Subgroups — #30624384

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462234	C₂	1	2	yes	maximal, complemented
2	#21462205	C₂	1	2	yes	maximal, complemented
3	#21462202	C₂	1	2	yes	maximal, complemented
4	#21462202	C₂	1	2	yes	maximal, complemented
5	#21462187	C₂	1	2	yes	maximal, complemented
6	#21462183	C₂	1	1	yes	maximal, characteristic, complemented
7	#21462029	C₂	1	2	yes	maximal, complemented
8	#21462026	C₂	1	1	yes	maximal, characteristic, complemented
9	#21454793	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519852	C₂²	1	2	yes	—
11	C₂×#14519852	C₂²	1	2	yes	—
12	C₂×D₆₁⋊C₁₆	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	1	yes	characteristic
17	#16840657	C₂²	1	1	yes	characteristic
18	#16840657	C₂²	1	2	yes	—
19	#16840657	C₂²	1	2	yes	—
20	#16840651	C₂²	1	2	yes	—
21	#16840650	C₂²	1	2	yes	—
22	#16840650	C₂²	1	2	yes	—
23	#16840648	C₂²	1	1	yes	characteristic
24	#16840648	C₂²	1	1	yes	characteristic
25	C₄×#11758187	C₂²	1	2	yes	—
26	#16840299	C₄	1	1	yes	characteristic
27	Q₈×#92418	C₂²	1	1	yes	characteristic
28	#16839841	C₄	1	1	yes	characteristic
29	#16839835	C₄	1	2	yes	—
30	#16839827	C₂²	1	1	yes	characteristic
31	#16839827	C₂²	1	2	yes	—
32	#16839822	C₄	1	2	yes	—
33	#16839818	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	#14519852	—	2	4	no	—
38	#14519852	—	2	4	no	—
39	D₆₁⋊C₁₆	—	2	4	no	—
40	C₂×#11758187	C₂³	1	2	yes	—
41	C₂×#11758187	C₂³	1	2	yes	—
42	C₂×#11758187	C₂³	1	2	yes	—
43	C₂×#11758187	C₂³	1	2	yes	—
44	C₂×#11758189	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₆₁	C₂³	1	2	yes	—
48	#14519772	C₈	1	2	yes	—
49	#14519768	C₈	1	1	yes	characteristic
50	#14519766	C₈	1	1	yes	characteristic
51	#14519762	C₈	1	2	yes	—
52	#14519753	C₈	1	1	yes	characteristic
53	#14519752	C₈	1	1	yes	characteristic
54	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
55	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
56	#14519734	C₂×C₄	1	2	yes	—
57	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#92418	C₂×C₄	1	2	yes	—
60	#14519689	C₂×C₄	1	2	yes	—
61	#14519689	C₂×C₄	1	2	yes	—
62	#14519689	C₂×C₄	1	2	yes	—
63	#14519687	C₂×C₄	1	2	yes	—
64	C₄×#92418	C₂³	1	1	yes	characteristic
65	C₄×#92418	C₂³	1	2	yes	—
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
71	#97×C₆₁	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
75	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
85	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
87	#11758198	C₂×C₈	1	2	yes	—
88	#11758198	C₂×C₈	1	2	yes	—
89	#11758198	C₂×C₈	1	2	yes	—
90	#11758198	C₂×C₈	1	2	yes	—
91	#11758198	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
94	#11758196	C₂×C₈	1	2	yes	—
95	#11758196	C₂×C₈	1	2	yes	—
96	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
100	C₂×#92418	C₂²×C₄	1	2	yes	—
101	C₂×#92418	C₂⁴	1	1	yes	characteristic
102	#11758189	—	2	4	no	—
103	C₈×D₆₁	—	2	2	no	—
104	C₈×D₆₁	—	2	2	no	—
105	C₈×D₆₁	—	2	2	no	—
