Subgroups — #29663216

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	#21456760	C₂	1	2	yes	maximal, complemented
2	#21454753	C₂	1	1	yes	maximal, characteristic, complemented
3	#21445300	C₂	1	4	yes	maximal, complemented
4	#21445176	C₂	1	4	yes	maximal, complemented
5	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444487	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic
8	#16840319	C₂²	1	1	yes	characteristic
9	#16840134	—	2	2	no	—
10	#16840108	—	2	2	no	—
11	#16839959	C₂²	1	8	yes	complemented
12	#16839913	—	2	4	no	—
13	#16839892	—	2	1	no	—
14	#16839892	C₂²	1	2	yes	complemented
15	C₈×Dic₁₂₂	—	2	1	no	—
16	C₈×Dic₁₂₂	C₂²	1	2	yes	complemented
17	#16839815	C₂²	1	2	yes	—
18	#16839788	—	2	1	no	—
19	#16839786	C₂²	1	1	yes	characteristic
20	#16839717	—	2	1	no	—
21	#16839515	C₂²	1	2	yes	complemented
22	#16839513	C₂²	1	2	yes	complemented
23	#16839506	C₂²	1	8	yes	complemented
24	C₂²×D₈×C₆₁	—	2	1	no	—
25	C₂²×D₈×C₆₁	—	2	1	no	—
26	C₂×#127×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#493	—	2	1	no	—
28	C₆₁×#493	—	2	1	no	—
29	C₆₁×#493	C₂²	1	2	yes	—
30	C₆₁×#493	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#11758200	—	2	8	no	—
33	C₂×#11758194	—	2	1	no	—
34	#14519777	D₄	1	1	yes	characteristic
35	#14519777	D₄	1	1	yes	characteristic
36	#14519769	—	2	1	no	—
37	#14519756	—	2	1	no	—
38	#14519756	D₄	1	2	yes	—
39	C₄×Dic₁₂₂	—	2	1	no	—
40	C₄×Dic₁₂₂	C₂³	1	2	yes	—
41	#14519736	—	2	1	no	—
42	C₄⋊#92418	—	2	1	no	—
43	C₄⋊#92418	C₂³	1	4	yes	—
44	C₄⋊#92418	C₂³	1	2	yes	—
45	#14519706	—	2	2	no	—
46	#14519706	—	2	2	no	—
47	#14519698	—	2	8	no	—
48	C₂²⋊#92418	—	2	2	no	—
49	#14519687	—	2	2	no	—
50	C₄×#92418	—	2	2	no	—
51	C₈×Dic₆₁	—	2	2	no	—
52	C₂²×D₄×C₆₁	—	2	2	no	—
53	C₂²×D₄×C₆₁	—	2	2	no	—
54	C₂²×D₄×C₆₁	—	2	2	no	—
55	C₂×D₈×C₆₁	—	2	2	no	—
56	C₂×D₈×C₆₁	—	2	2	no	—
57	C₂×D₈×C₆₁	—	2	2	no	—
58	C₂×D₈×C₆₁	—	2	2	no	—
59	C₂×D₈×C₆₁	—	4	4	no	—
60	C₂×D₈×C₆₁	—	4	4	no	—
61	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
63	#127×C₆₁	—	2	2	no	—
64	#127×C₆₁	—	2	2	no	—
65	#127×C₆₁	—	2	2	no	—
66	#127×C₆₁	C₂³	1	4	yes	—
67	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
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	C₂⁴×C₆₁	—	4	2	no	—
73	C₂×D₄×C₆₁	—	2	4	no	—
74	C₂×D₄×C₆₁	—	2	4	no	—
75	C₂×D₄×C₆₁	—	2	4	no	—
76	C₂×D₄×C₆₁	—	4	2	no	—
77	C₂×D₄×C₆₁	—	4	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	4	4	no	—
80	C₂×D₄×C₆₁	—	4	4	no	—
81	C₂×D₄×C₆₁	—	8	4	no	—
82	C₂×D₄×C₆₁	—	8	4	no	—
83	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
86	C₂×Dic₁₂₂	—	2	1	no	—
87	C₂×Dic₁₂₂	C₂×D₄	1	2	yes	—
88	D₈×C₆₁	—	4	4	no	—
89	D₈×C₆₁	—	4	4	no	—
90	D₈×C₆₁	—	4	4	no	—
91	D₈×C₆₁	—	4	4	no	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	SD₁₆	1	4	yes	complemented
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₁	#41	1	1	yes	characteristic
101	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
102	C₂²⋊Dic₆₁	—	2	2	no	—
103	#11758200	—	8	8	no	—
104	C₄⋊Dic₆₁	—	2	1	no	—
