Subgroups — #29578887

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446614	C₂	1	2	yes	maximal, complemented
2	C₂×#16839513	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊#92418	C₂	1	2	yes	maximal, complemented
4	C₂×#16839483	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₂²⋊#92418	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839513	—	2	2	no	—
11	D₄⋊#92418	—	2	4	no	—
12	#16839483	—	2	2	no	—
13	C₂²×D₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
15	C₄×D₈×C₆₁	—	2	2	no	—
16	C₄×D₈×C₆₁	—	2	2	no	—
17	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
18	C₂×#102×C₆₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	1	no	—
21	C₂²×#92418	—	2	2	no	—
22	C₂²×#92418	D₄	1	2	yes	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₂²⋊#92418	—	4	8	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₄×#92418	D₄	1	2	yes	—
30	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₁	—	2	2	no	—
32	C₂×D₈×C₆₁	—	2	2	no	—
33	C₂×D₈×C₆₁	—	2	2	no	—
34	C₂×D₈×C₆₁	C₈	1	4	yes	complemented
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₆₁	—	2	2	no	—
38	C₄×D₄×C₆₁	—	2	2	no	—
39	C₄×D₄×C₆₁	—	2	2	no	—
40	C₄×D₄×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	#107×C₆₁	—	2	2	no	—
45	#102×C₆₁	—	2	4	no	—
46	C₄×C₈×C₆₁	D₄	1	2	yes	—
47	C₄×C₈×C₆₁	D₄	1	2	yes	—
48	C₂⁴×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	2	2	no	—
50	C₂×D₄×C₆₁	—	2	2	no	—
51	C₂×D₄×C₆₁	—	2	2	no	—
52	C₂×D₄×C₆₁	—	2	2	no	—
53	C₂×D₄×C₆₁	—	2	2	no	—
54	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	—	4	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	—	4	4	no	—
59	C₂²×C₄×C₆₁	—	4	4	no	—
60	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
63	D₈×C₆₁	—	2	8	no	—
64	C₂×C₈×C₆₁	—	2	1	no	—
65	C₂×C₈×C₆₁	—	2	1	no	—
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
70	C₄⋊C₄×C₆₁	—	2	2	no	—
71	C₄⋊C₄×C₆₁	—	2	2	no	—
72	C₂²⋊C₄×C₆₁	—	4	2	no	—
73	C₂²⋊C₄×C₆₁	—	4	2	no	—
74	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₁	#41	1	1	yes	characteristic
78	C₂×#92418	—	2	2	no	—
79	C₂×#92418	—	2	2	no	—
80	C₂×#92418	—	2	4	no	—
81	C₂×#92418	—	4	2	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂×#92418	—	4	4	no	—
84	C₂³×C₆₁	—	2	4	no	—
85	C₂³×C₆₁	—	4	2	no	—
86	C₂³×C₆₁	—	4	2	no	—
87	C₂³×C₆₁	—	4	4	no	—
88	C₂³×C₆₁	—	4	4	no	—
89	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
90	D₄×C₆₁	—	2	4	no	—
91	D₄×C₆₁	—	2	4	no	—
92	D₄×C₆₁	—	4	2	no	—
93	D₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	4	2	no	—
96	C₂×C₄×C₆₁	—	4	4	no	—
97	C₂×C₄×C₆₁	—	8	4	no	—
98	C₂×C₄×C₆₁	—	8	2	no	—
99	C₂×C₄×C₆₁	—	8	2	no	—
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
109	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
110	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
111	C₈×C₆₁	—	2	2	no	—
112	C₈×C₆₁	C₂²⋊C₈	1	4	yes	complemented
113	#92418	—	4	2	no	—
114	#92418	—	4	2	no	—
115	#92418	—	8	2	no	—
116	#9659	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	—	4	8	no	—
118	C₂²×C₆₁	—	4	4	no	—
119	C₂²×C₆₁	—	8	4	no	—
120	C₂²×C₆₁	—	8	2	no	—
121	C₂²×C₆₁	—	8	2	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	C₂²×C₆₁	#442	1	2	yes	—
126	C₂²×C₆₁	#444	1	2	yes	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	—	2	1	no	—
129	C₄×C₆₁	—	8	2	no	—
130	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₆₁	#408	1	1	yes	characteristic
132	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
133	C₄×C₆₁	#436	1	1	yes	characteristic
134	C₄×C₆₁	#469	1	2	yes	—
135	C₄×C₆₁	#496	1	2	yes	—
136	C₂×C₄×D₈	D₆₁	1	1	yes	characteristic
137	#1954	—	61	2	no	—
138	C₂×#334	—	61	1	no	—
139	C₂×D₄⋊C₈	—	61	2	no	—
140	C₂×#322	—	61	1	no	—
141	C₂×C₆₁	—	8	4	no	—
142	C₂×C₆₁	#1580	1	2	yes	—
143	C₂×C₆₁	#1594	1	2	yes	—
