Subgroups — #29833660

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	2	yes	maximal, complemented
3	#21447726	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
5	#21447622	C₂	1	1	yes	maximal, characteristic
6	#21447619	C₂	1	2	yes	maximal
7	#21447615	C₂	1	2	yes	maximal
8	C₆₁×#2162	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
10	C₈×Dic₁₂₂	C₄	1	2	yes	—
11	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
12	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
13	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₂₂	C₂²	1	2	yes	—
15	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
17	C₂×#14519706	C₄	1	2	yes	—
18	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
19	C₄⋊#11758185	C₂²	1	2	yes	—
20	#16839580	C₂²	1	4	yes	—
21	#16839580	C₂²	1	4	yes	—
22	C₂²⋊#11758185	C₂²	1	2	yes	—
23	#16839538	C₂²	1	2	yes	—
24	#16839537	C₂²	1	4	yes	—
25	C₁₆×Dic₆₁	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₆₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₆₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
31	C₂²×Dic₁₂₂	C₈	1	1	yes	characteristic
32	C₂²×#92418	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
34	C₄×Dic₁₂₂	C₈	1	2	yes	—
35	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
36	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
40	C₂×#11758196	C₈	1	2	yes	—
41	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
42	C₂×#11758185	C₂³	1	4	yes	—
43	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
44	C₄⋊#92418	C₂×C₄	1	2	yes	—
45	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
46	#14519706	C₂×C₄	1	2	yes	—
47	#14519706	C₂×C₄	1	2	yes	—
48	#14519706	C₂×C₄	1	2	yes	—
49	#14519706	C₂×C₄	1	2	yes	—
50	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
51	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
52	C₈×Dic₆₁	C₂³	1	2	yes	—
53	C₈×Dic₆₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
58	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
59	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
67	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
68	C₂×Dic₁₂₂	—	2	4	no	—
69	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
70	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	4	1	no	—
74	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
83	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
85	#11758196	C₂×C₈	1	2	yes	—
86	#11758196	C₂×C₈	1	2	yes	—
87	#11758196	C₂×C₈	1	2	yes	—
88	#11758196	C₂×C₈	1	2	yes	—
89	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
90	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
91	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
92	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
93	C₂×#92418	—	4	2	no	—
94	C₂×#92418	C₂²×C₄	1	2	yes	—
95	C₂×#92418	C₂²×C₄	1	2	yes	—
96	C₁₆×C₆₁	—	2	2	no	—
97	C₁₆×C₆₁	—	2	2	no	—
98	#11758185	—	2	4	no	—
99	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	—	2	2	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₆₁	—	2	4	no	—
111	C₂×Dic₆₁	—	2	4	no	—
112	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
113	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
114	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
115	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
116	Dic₁₂₂	—	2	2	no	—
117	Dic₁₂₂	—	2	4	no	—
118	Dic₁₂₂	M₃₂	1	4	yes	—
119	C₈×C₆₁	—	2	1	no	—
120	C₈×C₆₁	—	2	1	no	—
121	C₈×C₆₁	—	2	1	no	—
122	C₈×C₆₁	2₋¹⁺⁴	1	2	yes	—
123	#92418	—	2	2	no	—
124	#92418	—	2	2	no	—
125	#33498	—	61	1	no	maximal, Sylow
126	C₂²×C₆₁	—	2	1	no	—
127	C₂²×C₆₁	—	2	1	no	—
128	C₂²×C₆₁	#504	1	2	yes	—
129	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₆₁	—	2	1	no	—
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
135	C₄×C₆₁	#519	1	1	yes	characteristic
136	C₄×C₆₁	#568	1	1	yes	characteristic
137	Dic₆₁	—	2	2	no	—
138	Dic₆₁	—	2	2	no	—
139	Dic₆₁	—	2	2	no	—
140	Dic₆₁	C₂×M₃₂	1	4	yes	—
141	C₂×C₈×Q₈	—	61	1	no	—
