Subgroups — #29833511

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454751	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	4	yes	maximal, complemented
3	#21447725	C₂	1	4	yes	maximal, complemented
4	#21447579	C₂	1	1	yes	maximal, characteristic
5	#21447578	C₂	1	1	yes	maximal, characteristic
6	#21447574	C₂	1	1	yes	maximal, characteristic
7	#21447415	C₂	1	1	yes	maximal, characteristic
8	C₂×#16839537	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic
10	#16840315	C₄	1	1	yes	characteristic
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂²×#11758185	C₂²	1	1	yes	characteristic
13	#16839890	C₄	1	4	yes	—
14	C₈×Dic₁₂₂	C₂²	1	4	yes	—
15	#16839790	C₄	1	1	yes	characteristic
16	#16839789	C₄	1	1	yes	characteristic
17	#16839785	C₄	1	1	yes	characteristic
18	#16839717	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₄⋊#11758185	C₂²	1	4	yes	—
21	#16839580	C₂²	1	4	yes	—
22	#16839580	C₂²	1	4	yes	—
23	#16839537	C₂²	1	2	yes	—
24	#16839537	C₂²	1	2	yes	—
25	#16839537	C₂²	1	2	yes	—
26	C₁₆×Dic₆₁	C₂²	1	2	yes	—
27	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#405	C₂²	1	1	yes	characteristic
30	C₆₁×#346	C₂²	1	2	yes	—
31	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
32	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
33	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
34	#14519776	C₈	1	2	yes	—
35	#14519768	C₈	1	2	yes	—
36	#14519754	C₈	1	4	yes	—
37	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
38	#14519736	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	#14519717	—	2	2	no	—
41	C₂×#11758185	—	2	2	no	—
42	C₂×#11758185	C₂³	1	2	yes	—
43	C₂×#11758185	C₂³	1	2	yes	—
44	C₄⋊#92418	C₂×C₄	1	4	yes	—
45	#14519706	C₂×C₄	1	4	yes	—
46	#14519706	C₂×C₄	1	4	yes	—
47	#14519687	C₂×C₄	1	2	yes	—
48	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
49	C₈×Dic₆₁	C₂³	1	2	yes	—
50	C₈×Dic₆₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₆₁	—	2	2	no	—
55	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
56	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
59	#97×C₆₁	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₆₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
63	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
64	M₁₆×C₆₁	—	2	2	no	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₆₁	#41	1	2	yes	—
69	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
71	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
72	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
74	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
75	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
76	#11758196	C₂×C₈	1	4	yes	—
77	#11758196	C₂×C₈	1	4	yes	—
78	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
80	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
81	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
82	#11758194	—	2	2	no	—
83	C₂×#92418	—	2	2	no	—
84	C₂×#92418	C₂²×C₄	1	2	yes	—
85	C₂×#92418	C₂²×C₄	1	2	yes	—
86	C₁₆×C₆₁	—	2	2	no	—
87	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
88	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
89	#11758185	—	2	2	no	—
90	#11758185	—	2	2	no	—
91	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#131	1	2	yes	—
96	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₆₁	—	4	2	no	—
98	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
99	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
100	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
101	Dic₁₂₂	—	4	4	no	—
102	C₈×C₆₁	—	2	2	no	—
103	C₈×C₆₁	C₄×Q₈	1	2	yes	—
104	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
105	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
106	#92418	—	2	2	no	—
107	#92418	—	2	2	no	—
108	#33475	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	2	no	—
110	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
111	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
112	C₄×C₆₁	—	2	4	no	—
113	C₄×C₆₁	C₈×Q₈	1	2	yes	—
114	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
116	Dic₆₁	—	2	2	no	—
117	Dic₆₁	—	2	4	no	—
118	Dic₆₁	—	2	2	no	—
