Subgroups — #30606339

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic
3	(C₂×#6923)⋊C₁₆	C₂	1	4	yes	maximal
4	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
6	C₄×D₆₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×#6923	C₄	1	2	yes	—
8	C₂×C₈×#6923	C₂²	1	2	yes	—
9	C₄²×#6923	C₄	1	1	yes	characteristic
10	#16840648	C₄	1	2	yes	complemented
11	C₄×#11758187	C₄	1	2	yes	complemented
12	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
13	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
14	C₄×#11758185	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₆₁	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
17	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×#6923	C₈	1	2	yes	—
19	C₂×C₄×#6923	C₈	1	2	yes	—
20	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
21	D₆₁⋊C₁₆	—	2	4	no	—
22	C₂×#11758187	C₂×C₄	1	4	yes	—
23	C₄×#92420	C₈	1	2	yes	—
24	C₄×#92420	C₈	1	2	yes	—
25	C₈×#6923	—	2	4	no	—
26	C₈×#6923	—	2	4	no	—
27	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
29	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×#11758185	C₂×C₄	1	2	yes	—
31	C₁₆×D₆₁	—	2	4	no	—
32	C₄×#92418	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×#6923	C₂×C₈	1	4	yes	—
38	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×#92420	C₂×C₈	1	4	yes	—
42	C₄×#6923	—	2	4	no	—
43	C₄×#6923	—	2	2	no	—
44	C₄×#6923	—	2	2	no	—
45	C₄×#6923	C₁₆	1	8	yes	complemented
46	D₆₁⋊C₈	—	2	2	no	—
47	D₆₁⋊C₈	—	2	2	no	—
48	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₄²	1	1	yes	characteristic
51	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
52	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
53	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
54	C₂×#92418	C₄²	1	1	yes	characteristic
55	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
56	C₈×D₆₁	M₁₆	1	4	yes	—
57	C₈×D₆₁	#41	1	4	yes	—
58	#11758187	—	2	4	no	—
59	C₁₆×C₆₁	—	2	2	no	—
60	#11758185	—	2	2	no	—
61	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
62	C₂×#6923	—	2	4	no	—
63	C₂×#6923	—	2	4	no	—
64	C₂×#6923	C₂×C₁₆	1	8	yes	—
65	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
69	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
70	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
71	C₄×D₆₁	C₂×C₁₆	1	4	yes	—
72	C₄×D₆₁	M₃₂	1	4	yes	—
73	C₄×D₆₁	#117	1	2	yes	—
74	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
75	#92420	—	2	4	no	—
76	C₈×C₆₁	#97	1	2	yes	—
77	C₈×C₆₁	#117	1	2	yes	—
78	#92418	#97	1	2	yes	—
79	#92418	#117	1	2	yes	—
80	#12590	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂×D₆₁	#431	1	2	yes	—
83	C₂×D₆₁	C₂²×C₁₆	1	2	yes	—
84	C₂×D₆₁	C₂×M₃₂	1	2	yes	—
85	#6923	—	2	8	no	—
86	C₄×C₆₁	C₄×C₁₆	1	2	yes	—
87	C₄×C₆₁	#346	1	2	yes	—
88	C₄×C₆₁	#376	1	1	yes	characteristic
89	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
90	Dic₆₁	C₄×C₁₆	1	2	yes	—
91	Dic₆₁	#346	1	2	yes	—
92	Dic₆₁	#431	1	2	yes	—
93	C₂×C₄×C₁₆	—	61	1	no	—
94	C₂×C₄×C₁₆	—	61	1	no	—
95	C₄²×C₈	—	61	1	no	—
96	#1374	—	61	4	no	—
97	C₂×C₆₁	#1744	1	1	yes	characteristic
98	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
