Subgroups — #30624469

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	#21462234	C₂	1	2	yes	maximal, complemented
4	#21462233	C₂	1	2	yes	maximal, complemented
5	#21462187	C₂	1	2	yes	maximal
6	#21462186	C₂	1	2	yes	maximal
7	#21462182	C₂	1	1	yes	maximal, characteristic
8	#21462179	C₂	1	1	yes	maximal, characteristic
9	#21454750	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	C₄⋊#11758187	C₂²	1	2	yes	—
14	C₄⋊#11758187	C₂²	1	2	yes	—
15	C₄⋊#11758187	C₂²	1	2	yes	—
16	#16840657	C₂²	1	2	yes	—
17	#16840657	C₂²	1	1	yes	characteristic
18	#16840657	C₂²	1	1	yes	characteristic
19	#16840651	C₂²	1	2	yes	—
20	#16840650	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	#16840316	C₄	1	1	yes	characteristic
27	#16839892	C₄	1	1	yes	characteristic
28	#16839890	C₄	1	2	yes	—
29	Q₈×#92418	C₂²	1	1	yes	characteristic
30	#16839886	C₂²	1	2	yes	—
31	#16839886	C₂²	1	1	yes	characteristic
32	#16839787	C₄	1	2	yes	—
33	#16839784	C₄	1	1	yes	characteristic
34	#16839719	C₂²	1	2	yes	—
35	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#446	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×#11758189	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
44	#14519776	C₈	1	1	yes	characteristic
45	#14519775	C₈	1	1	yes	characteristic
46	#14519767	C₈	1	2	yes	—
47	#14519757	C₈	1	1	yes	characteristic
48	#14519755	C₈	1	2	yes	—
49	#14519753	C₈	1	1	yes	characteristic
50	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
52	#14519734	C₂×C₄	1	2	yes	—
53	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
54	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
55	C₄⋊#92418	C₂×C₄	1	2	yes	—
56	#14519706	C₂×C₄	1	2	yes	—
57	#14519706	C₂×C₄	1	2	yes	—
58	#14519706	C₂×C₄	1	2	yes	—
59	#14519687	C₂×C₄	1	2	yes	—
60	C₄×#92418	C₂³	1	1	yes	characteristic
61	C₄×#92418	C₂³	1	2	yes	—
62	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	2	yes	—
65	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
68	#97×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
75	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
80	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
83	#11758198	C₂×C₈	1	2	yes	—
84	#11758198	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
86	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
87	#11758196	C₂×C₈	1	2	yes	—
88	#11758196	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	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
96	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
97	C₂×#92418	C₂²×C₄	1	2	yes	—
98	C₂×#92418	C₂⁴	1	1	yes	characteristic
99	#11758189	—	4	2	no	—
100	C₈×D₆₁	—	2	2	no	—
101	C₈×D₆₁	—	4	1	no	—
102	#11758187	—	2	2	no	—
103	#11758187	—	2	2	no	—
104	#11758187	—	2	2	no	—
105	#11758187	—	2	2	no	—
106	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
107	Q₈×C₆₁	—	4	1	no	—
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
116	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
117	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
118	C₄×D₆₁	—	2	1	no	—
119	C₄×D₆₁	—	2	1	no	—
120	C₄×D₆₁	—	2	2	no	—
121	C₄×D₆₁	—	4	1	no	—
122	Dic₁₂₂	—	2	4	no	—
123	Dic₁₂₂	—	4	1	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	C₂³×C₈	1	1	yes	characteristic
132	C₂×D₆₁	—	2	1	no	—
133	C₂×D₆₁	—	2	1	no	—
134	C₂×D₆₁	—	2	1	no	—
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	—	2	1	no	—
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	#519	1	1	yes	characteristic
140	C₄×C₆₁	#568	1	1	yes	characteristic
141	Dic₆₁	—	2	2	no	—
142	Dic₆₁	—	2	2	no	—
