Subgroups — #29578944

Order 15616.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	#21444213	C₂	1	2	yes	maximal
4	#21444142	C₂	1	2	yes	maximal
5	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₂	C₂²	1	2	yes	—
8	C₄×Dic₂₄₄	—	2	1	no	—
9	C₄×Dic₂₄₄	—	2	1	no	—
10	C₄×Dic₂₄₄	—	2	1	no	—
11	C₄×Dic₂₄₄	—	2	1	no	—
12	#16839815	C₂²	1	2	yes	—
13	C₂×#14519711	C₄	1	1	yes	characteristic
14	C₂×#14519702	C₄	1	2	yes	—
15	#16839515	—	2	1	no	—
16	#16839513	—	2	1	no	—
17	#16839492	—	2	4	no	—
18	#16839487	—	2	4	no	—
19	C₆₁×#431	C₂²	1	1	yes	characteristic
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#322	—	2	1	no	—
23	C₈²×C₆₁	—	2	1	no	—
24	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₄₄	—	2	1	no	—
26	C₂×Dic₂₄₄	—	2	1	no	—
27	C₂×Dic₂₄₄	—	2	2	no	—
28	C₂×Dic₂₄₄	C₈	1	4	yes	complemented
29	C₄×Dic₁₂₂	—	2	2	no	—
30	C₄×Dic₁₂₂	—	2	2	no	—
31	C₄×Dic₁₂₂	—	2	2	no	—
32	C₄×Dic₁₂₂	—	2	2	no	—
33	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
34	C₂×#11758196	—	2	2	no	—
35	C₂×C₄×Dic₆₁	—	2	2	no	—
36	#14519711	—	2	1	no	—
37	#14519711	—	2	1	no	—
38	C₄⋊#92418	—	2	2	no	—
39	#14519702	—	2	2	no	—
40	#14519702	—	2	2	no	—
41	C₂²⋊#92418	—	2	2	no	—
42	C₄×#92418	—	2	2	no	—
43	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₁	—	2	1	no	—
47	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
48	C₂²⋊C₈×C₆₁	—	2	1	no	—
49	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
50	C₄×C₈×C₆₁	—	2	1	no	—
51	C₄×C₈×C₆₁	—	2	1	no	—
52	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	2	2	no	—
60	C₂²×Dic₆₁	—	2	2	no	—
61	C₂×Dic₁₂₂	—	2	2	no	—
62	C₂×Dic₁₂₂	—	2	2	no	—
63	C₂×Dic₁₂₂	—	2	2	no	—
64	C₂×Dic₁₂₂	—	2	2	no	—
65	C₂×Dic₁₂₂	—	2	2	no	—
66	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₆₁	—	2	1	no	—
68	C₂×C₈×C₆₁	—	2	1	no	—
69	C₂×C₈×C₆₁	—	2	2	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
77	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	#41	1	1	yes	characteristic
81	C₄⋊Dic₆₁	—	2	2	no	—
82	C₄⋊Dic₆₁	—	2	2	no	—
83	#11758196	—	4	2	no	—
84	#11758196	—	4	2	no	—
85	C₄×Dic₆₁	—	4	2	no	—
86	C₄×Dic₆₁	—	4	2	no	—
87	C₂×#92418	—	4	2	no	—
88	Dic₂₄₄	—	2	4	no	—
89	Dic₂₄₄	—	2	4	no	—
90	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	4	1	no	—
93	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
103	C₂×Dic₆₁	—	2	4	no	—
104	C₂×Dic₆₁	—	4	2	no	—
105	C₂×Dic₆₁	—	4	2	no	—
106	C₂×Dic₆₁	—	4	2	no	—
107	C₂×Dic₆₁	—	8	2	no	—
108	Dic₁₂₂	—	2	4	no	—
109	Dic₁₂₂	—	2	4	no	—
110	Dic₁₂₂	—	4	2	no	—
111	Dic₁₂₂	—	4	2	no	—
112	C₈×C₆₁	—	2	1	no	—
113	C₈×C₆₁	—	2	1	no	—
114	C₈×C₆₁	—	4	2	no	—
115	C₈×C₆₁	—	4	2	no	—
116	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
117	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
118	#92418	—	8	2	no	—
119	#9669	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	—	2	1	no	—
121	C₂²×C₆₁	—	2	1	no	—
122	C₂²×C₆₁	#390	1	1	yes	characteristic
123	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₆₁	#436	1	1	yes	characteristic
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₆₁	#408	1	1	yes	characteristic
130	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
131	C₄×C₆₁	#436	1	1	yes	characteristic
132	C₄×C₆₁	#473	1	2	yes	—
133	Dic₆₁	—	4	4	no	—
134	Dic₆₁	—	8	2	no	—
135	C₂×C₄×Q₁₆	—	61	1	no	—
136	#1564	—	61	1	no	—
137	#1479	—	61	2	no	—
138	#1461	—	61	2	no	—
139	#1450	D₆₁	1	1	yes	characteristic
140	C₂×C₆₁	—	2	1	no	—
141	C₂×C₆₁	—	2	1	no	—
142	C₂×C₆₁	#1954	1	1	yes	characteristic
143	C₂×C₆₁	#1976	1	1	yes	characteristic
144	C₂×C₆₁	#1979	1	1	yes	characteristic
145	C₂²×Q₁₆	—	61	1	no	—
146	C₂×C₄×Q₈	—	61	2	no	—
147	C₄×Q₁₆	—	122	1	no	—
148	C₄×Q₁₆	—	122	1	no	—
149	C₄×Q₁₆	—	122	1	no	—
150	C₄×Q₁₆	—	122	1	no	—
151	#431	—	61	2	no	—
152	#431	C₂×D₆₁	1	1	yes	characteristic
153	#431	C₂×D₆₁	1	1	yes	characteristic
154	C₂×#107	—	61	1	no	—
155	C₂×#103	—	61	2	no	—