106	C₈×D₆₁	—	2	2	no	—
107	#11758187	—	2	2	no	—
108	#11758187	—	2	2	no	—
109	#11758187	—	2	2	no	—
110	#11758187	#41	1	4	yes	—
111	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
112	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
123	D₂₄₄	—	4	1	no	—
124	D₂₄₄	—	4	2	no	—
125	C₄×D₆₁	—	2	2	no	—
126	C₄×D₆₁	—	2	1	no	—
127	C₄×D₆₁	—	2	1	no	—
128	C₄×D₆₁	—	2	2	no	—
129	C₄×D₆₁	—	4	1	no	—
130	C₄×D₆₁	—	4	1	no	—
131	C₈×C₆₁	—	2	2	no	—
132	C₈×C₆₁	—	2	1	no	—
133	C₈×C₆₁	#131	1	2	yes	—
134	#92418	—	2	1	no	—
135	#92418	—	2	2	no	—
136	#92418	C₂×#41	1	2	yes	—
137	#33481	—	61	1	no	maximal, Sylow
138	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
139	C₂×D₆₁	—	2	1	no	—
140	C₂×D₆₁	—	2	1	no	—
141	C₂×D₆₁	—	2	2	no	—
142	C₂×D₆₁	—	2	2	no	—
143	C₂×D₆₁	—	2	1	no	—
144	C₂×D₆₁	—	2	2	no	—
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
150	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
151	Dic₆₁	—	2	1	no	—
152	Dic₆₁	—	2	1	no	—
153	Dic₆₁	—	2	1	no	—
154	Dic₆₁	—	2	2	no	—
155	#2966	—	61	1	no	—
156	#2183	—	61	2	no	—
157	#2169	—	61	2	no	—
158	#2168	—	61	2	no	—
159	#2168	—	61	2	no	—
160	#2164	—	61	2	no	—
161	#2163	—	61	1	no	—
162	#2108	—	61	2	no	—
163	C₄×M₃₂	—	61	1	no	—
164	C₂×C₆₁	C₈×#41	1	1	yes	characteristic
165	C₂×C₆₁	#3407	1	1	yes	characteristic
166	C₂×C₆₁	#3407	1	1	yes	characteristic
167	D₆₁	—	4	1	no	—
168	D₆₁	—	4	2	no	—
169	#529	—	61	1	no	—
170	C₂×M₃₂	—	61	2	no	—
171	C₂×M₃₂	—	61	2	no	—
172	C₂²×C₁₆	—	61	2	no	—
173	#446	#6923	1	1	yes	characteristic
174	C₈×Q₈	—	61	1	no	—
175	#436	—	61	1	no	—
176	#435	—	61	2	no	—
177	C₈×D₄	—	61	1	no	—
178	C₈×D₄	—	61	2	no	—
179	#433	—	61	2	no	—
180	#432	—	61	1	no	—
181	#405	—	61	2	no	—
182	C₂×C₄×C₈	—	61	1	no	—
183	C₄⋊C₁₆	—	61	2	no	—
184	C₄⋊C₁₆	—	61	2	no	—
185	C₄⋊C₁₆	—	61	2	no	—
186	C₂²⋊C₁₆	—	61	1	no	—
187	C₂²⋊C₁₆	—	61	1	no	—
188	C₂²⋊C₁₆	—	61	2	no	—
189	C₂²⋊C₁₆	—	61	2	no	—
190	#346	—	61	2	no	—
191	#346	—	61	2	no	—
192	#346	—	61	2	no	—
193	C₄×C₁₆	—	61	1	no	—
194	C₄×C₁₆	—	61	1	no	—
195	C₄×C₁₆	—	61	2	no	—
196	C₆₁	#33481	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	61	2	no	—
198	C₂²×C₈	—	61	1	no	—
199	C₂²×C₈	—	61	2	no	—
200	C₂²×C₈	—	61	1	no	—
201	#126	—	61	2	no	—
202	#125	—	61	1	no	—
203	#124	—	61	1	no	—
204	#123	—	61	2	no	—
205	C₂²⋊Q₈	—	61	1	no	—
206	#121	—	61	1	no	—
207	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
208	C₄×D₄	—	61	2	no	—
209	C₄×D₄	—	61	1	no	—
210	#117	—	61	2	no	—
211	C₂×C₄²	—	61	1	no	—
212	M₃₂	—	122	4	no	—
213	M₃₂	—	122	4	no	—
214	C₂×C₁₆	—	61	2	no	—
215	C₂×C₁₆	—	61	2	no	—
216	C₂×C₁₆	—	61	2	no	—
217	C₂×C₁₆	—	61	2	no	—
218	C₂×C₁₆	—	122	4	no	—
219	C₄⋊C₈	—	61	1	no	—
220	C₄⋊C₈	—	61	2	no	—
221	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
222	C₄⋊C₈	C₂×#6923	1	2	yes	—
223	C₂²⋊C₈	—	61	2	no	—
224	C₂²⋊C₈	—	61	2	no	—
225	C₂²⋊C₈	—	61	2	no	—
226	#97	—	61	2	no	—
227	#97	C₂×#6923	1	2	yes	—
228	C₄×C₈	—	61	1	no	—
229	C₄×C₈	—	61	1	no	—
230	C₄×C₈	—	61	2	no	—
