Subgroups — #29699821

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448776	C₂	1	2	yes	maximal
2	#21446691	C₂	1	2	yes	maximal
3	#21446455	C₂	1	4	yes	maximal
4	#21445976	C₂	1	1	yes	maximal, characteristic
5	#21445824	C₂	1	2	yes	maximal
6	#21445718	C₂	1	2	yes	maximal
7	#21445686	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519709	C₂²	1	2	yes	—
10	C₂×#14519709	C₂²	1	2	yes	—
11	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₂×#14519701	C₂²	1	4	yes	—
13	C₂×#14519687	C₂²	1	2	yes	—
14	#16839669	C₂²	1	4	yes	—
15	#16839648	—	2	4	no	—
16	#16839632	C₂²	1	2	yes	—
17	#16839628	—	2	2	no	—
18	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₆₁	—	2	2	no	—
20	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
21	#16839524	C₂²	1	4	yes	—
22	#16839516	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₁	C₄	1	8	yes	complemented
26	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₆₁	—	2	2	no	—
31	C₂×C₄⋊Dic₆₁	—	2	2	no	—
32	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
33	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₁	—	2	4	no	—
35	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
36	#14519709	—	2	4	no	—
37	#14519709	—	4	2	no	—
38	C₄⋊#92418	—	2	2	no	—
39	#14519701	—	4	4	no	—
40	#14519687	—	2	4	no	—
41	#14519684	—	2	4	no	—
42	#14519684	—	2	4	no	—
43	#14519684	—	2	4	no	—
44	#14519684	D₄	1	2	yes	—
45	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₆₁	—	2	2	no	—
47	C₂×Q₁₆×C₆₁	—	2	2	no	—
48	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₁	—	2	4	no	—
52	C₄×Q₈×C₆₁	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₆₁	—	2	2	no	—
56	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	#107×C₆₁	C₂×C₄	1	4	yes	—
58	#103×C₆₁	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
60	C₂×Q₈×C₆₁	—	2	2	no	—
61	C₂×Q₈×C₆₁	—	2	2	no	—
62	C₂×Q₈×C₆₁	—	2	2	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	2	no	—
65	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	—	2	2	no	—
68	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₆₁	—	2	4	no	—
72	C₂²×Dic₆₁	C₂×D₄	1	4	yes	—
73	Q₁₆×C₆₁	—	2	8	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₁	#41	1	2	yes	—
79	C₂×C₈×C₆₁	#41	1	2	yes	—
80	C₄⋊C₄×C₆₁	—	2	8	no	—
81	C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
83	C₄²×C₆₁	—	2	8	no	—
84	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₆₁	#41	1	2	yes	—
86	C₄⋊Dic₆₁	—	2	4	no	—
87	C₄⋊Dic₆₁	—	2	4	no	—
88	C₄⋊Dic₆₁	—	2	4	no	—
89	C₄⋊Dic₆₁	—	4	4	no	—
90	C₄⋊Dic₆₁	—	8	2	no	—
91	C₄×Dic₆₁	—	2	2	no	—
92	C₄×Dic₆₁	—	2	2	no	—
93	C₄×Dic₆₁	—	8	4	no	—
94	C₂×#92418	—	2	2	no	—
95	C₂×#92418	—	2	2	no	—
96	C₂×#92418	—	2	2	no	—
97	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₆₁	—	2	4	no	—
99	Q₈×C₆₁	—	2	4	no	—
100	Q₈×C₆₁	—	4	2	no	—
101	Q₈×C₆₁	—	4	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	4	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₁	#137	1	4	yes	—
112	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×Dic₆₁	—	2	4	no	—
119	C₂×Dic₆₁	—	2	4	no	—
120	C₂×Dic₆₁	—	2	4	no	—
121	C₂×Dic₆₁	—	4	4	no	—
122	C₂×Dic₆₁	—	4	4	no	—
123	C₂×Dic₆₁	—	4	4	no	—
124	C₈×C₆₁	—	2	2	no	—
125	C₈×C₆₁	#117	1	4	yes	—
126	#92418	—	4	2	no	—
127	#22814	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₆₁	#547	1	1	yes	characteristic
130	C₂²×C₆₁	#550	1	1	yes	characteristic
131	C₂²×C₆₁	C₂×#137	1	2	yes	—
132	C₂²×C₆₁	#577	1	2	yes	—
133	C₄×C₆₁	—	2	2	no	—
134	C₄×C₆₁	—	4	4	no	—
135	C₄×C₆₁	—	4	4	no	—
136	C₄×C₆₁	#441	1	4	yes	—
137	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
138	C₄×C₆₁	#520	1	1	yes	characteristic
139	C₄×C₆₁	#555	1	1	yes	characteristic
140	C₄×C₆₁	#556	1	1	yes	characteristic
141	Dic₆₁	—	4	4	no	—
142	Dic₆₁	—	8	4	no	—
143	C₂×C₄×Q₁₆	D₆₁	1	1	yes	characteristic
144	#2355	—	61	2	no	—
145	#1971	—	61	2	no	—
146	#1927	—	61	4	no	—
147	#1844	—	61	1	no	—
148	#1803	—	61	2	no	—
149	#1776	—	61	2	no	—
150	#1770	—	61	1	no	—
151	C₂×C₆₁	#2354	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
153	C₂×C₆₁	#2952	1	1	yes	characteristic
154	C₂×C₆₁	#3356	1	2	yes	—
155	C₂×C₆₁	#3400	1	2	yes	—
