Subgroups — #29585698

Order 15616.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445551	C₂	1	2	yes	maximal
2	#21445530	C₂	1	2	yes	maximal
3	C₂×#16839519	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839519	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2873	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519736	C₂²	1	2	yes	—
7	C₂×#14519736	C₂²	1	2	yes	—
8	#16839528	—	2	4	no	—
9	#16839523	C₄	1	8	yes	complemented
10	#16839519	—	2	2	no	—
11	#16839519	—	2	2	no	—
12	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#405	—	2	2	no	—
16	C₆₁×#405	C₄	1	4	yes	complemented
17	C₄×M₁₆×C₆₁	—	2	2	no	—
18	#14519736	—	2	2	no	—
19	#14519736	—	2	2	no	—
20	#14519736	—	2	2	no	—
21	#14519736	—	2	2	no	—
22	#14519736	—	2	2	no	—
23	#14519736	—	2	2	no	—
24	#14519736	—	2	2	no	—
25	#14519736	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₆₁	—	2	2	no	—
27	C₂×C₄⋊Dic₆₁	—	2	2	no	—
28	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
29	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₁	—	2	1	no	—
34	C₂×M₁₆×C₆₁	—	2	2	no	—
35	C₂×M₁₆×C₆₁	—	2	2	no	—
36	C₂×M₁₆×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
40	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
43	#117×C₆₁	—	2	1	no	—
44	#117×C₆₁	—	2	1	no	—
45	#117×C₆₁	C₂×C₄	1	2	yes	—
46	#117×C₆₁	Q₈	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
49	C₂×C₄²×C₆₁	—	2	1	no	—
50	#97×C₆₁	—	2	4	no	—
51	C₄×C₈×C₆₁	—	2	4	no	—
52	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	#41	1	2	yes	—
62	C₂²×C₄×C₆₁	#41	1	2	yes	—
63	C₂²×C₄×C₆₁	#41	1	2	yes	—
64	C₂²×Dic₆₁	—	2	4	no	—
65	C₂²×Dic₆₁	—	4	2	no	—
66	M₁₆×C₆₁	—	2	2	no	—
67	M₁₆×C₆₁	—	2	4	no	—
68	M₁₆×C₆₁	—	2	4	no	—
69	M₁₆×C₆₁	—	2	4	no	—
70	M₁₆×C₆₁	C₄²	1	4	yes	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
76	C₄⋊C₄×C₆₁	—	2	2	no	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₂²⋊C₄×C₆₁	—	2	2	no	—
79	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	2	2	no	—
82	C₂²⋊Dic₆₁	—	2	4	no	—
83	C₂²⋊Dic₆₁	—	4	2	no	—
84	C₂²⋊Dic₆₁	—	4	2	no	—
85	C₂²⋊Dic₆₁	—	4	2	no	—
86	C₄⋊Dic₆₁	—	2	4	no	—
87	C₄⋊Dic₆₁	—	4	2	no	—
88	C₄⋊Dic₆₁	—	4	2	no	—
89	C₄⋊Dic₆₁	—	4	2	no	—
90	C₄×Dic₆₁	—	2	4	no	—
91	C₄×Dic₆₁	—	2	4	no	—
92	C₄×Dic₆₁	—	2	4	no	—
93	C₄×Dic₆₁	—	4	2	no	—
94	C₂³×C₆₁	—	8	1	no	—
95	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₁	C₄×D₄	1	2	yes	—
97	C₂³×C₆₁	#121	1	1	yes	characteristic
98	C₂³×C₆₁	#121	1	1	yes	characteristic
99	C₂³×C₆₁	#124	1	2	yes	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	—	8	2	no	—
102	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#117	1	2	yes	—
107	C₂×C₄×C₆₁	#117	1	2	yes	—
108	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	#123	1	2	yes	—
116	C₂×C₄×C₆₁	#123	1	2	yes	—
117	C₂×C₄×C₆₁	#125	1	2	yes	—
118	C₂×C₄×C₆₁	#126	1	2	yes	—
119	C₂×C₄×C₆₁	#126	1	2	yes	—
120	C₂×Dic₆₁	—	4	4	no	—
121	C₂×Dic₆₁	—	4	4	no	—
122	C₂×Dic₆₁	—	8	4	no	—
123	C₂×Dic₆₁	—	8	4	no	—
124	C₈×C₆₁	—	2	4	no	—
125	C₈×C₆₁	—	2	4	no	—
126	#10811	—	61	1	no	maximal, Sylow
127	C₂²×C₆₁	—	4	2	no	—
128	C₂²×C₆₁	—	4	2	no	—
129	C₂²×C₆₁	—	4	1	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	1	no	—
132	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₆₁	#380	1	1	yes	characteristic
134	C₂²×C₆₁	#382	1	1	yes	characteristic
135	C₂²×C₆₁	#388	1	2	yes	—
136	C₂²×C₆₁	#388	1	2	yes	—
137	C₄×C₆₁	—	4	4	no	—
138	C₄×C₆₁	#376	1	1	yes	characteristic
139	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₆₁	#381	1	1	yes	characteristic
141	C₄×C₆₁	#383	1	1	yes	characteristic
142	C₄×C₆₁	#387	1	2	yes	—
143	C₄×C₆₁	#387	1	2	yes	—
144	Dic₆₁	—	8	4	no	—
145	Dic₆₁	—	8	4	no	—
146	#2873	D₆₁	1	1	yes	characteristic
147	#1740	—	61	2	no	—
148	#1733	—	61	2	no	—
149	C₂×H(3,ℤ/4)	—	61	1	no	—
150	C₂×H(3,ℤ/4)	—	61	1	no	—
151	C₂×C₆₁	—	2	2	no	—
152	C₂×C₆₁	—	2	2	no	—
153	C₂×C₆₁	—	2	1	no	—
154	C₂×C₆₁	—	2	1	no	—
155	C₂×C₆₁	—	2	1	no	—
156	C₂×C₆₁	#1434	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
158	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
159	C₂×#117	—	61	2	no	—
160	C₂×#117	—	61	2	no	—
161	C₂×#117	C₂×D₆₁	1	1	yes	characteristic
