Subgroups — #29833391

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454687	C₂	1	2	yes	maximal
2	#21447690	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₆₁	C₂	1	4	yes	maximal, complemented
4	#21447603	C₂	1	1	yes	maximal, characteristic
5	#21447539	C₂	1	2	yes	maximal
6	#21447415	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
8	#16840240	—	2	2	no	—
9	C₂×#14519717	C₂²	1	1	yes	characteristic
10	#16839869	—	2	4	no	—
11	Q₁₆×Dic₆₁	C₂²	1	8	yes	—
12	#16839800	C₂²	1	2	yes	—
13	#16839771	—	2	2	no	—
14	#16839717	C₂²	1	1	yes	characteristic
15	#16839590	C₂²	1	2	yes	—
16	#16839588	C₂²	1	2	yes	—
17	#16839566	C₂²	1	8	yes	—
18	#16839537	C₂²	1	2	yes	—
19	C₁₆×Dic₆₁	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₆₁	C₄	1	8	yes	complemented
22	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
23	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#11758194	D₄	1	1	yes	characteristic
26	Q₈×Dic₆₁	—	2	8	no	—
27	Q₈×Dic₆₁	—	4	4	no	—
28	#14519736	—	2	2	no	—
29	#14519736	—	4	4	no	—
30	#14519736	D₄	1	1	yes	characteristic
31	#14519717	—	2	2	no	—
32	C₂×#11758185	C₂³	1	2	yes	—
33	#14519711	C₂³	1	4	yes	—
34	#14519709	—	2	2	no	—
35	#14519701	—	2	8	no	—
36	#14519687	D₄	1	2	yes	—
37	C₈×Dic₆₁	C₂³	1	2	yes	—
38	C₂²×Q₈×C₆₁	—	2	2	no	—
39	C₂×Q₁₆×C₆₁	—	2	2	no	—
40	C₂×Q₁₆×C₆₁	—	2	4	no	—
41	C₂×Q₁₆×C₆₁	C₂×C₄	1	8	yes	—
42	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	Q₃₂×C₆₁	C₂×C₄	1	8	yes	—
45	Q₃₂×C₆₁	C₂×C₄	1	8	yes	—
46	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₆₁	—	2	8	no	—
50	C₂×Q₈×C₆₁	—	2	4	no	—
51	C₂×Q₈×C₆₁	—	4	2	no	—
52	C₂×Q₈×C₆₁	—	4	4	no	—
53	C₂²×C₄×C₆₁	—	4	2	no	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₆₁	—	2	8	no	—
56	Q₁₆×C₆₁	—	2	4	no	—
57	Q₁₆×C₆₁	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	#41	1	2	yes	—
61	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₆₁	—	4	2	no	—
63	C₂²⋊Dic₆₁	D₈	1	2	yes	—
64	C₄⋊Dic₆₁	—	2	4	no	—
65	C₄⋊Dic₆₁	—	4	8	no	—
66	C₄⋊Dic₆₁	D₈	1	2	yes	—
67	C₄×Dic₆₁	—	4	2	no	—
68	C₄×Dic₆₁	—	4	8	no	—
69	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
70	#11758194	—	2	2	no	—
71	C₂×#92418	C₂×D₄	1	2	yes	—
72	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
73	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
74	#11758185	—	2	2	no	—
75	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₆₁	—	2	8	no	—
77	Q₈×C₆₁	—	4	4	no	—
78	Q₈×C₆₁	—	4	8	no	—
79	C₂×C₄×C₆₁	—	4	4	no	—
80	C₂×C₄×C₆₁	—	4	8	no	—
81	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#135	1	2	yes	—
84	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₆₁	—	4	4	no	—
86	C₂×Dic₆₁	C₂×D₈	1	4	yes	—
87	C₈×C₆₁	C₄×D₄	1	2	yes	—
88	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
90	#92418	—	2	2	no	—
91	#33458	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	2	no	—
93	C₂²×C₆₁	C₄×D₈	1	2	yes	—
94	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₆₁	—	4	8	no	—
96	C₄×C₆₁	C₄×D₈	1	2	yes	—
97	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
99	Dic₆₁	—	2	4	no	—
100	Dic₆₁	—	8	4	no	—
101	C₂²×Q₃₂	D₆₁	1	1	yes	characteristic
102	#2950	—	61	2	no	—
103	#2176	—	61	4	no	—
104	C₄×Q₃₂	—	61	4	no	—