105	C₄⋊Dic₆₁	—	2	1	no	—
106	C₄⋊Dic₆₁	—	2	1	no	—
107	C₄⋊Dic₆₁	C₂×D₄	1	2	yes	—
108	#11758196	—	2	2	no	—
109	#11758196	—	2	2	no	—
110	C₄×Dic₆₁	—	2	2	no	—
111	#11758194	—	4	2	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	—	2	4	no	—
114	C₂³×C₆₁	—	4	4	no	—
115	C₂³×C₆₁	—	8	4	no	—
116	C₂³×C₆₁	—	8	4	no	—
117	C₂³×C₆₁	—	8	4	no	—
118	C₂³×C₆₁	#121	1	1	yes	characteristic
119	D₄×C₆₁	—	4	4	no	—
120	D₄×C₆₁	—	4	4	no	—
121	D₄×C₆₁	—	4	8	no	—
122	D₄×C₆₁	—	8	4	no	—
123	D₄×C₆₁	—	8	4	no	—
124	D₄×C₆₁	—	8	4	no	—
125	D₄×C₆₁	—	8	4	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
136	C₂×Dic₆₁	—	2	2	no	—
137	C₂×Dic₆₁	—	2	2	no	—
138	Dic₁₂₂	—	2	4	no	—
139	Dic₁₂₂	—	4	2	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
143	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
144	#92418	—	4	2	no	—
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	—	8	4	no	—
150	C₂²×C₆₁	—	8	4	no	—
151	C₂²×C₆₁	—	8	4	no	—
152	C₂²×C₆₁	—	16	8	no	—
153	C₂²×C₆₁	#460	1	2	yes	—
154	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	#460	1	2	yes	—
158	C₄×C₆₁	#472	1	2	yes	—
159	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
160	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄×C₆₁	#576	1	2	yes	—
162	Dic₆₁	—	4	2	no	—
163	Dic₆₁	—	4	2	no	—
164	#3208	—	61	2	no	—
165	C₂×#493	D₆₁	1	1	yes	characteristic
166	#2960	—	61	1	no	—
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	—	16	4	no	—
173	C₂×C₆₁	C₂×#460	1	1	yes	characteristic
174	C₂×C₆₁	#3058	1	1	yes	characteristic
175	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
176	C₂²×D₈	—	2	1	no	—
177	C₂²×D₈	—	2	1	no	—
178	#533	—	61	1	no	—
179	C₂×#127	C₂×D₆₁	1	2	yes	—
180	#493	—	2	1	no	—
181	#493	—	2	1	no	—
182	#493	C₂×D₆₁	1	2	yes	—
183	#493	C₂×D₆₁	1	2	yes	—
184	#492	—	122	2	no	—
185	#486	—	122	2	no	—
186	#460	—	61	8	no	—
187	#450	—	122	4	no	—
188	#446	—	61	2	no	—
189	#446	—	122	1	no	—
190	C₈×Q₈	—	61	2	no	—
191	C₈×Q₈	—	122	1	no	—
192	#431	—	61	2	no	—
193	#424	—	122	1	no	—
194	#423	—	61	1	no	—
195	#405	—	122	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	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	—	2	2	no	—
204	C₂×SD₁₆	—	122	8	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	4	4	no	—
210	C₂×D₈	—	4	4	no	—
211	C₂×M₁₆	—	122	1	no	—
212	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
214	C₄⋊Q₈	—	61	1	no	—
215	C₄⋊Q₈	—	61	1	no	—
216	#127	—	2	2	no	—
217	#127	—	2	2	no	—
218	#127	—	2	2	no	—
219	#127	C₂²×D₆₁	1	4	yes	—
220	#125	—	122	1	no	—
221	C₂²⋊Q₈	—	61	2	no	—
222	C₂²⋊Q₈	—	122	1	no	—
223	C₄×Q₈	—	61	2	no	—
224	C₄×Q₈	—	122	1	no	—
225	#117	—	122	1	no	—
226	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
227	C₄⋊C₈	—	61	2	no	—
228	C₄⋊C₈	—	61	4	no	—
229	C₄⋊C₈	—	122	2	no	—
230	C₄⋊C₈	—	122	2	no	—
231	C₄⋊C₈	—	122	1	no	—
232	#102	—	122	8	no	—
233	C₂²⋊C₈	—	122	2	no	—
234	#97	—	122	2	no	—
235	C₄×C₈	—	122	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	—	4	2	no	—
242	C₂×Q₈	—	61	2	no	—
