Subgroups — #29573752

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445970	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839514	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519706	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
13	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
14	#16839515	—	2	2	no	—
15	#16839514	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#335	—	2	2	no	—
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
25	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	—	2	2	no	—
28	#14519706	—	4	8	no	—
29	C₄×#92418	D₄	1	2	yes	—
30	C₄×#92418	Q₈	1	2	yes	—
31	C₈×Dic₆₁	—	2	2	no	—
32	C₈×Dic₆₁	—	2	2	no	—
33	C₈×Dic₆₁	—	2	2	no	—
34	C₈×Dic₆₁	C₈	1	4	yes	complemented
35	C₂²×C₈×C₆₁	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	—	2	2	no	—
40	C₄⋊C₈×C₆₁	—	2	2	no	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₄×C₈×C₆₁	Q₈	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	—	4	4	no	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
56	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₆₁	—	2	1	no	—
61	C₄×Dic₆₁	—	2	1	no	—
62	C₄×Dic₆₁	—	2	4	no	—
63	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
64	C₄×Dic₆₁	M₁₆	1	2	yes	—
65	C₄×Dic₆₁	D₈	1	4	yes	—
66	C₄×Dic₆₁	Q₁₆	1	4	yes	—
67	C₂×#92418	—	2	1	no	—
68	C₂×#92418	—	2	1	no	—
69	C₂×#92418	—	4	2	no	—
70	C₂×#92418	—	4	4	no	—
71	C₂×#92418	C₄⋊C₄	1	2	yes	—
72	C₂×#92418	C₄⋊C₄	1	2	yes	—
73	C₂×#92418	C₂×C₈	1	2	yes	—
74	C₂×#92418	M₁₆	1	2	yes	—
75	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
89	C₂×Dic₆₁	—	2	2	no	—
90	C₂×Dic₆₁	—	2	2	no	—
91	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
92	C₂×Dic₆₁	#107	1	4	yes	—
93	C₂×Dic₆₁	#107	1	4	yes	—
94	C₂×Dic₆₁	#108	1	4	yes	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	—	8	2	no	—
97	C₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
98	#92418	—	2	2	no	—
99	#92418	—	8	2	no	—
100	#92418	C₄⋊C₈	1	4	yes	complemented
101	#8806	—	61	1	no	maximal, Sylow
102	C₂²×C₆₁	#384	1	1	yes	characteristic
103	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₆₁	#423	1	1	yes	characteristic
105	C₂²×C₆₁	C₂×#107	1	2	yes	—
106	C₂²×C₆₁	C₂×#108	1	2	yes	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₆₁	#423	1	1	yes	characteristic
112	C₄×C₆₁	#423	1	1	yes	characteristic
113	C₄×C₆₁	#493	1	1	yes	characteristic
114	C₄×C₆₁	#494	1	1	yes	characteristic
115	C₄×C₆₁	#500	1	2	yes	—
116	Dic₆₁	—	2	4	no	—
117	Dic₆₁	#335	1	8	yes	complemented
118	#1842	—	61	2	no	—
119	C₄²×C₈	—	61	1	no	—
120	C₂×#335	—	61	2	no	—
121	C₂×#335	—	61	1	no	—
122	C₂×#335	D₆₁	1	1	yes	characteristic
123	C₂×C₆₁	C₂×#335	1	2	yes	—
124	C₂×C₆₁	#1567	1	2	yes	—
125	C₂×C₆₁	#1842	1	1	yes	characteristic
126	C₂×C₆₁	#1845	1	1	yes	characteristic
127	C₂×C₆₁	#1848	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	61	2	no	—
129	C₂×C₄⋊C₈	—	122	4	no	—
130	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
131	C₂×C₄×C₈	—	61	1	no	—
132	C₂×C₄×C₈	—	61	1	no	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
137	C₄³	—	61	1	no	—
138	#335	—	2	2	no	—
139	#335	—	122	4	no	—
140	#335	—	122	2	no	—
141	C₆₁	#8806	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	1	no	—
145	C₂²×C₈	—	122	2	no	—
146	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
148	C₂×C₄²	—	61	1	no	—
149	C₂×C₄²	—	61	2	no	—