144	C₂×C₆₁	#1954	1	1	yes	characteristic
145	C₂×C₆₁	#1972	1	1	yes	characteristic
146	C₂×C₆₁	#1975	1	1	yes	characteristic
147	C₂²×D₈	Dic₆₁	1	1	yes	characteristic
148	C₂×C₄×D₄	C₂×D₆₁	1	2	yes	—
149	C₄×D₈	—	2	2	no	—
150	C₄×D₈	—	2	2	no	—
151	C₂×#107	Dic₆₁	1	1	yes	characteristic
152	C₂×C₄⋊C₈	—	61	2	no	—
153	C₂×#102	Dic₆₁	1	2	yes	—
154	C₂×C₂²⋊C₈	—	122	4	no	—
155	C₂×C₄×C₈	—	61	1	no	—
156	C₂×C₄×C₈	—	61	1	no	—
157	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
158	#334	—	122	2	no	—
159	D₄⋊C₈	—	122	4	no	—
160	#322	—	122	2	no	—
161	C₆₁	#9659	1	1	yes	characteristic, complemented, Sylow
162	C₂²×D₄	C₂×Dic₆₁	1	2	yes	—
163	C₂³×C₄	—	2	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	#92418	1	4	yes	complemented
167	C₂²×C₈	—	61	2	no	—
168	C₂²×C₈	—	122	1	no	—
169	C₂²×C₈	—	122	2	no	—
170	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
179	#107	—	2	2	no	—
180	C₄⋊C₈	—	122	2	no	—
181	C₄⋊C₈	—	122	2	no	—
182	#102	—	2	4	no	—
183	C₂²⋊C₈	—	244	8	no	—
184	C₄×C₈	—	61	2	no	—
185	C₄×C₈	—	122	1	no	—
186	C₄×C₈	—	122	1	no	—
187	C₄×C₈	—	122	1	no	—
188	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
189	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂×#92418	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
203	C₂²×C₄	#11758224	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
205	D₈	—	2	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	122	2	no	—
209	C₂×C₈	—	122	2	no	—
210	C₂×C₈	—	122	4	no	—
211	C₂×C₈	—	244	2	no	—
212	C₂×C₈	—	244	4	no	—
213	C₂×C₈	—	244	2	no	—
214	C₂×C₈	C₂×#92418	1	2	yes	—
215	C₂×C₈	#11758194	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
217	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	D₄×D₆₁	1	1	yes	characteristic
223	C₄²	#11758224	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
244	C₂×C₄	#14519748	1	1	yes	characteristic
245	C₂×C₄	#14519748	1	1	yes	characteristic
246	C₂×C₄	#14519764	1	1	yes	characteristic
247	C₂×C₄	#14519773	1	1	yes	characteristic
248	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
249	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
250	C₂×C₄	#14519784	1	1	yes	characteristic
251	C₂×C₄	#14519784	1	1	yes	characteristic
252	C₂×C₄	#14519801	1	4	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	488	2	no	—
255	C₈	—	244	2	no	—
256	C₈	—	244	2	no	—
257	C₈	C₂²⋊#92418	1	4	yes	complemented
258	C₂²	—	4	8	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	4	no	—
263	C₂²	#16839665	1	1	yes	characteristic
264	C₂²	D₄×#92418	1	1	yes	characteristic
265	C₂²	#16839840	1	1	yes	characteristic
266	C₂²	#16839874	1	2	yes	—
267	C₂²	#16839884	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	C₂×C₂²⋊#92418	1	1	yes	characteristic
272	C₄	#16839729	1	1	yes	characteristic
273	C₄	D₄×#92418	1	1	yes	characteristic
274	C₄	#16839840	1	1	yes	characteristic
275	C₄	#16840012	1	2	yes	—
276	C₄	#16840150	1	2	yes	—
277	C₂	—	8	4	no	—
278	C₂	#21444643	1	2	yes	—
279	C₂	#21444733	1	2	yes	—
280	C₂	#21446614	1	1	yes	characteristic
281	C₂	#21446696	1	1	yes	characteristic
282	C₂	#21446713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₄×D₈×C₆₁	C₂
socle	C₂³×C₆₁	C₄×D₄
prime radical	C₂²×D₈×C₆₁	C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₄×D₈	D₆₁
hypocenter	C₆₁	#9659
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #21446614	C₂	semidirect
2. C₂×#16839513	C₂	semidirect
3. C₂×D₄⋊#92418	C₂	semidirect
4. C₂×#16839483	C₂	semidirect
34. C₂×D₈×C₆₁	C₈	semidirect
112. C₈×C₆₁	C₂²⋊C₈	semidirect
161. C₆₁	#9659	semidirect
166. C₂×D₈	#92418	semidirect
257. C₈	C₂²⋊#92418	semidirect

Maximal subgroups

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

subgroup	index
#21446614	2
C₂×#16839513	2
C₂×D₄⋊#92418	2
C₂×#16839483	2
C₂×C₄×D₈×C₆₁	2
#9659	61