142	#2183	—	61	4	no	—
143	#2183	—	61	2	no	—
144	Q₈×C₁₆	—	61	2	no	—
145	#2163	—	61	1	no	—
146	#2163	—	61	2	no	—
147	#2162	—	61	2	no	—
148	#2162	D₆₁	1	1	yes	characteristic
149	C₂×C₆₁	—	2	2	no	—
150	C₂×C₆₁	#2982	1	1	yes	characteristic
151	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	61	1	no	—
154	C₈×Q₈	—	61	2	no	—
155	C₈×Q₈	—	61	1	no	—
156	C₈×Q₈	—	61	1	no	—
157	C₈×Q₈	—	61	1	no	—
158	C₈×Q₈	—	61	2	no	—
159	C₈×Q₈	—	61	1	no	—
160	C₂×C₄⋊C₈	—	61	2	no	—
161	C₂×C₄⋊C₈	—	61	1	no	—
162	C₂×C₄×C₈	—	61	2	no	—
163	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
164	C₄⋊C₁₆	—	61	4	no	—
165	C₄⋊C₁₆	—	61	4	no	—
166	C₄⋊C₁₆	—	61	2	no	—
167	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
168	C₂²⋊C₁₆	—	61	2	no	—
169	C₂²⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
170	C₂²⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
171	#346	—	61	4	no	—
172	#346	—	61	2	no	—
173	C₄×C₁₆	—	61	4	no	—
174	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
175	C₆₁	#33498	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	61	1	no	—
177	C₂²×C₈	—	61	2	no	—
178	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₄×Q₈	—	61	1	no	—
181	C₄×Q₈	—	61	2	no	—
182	C₄×Q₈	—	61	1	no	—
183	C₄×Q₈	—	61	1	no	—
184	C₄×Q₈	—	61	2	no	—
185	C₄×Q₈	—	61	1	no	—
186	C₂×C₄⋊C₄	—	61	2	no	—
187	C₂×C₄⋊C₄	—	61	1	no	—
188	C₂×C₄²	—	61	2	no	—
189	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
190	C₂×C₁₆	—	61	4	no	—
191	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
192	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
193	C₄⋊C₈	—	61	2	no	—
194	C₄⋊C₈	—	61	2	no	—
195	C₄⋊C₈	—	61	2	no	—
196	C₄⋊C₈	—	61	2	no	—
197	C₄⋊C₈	—	61	1	no	—
198	C₄⋊C₈	—	61	2	no	—
199	C₄⋊C₈	—	61	1	no	—
200	C₄×C₈	—	61	2	no	—
201	C₄×C₈	—	61	2	no	—
202	C₄×C₈	—	61	2	no	—
203	C₄×C₈	—	61	2	no	—
204	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
205	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
208	C₂×Q₈	—	61	1	no	—
209	C₂×Q₈	—	61	2	no	—
210	C₂×Q₈	—	61	1	no	—
211	C₂×Q₈	—	122	4	no	—
212	C₂²×C₄	—	61	2	no	—
213	C₂²×C₄	—	61	2	no	—
214	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
215	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	61	2	no	—
220	C₂×C₈	—	61	2	no	—
221	C₂×C₈	—	244	2	no	—
222	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
226	C₄⋊C₄	—	61	2	no	—
227	C₄⋊C₄	—	61	2	no	—
228	C₄⋊C₄	—	61	2	no	—
229	C₄⋊C₄	—	61	2	no	—
230	C₄⋊C₄	—	61	2	no	—
231	C₄⋊C₄	—	61	1	no	—
232	C₄⋊C₄	—	61	1	no	—
233	C₄²	—	61	2	no	—
234	C₄²	—	61	2	no	—
235	C₄²	—	61	2	no	—
236	C₄²	—	61	2	no	—
237	C₄²	C₈×D₆₁	1	1	yes	characteristic
238	C₄²	C₈×D₆₁	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	122	4	no	—
244	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
245	Q₈	—	61	4	no	—
246	Q₈	—	122	4	no	—
247	Q₈	—	122	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	61	2	no	—
253	C₂×C₄	—	61	2	no	—
254	C₂×C₄	—	61	2	no	—
255	C₂×C₄	—	61	2	no	—
256	C₂×C₄	—	122	4	no	—
257	C₂×C₄	—	122	4	no	—
258	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	122	2	no	—
268	C₈	—	122	2	no	—
269	C₈	#14519824	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	#16840178	1	2	yes	—
273	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	61	4	no	—
279	C₄	—	122	2	no	—
280	C₄	—	122	2	no	—
281	C₄	—	122	2	no	—
282	C₄	M₃₂×D₆₁	1	2	yes	—
283	C₄	#16840239	1	1	yes	characteristic
284	C₄	#16840528	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	#21454899	1	1	yes	characteristic
287	C₂	C₂×M₃₂×D₆₁	1	1	yes	characteristic
288	C₂	C₂×#16840178	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2162	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2162	D₆₁
hypocenter	C₆₁	#33498
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447726	C₂	semidirect
4. C₁₆×Dic₁₂₂	C₂	semidirect
175. C₆₁	#33498	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₂₂	2
#21447727	2
#21447726	2
C₁₆×Dic₁₂₂	2
#21447622	2
#21447619	2
#21447615	2
C₆₁×#2162	2
#33498	61