Subgroups — #29588388

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445552	C₂	1	2	yes	maximal
2	#21445535	C₂	1	2	yes	maximal
3	C₂×#16839519	C₂	1	1	yes	maximal, characteristic
4	#21445517	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2911	C₂	1	1	yes	maximal, characteristic
6	C₂³×#92418	C₂²	1	1	yes	characteristic
7	C₂×#14519736	C₂²	1	2	yes	—
8	C₂×#14519736	C₂²	1	1	yes	characteristic
9	#16839529	—	2	4	no	—
10	#16839524	—	2	4	no	—
11	#16839519	C₄	1	4	yes	complemented
12	#16839516	—	2	2	no	—
13	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#428	—	2	4	no	—
16	C₆₁×#427	—	2	2	no	—
17	C₂²×#92418	—	2	1	no	—
18	C₂²×#92418	—	2	2	no	—
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	—	2	1	no	—
21	C₂²×#92418	Q₈	1	2	yes	—
22	#14519736	—	2	2	no	—
23	#14519736	—	2	2	no	—
24	#14519736	—	2	2	no	—
25	#14519736	—	2	2	no	—
26	#14519736	—	2	2	no	—
27	#14519736	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊Dic₆₁	—	2	2	no	—
29	C₂×C₄⋊Dic₆₁	—	2	1	no	—
30	C₂×C₂²⋊Dic₆₁	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₁	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₁	—	2	2	no	—
33	C₂×C₄×Dic₆₁	—	2	1	no	—
34	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	—	2	2	no	—
36	C₂×M₁₆×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
39	C₂²×C₈×C₆₁	D₄	1	2	yes	—
40	#117×C₆₁	—	2	2	no	—
41	#117×C₆₁	—	2	2	no	—
42	#117×C₆₁	—	2	2	no	—
43	#117×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	D₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₄²×C₆₁	—	2	2	no	—
47	#107×C₆₁	—	2	4	no	—
48	#106×C₆₁	—	2	4	no	—
49	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	—	2	4	no	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	#41	1	2	yes	—
57	C₂²×C₄×C₆₁	#41	1	2	yes	—
58	C₂²×C₄×C₆₁	#41	1	2	yes	—
59	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
61	C₂²×Dic₆₁	—	2	2	no	—
62	C₂²×Dic₆₁	—	4	2	no	—
63	M₁₆×C₆₁	—	2	4	no	—
64	M₁₆×C₆₁	—	2	4	no	—
65	C₂×C₈×C₆₁	—	2	2	no	—
66	C₂×C₈×C₆₁	—	2	2	no	—
67	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
68	C₄⋊C₄×C₆₁	—	2	4	no	—
69	C₄⋊C₄×C₆₁	—	2	4	no	—
70	C₂²⋊C₄×C₆₁	—	4	2	no	—
71	C₂²⋊C₄×C₆₁	—	4	2	no	—
72	C₄²×C₆₁	—	4	2	no	—
73	C₄²×C₆₁	—	4	2	no	—
74	C₂²⋊Dic₆₁	—	2	4	no	—
75	C₂²⋊Dic₆₁	—	2	4	no	—
76	C₂²⋊Dic₆₁	—	4	2	no	—
77	C₄⋊Dic₆₁	—	2	4	no	—
78	C₄⋊Dic₆₁	—	4	2	no	—
79	C₄⋊Dic₆₁	—	4	2	no	—
80	C₄×Dic₆₁	—	2	4	no	—
81	C₄×Dic₆₁	—	4	1	no	—
82	C₄×Dic₆₁	—	4	1	no	—
83	C₄×Dic₆₁	—	4	2	no	—
84	C₂×#92418	—	2	4	no	—
85	C₂×#92418	—	2	2	no	—
86	C₂×#92418	—	4	1	no	—
87	C₂×#92418	—	4	1	no	—
88	C₂×#92418	—	4	2	no	—
89	C₂³×C₆₁	—	4	2	no	—
90	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
91	C₂³×C₆₁	#121	1	2	yes	—
92	C₂³×C₆₁	#121	1	1	yes	characteristic
93	C₂³×C₆₁	#123	1	2	yes	—
94	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	8	4	no	—
97	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#117	1	2	yes	—
