Subgroups — #29832586

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840165	C₂	1	2	yes	maximal
3	#21454215	C₂	1	2	yes	maximal
4	#21445977	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519777	C₂²	1	2	yes	—
8	C₂×#14519775	—	2	4	no	—
9	#16840167	—	2	4	no	—
10	#16840165	—	2	8	no	—
11	C₂×#14519711	C₂²	1	4	yes	—
12	C₂×#14519710	C₂²	1	8	yes	—
13	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₂²	1	4	yes	—
15	#16839631	—	2	4	no	—
16	#16839631	C₂²	1	8	yes	—
17	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#500	—	2	4	no	—
20	C₂×#107×C₆₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₂₂	—	2	2	no	—
23	C₂²×#92418	C₂³	1	2	yes	—
24	#14519777	—	2	4	no	—
25	#14519777	—	2	4	no	—
26	#14519775	—	4	16	no	—
27	C₂×C₄⋊Dic₆₁	—	2	2	no	—
28	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
29	C₂×#11758196	—	2	16	no	—
30	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
32	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
33	#14519711	—	2	8	no	—
34	#14519710	—	2	16	no	—
35	C₄×#92418	D₄	1	4	yes	—
36	C₈×Dic₆₁	Q₈	1	16	yes	complemented
37	C₂²×Q₈×C₆₁	—	2	2	no	—
38	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₆₁	—	2	4	no	—
40	C₄⋊Q₈×C₆₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	#107×C₆₁	—	2	8	no	—
45	C₄×C₈×C₆₁	D₄	1	4	yes	—
46	C₂×Q₈×C₆₁	—	4	4	no	—
47	C₂×Q₈×C₆₁	—	4	8	no	—
48	C₂²×C₄×C₆₁	—	2	4	no	—
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₆₁	—	2	4	no	—
53	C₂²×Dic₆₁	C₂×D₄	1	4	yes	—
54	C₂×Dic₁₂₂	—	4	4	no	—
55	C₂×Dic₁₂₂	—	4	8	no	—
56	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₆₁	C₂×Q₈	1	8	yes	—
58	C₄⋊C₄×C₆₁	—	2	4	no	—
59	C₄⋊C₄×C₆₁	—	2	4	no	—
60	C₄⋊C₄×C₆₁	—	4	4	no	—
61	C₄²×C₆₁	C₂×D₄	1	2	yes	—
62	C₄²×C₆₁	C₂×D₄	1	2	yes	—
63	C₄⋊Dic₆₁	—	2	4	no	—
64	C₄⋊Dic₆₁	—	2	4	no	—
65	C₄⋊Dic₆₁	—	4	4	no	—
66	#11758196	—	4	32	no	—
67	C₄×Dic₆₁	D₈	1	8	yes	—
68	C₄×Dic₆₁	Q₁₆	1	8	yes	—
69	C₄×Dic₆₁	C₂×Q₈	1	8	yes	—
70	C₂×#92418	C₂×D₄	1	4	yes	—
71	C₂×#92418	C₂×Q₈	1	8	yes	—
72	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
73	Q₈×C₆₁	—	8	8	no	—
74	C₂×C₄×C₆₁	—	4	4	no	—
75	C₂×C₄×C₆₁	—	4	8	no	—
76	C₂×C₄×C₆₁	#127	1	2	yes	—
77	C₂×C₄×C₆₁	#127	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
80	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²×Q₈	1	4	yes	—
83	C₂×Dic₆₁	—	4	4	no	—
84	C₂×Dic₆₁	—	4	8	no	—
85	C₂×Dic₆₁	C₄⋊Q₈	1	8	yes	—
86	C₂×Dic₆₁	C₂×D₈	1	8	yes	—
87	C₂×Dic₆₁	C₂×Q₁₆	1	8	yes	—
88	Dic₁₂₂	—	8	8	no	—
89	C₈×C₆₁	C₄⋊Q₈	1	8	yes	complemented
90	#92418	C₄⋊Q₈	1	8	yes	complemented
91	#33312	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
93	C₂²×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
94	C₂²×C₆₁	C₂²×D₈	1	2	yes	—
95	C₂²×C₆₁	C₂²×Q₁₆	1	2	yes	—
96	C₄×C₆₁	—	8	4	no	—
97	C₄×C₆₁	#493	1	4	yes	—
98	C₄×C₆₁	#494	1	4	yes	—
99	C₄×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
100	C₄×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
101	Dic₆₁	—	8	4	no	—
102	Dic₆₁	#500	1	16	yes	complemented
103	C₂×#500	—	61	1	no	—
104	C₂×#500	—	61	2	no	—
105	C₂×#500	D₆₁	1	1	yes	characteristic
106	#2868	—	61	2	no	—
107	#1845	—	61	8	no	—
108	C₄²×C₈	—	61	1	no	—
109	C₂×C₆₁	#2868	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
111	C₂×C₆₁	C₂×#494	1	1	yes	characteristic
