Subgroups — #30606406

Order 15616.

490 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	C₂×#16840660	C₂	1	2	yes	maximal
2	#21462025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16840648	C₂	1	1	yes	maximal, characteristic, complemented
4	#21460874	C₂	1	2	yes	maximal, complemented
5	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	2	yes	—
7	C₂²×#11758187	C₂²	1	2	yes	—
8	#16840660	C₄	1	8	yes	complemented
9	#16840651	—	2	1	no	—
10	#16840648	—	2	1	no	—
11	#16840648	—	2	1	no	—
12	#16840648	—	2	1	no	—
13	C₂×#14519784	C₂²	1	1	yes	characteristic
14	#41×Dic₆₁	C₄	1	1	yes	characteristic
15	#16839720	—	2	1	no	—
16	#16839720	C₂²	1	1	yes	characteristic
17	#16839717	—	2	1	no	—
18	#16839717	C₄	1	1	yes	characteristic
19	M₁₆×Dic₆₁	—	2	1	no	—
20	M₁₆×Dic₆₁	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₆₁	—	2	1	no	—
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
24	#14519851	—	2	2	no	—
25	#14519851	C₂×C₄	1	4	yes	complemented
26	C₂×#11758187	—	2	2	no	—
27	C₂×#11758187	—	2	2	no	—
28	C₂×#11758187	—	2	2	no	—
29	C₂×#11758187	C₂×C₄	1	4	yes	complemented
30	#14519784	—	2	2	no	—
31	#14519784	C₂×C₄	1	4	yes	—
32	C₂×#11758194	—	2	1	no	—
33	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
34	C₂²×#92418	—	2	1	no	—
35	C₂²×#92418	C₂³	1	1	yes	characteristic
36	Q₈×Dic₆₁	C₈	1	2	yes	—
37	D₄×Dic₆₁	—	2	2	no	—
38	D₄×Dic₆₁	C₈	1	2	yes	—
39	#14519736	—	2	1	no	—
40	#14519736	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₆₁	—	2	1	no	—
42	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	#14519687	—	2	1	no	—
44	#14519687	—	2	1	no	—
45	#14519687	—	2	1	no	—
46	C₈×Dic₆₁	—	2	1	no	—
47	C₈×Dic₆₁	—	2	1	no	—
48	C₈×Dic₆₁	—	2	1	no	—
49	C₈×Dic₆₁	D₄	1	1	yes	characteristic
50	C₈×Dic₆₁	D₄	1	1	yes	characteristic
51	C₈×Dic₆₁	D₄	1	1	yes	characteristic
52	C₈×Dic₆₁	D₄	1	1	yes	characteristic
53	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
54	#131×C₆₁	—	2	2	no	—
55	#131×C₆₁	C₈	1	4	yes	—
56	C₂×M₁₆×C₆₁	—	2	1	no	—
57	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₁	—	2	1	no	—
59	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	#41×C₆₁	—	2	1	no	—
61	#41×C₆₁	—	2	1	no	—
62	#41×C₆₁	C₄²	1	2	yes	—
63	#41×C₆₁	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₆₁	—	2	1	no	—
66	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₆₁	—	2	2	no	—
70	C₂²×Dic₆₁	M₁₆	1	2	yes	—
71	M₁₆×C₆₁	—	2	1	no	—
72	M₁₆×C₆₁	—	2	4	no	—
73	M₁₆×C₆₁	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
80	C₂²⋊Dic₆₁	—	2	1	no	—
81	C₂²⋊Dic₆₁	—	2	2	no	—
82	C₄⋊Dic₆₁	—	2	2	no	—
83	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
84	C₄×Dic₆₁	—	2	1	no	—
85	C₄×Dic₆₁	—	2	2	no	—
86	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
89	#11758194	—	2	1	no	—
90	#11758194	—	2	4	no	—
91	#11758194	C₄²	1	2	yes	—
92	C₂×#92418	—	2	1	no	—
93	C₂×#92418	—	2	4	no	—
94	C₂×#92418	C₂²×C₄	1	2	yes	—
95	C₂×#92418	C₂×D₄	1	1	yes	characteristic
96	C₂×#92418	C₂×D₄	1	1	yes	characteristic
97	C₂×#92418	#41	1	2	yes	—
98	#11758187	—	2	4	no	—
99	#11758187	—	2	4	no	—
100	C₂³×C₆₁	—	2	1	no	—
101	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₆₁	—	2	1	no	—
103	Q₈×C₆₁	C₄×C₈	1	2	yes	—
104	D₄×C₆₁	—	2	2	no	—
105	D₄×C₆₁	—	2	2	no	—
106	D₄×C₆₁	—	2	1	no	—
107	D₄×C₆₁	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
117	C₂×Dic₆₁	—	2	4	no	—
118	C₂×Dic₆₁	—	2	2	no	—
119	C₂×Dic₆₁	—	4	4	no	—
120	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
121	C₂×Dic₆₁	C₂²⋊C₈	1	2	yes	—
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
124	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
125	#92418	—	2	2	no	—
126	#92418	C₄×D₄	1	2	yes	—
127	#92418	C₄×D₄	1	2	yes	—
128	C₄×#350	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₆₁	#431	1	1	yes	characteristic
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₆₁	#431	1	1	yes	characteristic
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	—	4	2	no	—
142	Dic₆₁	#350	1	4	yes	complemented
143	C₄×#131	—	61	1	no	—
144	C₂×#350	—	61	2	no	—
145	C₄×M₃₂	—	61	1	no	—
146	C₂×C₄×C₁₆	—	61	1	no	—
147	#1379	—	61	2	no	—
