Subgroups — #29832593

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840166	C₂	1	2	yes	maximal
2	C₂×#16840159	C₂	1	2	yes	maximal
3	#21454195	C₂	1	2	yes	maximal
4	#21445983	C₂	1	1	yes	maximal, characteristic
5	#21445982	C₂	1	2	yes	maximal
6	#21445981	C₂	1	2	yes	maximal
7	#21445980	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519776	C₂²	1	4	yes	—
11	#16840166	—	2	8	no	—
12	#16840159	—	2	8	no	—
13	C₂×#14519711	C₂²	1	2	yes	—
14	C₂×#14519710	C₂²	1	4	yes	—
15	C₂×#14519709	C₂²	1	2	yes	—
16	C₂×#14519708	C₂²	1	4	yes	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	#16839666	—	2	8	no	—
23	#16839628	C₂²	1	2	yes	—
24	#16839627	C₂²	1	4	yes	—
25	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₁	C₂²	1	2	yes	—
27	C₂×#106×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#382	C₂²	1	2	yes	—
30	C₂²×Dic₁₂₂	—	2	4	no	—
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	C₂²×#92418	C₂³	1	1	yes	characteristic
33	#14519776	—	2	8	no	—
34	#14519776	—	2	8	no	—
35	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
36	C₂×#11758196	—	2	4	no	—
37	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
38	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	#14519711	—	2	4	no	—
41	#14519710	—	2	8	no	—
42	#14519709	—	2	4	no	—
43	#14519708	—	2	8	no	—
44	C₄×#92418	Q₈	1	4	yes	complemented
45	C₈×Dic₆₁	D₄	1	4	yes	—
46	C₈×Dic₆₁	D₄	1	4	yes	—
47	C₈×Dic₆₁	Q₈	1	4	yes	complemented
48	C₈×Dic₆₁	Q₈	1	4	yes	complemented
49	#14519684	—	2	4	no	—
50	#14519684	—	2	8	no	—
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	#107×C₆₁	—	2	4	no	—
56	#106×C₆₁	—	2	4	no	—
57	C₄×C₈×C₆₁	Q₈	1	4	yes	complemented
58	#95×C₆₁	—	2	4	no	—
59	C₂²×C₄×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₆₁	—	2	4	no	—
63	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
64	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₂₂	—	4	8	no	—
66	C₂×Dic₁₂₂	—	4	16	no	—
67	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
73	C₄⋊C₄×C₆₁	—	2	4	no	—
74	C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
76	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
77	C₄⋊Dic₆₁	—	2	4	no	—
78	C₄⋊Dic₆₁	—	2	4	no	—
79	#11758196	—	4	8	no	—
80	C₄×Dic₆₁	—	2	8	no	—
81	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
82	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
83	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
84	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
85	C₂×#92418	C₂×D₄	1	2	yes	—
86	C₂×#92418	C₂×D₄	1	2	yes	—
87	C₂×#92418	C₂×Q₈	1	2	yes	—
88	C₂×#92418	C₂×Q₈	1	2	yes	—
89	C₂×#92418	C₂×Q₈	1	2	yes	—
90	C₂×#92418	C₂×Q₈	1	2	yes	—
91	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	—	2	4	no	—
93	C₂×C₄×C₆₁	—	4	4	no	—
94	C₂×C₄×C₆₁	—	4	8	no	—
95	C₂×C₄×C₆₁	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂×Dic₆₁	—	2	4	no	—
103	C₂×Dic₆₁	—	4	4	no	—
104	C₂×Dic₆₁	—	4	8	no	—
105	C₂×Dic₆₁	#127	1	4	yes	—
106	C₂×Dic₆₁	C₄⋊Q₈	1	4	yes	—
107	C₂×Dic₆₁	#135	1	8	yes	—
108	Dic₁₂₂	—	8	16	no	—
109	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
110	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
111	#92418	C₄⋊Q₈	1	4	yes	—
112	#92418	C₄⋊Q₈	1	4	yes	—
113	#33313	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₆₁	C₂×#135	1	4	yes	—
118	C₄×C₆₁	—	2	4	no	—
119	C₄×C₆₁	—	8	4	no	—
120	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	Dic₆₁	—	2	4	no	—
125	Dic₆₁	—	8	4	no	—
126	Dic₆₁	#495	1	8	yes	—
127	C₂×#500	—	61	2	no	—
128	C₂×#498	—	61	2	no	—
129	#2862	—	61	2	no	—
130	#1846	—	61	2	no	—
131	#1846	—	61	2	no	—
132	#1846	—	61	1	no	—
133	#1846	—	61	2	no	—
134	#1846	D₆₁	1	1	yes	characteristic
135	C₄²×C₈	—	61	1	no	—
136	C₂×C₆₁	#2868	1	1	yes	characteristic
137	C₂×C₆₁	C₂×#495	1	2	yes	—
138	C₂×C₆₁	#3160	1	2	yes	—
139	C₂×C₆₁	#3160	1	2	yes	—
140	C₂×C₄⋊Q₈	—	61	4	no	—
