	1	no	—
277	C₂×C₄	—	244	1	no	—
278	C₂×C₄	—	244	1	no	—
279	C₂×C₄	—	244	1	no	—
280	C₂×C₄	—	244	1	no	—
281	C₂×C₄	#14519746	1	1	yes	characteristic
282	C₂×C₄	#14519749	1	2	yes	—
283	C₂×C₄	#14519758	1	2	yes	—
284	C₂×C₄	#14519764	1	1	yes	characteristic
285	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
287	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
288	C₂×C₄	#14519796	1	1	yes	characteristic
289	C₂×C₄	#14519801	1	2	yes	—
290	C₂×C₄	#14519803	1	1	yes	characteristic
291	C₂×C₄	#14519803	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	488	2	no	—
295	C₈	—	488	1	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	16	2	no	—
300	C₂²	—	16	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	#16839676	1	1	yes	characteristic
304	C₂²	#16839954	1	1	yes	characteristic
305	C₂²	#16839959	1	1	yes	characteristic
306	C₂²	#16839995	1	1	yes	characteristic
307	C₂²	#16840012	1	1	yes	characteristic
308	C₂²	#16840029	1	1	yes	characteristic
309	C₂²	#16840029	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	488	1	no	—
312	C₄	—	244	2	no	—
313	C₄	#16839896	1	1	yes	characteristic
314	C₄	#16839913	1	1	yes	characteristic
315	C₄	#16839955	1	2	yes	—
316	C₄	#16839975	1	2	yes	—
317	C₄	#16840108	1	1	yes	characteristic
318	C₄	#16840126	1	1	yes	characteristic
319	C₂	—	16	4	no	—
320	C₂	#21445171	1	1	yes	characteristic
321	C₂	#21445176	1	1	yes	characteristic
322	C₂	#21445295	1	1	yes	characteristic
323	C₂	#21445300	1	1	yes	characteristic
324	C₂	#21447050	1	1	yes	characteristic
325	C₂	#21447072	1	1	yes	characteristic
326	C₂	#21447127	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting, prime radical	C₂×C₆₁×#493	C₂
socle	C₂³×C₆₁	#121
hypercenter	C₂×#493	D₆₁
hypocenter	C₆₁	#10536
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #21446747	C₂	semidirect
2. #21446537	C₂	semidirect
3. C₂×#16839506	C₂	semidirect
4. #21443654	C₂	semidirect
82. C₄⋊Dic₆₁	D₈	semidirect
182. C₆₁	#10536	semidirect

Maximal subgroups

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

subgroup	index
#21446747	2
#21446537	2
C₂×#16839506	2
#21443654	2
C₂×C₆₁×#493	2
#10536	61