Subgroups — #29663220

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456757	C₂	1	2	yes	maximal, complemented
2	C₂×#16840139	C₂	1	1	yes	maximal, characteristic
3	#21445303	C₂	1	4	yes	maximal, complemented
4	#21445179	C₂	1	4	yes	maximal, complemented
5	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444488	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₈₈	—	2	1	no	—
9	C₂²×D₄₈₈	—	2	1	no	—
10	C₂×#14519774	C₂²	1	2	yes	—
11	#16840139	—	2	1	no	—
12	#16840139	—	2	1	no	—
13	#16840139	C₂²	1	2	yes	—
14	#16840139	C₂²	1	2	yes	—
15	#16840136	—	2	2	no	—
16	#16840110	—	2	2	no	—
17	#16839962	C₂²	1	8	yes	complemented
18	#16839912	—	2	4	no	—
19	#16839815	C₂²	1	2	yes	—
20	#16839515	C₂²	1	2	yes	complemented
21	#16839513	C₂²	1	2	yes	complemented
22	#16839505	C₂²	1	8	yes	complemented
23	C₆₁×#533	C₂²	1	1	yes	characteristic
24	C₆₁×#446	—	2	1	no	—
25	C₆₁×#446	C₂²	1	2	yes	complemented
26	C₈×Q₈×C₆₁	—	2	1	no	—
27	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
28	C₆₁×#424	—	2	1	no	—
29	C₆₁×#423	C₂²	1	1	yes	characteristic
30	C₆₁×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂²×D₂₄₄	—	2	2	no	—
33	C₂²×D₂₄₄	—	2	2	no	—
34	C₂²×D₂₄₄	—	2	2	no	—
35	C₂×#11758201	—	2	8	no	—
36	C₂×D₄₈₈	—	2	2	no	—
37	C₂×D₄₈₈	—	2	2	no	—
38	C₂×D₄₈₈	—	2	2	no	—
39	C₂×D₄₈₈	—	2	2	no	—
40	C₂×D₄₈₈	—	4	4	no	—
41	C₂×D₄₈₈	—	4	4	no	—
42	#14519774	—	2	2	no	—
43	#14519774	—	2	2	no	—
44	#14519774	—	2	2	no	—
45	#14519774	C₂³	1	4	yes	—
46	C₄⋊#92418	C₂³	1	4	yes	—
47	#14519697	—	2	8	no	—
48	C₂²⋊#92418	—	2	2	no	—
49	C₄×#92418	—	2	2	no	—
50	C₂×M₁₆×C₆₁	—	2	1	no	—
51	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
55	#125×C₆₁	—	2	1	no	—
56	C₂²⋊Q₈×C₆₁	—	2	1	no	—
57	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
58	C₄×Q₈×C₆₁	—	2	1	no	—
59	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
60	#117×C₆₁	—	2	1	no	—
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₁	—	2	2	no	—
63	C₄⋊C₈×C₆₁	—	2	2	no	—
64	C₄⋊C₈×C₆₁	—	2	1	no	—
65	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
66	#97×C₆₁	—	2	2	no	—
67	C₄×C₈×C₆₁	—	2	2	no	—
68	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂³×D₆₁	—	4	2	no	—
73	C₂×Q₈×C₆₁	—	2	1	no	—
74	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
78	C₂×D₂₄₄	—	2	4	no	—
79	C₂×D₂₄₄	—	2	4	no	—
80	C₂×D₂₄₄	—	2	4	no	—
81	C₂×D₂₄₄	—	4	4	no	—
82	C₂×D₂₄₄	—	4	4	no	—
83	C₂×D₂₄₄	—	4	2	no	—
84	C₂×D₂₄₄	—	4	2	no	—
85	C₂×D₂₄₄	—	4	2	no	—
86	C₂×D₂₄₄	—	8	4	no	—
87	C₂×D₂₄₄	—	8	4	no	—
88	M₁₆×C₆₁	—	4	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	SD₁₆	1	4	yes	complemented
92	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
102	C₂²⋊C₄×C₆₁	—	2	2	no	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₁	#41	1	1	yes	characteristic
107	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
108	#11758201	—	8	8	no	—
109	C₂×#92418	—	2	4	no	—
110	D₄₈₈	—	4	4	no	—
111	D₄₈₈	—	4	4	no	—
112	D₄₈₈	—	4	4	no	—
113	D₄₈₈	—	4	4	no	—
114	C₂³×C₆₁	#121	1	1	yes	characteristic
115	C₂²×D₆₁	—	4	4	no	—
116	C₂²×D₆₁	—	8	4	no	—
117	C₂²×D₆₁	—	8	4	no	—
118	C₂²×D₆₁	—	8	4	no	—
