Subgroups — #29596370

Order 15616.

314 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	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519687	C₂	1	1	yes	maximal, characteristic
3	#21443902	C₂	1	2	yes	maximal
4	#21443898	C₂	1	2	yes	maximal
5	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×#14519687	C₄	1	1	yes	characteristic
8	C₂×#14519687	C₄	1	1	yes	characteristic
9	C₂×#14519687	C₂²	1	1	yes	characteristic
10	C₂×#14519687	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92418	C₄	1	1	yes	characteristic
12	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
13	#16839583	—	2	2	no	—
14	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#364	—	2	2	no	—
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₁	—	2	2	no	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
22	#14519717	—	2	2	no	—
23	#14519717	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519687	—	2	2	no	—
26	#14519687	—	2	1	no	—
27	#14519687	—	2	1	no	—
28	#14519687	—	2	1	no	—
29	#14519687	—	2	1	no	—
30	#14519687	C₈	1	4	yes	—
31	C₄×#92418	—	2	2	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₆₁	—	2	2	no	—
36	M₃₂×C₆₁	—	2	2	no	—
37	C₂×C₁₆×C₆₁	—	2	2	no	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₄×C₈×C₆₁	Q₈	1	2	yes	—
40	C₂²×C₄×C₆₁	—	2	1	no	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	2	1	no	—
43	C₂²×Dic₆₁	M₁₆	1	2	yes	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
50	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
51	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
53	C₄²×C₆₁	—	2	1	no	—
54	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₆₁	—	2	1	no	—
56	C₄×Dic₆₁	—	2	1	no	—
57	C₄×Dic₆₁	—	2	1	no	—
58	C₄×Dic₆₁	—	4	4	no	—
59	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	—	2	1	no	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	C₂×C₈	1	2	yes	—
65	C₂×#92418	M₁₆	1	2	yes	—
66	C₁₆×C₆₁	—	4	2	no	—
67	#11758185	—	4	2	no	—
68	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₆₁	—	2	2	no	—
78	C₂×Dic₆₁	—	2	2	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
82	C₈×C₆₁	#124	1	1	yes	characteristic
83	C₈×C₆₁	#124	1	1	yes	characteristic
84	C₈×C₆₁	#125	1	1	yes	characteristic
85	C₈×C₆₁	#125	1	1	yes	characteristic
86	#92418	—	2	2	no	—
87	#92418	—	2	2	no	—
88	#12882	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	—	2	2	no	—
90	C₂²×C₆₁	#423	1	1	yes	characteristic
91	C₂²×C₆₁	#431	1	1	yes	characteristic
92	C₂²×C₆₁	#432	1	1	yes	characteristic
93	C₄×C₆₁	—	2	4	no	—
94	C₄×C₆₁	#382	1	1	yes	characteristic
95	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₆₁	#431	1	1	yes	characteristic
97	C₄×C₆₁	#432	1	1	yes	characteristic
98	Dic₆₁	—	4	2	no	—
99	Dic₆₁	—	4	2	no	—
100	C₂×#364	—	61	1	no	—
101	C₂×#364	D₆₁	1	1	yes	characteristic
102	C₄×#97	—	61	1	no	—
103	#1377	—	61	2	no	—
104	#1375	—	61	2	no	—
105	C₂×C₆₁	—	2	1	no	—
106	C₂×C₆₁	—	2	1	no	—
107	C₂×C₆₁	#1831	1	1	yes	characteristic
108	C₂×C₆₁	#2165	1	2	yes	—
109	C₂×M₃₂	—	61	2	no	—
110	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
111	C₂×#97	—	61	1	no	—
112	C₂×#97	—	61	1	no	—
113	C₂×#97	—	61	1	no	—
114	C₂×#97	—	61	1	no	—
115	C₂×C₄×C₈	—	61	1	no	—
116	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
117	C₄³	—	61	1	no	—
118	#364	—	2	2	no	—
119	#364	—	122	2	no	—
120	C₆₁	#12882	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	61	1	no	—
122	C₂²×C₈	—	61	1	no	—
123	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
125	C₂×C₄²	—	61	1	no	—
126	C₂×C₄²	—	61	1	no	—
127	C₂×C₄²	—	122	2	no	—
128	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	122	2	no	—
132	M₃₂	—	122	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	122	2	no	—
135	#97	—	61	4	no	—
136	#97	—	122	2	no	—
137	#97	—	122	1	no	—
138	#97	—	122	1	no	—
139	#97	—	122	1	no	—
140	#97	—	122	1	no	—
141	C₄×C₈	—	122	2	no	—
142	C₄×C₈	Dic₁₂₂	1	2	yes	—
143	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	61	2	no	—
146	C₂²×C₄	—	122	1	no	—
147	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
148	C₂×C₈	—	61	2	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	122	1	no	—
151	C₂×C₈	—	122	1	no	—
152	C₂×C₈	—	122	1	no	—
153	C₂×C₈	—	122	1	no	—
154	C₂×C₈	C₈×D₆₁	1	2	yes	—
155	C₂×C₈	#11758189	1	2	yes	—
156	C₂×C₈	#11758196	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
158	C₂×C₈	#11758224	1	1	yes	characteristic
159	C₂×C₈	#11758224	1	1	yes	characteristic
160	C₂×C₈	#11758226	1	1	yes	characteristic
161	C₂×C₈	#11758226	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
163	C₄²	—	2	1	no	—
164	C₄²	—	61	2	no	—
165	C₄²	—	122	1	no	—
166	C₄²	—	122	1	no	—
167	C₄²	—	122	1	no	—
168	C₄²	—	244	4	no	—
169	C₄²	#11758196	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₁₆	—	244	2	no	—
172	C₂³	M₁₆×D₆₁	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	122	2	no	—
176	C₂×C₄	—	122	2	no	—
177	C₂×C₄	—	244	2	no	—
178	C₂×C₄	—	244	2	no	—
179	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
180	C₂×C₄	#14519735	1	1	yes	characteristic
181	C₂×C₄	#14519735	1	1	yes	characteristic
182	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
184	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
185	C₈	—	122	2	no	—
186	C₈	—	122	2	no	—
187	C₈	#14519706	1	4	yes	—
188	C₈	#14519764	1	1	yes	characteristic
189	C₈	#14519766	1	1	yes	characteristic
190	C₈	#14519768	1	1	yes	characteristic
191	C₈	#14519768	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	#16839785	1	1	yes	characteristic
194	C₂²	#16839814	1	1	yes	characteristic
195	C₂²	#16839818	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	244	2	no	—
198	C₄	—	244	2	no	—
199	C₄	#16839627	1	1	yes	characteristic
200	C₄	C₂×#14519706	1	1	yes	characteristic
201	C₄	#16839814	1	1	yes	characteristic
202	C₄	#16839818	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#21445935	1	1	yes	characteristic
206	C₂	#21447630	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14519735
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#364	C₂
socle	C₂²×C₆₁	#432
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#364	D₆₁
hypocenter	C₆₁	#12882
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
120. C₆₁	#12882	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839583	2
C₄×#14519687	2
#21443902	2
#21443898	2
C₂×C₆₁×#364	2
#12882	61