Subgroups — #30612918

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462608	C₂	1	2	yes	maximal
2	#21461459	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#92420	C₂	1	4	yes	maximal, complemented
4	#21461377	C₂	1	1	yes	maximal, characteristic
5	#21461168	C₂	1	2	yes	maximal
6	C₈×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#21456403	C₂	1	1	yes	maximal, characteristic
8	#16840813	C₂²	1	4	yes	—
9	#16840812	—	2	2	no	—
10	#16840811	—	2	2	no	—
11	Q₈×#92420	C₂²	1	4	yes	—
12	#16840769	—	2	2	no	—
13	#16840766	C₂²	1	2	yes	—
14	#16840727	—	2	2	no	—
15	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
17	#16840625	C₂²	1	4	yes	—
18	Q₈⋊#92420	C₂²	1	8	yes	—
19	C₈×#92420	C₂²	1	2	yes	—
20	C₈×#92420	C₂²	1	2	yes	—
21	#16840317	C₂²	1	2	yes	—
22	#16840167	C₄	1	1	yes	characteristic
23	#16840112	C₄	1	2	yes	—
24	#16840000	C₄	1	4	yes	—
25	Q₁₆×Dic₆₁	C₂²	1	4	yes	—
26	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#494	C₄	1	1	yes	characteristic
28	C₂×#11758213	—	2	2	no	—
29	C₂×D₆₁⋊C₈	—	2	1	no	—
30	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
31	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
32	#14519844	—	2	4	no	—
33	C₄⋊#92420	—	2	4	no	—
34	C₄⋊#92420	—	2	2	no	—
35	C₄⋊#92420	C₂³	1	4	yes	—
36	#14519832	—	2	4	no	—
37	C₄×#92420	—	2	4	no	—
38	C₄×#92420	—	2	1	no	—
39	C₄×#92420	D₄	1	1	yes	characteristic
40	C₄×#92420	D₄	1	1	yes	characteristic
41	C₄×#92420	C₂³	1	2	yes	—
42	C₄×#92420	C₂³	1	1	yes	characteristic
43	C₄×#92420	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
45	#14519777	C₂×C₄	1	2	yes	—
46	#14519768	—	2	2	no	—
47	#14519757	—	2	2	no	—
48	#14519753	C₂×C₄	1	4	yes	—
49	Q₈×Dic₆₁	C₂³	1	4	yes	—
50	#14519742	—	2	2	no	—
51	#14519735	—	2	2	no	—
52	C₄²×D₆₁	C₂³	1	1	yes	characteristic
53	#14519711	C₂×C₄	1	4	yes	—
54	#14519701	C₂×C₄	1	8	yes	—
55	C₄×#92418	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₆₁	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
58	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
61	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
64	C₂×Dic₁₂₂	—	2	2	no	—
65	C₂×#92420	—	2	2	no	—
66	C₂×#92420	—	2	4	no	—
67	C₂×#92420	C₂×D₄	1	1	yes	characteristic
68	C₂×#92420	C₂×D₄	1	1	yes	characteristic
69	C₂×#92420	C₂×D₄	1	1	yes	characteristic
70	C₂×#92420	C₂×D₄	1	1	yes	characteristic
71	#11758213	—	4	4	no	—
72	D₆₁⋊C₈	—	2	4	no	—
73	D₆₁⋊C₈	—	2	2	no	—
74	D₆₁⋊C₈	Q₁₆	1	4	yes	complemented
75	Q₁₆×C₆₁	—	2	8	no	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
77	C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	#11758198	—	2	4	no	—
81	C₄⋊Dic₆₁	—	2	2	no	—
82	C₄⋊Dic₆₁	—	2	4	no	—
