Subgroups — #30606341

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840648	C₂	1	2	yes	maximal
2	#21460858	C₂	1	2	yes	maximal
3	#21460857	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758187	C₂²	1	4	yes	—
6	#16840648	C₄	1	8	yes	complemented
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×#11758187	—	2	4	no	—
15	C₂×#11758187	—	2	4	no	—
16	C₂×#11758187	C₂×C₄	1	8	yes	—
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂²×#92418	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	C₈	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₈	1	2	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×#92418	—	2	2	no	—
24	C₈×Dic₆₁	—	2	2	no	—
25	C₈×Dic₆₁	—	2	2	no	—
26	C₈×Dic₆₁	C₈	1	4	yes	—
27	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
35	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	C₄²	1	2	yes	—
40	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
43	C₄²×C₆₁	—	2	1	no	—
44	C₄²×C₆₁	—	2	1	no	—
45	C₄×Dic₆₁	—	2	1	no	—
46	C₄×Dic₆₁	—	2	1	no	—
47	C₄×Dic₆₁	—	2	4	no	—
48	C₄×Dic₆₁	C₁₆	1	8	yes	complemented
49	C₄×Dic₆₁	C₄²	1	2	yes	—
50	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
51	C₂×#92418	—	2	1	no	—
52	C₂×#92418	—	2	1	no	—
53	C₂×#92418	C₂×C₈	1	2	yes	—
54	C₂×#92418	C₂²×C₄	1	2	yes	—
55	C₂×#92418	#41	1	2	yes	—
56	C₂×#92418	#41	1	2	yes	—
57	#11758187	—	2	8	no	—
58	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
62	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
71	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
72	C₂×Dic₆₁	C₂×C₁₆	1	4	yes	—
73	C₂×Dic₆₁	C₂×C₁₆	1	4	yes	—
74	C₂×Dic₆₁	M₃₂	1	4	yes	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	#97	1	4	yes	—
77	#92418	—	2	2	no	—
78	#92418	#117	1	4	yes	—
79	#12590	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
81	C₂²×C₆₁	#431	1	1	yes	characteristic
82	C₂²×C₆₁	#431	1	1	yes	characteristic
83	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
84	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
85	C₄×C₆₁	—	2	4	no	—
86	C₄×C₆₁	#376	1	1	yes	characteristic
87	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
88	C₄×C₆₁	#431	1	1	yes	characteristic
89	C₄×C₆₁	#431	1	1	yes	characteristic
90	Dic₆₁	—	2	4	no	—
91	Dic₆₁	C₄×C₁₆	1	4	yes	—
92	Dic₆₁	#346	1	4	yes	—
93	C₂×C₄×C₁₆	—	61	2	no	—
94	C₄²×C₈	—	61	1	no	—
95	#1374	—	61	2	no	—
96	#1374	—	61	2	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	C₂×C₆₁	#2162	1	2	yes	—
101	C₂×C₆₁	#2162	1	2	yes	—
102	C₂²×C₁₆	—	61	4	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	—	61	1	no	—
106	C₂×C₄×C₈	—	61	1	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	C₂×C₄×C₈	#6923	1	1	yes	characteristic
109	C₄³	—	61	1	no	—
110	C₄×C₁₆	—	61	8	no	—
111	C₆₁	#12590	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	61	1	no	—
113	C₂²×C₈	—	61	1	no	—
114	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
115	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
116	C₂×C₄²	—	61	2	no	—
117	C₂×C₄²	—	61	1	no	—
118	C₂×C₄²	—	61	2	no	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
121	C₂×C₁₆	—	61	8	no	—
122	C₂×C₁₆	—	122	4	no	—
123	C₂×C₁₆	—	122	4	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	61	4	no	—
126	C₄×C₈	—	61	4	no	—
127	C₄×C₈	—	122	2	no	—
128	C₄×C₈	—	122	2	no	—
129	C₄×C₈	—	122	2	no	—
130	C₂²×C₄	—	61	1	no	—
131	C₂²×C₄	—	61	1	no	—
132	C₂²×C₄	—	61	2	no	—
133	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
134	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	61	2	no	—
138	C₂×C₈	—	61	2	no	—
139	C₂×C₈	—	61	2	no	—
140	C₂×C₈	—	61	2	no	—
141	C₂×C₈	—	122	1	no	—
142	C₂×C₈	—	122	1	no	—
143	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
144	C₂×C₈	#11758213	1	2	yes	—
145	C₂×C₈	C₄×#6923	1	2	yes	—
146	C₂×C₈	#11758217	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	61	8	no	—
150	C₄²	—	61	2	no	—
151	C₄²	—	61	2	no	—
152	C₄²	—	122	4	no	—
153	C₄²	—	122	1	no	—
154	C₄²	—	122	1	no	—
155	C₁₆	—	122	8	no	—
156	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	61	2	no	—
160	C₂×C₄	—	61	2	no	—
161	C₂×C₄	—	61	4	no	—
162	C₂×C₄	—	61	4	no	—
163	C₂×C₄	—	61	4	no	—
164	C₂×C₄	—	122	2	no	—
165	C₂×C₄	—	122	2	no	—
166	C₂×C₄	D₆₁⋊C₁₆	1	4	yes	—
167	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
168	C₂×C₄	#14519856	1	1	yes	characteristic
169	C₂×C₄	#14519856	1	1	yes	characteristic
170	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
172	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	61	4	no	—
175	C₈	—	122	2	no	—
176	C₈	#14519832	1	4	yes	—
177	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
178	C₂²	#16840779	1	1	yes	characteristic
179	C₂²	#16840781	1	1	yes	characteristic
180	C₂²	C₂×D₆₁⋊C₁₆	1	2	yes	—
181	C₂²	D₆₁⋊M₃₂	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	61	4	no	—
184	C₄	—	61	4	no	—
185	C₄	—	122	4	no	—
186	C₄	#16840698	1	1	yes	characteristic
187	C₄	C₂×#14519832	1	1	yes	characteristic
188	C₄	#16840779	1	1	yes	characteristic
189	C₄	#16840781	1	1	yes	characteristic
190	C₂	#21461534	1	1	yes	characteristic
191	C₂	C₄×D₆₁⋊C₁₆	1	1	yes	characteristic
192	C₂	#21462020	1	1	yes	characteristic
193	C₂	#21462177	1	2	yes	—
194	C₂	#21462179	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#6923
commutator	C₂×C₆₁	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#12590
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #16840648	C₄	semidirect
48. C₄×Dic₆₁	C₁₆	semidirect
111. C₆₁	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840648	2
#21460858	2
#21460857	2
C₄×C₈×Dic₆₁	2
#12590	61