112	C₂×C₆₁	C₂×#500	1	4	yes	—
113	C₂×C₄⋊Q₈	—	61	2	no	—
114	C₂×C₄⋊Q₈	—	122	4	no	—
115	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
116	#500	—	2	4	no	—
117	#500	—	122	8	no	—
118	#500	—	122	4	no	—
119	C₂×#107	—	61	4	no	—
120	C₂×#107	—	61	8	no	—
121	C₂×#107	C₂×D₆₁	1	4	yes	—
122	C₂×C₄×C₈	—	61	4	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
125	#384	—	61	8	no	—
126	#384	—	122	4	no	—
127	C₄³	—	61	1	no	—
128	C₆₁	#33312	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	2	no	—
130	C₂²×Q₈	—	122	2	no	—
131	C₂²×C₈	—	61	2	no	—
132	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	2	4	no	—
135	C₄⋊Q₈	—	122	4	no	—
136	C₄⋊Q₈	—	122	4	no	—
137	C₄⋊Q₈	—	244	16	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	61	4	no	—
140	C₂×C₄⋊C₄	—	122	16	no	—
141	C₂×C₄⋊C₄	—	122	2	no	—
142	C₂×C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
143	C₂×C₄²	—	61	2	no	—
144	C₂×C₄²	—	61	2	no	—
145	C₂×C₄²	—	61	2	no	—
146	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
147	#107	—	2	8	no	—
148	#107	—	122	16	no	—
149	#107	—	122	8	no	—
150	C₄×C₈	—	61	16	no	—
151	C₄×C₈	—	61	4	no	—
152	C₄×C₈	C₂²⋊D₆₁	1	4	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	8	no	—
155	C₂×Q₈	—	244	4	no	—
156	C₂×Q₈	—	244	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	61	4	no	—
159	C₂²×C₄	—	122	4	no	—
160	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
162	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
163	C₂×C₈	—	61	4	no	—
164	C₂×C₈	—	61	8	no	—
165	C₂×C₈	Q₈×D₆₁	1	8	yes	—
166	C₂×C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	122	4	no	—
171	C₄⋊C₄	—	122	4	no	—
172	C₄⋊C₄	—	244	32	no	—
173	C₄⋊C₄	—	244	4	no	—
174	C₄²	—	61	8	no	—
175	C₄²	—	61	8	no	—
176	C₄²	—	61	8	no	—
177	C₄²	D₄×D₆₁	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
179	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	Q₈	—	488	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	61	8	no	—
185	C₂×C₄	—	61	8	no	—
186	C₂×C₄	—	61	8	no	—
187	C₂×C₄	—	244	8	no	—
188	C₂×C₄	—	244	4	no	—
189	C₂×C₄	#14519773	1	2	yes	—
190	C₂×C₄	#14519773	1	2	yes	—
191	C₂×C₄	D₈×D₆₁	1	4	yes	—
192	C₂×C₄	Q₁₆×D₆₁	1	4	yes	—
193	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
195	C₂×C₄	C₂×Q₈×D₆₁	1	4	yes	—
196	C₈	—	61	8	no	—
197	C₈	#14519775	1	8	yes	complemented
198	C₂²	C₂×#14519773	1	1	yes	characteristic
199	C₂²	C₄⋊Q₈×D₆₁	1	2	yes	—
200	C₂²	C₂×D₈×D₆₁	1	2	yes	—
201	C₂²	C₂×Q₁₆×D₆₁	1	2	yes	—
202	C₄	—	8	4	no	—
203	C₄	—	61	16	no	—
204	C₄	—	488	4	no	—
205	C₄	#16840137	1	4	yes	—
206	C₄	#16840140	1	4	yes	—
207	C₄	C₄⋊Q₈×D₆₁	1	2	yes	—
208	C₄	C₂×#14519775	1	2	yes	—
209	C₂	#21454215	1	1	yes	characteristic
210	C₂	C₂×#16840137	1	1	yes	characteristic
211	C₂	C₂×#16840140	1	1	yes	characteristic
212	C₂	#500×D₆₁	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator, cosocle	C₂²×C₄×C₆₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#33312
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
36. C₈×Dic₆₁	Q₈	semidirect
89. C₈×C₆₁	C₄⋊Q₈	semidirect
90. #92418	C₄⋊Q₈	semidirect
102. Dic₆₁	#500	semidirect
128. C₆₁	#33312	semidirect
197. C₈	#14519775	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
C₂×#16840165	2
#21454215	2
#21445977	2
C₄×C₈×Dic₆₁	2
C₂×C₆₁×#500	2
#33312	61