162	#405	—	2	2	no	—
163	#405	Dic₆₁	1	4	yes	complemented
164	C₄×M₁₆	—	2	2	no	—
165	#342	—	122	4	no	—
166	#339	—	61	8	no	—
167	H(3,ℤ/4)	—	122	2	no	—
168	H(3,ℤ/4)	—	122	2	no	—
169	C₆₁	#10811	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	C₄×D₆₁	1	4	yes	—
176	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
177	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
178	C₂×M₁₆	C₂²⋊D₆₁	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
181	#117	—	2	1	no	—
182	#117	—	2	1	no	—
183	#117	—	61	4	no	—
184	#117	—	122	2	no	—
185	#117	—	122	2	no	—
186	#117	—	122	2	no	—
187	#117	—	122	2	no	—
188	#117	—	122	2	no	—
189	#117	—	122	2	no	—
190	#117	—	122	2	no	—
191	#117	Dic₁₂₂	1	2	yes	—
192	#117	C₂×Dic₆₁	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	122	2	no	—
195	C₂×C₄⋊C₄	—	122	2	no	—
196	C₂×C₂²⋊C₄	—	122	2	no	—
197	C₂×C₂²⋊C₄	—	122	2	no	—
198	C₂×C₂²⋊C₄	D₂₄₄	1	2	yes	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	61	4	no	—
201	C₂×C₄²	—	122	2	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₂⁴	C₂×D₂₄₄	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	122	4	no	—
207	C₂²×C₄	—	244	2	no	—
208	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
210	C₂²×C₄	#11758222	1	1	yes	characteristic
211	C₂²×C₄	#11758222	1	1	yes	characteristic
212	C₂²×C₄	#11758222	1	2	yes	—
213	C₂²×C₄	#11758222	1	2	yes	—
214	C₂²×C₄	#11758222	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	C₄×Dic₆₁	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	122	4	no	—
231	C₄⋊C₄	—	244	2	no	—
232	C₄⋊C₄	—	244	2	no	—
233	C₄⋊C₄	—	244	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	122	4	no	—
236	C₂²⋊C₄	—	244	2	no	—
237	C₂²⋊C₄	—	244	2	no	—
238	C₂²⋊C₄	—	244	2	no	—
239	C₂²⋊C₄	C₄⋊Dic₆₁	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	122	4	no	—
243	C₄²	—	122	4	no	—
244	C₄²	—	122	4	no	—
245	C₄²	—	244	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
248	C₂³	C₄×D₂₄₄	1	2	yes	—
249	C₂³	#14519751	1	1	yes	characteristic
250	C₂³	#14519751	1	1	yes	characteristic
251	C₂³	#14519765	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	488	4	no	—
255	C₂×C₄	—	488	4	no	—
256	C₂×C₄	—	244	4	no	—
257	C₂×C₄	—	244	4	no	—
258	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
260	C₂×C₄	#14519734	1	2	yes	—
261	C₂×C₄	#14519734	1	2	yes	—
262	C₂×C₄	#14519736	1	1	yes	characteristic
263	C₂×C₄	#14519736	1	1	yes	characteristic
264	C₂×C₄	#14519736	1	1	yes	characteristic
265	C₂×C₄	#14519736	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
267	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
268	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
269	C₂×C₄	#14519756	1	1	yes	characteristic
270	C₂×C₄	#14519756	1	1	yes	characteristic
271	C₂×C₄	#14519760	1	2	yes	—
272	C₂×C₄	#14519760	1	2	yes	—
273	C₂×C₄	#14519769	1	2	yes	—
274	C₂×C₄	#14519771	1	2	yes	—
275	C₂×C₄	#14519771	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
284	C₂²	#16839622	1	1	yes	characteristic
285	C₂²	#16839628	1	1	yes	characteristic
286	C₂²	#16839653	1	2	yes	—
287	C₂²	#16839653	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	—	488	4	no	—
290	C₄	—	488	4	no	—
291	C₄	#16839608	1	1	yes	characteristic
292	C₄	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
293	C₄	#16839625	1	1	yes	characteristic
294	C₄	#16839630	1	1	yes	characteristic
295	C₄	#16839646	1	2	yes	—
296	C₄	#16839646	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#21444046	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#2873	C₂
socle	C₂×C₆₁	#1434
prime radical	C₂⁴×C₆₁	C₂×D₄
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#2873	D₆₁
hypocenter	C₆₁	#10811
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ #2873

Complements

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

N	H ≅ G/N	type
9. #16839523	C₄	semidirect
16. C₆₁×#405	C₄	semidirect
163. #405	Dic₆₁	semidirect
169. C₆₁	#10811	semidirect

Maximal subgroups

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

subgroup	index
#21445551	2
#21445530	2
C₂×#16839519	2
C₆₁×#2873	2
#10811	61