243	C₂×Q₈	—	122	1	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	8	4	no	—
253	C₂×D₄	—	8	4	no	—
254	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
255	C₂²×C₄	#11758224	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
257	SD₁₆	—	488	8	no	—
258	D₈	—	4	4	no	—
259	D₈	—	4	4	no	—
260	D₈	—	4	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	244	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	122	4	no	—
265	C₂×C₈	—	122	2	no	—
266	C₂×C₈	#11758200	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	—	61	2	no	—
272	C₄⋊C₄	—	122	2	no	—
273	C₄⋊C₄	—	122	2	no	—
274	C₄⋊C₄	—	122	1	no	—
275	C₄⋊C₄	—	122	1	no	—
276	C₄⋊C₄	—	122	1	no	—
277	C₂²⋊C₄	—	122	2	no	—
278	C₄²	—	122	2	no	—
279	C₄²	D₄×D₆₁	1	1	yes	characteristic
280	C₄²	#11758224	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
282	C₄²	C₂³×D₆₁	1	1	yes	characteristic
283	C₂³	—	4	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	—	8	4	no	—
287	C₂³	#14519748	1	1	yes	characteristic
288	Q₈	—	122	4	no	—
289	Q₈	—	244	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	8	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	D₄	—	8	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	122	2	no	—
300	C₂×C₄	—	122	2	no	—
301	C₂×C₄	#14519748	1	1	yes	characteristic
302	C₂×C₄	#14519748	1	1	yes	characteristic
303	C₂×C₄	#14519748	1	1	yes	characteristic
304	C₂×C₄	C₂×#11758200	1	2	yes	—
305	C₂×C₄	#14519803	1	2	yes	—
306	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	244	2	no	—
312	C₈	—	244	4	no	—
313	C₈	C₂×#11758200	1	2	yes	—
314	C₈	C₂×#11758200	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	16	8	no	—
321	C₂²	#16839959	1	2	yes	—
322	C₂²	C₂×#14519748	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	244	2	no	—
326	C₄	—	244	2	no	—
327	C₄	#16839959	1	2	yes	—
328	C₄	#16840029	1	2	yes	—
329	C₄	C₂²×#11758200	1	1	yes	characteristic
330	C₄	C₂×#14519803	1	1	yes	characteristic
331	C₄	#16840563	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	16	4	no	—
334	C₂	C₂×#16839959	1	1	yes	characteristic
335	C₂	#21455463	1	1	yes	characteristic
336	C₂	C₂×#16840029	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14519748
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting, prime radical	C₂×C₆₁×#493	C₂
socle	C₂²×C₆₁	C₂×#121
hypercenter	C₂×#493	D₆₁
hypocenter	C₆₁	#19203
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #21456760	C₂	semidirect
2. #21454753	C₂	semidirect
3. #21445300	C₂	semidirect
4. #21445176	C₂	semidirect
5. #21444567	C₂	semidirect
6. #21444487	C₂	semidirect
11. #16839959	C₂²	semidirect
14. #16839892	C₂²	semidirect
16. C₈×Dic₁₂₂	C₂²	semidirect
21. #16839515	C₂²	semidirect
22. #16839513	C₂²	semidirect
23. #16839506	C₂²	semidirect
93. C₂×C₈×C₆₁	SD₁₆	semidirect
200. C₆₁	#19203	semidirect
266. C₂×C₈	#11758200	semidirect

Maximal subgroups

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

subgroup	index
#21456760	2
#21454753	2
#21445300	2
#21445176	2
#21444567	2
#21444487	2
C₂×C₆₁×#493	2
#19203	61