150	C₂×C₄²	—	61	1	no	—
151	C₂×C₄²	—	61	2	no	—
152	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	122	2	no	—
156	C₄⋊C₈	—	122	2	no	—
157	C₄⋊C₈	—	244	8	no	—
158	C₄×C₈	—	61	4	no	—
159	C₄×C₈	—	61	2	no	—
160	C₄×C₈	—	61	2	no	—
161	C₄×C₈	—	122	2	no	—
162	C₄×C₈	—	122	2	no	—
163	C₄×C₈	—	122	2	no	—
164	C₄×C₈	Dic₁₂₂	1	2	yes	—
165	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
166	C₂²×C₄	—	61	2	no	—
167	C₂²×C₄	—	61	2	no	—
168	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
170	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	61	2	no	—
176	C₂×C₈	—	61	2	no	—
177	C₂×C₈	—	61	2	no	—
178	C₂×C₈	—	61	2	no	—
179	C₂×C₈	—	122	1	no	—
180	C₂×C₈	—	122	1	no	—
181	C₂×C₈	—	244	2	no	—
182	C₂×C₈	—	244	4	no	—
183	C₂×C₈	C₈×D₆₁	1	2	yes	—
184	C₂×C₈	#11758189	1	2	yes	—
185	C₂×C₈	#11758196	1	2	yes	—
186	C₂×C₈	#11758196	1	2	yes	—
187	C₄²	—	61	2	no	—
188	C₄²	—	61	2	no	—
189	C₄²	—	61	4	no	—
190	C₄²	—	61	4	no	—
191	C₄²	—	122	4	no	—
192	C₄²	—	122	1	no	—
193	C₄²	—	122	1	no	—
194	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
195	C₄²	D₄×D₆₁	1	1	yes	characteristic
196	C₄²	Q₈×D₆₁	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
198	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	61	4	no	—
201	C₂×C₄	—	61	4	no	—
202	C₂×C₄	—	61	4	no	—
203	C₂×C₄	—	61	4	no	—
204	C₂×C₄	—	122	2	no	—
205	C₂×C₄	—	122	2	no	—
206	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
207	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
208	C₂×C₄	#14519773	1	1	yes	characteristic
209	C₂×C₄	#14519775	1	1	yes	characteristic
210	C₂×C₄	#14519776	1	1	yes	characteristic
211	C₂×C₄	#14519776	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
213	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
214	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
215	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
216	C₂×C₄	D₈×D₆₁	1	2	yes	—
217	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
218	C₈	—	2	2	no	—
219	C₈	—	8	2	no	—
220	C₈	—	61	4	no	—
221	C₈	—	122	2	no	—
222	C₈	—	488	2	no	—
223	C₈	#14519706	1	4	yes	complemented
224	C₂²	#16839631	1	1	yes	characteristic
225	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
226	C₂²	#16839784	1	1	yes	characteristic
227	C₂²	#107×D₆₁	1	2	yes	—
228	C₂²	#108×D₆₁	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	61	8	no	—
232	C₄	—	122	4	no	—
233	C₄	C₄⋊C₈×D₆₁	1	1	yes	characteristic
234	C₄	C₂×#14519706	1	1	yes	characteristic
235	C₄	#16839784	1	1	yes	characteristic
236	C₄	#16839785	1	1	yes	characteristic
237	C₄	#16840137	1	1	yes	characteristic
238	C₄	#16840140	1	1	yes	characteristic
239	C₄	#16840166	1	2	yes	—
240	C₂	#335×D₆₁	1	2	yes	—
241	C₂	#21444576	1	2	yes	—
242	C₂	#21445970	1	1	yes	characteristic
243	C₂	#21445977	1	1	yes	characteristic
244	C₂	#21445989	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×C₆₁×#335	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#8806
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₆₁	C₈	semidirect
97. C₈×C₆₁	C₄⋊C₈	semidirect
100. #92418	C₄⋊C₈	semidirect
117. Dic₆₁	#335	semidirect
141. C₆₁	#8806	semidirect
223. C₈	#14519706	semidirect

Maximal subgroups

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

subgroup	index
#21445970	2
C₄×C₈×Dic₆₁	2
C₂×#16839515	2
C₂×#16839514	2
C₂×C₆₁×#335	2
#8806	61