119	Q₈×C₆₁	—	2	4	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
133	D₂₄₄	—	4	4	no	—
134	D₂₄₄	—	4	4	no	—
135	D₂₄₄	—	4	8	no	—
136	D₂₄₄	—	8	4	no	—
137	D₂₄₄	—	8	4	no	—
138	D₂₄₄	—	8	4	no	—
139	D₂₄₄	—	8	4	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	4	2	no	—
143	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
144	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
145	#92418	—	4	4	no	—
146	#19203	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	—	2	1	no	—
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	#460	1	2	yes	—
150	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
151	C₂×D₆₁	—	8	4	no	—
152	C₂×D₆₁	—	8	4	no	—
153	C₂×D₆₁	—	8	4	no	—
154	C₂×D₆₁	—	16	8	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	4	2	no	—
158	C₄×C₆₁	—	4	2	no	—
159	C₄×C₆₁	#460	1	2	yes	—
160	C₄×C₆₁	#472	1	2	yes	—
161	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
162	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₆₁	#576	1	2	yes	—
164	#3208	—	61	2	no	—
165	C₂×#493	—	61	1	no	—
166	#2960	D₆₁	1	1	yes	characteristic, complemented
167	#1688	—	61	4	no	—
168	#1668	—	61	4	no	—
169	#1564	—	61	1	no	—
170	#1545	—	61	2	no	—
171	C₂×C₆₁	—	2	2	no	—
172	C₂×C₆₁	C₂×#460	1	1	yes	characteristic
173	C₂×C₆₁	#3058	1	1	yes	characteristic
174	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
175	D₆₁	—	16	4	no	—
176	C₂²×D₈	—	122	1	no	—
177	C₂²×D₈	—	122	1	no	—
178	#533	C₂×D₆₁	1	1	yes	characteristic
179	C₂×#127	—	61	2	no	—
180	#493	—	61	2	no	—
181	#493	—	61	2	no	—
182	#493	—	122	1	no	—
183	#493	—	122	1	no	—
184	#492	—	122	2	no	—
185	#486	—	122	2	no	—
186	#460	—	61	8	no	—
187	#450	—	122	4	no	—
188	#446	—	2	1	no	—
189	#446	C₂×D₆₁	1	2	yes	complemented
190	C₈×Q₈	—	2	1	no	—
191	C₈×Q₈	C₂×D₆₁	1	2	yes	complemented
192	#431	—	61	2	no	—
193	#424	—	2	1	no	—
194	#423	C₂×D₆₁	1	1	yes	characteristic
195	#405	—	2	1	no	—
196	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
197	#335	—	61	2	no	—
198	#334	—	61	2	no	—
199	#331	—	61	8	no	—
200	C₆₁	#19203	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	122	2	no	—
202	C₂²×D₄	—	122	2	no	—
203	C₂²×D₄	—	122	2	no	—
204	C₂×SD₁₆	—	122	8	no	—
205	C₂×D₈	—	122	2	no	—
206	C₂×D₈	—	122	2	no	—
207	C₂×D₈	—	122	2	no	—
208	C₂×D₈	—	122	2	no	—
209	C₂×D₈	—	244	4	no	—
210	C₂×D₈	—	244	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
214	C₄⋊Q₈	D₂₄₄	1	1	yes	characteristic
215	C₄⋊Q₈	D₂₄₄	1	1	yes	characteristic
216	#127	—	61	4	no	—
217	#127	—	122	2	no	—
218	#127	—	122	2	no	—
219	#127	—	122	2	no	—
220	#125	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	C₂²⋊Q₈	D₂₄₄	1	2	yes	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
225	#117	—	2	1	no	—
226	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	61	4	no	—
231	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
232	#102	—	122	8	no	—
233	C₂²⋊C₈	—	122	2	no	—
234	#97	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	122	2	no	—
237	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
241	C₂⁴	—	244	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	C₂×D₂₄₄	1	2	yes	—