83	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
84	#11758196	—	2	4	no	—
85	C₄×Dic₆₁	—	2	4	no	—
86	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
87	C₄×Dic₆₁	#41	1	1	yes	characteristic
88	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
89	C₂×#92418	C₂²×C₄	1	2	yes	—
90	C₈×D₆₁	—	2	4	no	—
91	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₆₁	—	2	8	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₆₁	—	2	4	no	—
98	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
99	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
101	C₄×D₆₁	—	2	2	no	—
102	C₄×D₆₁	—	2	2	no	—
103	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
104	C₄×D₆₁	#135	1	2	yes	—
105	Dic₁₂₂	—	4	4	no	—
106	#92420	—	2	4	no	—
107	#92420	—	2	1	no	—
108	#92420	—	2	1	no	—
109	#92420	—	4	4	no	—
110	#92420	C₂×Q₁₆	1	2	yes	—
111	#92420	C₂×Q₁₆	1	2	yes	—
112	C₈×C₆₁	#131	1	4	yes	—
113	#92418	—	2	2	no	—
114	#17958	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×D₆₁	—	2	1	no	—
117	C₂×D₆₁	—	2	1	no	—
118	C₂×D₆₁	C₄×Q₁₆	1	2	yes	—
119	C₄×C₆₁	—	4	4	no	—
120	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
121	C₄×C₆₁	#443	1	2	yes	—
122	C₄×C₆₁	C₂×#131	1	2	yes	—
123	Dic₆₁	—	2	1	no	—
124	Dic₆₁	—	2	1	no	—
125	Dic₆₁	—	4	4	no	—
126	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
127	Dic₆₁	C₂×#135	1	1	yes	characteristic
128	#3170	—	61	1	no	—
129	#2960	—	61	2	no	—
130	#1591	—	61	4	no	—
131	C₈×Q₁₆	—	61	4	no	—
132	#1567	—	61	1	no	—
133	#1493	—	61	2	no	—
134	C₂×C₈²	—	61	1	no	—
135	C₂×C₆₁	#2931	1	1	yes	characteristic
136	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
137	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
138	D₆₁	—	2	2	no	—
139	#533	—	61	2	no	—
140	#500	—	61	1	no	—
141	#494	#6923	1	1	yes	characteristic
142	#487	—	61	2	no	—
143	#467	—	61	4	no	—
144	#446	—	61	4	no	—
145	#446	—	122	2	no	—
146	C₈×Q₈	—	61	4	no	—
147	C₈×Q₈	—	122	2	no	—
148	C₄×Q₁₆	—	61	4	no	—
149	#424	—	122	2	no	—
150	#423	—	61	2	no	—
151	#405	—	122	2	no	—
152	C₂×C₄×C₈	—	61	1	no	—
153	C₂×C₄×C₈	—	61	1	no	—
154	C₂×C₄×C₈	—	61	1	no	—
155	#335	—	61	4	no	—
156	Q₈⋊C₈	—	61	8	no	—
157	C₈²	—	61	2	no	—
158	C₈²	—	61	2	no	—
159	C₆₁	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	C₂×#6923	1	4	yes	—
161	C₂×M₁₆	—	122	2	no	—
162	C₂²×C₈	—	61	1	no	—
163	C₂²×C₈	—	61	1	no	—
164	C₂²×C₈	—	61	2	no	—
165	C₂²×C₈	—	122	1	no	—
166	C₄⋊Q₈	—	61	2	no	—
167	C₄⋊Q₈	C₂×#6923	1	2	yes	—
168	#125	—	122	2	no	—
169	C₂²⋊Q₈	—	61	4	no	—
170	C₂²⋊Q₈	—	122	2	no	—
171	C₄×Q₈	—	61	4	no	—
172	C₄×Q₈	—	122	2	no	—
173	#117	—	122	2	no	—
174	C₂×C₄²	—	61	1	no	—
175	#107	—	61	4	no	—
176	C₄⋊C₈	—	61	4	no	—
177	C₄⋊C₈	—	122	2	no	—
178	C₄⋊C₈	—	122	4	no	—