99	C₂×C₆₁	C₂×#346	1	1	yes	characteristic
100	D₆₁	#2162	1	4	yes	—
101	C₂²×C₁₆	—	61	2	no	—
102	C₂²×C₁₆	—	61	2	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	—	61	2	no	—
106	C₂×C₄×C₈	—	61	2	no	—
107	C₄³	—	61	1	no	—
108	C₄×C₁₆	—	61	2	no	—
109	C₄×C₁₆	—	61	2	no	—
110	C₄×C₁₆	—	61	2	no	—
111	C₄×C₁₆	—	61	1	no	—
112	C₄×C₁₆	#6923	1	1	yes	characteristic, complemented
113	C₆₁	#12590	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	61	1	no	—
115	C₂²×C₈	—	61	1	no	—
116	C₂²×C₈	—	61	2	no	—
117	C₂×C₄²	—	61	2	no	—
118	C₂×C₄²	—	61	1	no	—
119	C₂×C₄²	—	61	2	no	—
120	C₂×C₄²	—	61	2	no	—
121	C₂×C₁₆	—	61	4	no	—
122	C₂×C₁₆	—	61	2	no	—
123	C₂×C₁₆	—	122	4	no	—
124	C₂×C₁₆	—	122	4	no	—
125	C₂×C₁₆	C₂×#6923	1	2	yes	—
126	C₄×C₈	—	61	2	no	—
127	C₄×C₈	—	61	2	no	—
128	C₄×C₈	—	61	1	no	—
129	C₄×C₈	—	61	1	no	—
130	C₄×C₈	—	61	1	no	—
131	C₄×C₈	—	122	4	no	—
132	C₄×C₈	—	122	4	no	—
133	C₄×C₈	C₂×#6923	1	1	yes	characteristic
134	C₂²×C₄	—	61	1	no	—
135	C₂²×C₄	—	61	1	no	—
136	C₂²×C₄	—	61	1	no	—
137	C₂²×C₄	—	61	4	no	—
138	C₂×C₈	—	61	1	no	—
139	C₂×C₈	—	61	1	no	—
140	C₂×C₈	—	61	4	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	—	61	4	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	122	2	no	—
145	C₂×C₈	C₄×#6923	1	1	yes	characteristic
146	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
147	C₄²	—	61	8	no	—
148	C₄²	—	61	1	no	—
149	C₄²	—	61	1	no	—
150	C₄²	—	61	1	no	—
151	C₄²	—	122	2	no	—
152	C₄²	—	122	2	no	—
153	C₄²	—	122	4	no	—
154	C₄²	C₄×#6923	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	—	122	4	no	—
157	C₁₆	—	122	2	no	—
158	C₂³	—	61	1	no	—
159	C₂×C₄	—	61	1	no	—
160	C₂×C₄	—	61	1	no	—
161	C₂×C₄	—	61	1	no	—
162	C₂×C₄	—	61	2	no	—
163	C₂×C₄	—	61	4	no	—
164	C₂×C₄	—	61	4	no	—
165	C₂×C₄	—	61	2	no	—
166	C₂×C₄	—	61	8	no	—
167	C₂×C₄	—	122	4	no	—
168	C₂×C₄	—	122	4	no	—
169	C₂×C₄	C₈×#6923	1	1	yes	characteristic
170	C₂×C₄	C₈×#6923	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
172	C₈	—	61	2	no	—
173	C₈	—	61	2	no	—
174	C₈	—	122	4	no	—
175	C₈	#14519831	1	2	yes	—
176	C₈	#14519856	1	2	yes	—
177	C₂²	—	61	2	no	—
178	C₂²	—	61	2	no	—
179	C₂²	—	61	2	no	—
180	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
181	C₄	—	61	2	no	—
182	C₄	—	61	2	no	—
183	C₄	—	61	2	no	—
184	C₄	—	122	8	no	—
185	C₄	C₁₆×#6923	1	2	yes	—
186	C₄	#16840649	1	2	yes	—
187	C₄	#16840698	1	1	yes	characteristic
188	C₄	C₂×#14519831	1	1	yes	characteristic
189	C₂	—	61	4	no	—
190	C₂	#21461532	1	1	yes	characteristic
191	C₂	C₂×C₁₆×#6923	1	1	yes	characteristic
192	C₂	C₂×#16840649	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6923
commutator	C₂×C₆₁	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12590
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16840648	C₄	semidirect
11. C₄×#11758187	C₄	semidirect
14. C₄×#11758185	C₄	semidirect
15. C₁₆×Dic₆₁	C₄	semidirect
16. C₄×C₁₆×C₆₁	C₄	semidirect
45. C₄×#6923	C₁₆	semidirect
112. C₄×C₁₆	#6923	semidirect
113. C₆₁	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₆₁⋊C₁₆	2
C₄×C₈×#6923	2
(C₂×#6923)⋊C₁₆	2
C₄×C₁₆×D₆₁	2
#12590	61