Subgroups — #30624468

Order 15616.

452 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462236	C₂	1	2	yes	maximal, complemented
2	#21462236	C₂	1	2	yes	maximal, complemented
3	#21462235	C₂	1	2	yes	maximal, complemented
4	#21462233	C₂	1	2	yes	maximal, complemented
5	#21462185	C₂	1	2	yes	maximal
6	#21462184	C₂	1	2	yes	maximal
7	#21462181	C₂	1	1	yes	maximal, characteristic
8	#21462177	C₂	1	1	yes	maximal, characteristic
9	#21454752	C₂	1	1	yes	maximal, characteristic
10	#16840683	C₂²	1	2	yes	—
11	#16840683	C₂²	1	2	yes	—
12	#16840683	C₂²	1	2	yes	—
13	#16840683	C₂²	1	2	yes	—
14	C₄⋊#11758187	C₂²	1	2	yes	—
15	C₄⋊#11758187	C₂²	1	2	yes	—
16	C₂²⋊#11758187	C₂²	1	2	yes	—
17	C₂²⋊#11758187	C₂²	1	2	yes	—
18	#16840651	C₂²	1	1	yes	characteristic
19	#16840651	C₂²	1	1	yes	characteristic
20	#16840651	C₂²	1	2	yes	—
21	#16840650	C₂²	1	2	yes	—
22	#16840648	C₂²	1	1	yes	characteristic
23	#16840648	C₂²	1	1	yes	characteristic
24	#16840648	C₂²	1	2	yes	—
25	C₄×#11758187	C₂²	1	2	yes	—
26	#16840315	C₄	1	1	yes	characteristic
27	#16839891	C₄	1	2	yes	—
28	#16839891	C₄	1	2	yes	—
29	#16839886	C₂²	1	2	yes	—
30	#16839886	C₂²	1	2	yes	—
31	#16839788	C₄	1	1	yes	characteristic
32	#16839785	C₄	1	1	yes	characteristic
33	#16839720	C₂²	1	1	yes	characteristic
34	#16839718	C₂²	1	1	yes	characteristic
35	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#424	C₄	1	1	yes	characteristic
37	C₂×#11758187	C₂³	1	2	yes	—
38	C₂×#11758187	C₂³	1	2	yes	—
39	C₂×#11758187	C₂³	1	2	yes	—
40	C₂×#11758187	C₂³	1	2	yes	—
41	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
42	C₂²×#92418	C₂³	1	1	yes	characteristic
43	#14519776	C₈	1	1	yes	characteristic
44	#14519776	C₈	1	1	yes	characteristic
45	#14519768	C₈	1	1	yes	characteristic
46	#14519768	C₈	1	1	yes	characteristic
47	#14519754	C₈	1	2	yes	—
48	#14519754	C₈	1	2	yes	—
49	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
51	#14519736	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
53	#14519706	C₂×C₄	1	2	yes	—
54	#14519706	C₂×C₄	1	2	yes	—
55	#14519706	C₂×C₄	1	2	yes	—
56	#14519706	C₂×C₄	1	2	yes	—
57	#14519688	C₂×C₄	1	2	yes	—
58	#14519687	C₂×C₄	1	2	yes	—
59	C₄×#92418	C₂³	1	2	yes	—
60	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
63	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	2	yes	—
65	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
72	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
73	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
75	M₁₆×C₆₁	—	4	1	no	—
76	C₂×C₈×C₆₁	—	4	1	no	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
81	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
83	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
84	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
86	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
87	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