179	C₄⋊C₈	—	122	4	no	—
180	#103	—	61	8	no	—
181	#97	—	122	4	no	—
182	C₄×C₈	—	61	1	no	—
183	C₄×C₈	—	61	1	no	—
184	C₄×C₈	—	61	1	no	—
185	C₄×C₈	—	61	1	no	—
186	C₄×C₈	—	61	2	no	—
187	C₄×C₈	—	61	2	no	—
188	C₄×C₈	—	61	1	no	—
189	C₄×C₈	—	122	1	no	—
190	C₄×C₈	—	122	4	no	—
191	C₄×C₈	C₂×#6923	1	1	yes	characteristic
192	C₂×Q₈	—	122	2	no	—
193	C₂×Q₈	C₂²×#6923	1	4	yes	—
194	C₂²×C₄	—	61	1	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	61	1	no	—
197	Q₁₆	—	2	8	no	—
198	M₁₆	—	244	4	no	—
199	C₂×C₈	—	61	1	no	—
200	C₂×C₈	—	61	1	no	—
201	C₂×C₈	—	61	4	no	—
202	C₂×C₈	—	61	1	no	—
203	C₂×C₈	—	61	1	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	—	122	2	no	—
206	C₂×C₈	—	122	2	no	—
207	C₂×C₈	—	122	4	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	—	122	4	no	—
210	C₂×C₈	C₂²×#6923	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	61	4	no	—
214	C₄⋊C₄	—	122	4	no	—
215	C₄⋊C₄	—	122	4	no	—
216	C₄⋊C₄	—	122	2	no	—
217	C₂²⋊C₄	—	122	4	no	—
218	C₄²	—	61	1	no	—
219	C₄²	—	61	1	no	—
220	C₄²	—	61	1	no	—
221	C₄²	—	122	4	no	—
222	C₄²	C₂²×#6923	1	1	yes	characteristic
223	C₂³	—	61	1	no	—
224	Q₈	—	2	8	no	—
225	Q₈	—	244	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	61	1	no	—
228	C₂×C₄	—	61	2	no	—
229	C₂×C₄	—	61	2	no	—
230	C₂×C₄	—	61	1	no	—
231	C₂×C₄	—	61	1	no	—
232	C₂×C₄	—	122	2	no	—
233	C₂×C₄	—	122	2	no	—
234	C₂×C₄	—	122	4	no	—
235	C₂×C₄	D₄×#6923	1	1	yes	characteristic
236	C₂×C₄	D₄×#6923	1	1	yes	characteristic
237	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
238	C₈	—	61	2	no	—
239	C₈	—	61	2	no	—
240	C₈	—	122	1	no	—
241	C₈	—	122	1	no	—
242	C₈	—	122	4	no	—
243	C₈	—	122	2	no	—
244	C₈	—	244	4	no	—
245	C₈	#14519864	1	4	yes	—
246	C₂²	—	61	2	no	—
247	C₂²	—	122	1	no	—
248	C₂²	—	122	1	no	—
249	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
250	C₄	—	4	4	no	—
251	C₄	—	61	1	no	—
252	C₄	—	61	1	no	—
253	C₄	—	122	1	no	—
254	C₄	—	122	1	no	—
255	C₄	—	244	4	no	—
256	C₄	Q₁₆×#6923	1	2	yes	—
257	C₄	#16840806	1	2	yes	—
258	C₄	C₂×#14519864	1	2	yes	—
259	C₂	—	122	2	no	—
260	C₂	#21462541	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×#6923	1	1	yes	characteristic
262	C₂	C₂×#16840806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#494	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#494	#6923
hypocenter	C₆₁	#17958
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #21461459	C₂	semidirect
3. Q₁₆×#92420	C₂	semidirect
74. D₆₁⋊C₈	Q₁₆	semidirect
159. C₆₁	#17958	semidirect

Maximal subgroups

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

subgroup	index
#21462608	2
#21461459	2
Q₁₆×#92420	2
#21461377	2
#21461168	2
C₈×D₆₁⋊C₈	2
#21456403	2
#17958	61