141	#500	—	122	8	no	—
142	#498	—	122	8	no	—
143	C₂×#107	—	61	2	no	—
144	C₂×#107	—	61	4	no	—
145	C₂×#107	C₂×D₆₁	1	2	yes	—
146	C₂×#106	—	61	2	no	—
147	C₂×#106	—	61	4	no	—
148	C₂×#106	C₂×D₆₁	1	2	yes	—
149	C₂×C₄×C₈	—	61	1	no	—
150	C₂×C₄×C₈	—	61	1	no	—
151	C₂×C₄×C₈	—	61	1	no	—
152	C₂×C₄×C₈	—	61	1	no	—
153	C₂×C₄×C₈	—	61	1	no	—
154	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
155	#391	—	122	8	no	—
156	#382	—	61	4	no	—
157	#382	—	61	2	no	—
158	#382	C₂×D₆₁	1	2	yes	—
159	C₄³	—	61	1	no	—
160	C₆₁	#33313	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	122	4	no	—
162	C₂²×C₈	—	61	1	no	—
163	C₂²×C₈	—	61	1	no	—
164	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
166	C₄⋊Q₈	—	122	8	no	—
167	C₄⋊Q₈	—	122	8	no	—
168	C₂×C₄⋊C₄	—	61	4	no	—
169	C₂×C₄⋊C₄	—	122	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
171	C₂×C₄²	—	61	1	no	—
172	C₂×C₄²	—	61	1	no	—
173	C₂×C₄²	—	61	4	no	—
174	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	—	122	8	no	—
177	#107	—	122	4	no	—
178	#106	—	2	4	no	—
179	#106	—	122	8	no	—
180	#106	—	122	4	no	—
181	C₄×C₈	—	61	4	no	—
182	C₄×C₈	—	61	4	no	—
183	C₄×C₈	—	61	4	no	—
184	C₄×C₈	—	61	4	no	—
185	C₄×C₈	—	61	4	no	—
186	C₄×C₈	Dic₁₂₂	1	4	yes	complemented
187	#95	—	2	4	no	—
188	#95	—	122	8	no	—
189	#95	—	122	4	no	—
190	C₂×Q₈	—	244	8	no	—
191	C₂×Q₈	—	244	16	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	61	2	no	—
194	C₂²×C₄	—	61	2	no	—
195	C₂²×C₄	—	122	4	no	—
196	C₂²×C₄	D₄×D₆₁	1	2	yes	—
197	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
198	C₂×C₈	—	61	2	no	—
199	C₂×C₈	—	61	2	no	—
200	C₂×C₈	—	61	2	no	—
201	C₂×C₈	—	61	2	no	—
202	C₂×C₈	—	61	2	no	—
203	C₂×C₈	—	61	2	no	—
204	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
205	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
206	C₂×C₈	D₄×D₆₁	1	2	yes	—
207	C₂×C₈	D₄×D₆₁	1	2	yes	—
208	C₂×C₈	Q₈×D₆₁	1	2	yes	—
209	C₂×C₈	Q₈×D₆₁	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	122	4	no	—
213	C₄⋊C₄	—	122	4	no	—
214	C₄⋊C₄	—	244	8	no	—
215	C₄²	—	61	2	no	—
216	C₄²	—	61	2	no	—
217	C₄²	—	61	2	no	—
218	C₄²	—	61	2	no	—
219	C₄²	—	122	8	no	—
220	C₄²	C₂×Dic₁₂₂	1	2	yes	—
221	C₄²	Q₈×D₆₁	1	2	yes	—
222	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
223	Q₈	—	488	16	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	61	8	no	—
228	C₂×C₄	—	61	4	no	—
229	C₂×C₄	—	61	4	no	—
230	C₂×C₄	—	122	4	no	—
231	C₂×C₄	—	244	8	no	—
232	C₂×C₄	—	244	4	no	—
233	C₂×C₄	#14519776	1	4	yes	—
234	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
240	C₈	—	61	4	no	—
241	C₈	—	61	4	no	—
242	C₈	#14519776	1	4	yes	—
243	C₈	#14519776	1	4	yes	—
244	C₂²	#127×D₆₁	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
246	C₂²	C₂×#14519776	1	1	yes	characteristic
247	C₂²	#135×D₆₁	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	8	4	no	—
250	C₄	—	61	8	no	—
251	C₄	—	122	4	no	—
252	C₄	—	488	4	no	—
253	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
254	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
255	C₄	C₂×#14519776	1	1	yes	characteristic
256	C₄	C₂×#14519776	1	1	yes	characteristic
257	C₂	#21454216	1	1	yes	characteristic
258	C₂	#495×D₆₁	1	2	yes	—
259	C₂	#21456331	1	2	yes	—
260	C₂	#21456332	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator, cosocle	C₂²×C₄×C₆₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1846	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
hypercenter	#1846	D₆₁
hypocenter	C₆₁	#33313
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
44. C₄×#92418	Q₈	semidirect
47. C₈×Dic₆₁	Q₈	semidirect
48. C₈×Dic₆₁	Q₈	semidirect
57. C₄×C₈×C₆₁	Q₈	semidirect
160. C₆₁	#33313	semidirect
186. C₄×C₈	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840166	2
C₂×#16840159	2
#21454195	2
#21445983	2
#21445982	2
#21445981	2
#21445980	2
C₄×C₈×Dic₆₁	2
C₆₁×#1846	2
#33313	61