231	C₄×C₈	—	61	1	no	—
232	C₄×C₈	C₂×#6923	1	1	yes	characteristic
233	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
234	C₂×D₄	—	61	2	no	—
235	C₂×D₄	—	61	1	no	—
236	C₂²×C₄	—	61	1	no	—
237	C₂²×C₄	—	61	2	no	—
238	C₂²×C₄	—	61	1	no	—
239	C₂²×C₄	—	61	1	no	—
240	M₁₆	—	122	4	no	—
241	C₂×C₈	—	61	1	no	—
242	C₂×C₈	—	61	1	no	—
243	C₂×C₈	—	61	2	no	—
244	C₂×C₈	—	122	2	no	—
245	C₂×C₈	—	122	2	no	—
246	C₂×C₈	—	122	2	no	—
247	C₂×C₈	—	122	2	no	—
248	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
249	C₂×C₈	C₂²×#6923	1	2	yes	—
250	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
251	C₄⋊C₄	—	61	2	no	—
252	C₄⋊C₄	—	61	2	no	—
253	C₄⋊C₄	—	61	1	no	—
254	C₄⋊C₄	—	61	2	no	—
255	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
256	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
257	C₂²⋊C₄	—	61	2	no	—
258	C₂²⋊C₄	—	61	2	no	—
259	C₂²⋊C₄	—	61	2	no	—
260	C₂²⋊C₄	—	61	2	no	—
261	C₂²⋊C₄	—	61	2	no	—
262	C₄²	—	61	1	no	—
263	C₄²	—	61	1	no	—
264	C₄²	—	61	1	no	—
265	C₄²	C₂²×#6923	1	1	yes	characteristic
266	C₄²	C₂²×#6923	1	2	yes	—
267	C₁₆	—	61	4	no	—
268	C₁₆	—	122	2	no	—
269	C₁₆	—	122	2	no	—
270	C₁₆	—	122	2	no	—
271	C₂³	—	61	2	no	—
272	C₂³	—	61	1	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	244	2	no	—
275	D₄	—	244	1	no	—
276	C₂×C₄	—	61	1	no	—
277	C₂×C₄	—	61	1	no	—
278	C₂×C₄	—	61	1	no	—
279	C₂×C₄	—	61	2	no	—
280	C₂×C₄	—	122	2	no	—
281	C₂×C₄	—	122	1	no	—
282	C₂×C₄	—	122	1	no	—
283	C₂×C₄	—	122	2	no	—
284	C₂×C₄	—	244	1	no	—
285	C₂×C₄	—	244	1	no	—
286	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
289	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
290	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	1	no	—
293	C₈	—	61	2	no	—
294	C₈	—	122	1	no	—
295	C₈	—	122	2	no	—
296	C₈	#14519865	1	2	yes	—
297	C₂²	—	122	1	no	—
298	C₂²	—	122	2	no	—
299	C₂²	—	122	1	no	—
300	C₂²	—	122	2	no	—
301	C₂²	—	122	1	no	—
302	C₂²	—	122	2	no	—
303	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	122	1	no	—
309	C₄	—	122	1	no	—
310	C₄	—	122	1	no	—
311	C₄	—	122	2	no	—
312	C₄	#41×#6923	1	1	yes	characteristic
313	C₄	C₂×#14519865	1	1	yes	characteristic
314	C₂	—	244	1	no	—
315	C₂	—	244	2	no	—
316	C₂	#21462625	1	1	yes	characteristic
317	C₂	#21462787	1	1	yes	characteristic
318	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	#14519752	C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#446	#6923
hypocenter	C₆₁	#33481
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. #21462234	C₂	semidirect
2. #21462205	C₂	semidirect
3. #21462202	C₂	semidirect
4. #21462202	C₂	semidirect
5. #21462187	C₂	semidirect
6. #21462183	C₂	semidirect
7. #21462029	C₂	semidirect
8. #21462026	C₂	semidirect
196. C₆₁	#33481	semidirect

Maximal subgroups

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

subgroup	index
#21462234	2
#21462205	2
#21462202	2
#21462187	2
#21462183	2
#21462029	2
#21462026	2
#21454793	2
#33481	61