156	C₂²×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂×D₆₁	1	2	yes	—
158	C₄×Q₁₆	Dic₆₁	1	8	yes	complemented
159	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
160	C₂×#106	—	61	2	no	—
161	C₂×#106	—	61	2	no	—
162	C₂×C₄⋊C₈	—	61	1	no	—
163	C₂×#103	—	61	4	no	—
164	C₂×#103	C₂×D₆₁	1	2	yes	—
165	C₂×#97	—	61	2	no	—
166	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
167	#391	—	61	4	no	—
168	#387	—	122	4	no	—
169	#384	—	61	2	no	—
170	#382	—	122	2	no	—
171	C₄×C₄⋊C₄	—	61	1	no	—
172	C₄×C₄⋊C₄	—	61	4	no	—
173	C₄×C₄⋊C₄	—	122	2	no	—
174	#340	—	61	4	no	—
175	#336	—	61	2	no	—
176	C₆₁	#22814	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	C₂²×D₆₁	1	2	yes	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	C₂×Dic₆₁	1	4	yes	—
181	C₂²×C₈	—	61	2	no	—
182	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	C₂×Dic₆₁	1	8	yes	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	61	2	no	—
188	C₂×C₄⋊C₄	—	61	4	no	—
189	C₂×C₄⋊C₄	—	122	2	no	—
190	C₂×C₄⋊C₄	—	122	2	no	—
191	C₂×C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	61	2	no	—
194	C₂×C₄²	—	122	4	no	—
195	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
196	#107	C₂×Dic₆₁	1	4	yes	—
197	#106	—	122	4	no	—
198	#106	—	244	2	no	—
199	C₄⋊C₈	—	122	2	no	—
200	#103	—	244	4	no	—
201	#103	C₂×Dic₆₁	1	8	yes	—
202	#97	—	122	4	no	—
203	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
204	#95	—	61	2	no	—
205	#95	—	122	4	no	—
206	#95	—	122	4	no	—
207	#95	—	122	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×Dic₆₁	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	61	4	no	—
217	C₂²×C₄	—	122	4	no	—
218	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
221	Q₁₆	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	122	2	no	—
225	C₂×C₈	—	122	2	no	—
226	C₂×C₈	—	122	2	no	—
227	C₂×C₈	#11758222	1	2	yes	—
228	C₂×C₈	#11758222	1	2	yes	—
229	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
230	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	122	4	no	—
234	C₄⋊C₄	—	122	4	no	—
235	C₄⋊C₄	—	122	4	no	—
236	C₄⋊C₄	—	488	2	no	—
237	C₄⋊C₄	—	244	4	no	—
238	C₄⋊C₄	C₂²×Dic₆₁	1	4	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	122	2	no	—
241	C₄²	—	122	2	no	—
242	C₄²	—	488	4	no	—
243	C₄²	#11758224	1	2	yes	—
244	C₄²	C₂²×Dic₆₁	1	2	yes	—
245	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	122	4	no	—
258	C₂×C₄	—	122	4	no	—
259	C₂×C₄	—	122	4	no	—
260	C₂×C₄	—	244	4	no	—
261	C₂×C₄	—	244	4	no	—
262	C₂×C₄	—	244	4	no	—
263	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
264	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
265	C₂×C₄	#14519809	1	4	yes	—
266	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
267	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
268	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
269	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
270	C₂×C₄	#14519823	1	1	yes	characteristic
271	C₂×C₄	#14519824	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	244	2	no	—
274	C₈	#14519736	1	4	yes	—
275	C₂²	C₂×D₄×Dic₆₁	1	1	yes	characteristic
276	C₂²	#16840417	1	1	yes	characteristic
277	C₂²	#16840433	1	1	yes	characteristic
278	C₂²	C₂×#14519809	1	2	yes	—
279	C₂²	#16840569	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	488	4	no	—
284	C₄	—	244	4	no	—
285	C₄	#16839869	1	4	yes	—
286	C₄	C₂×#14519736	1	1	yes	characteristic
287	C₄	#16840245	1	1	yes	characteristic
288	C₄	#16840477	1	1	yes	characteristic
289	C₄	#16840482	1	1	yes	characteristic
290	C₂	#21448768	1	1	yes	characteristic
291	C₂	C₂×#16839869	1	1	yes	characteristic
292	C₂	#21454703	1	1	yes	characteristic
293	C₂	#21458436	1	2	yes	—
294	C₂	#21458870	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₄×Q₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₁
hypocenter	C₆₁	#22814
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₆₁	C₄	semidirect
158. C₄×Q₁₆	Dic₆₁	semidirect
176. C₆₁	#22814	semidirect

Maximal subgroups

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

subgroup	index
#21448776	2
#21446691	2
#21446455	2
#21445976	2
#21445824	2
#21445718	2
#21445686	2
C₂×C₄×Q₁₆×C₆₁	2
#22814	61