99	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#123	1	2	yes	—
107	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#124	1	2	yes	—
110	C₂×C₄×C₆₁	#125	1	2	yes	—
111	C₂×C₄×C₆₁	#125	1	2	yes	—
112	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#126	1	2	yes	—
114	C₂×C₄×C₆₁	#126	1	2	yes	—
115	C₂×Dic₆₁	—	4	4	no	—
116	C₂×Dic₆₁	—	4	4	no	—
117	C₂×Dic₆₁	—	4	2	no	—
118	C₂×Dic₆₁	—	8	4	no	—
119	C₈×C₆₁	—	2	4	no	—
120	#92418	—	4	2	no	—
121	#11285	—	61	1	no	maximal, Sylow
122	C₂²×C₆₁	—	2	2	no	—
123	C₂²×C₆₁	—	2	4	no	—
124	C₂²×C₆₁	—	4	1	no	—
125	C₂²×C₆₁	—	4	2	no	—
126	C₂²×C₆₁	—	4	1	no	—
127	C₂²×C₆₁	#388	1	2	yes	—
128	C₂²×C₆₁	#389	1	1	yes	characteristic
129	C₂²×C₆₁	#394	1	1	yes	characteristic
130	C₂²×C₆₁	#397	1	1	yes	characteristic
131	C₂²×C₆₁	#398	1	2	yes	—
132	C₄×C₆₁	—	8	4	no	—
133	C₄×C₆₁	#382	1	1	yes	characteristic
134	C₄×C₆₁	#385	1	1	yes	characteristic
135	C₄×C₆₁	#387	1	2	yes	—
136	C₄×C₆₁	#393	1	1	yes	characteristic
137	C₄×C₆₁	#400	1	1	yes	characteristic
138	C₄×C₆₁	#400	1	2	yes	—
139	Dic₆₁	—	8	2	no	—
140	Dic₆₁	—	8	4	no	—
141	#2911	D₆₁	1	1	yes	characteristic
142	#1740	—	61	2	no	—
143	#1735	—	61	2	no	—
144	C₂×H(3,ℤ/4)	—	61	1	no	—
145	#1729	—	61	1	no	—
146	C₂×C₆₁	—	2	2	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	—	2	1	no	—
149	C₂×C₆₁	—	2	2	no	—
150	C₂×C₆₁	#1442	1	1	yes	characteristic
151	C₂×C₆₁	#2053	1	2	yes	—
152	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
153	C₂³×C₈	—	61	1	no	—
154	C₂×#117	—	61	1	no	—
155	C₂×#117	—	61	2	no	—
156	C₂×#117	C₂×D₆₁	1	2	yes	—
157	#428	—	2	4	no	—
158	#427	—	2	2	no	—
159	#342	—	122	4	no	—
160	#340	—	122	4	no	—
161	H(3,ℤ/4)	—	61	4	no	—
162	#336	—	122	2	no	—
163	C₆₁	#11285	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
168	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
169	C₂²×C₈	—	61	2	no	—
170	C₂²×C₈	—	122	2	no	—
171	C₂²×C₈	—	122	1	no	—
172	C₂²×C₈	—	122	1	no	—
173	C₂²×C₈	—	122	2	no	—
174	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	61	4	no	—
180	#117	—	122	2	no	—
181	#117	—	122	2	no	—
182	#117	—	122	2	no	—
183	#117	—	122	2	no	—
184	#117	—	122	2	no	—
185	C₂×C₄⋊C₄	—	122	2	no	—
186	C₂×C₄⋊C₄	—	122	1	no	—
187	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	61	2	no	—
190	C₂×C₂²⋊C₄	—	122	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	122	2	no	—
193	C₂×C₄²	—	122	1	no	—
194	#107	—	2	4	no	—
195	#106	—	2	4	no	—
196	C₂⁴	D₄×D₆₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	122	2	no	—
199	C₂²×C₄	—	244	2	no	—
200	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
202	C₂²×C₄	#11758222	1	2	yes	—
203	C₂²×C₄	#11758224	1	2	yes	—
204	C₂²×C₄	#11758224	1	1	yes	characteristic
205	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
206	C₂²×C₄	#11758226	1	2	yes	—
207	C₂²×C₄	#11758226	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	122	2	no	—