148	C₂×C₆₁	—	2	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	4	2	no	—
151	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#350	1	2	yes	—
153	C₂×#131	—	61	1	no	—
154	C₂×#131	#6923	1	1	yes	characteristic
155	C₄×#41	—	61	1	no	—
156	C₂×M₃₂	—	61	2	no	—
157	C₂²×C₁₆	—	61	2	no	—
158	#405	—	61	1	no	—
159	#405	—	61	1	no	—
160	#405	—	122	1	no	—
161	#405	—	122	1	no	—
162	C₄×M₁₆	—	61	1	no	—
163	C₄×M₁₆	—	122	1	no	—
164	C₂×C₄×C₈	—	61	1	no	—
165	C₂×C₄×C₈	—	122	1	no	—
166	#350	—	61	8	no	—
167	#346	—	122	1	no	—
168	C₄×C₁₆	—	122	1	no	—
169	C₄×C₁₆	—	122	1	no	—
170	C₄×C₁₆	—	122	1	no	—
171	C₆₁	C₄×#350	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂×#6923	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	61	4	no	—
175	#131	—	122	2	no	—
176	#131	#92420	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	61	1	no	—
179	C₂×M₁₆	—	122	1	no	—
180	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	61	1	no	—
183	C₂²×C₈	—	122	1	no	—
184	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
185	C₄×Q₈	—	61	2	no	—
186	C₄×D₄	—	61	2	no	—
187	C₄×D₄	—	122	2	no	—
188	#117	—	61	1	no	—
189	#117	—	122	1	no	—
190	C₂×C₄²	—	61	1	no	—
191	C₂×C₄²	—	122	1	no	—
192	M₃₂	—	61	4	no	—
193	M₃₂	—	122	2	no	—
194	C₂×C₁₆	—	61	4	no	—
195	C₂×C₁₆	—	122	2	no	—
196	C₂×C₁₆	—	122	2	no	—
197	C₂×C₁₆	—	122	2	no	—
198	#97	—	122	1	no	—
199	#97	—	122	1	no	—
200	#97	—	122	1	no	—
201	C₄×C₈	—	61	1	no	—
202	C₄×C₈	—	61	1	no	—
203	C₄×C₈	—	61	1	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	122	1	no	—
206	C₄×C₈	—	122	1	no	—
207	C₄×C₈	—	122	1	no	—
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄×#6923	1	2	yes	—
211	#41	C₂×#92420	1	2	yes	—
212	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	D₆₁⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	61	2	no	—
217	C₂²×C₄	—	122	2	no	—
218	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	61	2	no	—
222	M₁₆	—	122	4	no	—
223	M₁₆	—	122	1	no	—
224	M₁₆	C₂×#92420	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	61	2	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	122	4	no	—
232	C₂×C₈	—	122	1	no	—
233	C₂×C₈	C₂×#92420	1	2	yes	—
234	C₂×C₈	#11758217	1	2	yes	—
235	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
237	C₄⋊C₄	—	61	2	no	—
238	C₄⋊C₄	—	122	2	no	—
239	C₂²⋊C₄	—	122	2	no	—
240	C₂²⋊C₄	—	122	1	no	—
241	C₄²	—	61	1	no	—
242	C₄²	—	61	1	no	—
243	C₄²	—	61	2	no	—
244	C₄²	—	122	2	no	—
245	C₄²	—	122	1	no	—
246	C₁₆	—	122	4	no	—
247	C₁₆	—	122	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	D₆₁⋊M₁₆	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#92420	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#92420	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	61	2	no	—
260	C₂×C₄	—	61	2	no	—
261	C₂×C₄	—	122	4	no	—
262	C₂×C₄	—	122	2	no	—
263	C₂×C₄	—	244	4	no	—
264	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
266	C₂×C₄	#14519856	1	1	yes	characteristic
267	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
268	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	61	2	no	—
272	C₈	—	61	2	no	—
273	C₈	—	122	2	no	—
274	C₈	C₂²⋊#92420	1	2	yes	—
275	C₈	C₂²⋊#92420	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#11758217	1	1	yes	characteristic
280	C₂²	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
281	C₂²	#16840778	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	61	4	no	—
284	C₄	—	122	2	no	—
285	C₄	—	244	2	no	—
286	C₄	C₄×C₂²⋊#6923	1	1	yes	characteristic
287	C₄	C₂×C₄×#92420	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
289	C₄	#16840779	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#92420	1	1	yes	characteristic
294	C₂	D₆₁⋊#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	C₄×#350
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #21462025	C₂	semidirect
3. C₂×#16840648	C₂	semidirect
4. #21460874	C₂	semidirect
8. #16840660	C₄	semidirect
25. #14519851	C₂×C₄	semidirect
29. C₂×#11758187	C₂×C₄	semidirect
142. Dic₆₁	#350	semidirect
171. C₆₁	C₄×#350	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840660	2
#21462025	2
C₂×#16840648	2
#21460874	2
#131×Dic₆₁	2
C₄×#350	61