89	#11758196	C₂×C₈	1	2	yes	—
90	#11758196	C₂×C₈	1	2	yes	—
91	#11758196	C₂×C₈	1	2	yes	—
92	#11758196	C₂×C₈	1	2	yes	—
93	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
98	#11758194	—	4	1	no	—
99	C₂×#92418	—	2	2	no	—
100	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
101	C₂×#92418	C₂²×C₄	1	2	yes	—
102	C₂×#92418	C₂⁴	1	1	yes	characteristic
103	#11758187	—	2	2	no	—
104	#11758187	—	2	2	no	—
105	#11758187	—	2	2	no	—
106	#11758187	—	2	2	no	—
107	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₆₁	—	2	2	no	—
115	C₂×Dic₆₁	—	4	1	no	—
116	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
119	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
120	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
121	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
122	Dic₁₂₂	—	2	4	no	—
123	Dic₁₂₂	—	4	2	no	—
124	C₈×C₆₁	—	2	1	no	—
125	C₈×C₆₁	—	2	1	no	—
126	C₈×C₆₁	—	2	2	no	—
127	#92418	—	2	1	no	—
128	#92418	—	2	2	no	—
129	#92418	2₋¹⁺⁴	1	2	yes	—
130	#33499	—	61	1	no	maximal, Sylow
131	C₂²×C₆₁	—	2	1	no	—
132	C₂²×C₆₁	—	2	1	no	—
133	C₂²×C₆₁	—	2	1	no	—
134	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	2	2	no	—
137	C₄×C₆₁	—	2	1	no	—
138	C₄×C₆₁	#519	1	1	yes	characteristic
139	C₄×C₆₁	#568	1	1	yes	characteristic
140	Dic₆₁	—	2	2	no	—
141	Dic₆₁	—	2	1	no	—
142	Dic₆₁	—	2	1	no	—
143	Dic₆₁	—	2	2	no	—
144	Dic₆₁	—	2	1	no	—
145	Dic₆₁	#504	1	2	yes	—
146	#2960	—	61	1	no	—
147	#2183	—	61	2	no	—
148	#2183	—	61	2	no	—
149	#2183	—	61	2	no	—
150	Q₈×C₁₆	—	61	2	no	—
151	#2164	—	61	2	no	—
152	#2164	—	61	2	no	—
153	#2163	—	61	1	no	—
154	#2162	—	61	1	no	—
155	C₂×C₆₁	—	4	1	no	—
156	C₂×C₆₁	#2982	1	1	yes	characteristic
157	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
158	C₂×C₆₁	#3407	1	1	yes	characteristic
159	#533	—	61	1	no	—
160	#446	—	61	2	no	—
161	#446	—	61	2	no	—
162	C₈×Q₈	—	61	2	no	—
163	C₈×Q₈	—	61	2	no	—
164	#424	—	61	1	no	—
165	#424	#6923	1	1	yes	characteristic
166	#423	—	61	1	no	—
167	#405	—	61	1	no	—
168	#405	—	61	1	no	—
169	C₂×C₄×C₈	—	61	1	no	—
170	C₄⋊C₁₆	—	61	2	no	—
171	C₄⋊C₁₆	—	61	2	no	—
172	C₄⋊C₁₆	—	61	2	no	—
173	C₄⋊C₁₆	—	61	2	no	—
174	C₄⋊C₁₆	—	61	2	no	—
175	C₄⋊C₁₆	—	61	2	no	—
176	C₂²⋊C₁₆	—	61	2	no	—
177	C₂²⋊C₁₆	—	61	2	no	—
178	#346	—	61	1	no	—
179	#346	—	61	1	no	—
180	#346	—	61	2	no	—
181	#346	—	61	2	no	—
182	C₄×C₁₆	—	61	1	no	—
183	C₄×C₁₆	—	61	1	no	—
184	C₄×C₁₆	—	61	2	no	—
185	C₄×C₁₆	—	61	2	no	—
186	C₆₁	#33499	1	1	yes	characteristic, complemented, Sylow
187	C₂×M₁₆	—	61	1	no	—
188	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
189	C₂²×C₈	—	61	1	no	—
190	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
191	C₄⋊Q₈	—	61	1	no	—
192	C₄⋊Q₈	—	61	1	no	—
193	#125	—	61	1	no	—
194	#125	—	61	1	no	—
195	C₂²⋊Q₈	—	61	2	no	—
196	C₂²⋊Q₈	—	61	2	no	—
197	C₄×Q₈	—	61	2	no	—
198	C₄×Q₈	—	61	2	no	—
199	#117	—	61	1	no	—
200	#117	C₂×#6923	1	1	yes	characteristic
