Subgroups — #30624358

Order 15616.

498 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#21462155	C₂	1	2	yes	maximal
6	D₆₁⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
7	#21462029	C₂	1	2	yes	maximal
8	#21462026	C₂	1	1	yes	maximal, characteristic
9	#21454750	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519852	C₂²	1	1	yes	characteristic
11	C₂×#14519852	C₂²	1	1	yes	characteristic
12	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
13	#16840683	C₂²	1	1	yes	characteristic
14	#16840683	C₂²	1	1	yes	characteristic
15	#16840683	C₂²	1	2	yes	—
16	#16840683	C₂²	1	2	yes	—
17	C₄⋊#11758187	C₂²	1	2	yes	—
18	C₄⋊#11758187	C₂²	1	2	yes	—
19	C₄⋊#11758187	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	2	yes	—
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	#14519852	—	2	2	no	—
38	#14519852	Q₈	1	4	yes	complemented
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	#14519776	C₈	1	1	yes	characteristic
48	#14519775	C₈	1	1	yes	characteristic
49	#14519767	C₈	1	2	yes	—
50	#14519757	C₈	1	1	yes	characteristic
51	#14519755	C₈	1	2	yes	—
52	#14519753	C₈	1	1	yes	characteristic
53	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
55	#14519734	C₂×C₄	1	2	yes	—
56	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#92418	C₂×C₄	1	2	yes	—
59	#14519706	C₂×C₄	1	2	yes	—
60	#14519706	C₂×C₄	1	2	yes	—
61	#14519706	C₂×C₄	1	2	yes	—
62	#14519687	C₂×C₄	1	2	yes	—
63	C₄×#92418	C₂³	1	1	yes	characteristic
64	C₄×#92418	C₂³	1	2	yes	—
65	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂³	1	2	yes	—
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×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
78	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
86	#11758198	C₂×C₈	1	2	yes	—
87	#11758198	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₆₁	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	#11758196	C₂×C₈	1	2	yes	—
94	#11758196	C₂×C₈	1	2	yes	—
95	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
105	C₈×D₆₁	#41	1	2	yes	—
106	#11758187	—	2	2	no	—
107	#11758187	—	2	2	no	—
108	#11758187	—	2	2	no	—
109	#11758187	C₂×Q₈	1	4	yes	—
110	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
111	Q₈×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
119	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
120	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
122	C₄×D₆₁	—	2	2	no	—
123	C₄×D₆₁	—	4	1	no	—
124	C₄×D₆₁	C₄×Q₈	1	2	yes	—
125	C₄×D₆₁	#131	1	2	yes	—
126	Dic₁₂₂	—	2	4	no	—
127	Dic₁₂₂	—	4	1	no	—
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	—	2	2	no	—
130	C₈×C₆₁	C₄×Q₈	1	2	yes	—
131	#92418	—	2	1	no	—
132	#92418	—	2	2	no	—
133	#92418	C₂²×Q₈	1	1	yes	characteristic
134	#92418	C₂×#41	1	1	yes	characteristic
135	#33476	—	61	1	no	maximal, Sylow
136	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₆₁	—	2	1	no	—
138	C₂×D₆₁	—	2	1	no	—
139	C₂×D₆₁	C₈×Q₈	1	2	yes	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	1	no	—
142	C₄×C₆₁	—	2	1	no	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
146	Dic₆₁	—	2	2	no	—
147	Dic₆₁	—	2	2	no	—
148	Dic₆₁	—	2	1	no	—
149	Dic₆₁	C₈×Q₈	1	2	yes	—
150	Dic₆₁	#504	1	2	yes	—
151	#2960	—	61	1	no	—
152	#2183	—	61	2	no	—
153	#2183	—	61	2	no	—
154	#2183	—	61	2	no	—
155	Q₈×C₁₆	—	61	2	no	—
156	#2151	—	61	2	no	—
157	C₂×C₄⋊C₁₆	—	61	1	no	—
158	#2108	—	61	2	no	—
159	C₄×M₃₂	—	61	1	no	—
160	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
161	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
162	C₂×C₆₁	#3407	1	1	yes	characteristic
163	D₆₁	—	2	2	no	—
164	#533	—	61	1	no	—
165	C₂×M₃₂	—	61	1	no	—
166	C₂×M₃₂	—	61	1	no	—
167	C₂²×C₁₆	—	61	2	no	—
168	#446	—	61	2	no	—
169	#446	—	61	1	no	—
170	#446	#6923	1	1	yes	characteristic
171	C₈×Q₈	—	61	1	no	—