119	#2960	—	61	1	no	—
120	#2183	—	61	4	no	—
121	#2183	—	61	4	no	—
122	#2151	—	61	1	no	—
123	#2151	—	61	1	no	—
124	#2150	—	61	1	no	—
125	#2108	—	61	1	no	—
126	#2108	D₆₁	1	1	yes	characteristic
127	C₂×#346	—	61	1	no	—
128	C₂×C₆₁	—	2	2	no	—
129	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₆₁	#3407	1	2	yes	—
131	#533	—	61	1	no	—
132	C₂×M₃₂	—	61	1	no	—
133	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
134	C₂²×C₁₆	—	61	1	no	—
135	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
136	#446	—	61	4	no	—
137	C₈×Q₈	—	61	4	no	—
138	#424	—	61	1	no	—
139	#424	—	61	1	no	—
140	#423	—	61	1	no	—
141	#405	—	61	1	no	—
142	#405	C₂×D₆₁	1	1	yes	characteristic
143	C₂×C₄×C₈	—	61	1	no	—
144	C₄⋊C₁₆	—	61	4	no	—
145	C₄⋊C₁₆	—	61	4	no	—
146	C₄⋊C₁₆	—	61	4	no	—
147	#346	—	61	2	no	—
148	#346	—	61	2	no	—
149	#346	—	61	2	no	—
150	#346	C₂×D₆₁	1	2	yes	—
151	C₄×C₁₆	—	61	2	no	—
152	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
153	C₆₁	#33475	1	1	yes	characteristic, complemented, Sylow
154	C₂×M₁₆	—	61	1	no	—
155	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
156	C₂²×C₈	—	61	1	no	—
157	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
158	C₄⋊Q₈	—	61	2	no	—
159	#125	—	61	2	no	—
160	C₂²⋊Q₈	—	61	4	no	—
161	C₄×Q₈	—	61	4	no	—
162	#117	—	61	1	no	—
163	#117	C₄×D₆₁	1	1	yes	characteristic
164	C₂×C₄²	—	61	1	no	—
165	M₃₂	—	2	2	no	—
166	M₃₂	—	122	2	no	—
167	C₂×C₁₆	—	61	2	no	—
168	C₂×C₁₆	—	61	2	no	—
169	C₂×C₁₆	—	122	2	no	—
170	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
171	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
172	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
173	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
174	C₄⋊C₈	—	61	4	no	—
175	C₄⋊C₈	—	61	4	no	—
176	C₄⋊C₈	—	61	4	no	—
177	#97	—	61	2	no	—
178	#97	C₄×D₆₁	1	2	yes	—
179	C₄×C₈	—	61	2	no	—
180	C₄×C₈	—	61	2	no	—
181	C₄×C₈	—	61	2	no	—
182	C₄×C₈	C₂²×D₆₁	1	2	yes	—
183	C₂×Q₈	—	61	4	no	—
184	C₂²×C₄	—	61	2	no	—
185	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	122	2	no	—
188	C₂×C₈	—	61	2	no	—
189	C₂×C₈	—	61	2	no	—
190	C₂×C₈	—	122	2	no	—
191	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
194	C₂×C₈	#11758222	1	2	yes	—
195	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
196	C₄⋊C₄	—	61	4	no	—
197	C₄⋊C₄	—	61	4	no	—
198	C₄⋊C₄	—	61	2	no	—
199	C₄⋊C₄	—	61	4	no	—
200	C₄⋊C₄	C₈×D₆₁	1	2	yes	—
201	C₂²⋊C₄	—	61	2	no	—
202	C₂²⋊C₄	C₈×D₆₁	1	2	yes	—
203	C₄²	—	61	2	no	—
204	C₄²	—	61	2	no	—
205	C₄²	—	61	1	no	—
206	C₄²	—	61	1	no	—
207	C₄²	C₂×C₄×D₆₁	1	2	yes	—
208	C₁₆	—	2	2	no	—
209	C₁₆	—	122	2	no	—
210	C₁₆	—	122	2	no	—
211	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
212	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
213	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
214	Q₈	—	244	4	no	—
215	C₂×C₄	—	61	2	no	—
216	C₂×C₄	—	61	2	no	—
217	C₂×C₄	—	61	4	no	—
218	C₂×C₄	—	244	2	no	—
219	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₆₁	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
222	C₂×C₄	#14519783	1	2	yes	—
223	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	122	2	no	—
226	C₈	—	122	2	no	—
227	C₈	C₄×Dic₁₂₂	1	2	yes	—
228	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
229	C₈	C₂×#11758222	1	1	yes	characteristic
230	C₂²	—	2	2	no	—
231	C₂²	C₈×Dic₁₂₂	1	2	yes	—
232	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	122	2	no	—
235	C₄	—	122	2	no	—
236	C₄	—	122	4	no	—
237	C₄	C₈×Dic₁₂₂	1	2	yes	—
238	C₄	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
239	C₄	C₂×#14519783	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×C₈×Dic₁₂₂	1	1	yes	characteristic
242	C₂	#21458920	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2108	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2108	D₆₁
hypocenter	C₆₁	#33475
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447725	C₂	semidirect
55. C₂×C₁₆×C₆₁	Q₈	semidirect
153. C₆₁	#33475	semidirect
170. C₂×C₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21454751	2
#21447727	2
#21447725	2
#21447579	2
#21447578	2
#21447574	2
#21447415	2
C₂×#16839537	2
C₆₁×#2108	2
#33475	61