143	Dic₆₁	—	2	1	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	#2982	1	1	yes	characteristic
156	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
157	C₂×C₆₁	#3407	1	1	yes	characteristic
158	D₆₁	—	4	1	no	—
159	#533	—	61	1	no	—
160	#446	—	61	2	no	—
161	#446	—	61	1	no	—
162	#446	#6923	1	1	yes	characteristic
163	C₈×Q₈	—	61	1	no	—
164	C₈×Q₈	—	61	2	no	—
165	C₈×Q₈	—	61	1	no	—
166	#424	—	61	2	no	—
167	#423	—	61	1	no	—
168	#405	—	61	2	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	1	no	—
178	C₂²⋊C₁₆	—	61	1	no	—
179	#346	—	61	2	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	2	no	—
188	C₂²×C₈	—	61	1	no	—
189	C₂²×C₈	—	61	1	no	—
190	C₄⋊Q₈	—	61	1	no	—
191	C₄⋊Q₈	—	61	1	no	—
192	#125	—	61	2	no	—
193	C₂²⋊Q₈	—	61	1	no	—
194	C₂²⋊Q₈	—	61	2	no	—
195	C₂²⋊Q₈	—	61	1	no	—
196	C₄×Q₈	—	61	2	no	—
197	C₄×Q₈	—	61	1	no	—
198	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
199	#117	—	61	2	no	—
200	C₂×C₄²	—	61	1	no	—
201	C₂×C₁₆	—	61	2	no	—
202	C₂×C₁₆	—	61	2	no	—
203	C₂×C₁₆	—	61	2	no	—
204	C₂×C₁₆	—	61	2	no	—
205	C₄⋊C₈	—	61	2	no	—
206	C₄⋊C₈	—	61	2	no	—
207	C₄⋊C₈	—	61	2	no	—
208	C₄⋊C₈	—	61	1	no	—
209	C₄⋊C₈	—	61	2	no	—
210	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
211	C₄⋊C₈	C₂×#6923	1	2	yes	—
212	#97	—	61	2	no	—
213	#97	C₂×#6923	1	2	yes	—
214	C₄×C₈	—	61	1	no	—
215	C₄×C₈	—	61	2	no	—
216	C₄×C₈	—	61	1	no	—
217	C₄×C₈	—	61	2	no	—
218	C₄×C₈	—	61	1	no	—
219	C₄×C₈	C₂×#6923	1	1	yes	characteristic
220	C₂×Q₈	—	61	2	no	—
221	C₂×Q₈	—	61	1	no	—
222	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
223	C₂²×C₄	—	61	1	no	—
224	C₂²×C₄	—	61	1	no	—
225	C₂²×C₄	—	61	1	no	—
226	M₁₆	—	244	2	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	2	no	—
230	C₂×C₈	—	122	2	no	—
231	C₂×C₈	—	244	1	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	2	no	—
240	C₄⋊C₄	—	61	2	no	—
241	C₄⋊C₄	—	61	1	no	—
242	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
243	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
244	C₂²⋊C₄	—	61	2	no	—
245	C₂²⋊C₄	—	61	2	no	—
246	C₄²	—	61	1	no	—
247	C₄²	—	61	1	no	—
248	C₄²	—	61	1	no	—
249	C₄²	—	61	2	no	—
250	C₄²	C₂²×#6923	1	1	yes	characteristic
251	C₄²	C₂²×#6923	1	2	yes	—
252	C₁₆	—	122	2	no	—
253	C₁₆	—	122	2	no	—
254	C₁₆	—	122	2	no	—
255	C₁₆	—	122	2	no	—
256	C₂³	—	61	1	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	122	4	no	—
259	Q₈	—	244	1	no	—
260	C₂×C₄	—	61	1	no	—
261	C₂×C₄	—	61	1	no	—
262	C₂×C₄	—	61	1	no	—
263	C₂×C₄	—	61	2	no	—
264	C₂×C₄	—	61	2	no	—
265	C₂×C₄	—	122	2	no	—
266	C₂×C₄	—	122	1	no	—
267	C₂×C₄	—	122	1	no	—
268	C₂×C₄	—	244	1	no	—
269	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
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	—	122	1	no	—
281	C₂²	—	122	1	no	—
282	C₂²	—	122	1	no	—
283	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	61	2	no	—
289	C₄	—	122	2	no	—
290	C₄	—	122	1	no	—
291	C₄	—	122	2	no	—
292	C₄	—	122	1	no	—
293	C₄	#16840830	1	1	yes	characteristic
294	C₄	#16840841	1	1	yes	characteristic
295	C₂	—	244	1	no	—
296	C₂	#21462713	1	1	yes	characteristic
297	C₂	D₆₁⋊#504	1	1	yes	characteristic
298	C₂	#21462788	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#446	C₄
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#446	#6923
hypocenter	C₆₁	#33499
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

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. #21462234	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
#21462234	2
#21462233	2
#21462187	2
#21462186	2
#21462182	2
#21462179	2
#21454750	2
#33499	61