244	C₂×D₄	—	122	4	no	—
245	C₂×D₄	—	122	4	no	—
246	C₂×D₄	—	122	4	no	—
247	C₂×D₄	—	488	4	no	—
248	C₂×D₄	—	488	4	no	—
249	C₂×D₄	—	244	2	no	—
250	C₂×D₄	—	244	2	no	—
251	C₂×D₄	—	244	2	no	—
252	C₂×D₄	—	244	4	no	—
253	C₂×D₄	—	244	4	no	—
254	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
255	C₂²×C₄	#11758222	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
257	SD₁₆	—	488	8	no	—
258	D₈	—	244	4	no	—
259	D₈	—	244	4	no	—
260	D₈	—	244	4	no	—
261	D₈	—	244	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	122	4	no	—
266	C₂×C₈	#11758201	1	4	yes	complemented
267	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂×D₂₄₄	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
280	C₄²	#11758222	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
282	C₄²	C₂³×D₆₁	1	1	yes	characteristic
283	C₂³	—	488	4	no	—
284	C₂³	—	488	4	no	—
285	C₂³	—	488	4	no	—
286	C₂³	—	244	4	no	—
287	C₂³	#14519751	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	D₄	—	488	4	no	—
291	D₄	—	488	4	no	—
292	D₄	—	488	4	no	—
293	D₄	—	488	4	no	—
294	D₄	—	244	4	no	—
295	D₄	—	244	4	no	—
296	D₄	—	244	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	#14519751	1	1	yes	characteristic
302	C₂×C₄	#14519751	1	1	yes	characteristic
303	C₂×C₄	#14519751	1	1	yes	characteristic
304	C₂×C₄	C₂×#11758201	1	2	yes	—
305	C₂×C₄	#14519802	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
308	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	244	4	no	—
313	C₈	C₂×#11758201	1	2	yes	—
314	C₈	C₂×#11758201	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	976	8	no	—
318	C₂²	—	488	4	no	—
319	C₂²	—	488	4	no	—
320	C₂²	—	488	4	no	—
321	C₂²	#16839962	1	2	yes	—
322	C₂²	C₂×#14519751	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	#16839962	1	2	yes	—
328	C₄	#16840032	1	2	yes	—
329	C₄	C₂²×#11758201	1	1	yes	characteristic
330	C₄	C₂×#14519802	1	1	yes	characteristic
331	C₄	#16840567	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	976	4	no	—
334	C₂	C₂×#16839962	1	1	yes	characteristic
335	C₂	#21455467	1	1	yes	characteristic
336	C₂	C₂×#16840032	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14519751
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#2960	C₂
socle	C₂²×C₆₁	C₂×#121
prime radical	C₂×#16840139	C₂
hypercenter	#2960	D₆₁
hypocenter	C₆₁	#19203
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #21456757	C₂	semidirect
3. #21445303	C₂	semidirect
4. #21445179	C₂	semidirect
5. #21444567	C₂	semidirect
6. #21444488	C₂	semidirect
7. C₆₁×#2960	C₂	semidirect
17. #16839962	C₂²	semidirect
20. #16839515	C₂²	semidirect
21. #16839513	C₂²	semidirect
22. #16839505	C₂²	semidirect
25. C₆₁×#446	C₂²	semidirect
27. C₈×Q₈×C₆₁	C₂²	semidirect
91. C₂×C₈×C₆₁	SD₁₆	semidirect
166. #2960	D₆₁	semidirect
189. #446	C₂×D₆₁	semidirect
191. C₈×Q₈	C₂×D₆₁	semidirect
200. C₆₁	#19203	semidirect
266. C₂×C₈	#11758201	semidirect

Maximal subgroups

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

subgroup	index
#21456757	2
C₂×#16840139	2
#21445303	2
#21445179	2
#21444567	2
#21444488	2
C₆₁×#2960	2
#19203	61