Subgroups — #30624473

Order 15616.

460 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	#21462236	C₂	1	4	yes	maximal, complemented
2	#21462233	C₂	1	4	yes	maximal, complemented
3	#21462185	C₂	1	2	yes	maximal
4	#21462184	C₂	1	2	yes	maximal
5	#21462177	C₂	1	2	yes	maximal
6	#21454752	C₂	1	1	yes	maximal, characteristic
7	#16840683	C₂²	1	2	yes	—
8	#16840683	C₂²	1	2	yes	—
9	#16840683	C₂²	1	4	yes	—
10	C₄⋊#11758187	C₂²	1	4	yes	—
11	C₂²⋊#11758187	C₂²	1	4	yes	—
12	#16840651	C₂²	1	2	yes	—
13	#16840650	C₂²	1	2	yes	—
14	#16840648	C₂²	1	4	yes	—
15	#16840648	C₂²	1	2	yes	—
16	C₄×#11758187	C₂²	1	2	yes	—
17	#16840315	C₄	1	1	yes	characteristic
18	#16839891	C₄	1	4	yes	—
19	#16839886	C₂²	1	4	yes	—
20	#16839788	C₄	1	1	yes	characteristic
21	#16839785	C₄	1	1	yes	characteristic
22	#16839720	C₂²	1	1	yes	characteristic
23	#16839718	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#424	C₄	1	1	yes	characteristic
26	C₂×#11758187	C₂³	1	4	yes	—
27	C₂×#11758187	C₂³	1	4	yes	—
28	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
29	C₂²×#92418	C₂³	1	1	yes	characteristic
30	#14519776	C₈	1	2	yes	—
31	#14519768	C₈	1	2	yes	—
32	#14519754	C₈	1	4	yes	—
33	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
34	#14519736	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	#14519706	C₂×C₄	1	4	yes	—
37	#14519706	C₂×C₄	1	4	yes	—
38	#14519688	C₂×C₄	1	2	yes	—
39	#14519687	C₂×C₄	1	2	yes	—
40	C₄×#92418	C₂³	1	2	yes	—
41	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
42	C₈×Dic₆₁	C₂³	1	2	yes	—
43	C₈×Dic₆₁	C₂³	1	2	yes	—
44	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
50	C₂×Dic₁₂₂	C₂×C₈	1	4	yes	—
51	M₁₆×C₆₁	—	4	1	no	—
52	C₂×C₈×C₆₁	—	4	1	no	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
55	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
56	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
57	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
58	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
59	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
60	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
61	#11758196	C₂×C₈	1	4	yes	—
62	#11758196	C₂×C₈	1	4	yes	—
63	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
65	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
66	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
67	#11758194	—	4	1	no	—
68	C₂×#92418	—	2	2	no	—
69	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
70	C₂×#92418	C₂²×C₄	1	2	yes	—
71	C₂×#92418	C₂⁴	1	1	yes	characteristic
72	#11758187	—	2	4	no	—
73	#11758187	—	2	4	no	—
74	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
80	C₂×Dic₆₁	—	2	4	no	—
81	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
82	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
83	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
84	Dic₁₂₂	—	2	8	no	—
85	C₈×C₆₁	—	2	2	no	—
86	C₈×C₆₁	—	2	2	no	—
87	#92418	—	2	1	no	—
88	#92418	—	2	2	no	—
89	#92418	2₋¹⁺⁴	1	2	yes	—
90	#33500	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	2	no	—
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₆₁	—	2	4	no	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	#519	1	1	yes	characteristic