105	#2158	—	61	1	no	—
106	#2141	—	61	2	no	—
107	#2108	—	61	1	no	—
108	C₂×C₆₁	—	2	2	no	—
109	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₆₁	#3417	1	2	yes	—
111	C₂²×Q₁₆	C₂×D₆₁	1	2	yes	—
112	#519	—	122	2	no	—
113	C₂×Q₃₂	Dic₆₁	1	8	yes	complemented
114	C₂×Q₃₂	C₂×D₆₁	1	2	yes	—
115	C₂×Q₃₂	C₂×D₆₁	1	2	yes	—
116	C₂×M₃₂	—	61	1	no	—
117	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
118	#441	—	122	4	no	—
119	C₄×Q₁₆	—	61	8	no	—
120	#427	—	61	2	no	—
121	#419	—	122	2	no	—
122	#405	—	61	1	no	—
123	#367	—	61	2	no	—
124	#366	—	61	2	no	—
125	#358	—	61	8	no	—
126	#346	—	61	2	no	—
127	C₄×C₁₆	—	61	2	no	—
128	C₆₁	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×Dic₆₁	1	8	yes	—
133	C₂×Q₁₆	C₂²×D₆₁	1	4	yes	—
134	C₂×M₁₆	—	61	1	no	—
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	C₄×Q₈	—	122	8	no	—
137	C₄×Q₈	—	244	4	no	—
138	#117	—	61	1	no	—
139	#117	—	122	2	no	—
140	#117	—	244	4	no	—
141	Q₃₂	C₂×Dic₆₁	1	8	yes	—
142	Q₃₂	C₂×Dic₆₁	1	8	yes	—
143	M₃₂	—	122	2	no	—
144	C₂×C₁₆	—	61	2	no	—
145	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
146	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
148	#107	—	61	4	no	—
149	#106	—	122	2	no	—
150	#103	—	122	8	no	—
151	#97	—	61	2	no	—
152	C₄×C₈	—	61	2	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
159	Q₁₆	—	2	4	no	—
160	Q₁₆	—	2	8	no	—
161	Q₁₆	C₂²×Dic₆₁	1	8	yes	—
162	M₁₆	—	122	2	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
165	C₂×C₈	#11758224	1	2	yes	—
166	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
167	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
168	C₄⋊C₄	—	61	2	no	—
169	C₄⋊C₄	—	122	4	no	—
170	C₄⋊C₄	—	244	8	no	—
171	C₂²⋊C₄	—	61	2	no	—
172	C₂²⋊C₄	—	244	2	no	—
173	C₄²	—	61	2	no	—
174	C₄²	—	244	8	no	—
175	C₄²	—	244	2	no	—
176	C₁₆	—	122	2	no	—
177	C₁₆	C₂²×Dic₆₁	1	2	yes	—
178	C₁₆	C₂²×Dic₆₁	1	2	yes	—
179	C₂³	D₈×D₆₁	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	61	4	no	—
186	C₂×C₄	—	244	4	no	—
187	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
188	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
189	C₂×C₄	#14519796	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
191	C₈	—	122	2	no	—
192	C₈	D₄×Dic₆₁	1	2	yes	—
193	C₈	C₂³×Dic₆₁	1	1	yes	characteristic
194	C₈	C₂×#11758224	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₆₁	1	2	yes	—
197	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	122	4	no	—
200	C₄	—	488	4	no	—
201	C₄	D₈×Dic₆₁	1	2	yes	—
202	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
203	C₄	C₂×#14519796	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₆₁	1	1	yes	characteristic
206	C₂	#21458968	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂²×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂²×Q₃₂	D₆₁
hypocenter	C₆₁	#33458
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #21447690	C₂	semidirect
3. Q₃₂×Dic₆₁	C₂	semidirect
21. C₂×Q₃₂×C₆₁	C₄	semidirect
113. C₂×Q₃₂	Dic₆₁	semidirect
128. C₆₁	#33458	semidirect

Maximal subgroups

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

subgroup	index
#21454687	2
#21447690	2
Q₃₂×Dic₆₁	2
#21447603	2
#21447539	2
#21447415	2
C₂²×Q₃₂×C₆₁	2
#33458	61