Subgroups — #29589346

Order 15616.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447622	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447618	C₂	1	1	yes	maximal, characteristic, complemented
3	#21443745	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11758185	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₂²⋊#11758185	C₂²	1	2	yes	—
8	C₂²⋊#11758185	C₂²	1	2	yes	—
9	#16839538	—	2	1	no	—
10	C₄×#11758185	—	2	1	no	—
11	C₂×#131×C₆₁	C₄	1	2	yes	—
12	C₆₁×#532	C₄	1	1	yes	characteristic
13	C₆₁×#436	—	2	1	no	—
14	C₆₁×#436	—	2	1	no	—
15	C₈×D₄×C₆₁	—	2	1	no	—
16	C₈×D₄×C₆₁	—	2	1	no	—
17	C₆₁×#423	C₄	1	1	yes	characteristic
18	C₆₁×#408	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
23	#131×C₆₁	—	4	4	no	—
24	C₂×M₁₆×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
30	#127×C₆₁	C₈	1	1	yes	characteristic
31	#124×C₆₁	C₈	1	2	yes	—
32	#121×C₆₁	—	2	1	no	—
33	#121×C₆₁	—	2	1	no	—
34	C₄×D₄×C₆₁	—	2	1	no	—
35	C₄×D₄×C₆₁	—	2	1	no	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	—	2	1	no	—
38	C₄⋊C₈×C₆₁	—	2	1	no	—
39	C₂²⋊C₈×C₆₁	—	2	2	no	—
40	C₂²⋊C₈×C₆₁	—	2	2	no	—
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	#41×C₆₁	—	4	2	no	—
46	#41×C₆₁	—	4	2	no	—
47	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₆₁	—	2	2	no	—
49	C₂×D₄×C₆₁	—	2	1	no	—
50	C₂×D₄×C₆₁	—	2	1	no	—
51	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₆₁	—	4	4	no	—
57	M₁₆×C₆₁	—	4	2	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	4	1	no	—
61	C₂×C₈×C₆₁	—	4	4	no	—
62	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₆₁	—	2	1	no	—
67	C₄⋊C₄×C₆₁	—	2	1	no	—
68	C₂²⋊C₄×C₆₁	—	2	2	no	—
69	C₂²⋊C₄×C₆₁	—	2	2	no	—
70	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
74	#11758185	—	4	2	no	—
75	#11758185	—	4	2	no	—
76	C₂³×C₆₁	—	2	2	no	—
77	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
78	Q₈×C₆₁	—	4	2	no	—
79	D₄×C₆₁	—	4	2	no	—
80	D₄×C₆₁	—	4	2	no	—
81	D₄×C₆₁	—	4	4	no	—
82	D₄×C₆₁	—	4	2	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	4	1	no	—
88	C₂×C₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	—	4	2	no	—
90	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
96	C₈×C₆₁	—	2	1	no	—
97	C₈×C₆₁	—	2	1	no	—
98	C₈×C₆₁	—	2	1	no	—
99	C₈×C₆₁	—	4	2	no	—
100	C₈×C₆₁	#100	1	2	yes	—
101	#11492	—	61	1	no	maximal, Sylow
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	—	2	1	no	—
104	C₂²×C₆₁	—	2	1	no	—
105	C₂²×C₆₁	—	4	2	no	—
106	C₂²×C₆₁	—	4	2	no	—
107	C₂²×C₆₁	—	4	2	no	—
108	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	—	2	1	no	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	—	4	2	no	—
114	C₄×C₆₁	#350	1	2	yes	—
115	C₄×C₆₁	#410	1	1	yes	characteristic
116	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
117	#2931	D₆₁	1	1	yes	characteristic
118	#2163	—	61	1	no	—
119	#2162	—	61	1	no	—
120	#1315	—	61	4	no	—
121	C₂×C₆₁	—	4	1	no	—
122	C₂×C₆₁	—	8	2	no	—
123	C₂×C₆₁	#1460	1	1	yes	characteristic
124	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
125	C₂×C₆₁	#2117	1	1	yes	characteristic
126	C₂×#131	Dic₆₁	1	2	yes	—
127	#532	Dic₆₁	1	1	yes	characteristic
128	#436	—	2	1	no	—
129	#436	—	2	1	no	—
130	C₈×D₄	—	2	1	no	—
131	C₈×D₄	—	2	1	no	—
132	#423	Dic₆₁	1	1	yes	characteristic
133	#408	C₂×D₆₁	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
135	C₄⋊C₁₆	—	122	1	no	—
136	C₄⋊C₁₆	—	122	1	no	—
137	C₂²⋊C₁₆	—	61	2	no	—
138	C₂²⋊C₁₆	—	61	2	no	—
139	#346	—	122	1	no	—
140	C₄×C₁₆	—	122	1	no	—
141	C₆₁	#11492	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	C₂×Dic₆₁	1	2	yes	—
143	#131	—	4	4	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
149	C₄⋊Q₈	#92418	1	1	yes	characteristic
150	#127	#92418	1	1	yes	characteristic
151	#124	#92418	1	2	yes	—
152	#121	—	2	1	no	—
153	#121	—	2	1	no	—
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
157	C₂×C₁₆	—	122	2	no	—
158	C₂×C₁₆	—	122	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
167	#41	—	4	2	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	C₂×#92418	1	2	yes	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂×#92418	1	2	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	#11758194	1	1	yes	characteristic
176	C₂²×C₄	#11758194	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
178	M₁₆	—	4	4	no	—
179	M₁₆	—	4	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	C₂×#92418	1	1	yes	characteristic
193	C₄²	C₂×#92418	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
195	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
196	C₁₆	—	244	2	no	—
197	C₁₆	—	244	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂×#11758194	1	1	yes	characteristic
200	Q₈	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
217	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	#14519694	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#16839546	1	2	yes	—
236	C₄	#16839736	1	1	yes	characteristic
237	C₄	C₂×#14519694	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	#21444140	1	1	yes	characteristic
241	C₂	C₂×#16839546	1	1	yes	characteristic
242	C₂	#21447447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2931	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₆₁×#532	C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2931	D₆₁
hypocenter	C₆₁	#11492
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #21447622	C₂	semidirect
2. #21447618	C₂	semidirect
3. #21443745	C₂	semidirect
141. C₆₁	#11492	semidirect

Maximal subgroups

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

subgroup	index
#21447622	2
#21447618	2
#21443745	2
C₆₁×#2931	2
#11492	61