97	C₄×C₆₁	#568	1	1	yes	characteristic
98	Dic₆₁	—	2	4	no	—
99	Dic₆₁	—	2	2	no	—
100	Dic₆₁	#504	1	4	yes	—
101	#2960	—	61	1	no	—
102	#2183	—	61	4	no	—
103	Q₈×C₁₆	—	61	4	no	—
104	#2164	—	61	2	no	—
105	#2164	—	61	2	no	—
106	#2162	—	61	2	no	—
107	C₂×C₆₁	—	4	1	no	—
108	C₂×C₆₁	#2982	1	1	yes	characteristic
109	C₂×C₆₁	C₂×#504	1	2	yes	—
110	#533	—	61	1	no	—
111	#446	—	61	4	no	—
112	C₈×Q₈	—	61	4	no	—
113	#424	—	61	1	no	—
114	#424	#6923	1	1	yes	characteristic
115	#423	—	61	1	no	—
116	#405	—	61	1	no	—
117	#405	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₄⋊C₁₆	—	61	2	no	—
120	C₄⋊C₁₆	—	61	2	no	—
121	C₄⋊C₁₆	—	61	4	no	—
122	C₄⋊C₁₆	—	61	4	no	—
123	C₂²⋊C₁₆	—	61	4	no	—
124	#346	—	61	2	no	—
125	#346	—	61	2	no	—
126	C₄×C₁₆	—	61	4	no	—
127	C₄×C₁₆	—	61	2	no	—
128	C₄×C₁₆	—	61	2	no	—
129	C₆₁	#33500	1	1	yes	characteristic, complemented, Sylow
130	C₂×M₁₆	—	61	1	no	—
131	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
134	C₄⋊Q₈	—	61	2	no	—
135	#125	—	61	2	no	—
136	C₂²⋊Q₈	—	61	4	no	—
137	C₄×Q₈	—	61	4	no	—
138	#117	—	61	1	no	—
139	#117	C₂×#6923	1	1	yes	characteristic
140	C₂×C₄²	—	61	1	no	—
141	C₂×C₁₆	—	61	4	no	—
142	C₂×C₁₆	—	61	4	no	—
143	C₄⋊C₈	—	61	4	no	—
144	C₄⋊C₈	—	61	4	no	—
145	C₄⋊C₈	C₂×#6923	1	4	yes	—
146	#97	—	61	2	no	—
147	#97	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	2	no	—
150	C₄×C₈	—	61	2	no	—
151	C₄×C₈	—	61	2	no	—
152	C₂×Q₈	—	61	4	no	—
153	C₂²×C₄	—	61	2	no	—
154	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
155	M₁₆	—	4	1	no	—
156	M₁₆	—	244	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	61	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	C₂²×#6923	1	2	yes	—
163	C₂×C₈	C₂²×#6923	1	2	yes	—
164	C₄⋊C₄	—	61	4	no	—
165	C₄⋊C₄	—	61	4	no	—
166	C₄⋊C₄	—	61	2	no	—
167	C₄⋊C₄	—	61	4	no	—
168	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
169	C₂²⋊C₄	—	61	2	no	—
170	C₂²⋊C₄	D₆₁⋊C₈	1	2	yes	—
171	C₄²	—	61	1	no	—
172	C₄²	—	61	1	no	—
173	C₄²	—	61	2	no	—
174	C₄²	—	61	2	no	—
175	C₄²	C₂²×#6923	1	2	yes	—
176	C₁₆	—	122	4	no	—
177	C₁₆	—	122	4	no	—
178	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
179	Q₈	—	122	8	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	61	2	no	—
183	C₂×C₄	—	61	2	no	—
184	C₂×C₄	—	61	4	no	—
185	C₂×C₄	—	122	4	no	—
186	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₂×D₆₁⋊C₈	1	4	yes	—
188	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	61	2	no	—
192	C₈	—	122	1	no	—
193	C₈	—	122	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	1	no	—
199	C₄	—	61	4	no	—
200	C₄	—	122	2	no	—
201	C₄	—	122	4	no	—
202	C₄	#16840830	1	1	yes	characteristic
203	C₄	#16840840	1	1	yes	characteristic
204	C₂	—	4	1	no	—
205	C₂	#21462714	1	1	yes	characteristic
206	C₂	D₆₁⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#424	C₄
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#424	#6923
hypocenter	C₆₁	#33500
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #21462236	C₂	semidirect
2. #21462233	C₂	semidirect
129. C₆₁	#33500	semidirect

Maximal subgroups

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

subgroup	index
#21462236	2
#21462233	2
#21462185	2
#21462184	2
#21462177	2
#21454752	2
#33500	61