156	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
157	#335	—	122	1	no	—
158	#334	—	122	1	no	—
159	Q₈⋊C₈	—	122	4	no	—
160	#324	—	122	4	no	—
161	#322	—	2	1	no	—
162	C₈²	—	2	1	no	—
163	C₆₁	#9669	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	61	2	no	—
165	C₂×Q₁₆	—	61	4	no	—
166	C₂×Q₁₆	—	122	2	no	—
167	C₂×Q₁₆	—	122	1	no	—
168	C₂×Q₁₆	—	122	1	no	—
169	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
171	C₄×Q₈	—	122	2	no	—
172	C₄×Q₈	—	122	2	no	—
173	C₄×Q₈	—	122	2	no	—
174	C₄×Q₈	—	122	2	no	—
175	C₂×C₄⋊C₄	—	61	2	no	—
176	C₂×C₄⋊C₄	—	122	2	no	—
177	C₂×C₄²	—	122	2	no	—
178	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
179	#107	—	122	1	no	—
180	#107	—	122	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	122	2	no	—
183	C₄⋊C₈	D₂₄₄	1	2	yes	—
184	#103	—	122	2	no	—
185	#103	—	122	2	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	122	2	no	—
188	C₂²⋊C₈	D₂₄₄	1	2	yes	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	122	2	no	—
192	C₄×C₈	D₂₄₄	1	1	yes	characteristic
193	C₄×C₈	D₂₄₄	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
196	C₂×Q₈	—	61	4	no	—
197	C₂×Q₈	—	122	2	no	—
198	C₂×Q₈	—	122	2	no	—
199	C₂×Q₈	—	122	2	no	—
200	C₂×Q₈	—	122	2	no	—
201	C₂×Q₈	—	122	2	no	—
202	C₂²×C₄	—	122	2	no	—
203	C₂²×C₄	—	122	2	no	—
204	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
206	C₂²×C₄	#11758222	1	1	yes	characteristic
207	Q₁₆	—	122	4	no	—
208	Q₁₆	—	122	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	244	2	no	—
214	C₂×C₈	C₈×D₆₁	1	2	yes	—
215	C₂×C₈	#11758189	1	2	yes	—
216	C₂×C₈	#11758198	1	1	yes	characteristic
217	C₂×C₈	#11758198	1	1	yes	characteristic
218	C₂×C₈	#11758198	1	1	yes	characteristic
219	C₂×C₈	#11758198	1	1	yes	characteristic
220	C₄⋊C₄	—	122	2	no	—
221	C₄⋊C₄	—	122	2	no	—
222	C₄⋊C₄	—	244	2	no	—
223	C₄⋊C₄	—	244	2	no	—
224	C₄²	—	244	2	no	—
225	C₄²	—	244	2	no	—
226	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
227	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
228	C₄²	#11758222	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
230	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
231	Q₈	—	122	4	no	—
232	Q₈	—	122	4	no	—
233	Q₈	—	244	2	no	—
234	Q₈	—	244	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	122	4	no	—
238	C₂×C₄	—	488	2	no	—
239	C₂×C₄	—	244	2	no	—
240	C₂×C₄	—	244	2	no	—
241	C₂×C₄	—	244	2	no	—
242	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
243	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
244	C₂×C₄	#14519751	1	1	yes	characteristic
245	C₂×C₄	#14519751	1	1	yes	characteristic
246	C₂×C₄	#14519765	1	1	yes	characteristic
247	C₂×C₄	#14519774	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
249	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
250	C₂×C₄	#14519783	1	1	yes	characteristic
251	C₂×C₄	#14519783	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	488	2	no	—
257	C₈	#14519689	1	2	yes	—
258	C₈	#14519689	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#16839664	1	1	yes	characteristic
262	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
263	C₂²	#16839841	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	488	2	no	—
268	C₄	—	244	4	no	—
269	C₄	C₂×#14519689	1	1	yes	characteristic
270	C₄	#16839728	1	1	yes	characteristic
271	C₄	C₈×D₂₄₄	1	1	yes	characteristic
272	C₄	#16839841	1	1	yes	characteristic
273	C₄	#16840037	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#21446613	1	1	yes	characteristic
277	C₂	#21446717	1	1	yes	characteristic
278	C₂	#21446734	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519751
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#9669
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
28. C₂×Dic₂₄₄	C₈	semidirect
163. C₆₁	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
#21444567	2
#21444213	2
#21444142	2
C₆₁×#1450	2
#9669	61