215	C₂×C₈	—	122	4	no	—
216	C₂×C₈	—	244	1	no	—
217	C₂×C₈	—	244	2	no	—
218	C₂×C₈	—	244	1	no	—
219	C₂×C₈	#11758196	1	4	yes	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	122	4	no	—
223	C₄⋊C₄	—	244	2	no	—
224	C₄⋊C₄	—	244	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	122	4	no	—
228	C₂²⋊C₄	—	122	4	no	—
229	C₂²⋊C₄	—	244	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	122	4	no	—
233	C₄²	—	244	1	no	—
234	C₄²	—	244	1	no	—
235	C₄²	—	244	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	#14519739	1	1	yes	characteristic
238	C₂³	#14519748	1	1	yes	characteristic
239	C₂³	#14519749	1	2	yes	—
240	C₂³	#14519762	1	2	yes	—
241	C₂³	#14519776	1	1	yes	characteristic
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	488	4	no	—
245	C₂×C₄	—	244	4	no	—
246	C₂×C₄	—	244	2	no	—
247	C₂×C₄	—	244	4	no	—
248	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
249	C₂×C₄	#14519735	1	2	yes	—
250	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
251	C₂×C₄	#14519742	1	1	yes	characteristic
252	C₂×C₄	#14519746	1	1	yes	characteristic
253	C₂×C₄	#14519753	1	1	yes	characteristic
254	C₂×C₄	#14519753	1	2	yes	—
255	C₂×C₄	#14519756	1	2	yes	—
256	C₂×C₄	#14519758	1	2	yes	—
257	C₂×C₄	#14519760	1	1	yes	characteristic
258	C₂×C₄	#14519764	1	1	yes	characteristic
259	C₂×C₄	#14519766	1	2	yes	—
260	C₂×C₄	#14519766	1	1	yes	characteristic
261	C₂×C₄	#14519767	1	2	yes	—
262	C₂×C₄	#14519768	1	1	yes	characteristic
263	C₂×C₄	#14519768	1	2	yes	—
264	C₂×C₄	#14519772	1	2	yes	—
265	C₂×C₄	#14519772	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	244	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	#16839652	1	2	yes	—
274	C₂²	#16839658	1	1	yes	characteristic
275	C₂²	#16839676	1	1	yes	characteristic
276	C₂²	#16839688	1	1	yes	characteristic
277	C₂²	#16839695	1	2	yes	—
278	C₄	—	8	4	no	—
279	C₄	—	488	4	no	—
280	C₄	—	488	2	no	—
281	C₄	#16839627	1	1	yes	characteristic
282	C₄	#16839635	1	1	yes	characteristic
283	C₄	#16839643	1	2	yes	—
284	C₄	#16839673	1	1	yes	characteristic
285	C₄	#16839700	1	1	yes	characteristic
286	C₄	#16839701	1	2	yes	—
287	C₂	—	2	1	no	—
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#21444078	1	1	yes	characteristic
292	C₂	#21447122	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519764
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#2911	C₂
socle	C₂²×C₆₁	#394
prime radical	C₂⁴×C₆₁	C₂×D₄
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#2911	D₆₁
hypocenter	C₆₁	#11285
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2911

Complements

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

N	H ≅ G/N	type
11. #16839519	C₄	semidirect
163. C₆₁	#11285	semidirect

Maximal subgroups

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

subgroup	index
#21445552	2
#21445535	2
C₂×#16839519	2
#21445517	2
C₆₁×#2911	2
#11285	61