201	C₂×C₄²	—	61	1	no	—
202	C₂×C₁₆	—	61	2	no	—
203	C₂×C₁₆	—	61	2	no	—
204	C₂×C₁₆	—	61	2	no	—
205	C₂×C₁₆	—	61	2	no	—
206	C₄⋊C₈	—	61	2	no	—
207	C₄⋊C₈	—	61	2	no	—
208	C₄⋊C₈	—	61	2	no	—
209	C₄⋊C₈	—	61	2	no	—
210	C₄⋊C₈	C₂×#6923	1	2	yes	—
211	C₄⋊C₈	C₂×#6923	1	2	yes	—
212	#97	—	61	2	no	—
213	#97	—	61	2	no	—
214	C₄×C₈	—	61	1	no	—
215	C₄×C₈	—	61	2	no	—
216	C₄×C₈	—	61	1	no	—
217	C₄×C₈	—	61	1	no	—
218	C₄×C₈	—	61	1	no	—
219	C₄×C₈	—	61	2	no	—
220	C₂×Q₈	—	61	2	no	—
221	C₂×Q₈	—	61	2	no	—
222	C₂²×C₄	—	61	1	no	—
223	C₂²×C₄	—	61	1	no	—
224	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
225	M₁₆	—	4	1	no	—
226	M₁₆	—	244	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	122	2	no	—
232	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
233	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
234	C₂×C₈	C₂²×#6923	1	2	yes	—
235	C₄⋊C₄	—	61	2	no	—
236	C₄⋊C₄	—	61	2	no	—
237	C₄⋊C₄	—	61	2	no	—
238	C₄⋊C₄	—	61	2	no	—
239	C₄⋊C₄	—	61	1	no	—
240	C₄⋊C₄	—	61	1	no	—
241	C₄⋊C₄	—	61	2	no	—
242	C₄⋊C₄	—	61	2	no	—
243	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
244	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
245	C₂²⋊C₄	—	61	2	no	—
246	C₂²⋊C₄	D₆₁⋊C₈	1	2	yes	—
247	C₄²	—	61	1	no	—
248	C₄²	—	61	1	no	—
249	C₄²	—	61	1	no	—
250	C₄²	—	61	1	no	—
251	C₄²	—	61	2	no	—
252	C₄²	C₂²×#6923	1	2	yes	—
253	C₁₆	—	122	2	no	—
254	C₁₆	—	122	2	no	—
255	C₁₆	—	122	2	no	—
256	C₁₆	—	122	2	no	—
257	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
258	Q₈	—	122	4	no	—
259	Q₈	—	244	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	61	1	no	—
263	C₂×C₄	—	61	1	no	—
264	C₂×C₄	—	61	1	no	—
265	C₂×C₄	—	61	1	no	—
266	C₂×C₄	—	61	2	no	—
267	C₂×C₄	—	61	2	no	—
268	C₂×C₄	—	122	2	no	—
269	C₂×C₄	—	244	1	no	—
270	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
272	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
273	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	2	no	—
277	C₈	—	61	2	no	—
278	C₈	—	122	1	no	—
279	C₈	—	122	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	61	2	no	—
288	C₄	—	122	1	no	—
289	C₄	—	122	2	no	—
290	C₄	—	122	1	no	—
291	C₄	—	122	1	no	—
292	C₄	—	122	2	no	—
293	C₄	#16840830	1	1	yes	characteristic
294	C₄	#16840840	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	#21462714	1	1	yes	characteristic
297	C₂	D₆₁⋊#504	1	1	yes	characteristic
298	C₂	#21462787	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#424	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#424	#6923
hypocenter	C₆₁	#33499
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
2. #21462236	C₂	semidirect
3. #21462235	C₂	semidirect
4. #21462233	C₂	semidirect
186. C₆₁	#33499	semidirect

Maximal subgroups

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

subgroup	index
#21462236	2
#21462235	2
#21462233	2
#21462185	2
#21462184	2
#21462181	2
#21462177	2
#21454752	2
#33499	61