172	C₈×Q₈	—	61	2	no	—
173	C₈×Q₈	—	61	1	no	—
174	#424	—	61	2	no	—
175	#423	—	61	1	no	—
176	#405	—	61	2	no	—
177	C₂×C₄×C₈	—	61	1	no	—
178	C₄⋊C₁₆	—	61	1	no	—
179	C₄⋊C₁₆	—	61	2	no	—
180	C₄⋊C₁₆	—	61	1	no	—
181	C₄⋊C₁₆	—	61	2	no	—
182	C₄⋊C₁₆	—	61	2	no	—
183	C₄⋊C₁₆	—	61	2	no	—
184	C₄⋊C₁₆	—	61	2	no	—
185	#346	—	61	2	no	—
186	#346	—	61	2	no	—
187	#346	—	61	2	no	—
188	C₄×C₁₆	—	61	2	no	—
189	C₄×C₁₆	—	61	2	no	—
190	C₄×C₁₆	—	61	2	no	—
191	C₆₁	#33476	1	1	yes	characteristic, complemented, Sylow
192	C₂×M₁₆	—	61	2	no	—
193	C₂²×C₈	—	61	1	no	—
194	C₂²×C₈	—	61	1	no	—
195	C₄⋊Q₈	—	61	1	no	—
196	C₄⋊Q₈	—	61	1	no	—
197	#125	—	61	2	no	—
198	C₂²⋊Q₈	—	61	1	no	—
199	C₂²⋊Q₈	—	61	2	no	—
200	C₂²⋊Q₈	—	61	1	no	—
201	C₄×Q₈	—	61	2	no	—
202	C₄×Q₈	—	61	1	no	—
203	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
204	#117	—	61	2	no	—
205	C₂×C₄²	—	61	1	no	—
206	M₃₂	—	61	4	no	—
207	M₃₂	—	122	2	no	—
208	C₂×C₁₆	—	61	2	no	—
209	C₂×C₁₆	—	61	2	no	—
210	C₂×C₁₆	—	61	2	no	—
211	C₂×C₁₆	—	61	2	no	—
212	C₂×C₁₆	—	122	4	no	—
213	C₄⋊C₈	—	61	2	no	—
214	C₄⋊C₈	—	61	2	no	—
215	C₄⋊C₈	—	61	2	no	—
216	C₄⋊C₈	—	61	1	no	—
217	C₄⋊C₈	—	61	2	no	—
218	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
219	C₄⋊C₈	C₂×#6923	1	2	yes	—
220	#97	—	61	2	no	—
221	#97	C₂×#6923	1	2	yes	—
222	C₄×C₈	—	61	1	no	—
223	C₄×C₈	—	61	2	no	—
224	C₄×C₈	—	61	1	no	—
225	C₄×C₈	—	61	2	no	—
226	C₄×C₈	—	61	1	no	—
227	C₄×C₈	C₂×#6923	1	1	yes	characteristic
228	C₂×Q₈	—	61	2	no	—
229	C₂×Q₈	—	61	1	no	—
230	C₂×Q₈	D₆₁⋊C₈	1	1	yes	characteristic
231	C₂²×C₄	—	61	1	no	—
232	C₂²×C₄	—	61	1	no	—
233	C₂²×C₄	—	61	1	no	—
234	M₁₆	—	122	4	no	—
235	C₂×C₈	—	61	1	no	—
236	C₂×C₈	—	61	2	no	—
237	C₂×C₈	—	61	2	no	—
238	C₂×C₈	—	61	1	no	—
239	C₂×C₈	—	61	2	no	—
240	C₂×C₈	—	122	2	no	—
241	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
242	C₂×C₈	C₂²×#6923	1	2	yes	—
243	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
244	C₄⋊C₄	—	61	2	no	—
245	C₄⋊C₄	—	61	2	no	—
246	C₄⋊C₄	—	61	2	no	—
247	C₄⋊C₄	—	61	2	no	—
248	C₄⋊C₄	—	61	2	no	—
249	C₄⋊C₄	—	61	2	no	—
250	C₄⋊C₄	—	61	1	no	—
251	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
252	C₄⋊C₄	D₆₁⋊C₈	1	1	yes	characteristic
253	C₂²⋊C₄	—	61	2	no	—
254	C₂²⋊C₄	—	61	2	no	—
255	C₄²	—	61	1	no	—
256	C₄²	—	61	1	no	—
257	C₄²	—	61	1	no	—
258	C₄²	—	61	2	no	—
259	C₄²	C₂²×#6923	1	1	yes	characteristic
260	C₄²	C₂²×#6923	1	2	yes	—
261	C₁₆	—	61	4	no	—
262	C₁₆	—	122	2	no	—
263	C₁₆	—	122	2	no	—
264	C₁₆	—	122	2	no	—
265	C₂³	—	61	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	122	4	no	—
268	Q₈	—	244	1	no	—
269	C₂×C₄	—	61	1	no	—
270	C₂×C₄	—	61	1	no	—
271	C₂×C₄	—	61	2	no	—
272	C₂×C₄	—	61	2	no	—
273	C₂×C₄	—	61	1	no	—
274	C₂×C₄	—	61	2	no	—
275	C₂×C₄	—	61	2	no	—
276	C₂×C₄	—	122	2	no	—
277	C₂×C₄	—	244	1	no	—
278	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
281	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
282	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	2	no	—
285	C₈	—	61	1	no	—
286	C₈	—	61	1	no	—
287	C₈	—	122	1	no	—
288	C₈	—	122	2	no	—
289	C₈	Q₈×#6923	1	2	yes	—
290	C₂²	—	61	2	no	—
291	C₂²	—	122	1	no	—
292	C₂²	—	122	1	no	—
293	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	61	2	no	—
299	C₄	—	61	2	no	—
300	C₄	—	122	2	no	—
301	C₄	—	122	1	no	—
302	C₄	—	122	2	no	—
303	C₄	C₂×Q₈×#6923	1	1	yes	characteristic
304	C₄	D₆₁⋊#131	1	1	yes	characteristic
305	C₂	—	122	2	no	—
306	C₂	Q₈×D₆₁⋊C₈	1	1	yes	characteristic
307	C₂	D₆₁⋊#504	1	1	yes	characteristic
308	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	#33476
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
38. #14519852	Q₈	semidirect
191. C₆₁	#33476	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
#21462155	2
D₆₁⋊(C₄⋊C₁₆)	2
#21462029	2
#21462026	2
#21454750	2
#33476	61