Subgroups — #29584925

Order 15616.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446575	C₂	1	2	yes	maximal
2	#21446537	C₂	1	1	yes	maximal, characteristic
3	#21443702	C₂	1	2	yes	maximal
4	#21443654	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519710	—	2	2	no	—
7	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	C₂×#14519706	—	2	1	no	—
10	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839524	—	2	2	no	—
12	#16839516	—	2	1	no	—
13	C₂×#107×C₆₁	—	2	1	no	—
14	C₂×#107×C₆₁	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#384	C₂²	1	2	yes	—
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	—	2	1	no	—
19	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	—	2	1	no	—
22	#14519710	—	4	4	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	Q₈	1	4	yes	—
26	#14519706	—	4	2	no	—
27	#14519684	—	2	1	no	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#107×C₆₁	—	4	2	no	—
35	#107×C₆₁	—	4	2	no	—
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	—	4	2	no	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	1	no	—
42	C₂²×Dic₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	D₈	1	2	yes	—
48	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
49	C₄⋊C₄×C₆₁	—	4	2	no	—
50	C₄⋊C₄×C₆₁	—	4	2	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	4	2	no	—
53	C₄⋊C₄×C₆₁	—	8	2	no	—
54	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
55	C₄²×C₆₁	#41	1	2	yes	—
56	C₄⋊Dic₆₁	—	2	1	no	—
57	C₄⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	Q₁₆	1	2	yes	—
59	C₄×Dic₆₁	—	4	1	no	—
60	C₄×Dic₆₁	—	4	1	no	—
61	C₂×#92418	—	2	4	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	1	no	—
65	C₂×#92418	—	4	1	no	—
66	C₂×#92418	—	4	1	no	—
67	C₂³×C₆₁	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	8	2	no	—
71	C₂×C₄×C₆₁	—	8	2	no	—
72	C₂×C₄×C₆₁	—	8	2	no	—
73	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#125	1	2	yes	—
76	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
81	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	4	1	no	—
86	C₂×Dic₆₁	—	4	1	no	—
87	C₂×Dic₆₁	—	4	1	no	—
88	C₂×Dic₆₁	—	4	1	no	—
89	C₂×Dic₆₁	—	4	1	no	—
90	C₈×C₆₁	—	2	2	no	—
91	C₈×C₆₁	—	2	2	no	—
92	#92418	—	4	4	no	—
93	#92418	—	8	1	no	—
94	#10650	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	#398	1	1	yes	characteristic
96	C₂²×C₆₁	#459	1	1	yes	characteristic
97	C₂²×C₆₁	#462	1	1	yes	characteristic
98	C₂²×C₆₁	#467	1	1	yes	characteristic
99	C₂²×C₆₁	#469	1	1	yes	characteristic
100	C₂²×C₆₁	#472	1	1	yes	characteristic
101	C₂²×C₆₁	#473	1	1	yes	characteristic
102	C₄×C₆₁	—	2	2	no	—
103	C₄×C₆₁	—	16	2	no	—
104	C₄×C₆₁	#477	1	2	yes	—
105	C₄×C₆₁	#478	1	2	yes	—
106	C₄×C₆₁	#480	1	1	yes	characteristic
107	C₄×C₆₁	#484	1	1	yes	characteristic
108	C₄×C₆₁	#500	1	1	yes	characteristic
109	C₄×C₆₁	#501	1	1	yes	characteristic
110	Dic₆₁	—	4	2	no	—
111	Dic₆₁	—	8	1	no	—
112	#1947	—	61	2	no	—
113	#1939	—	61	1	no	—
114	#1845	D₆₁	1	1	yes	characteristic
115	#1296	—	61	2	no	—
116	#1277	—	61	1	no	—
117	C₂×C₆₁	#1669	1	1	yes	characteristic
118	C₂×C₆₁	#1676	1	1	yes	characteristic
119	C₂×C₆₁	#1687	1	1	yes	characteristic
120	C₂×C₆₁	#1694	1	1	yes	characteristic
121	C₂×C₆₁	#2083	1	1	yes	characteristic
122	C₂×C₆₁	#2087	1	1	yes	characteristic
123	C₂×C₆₁	#2089	1	1	yes	characteristic
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	122	2	no	—
127	C₂×C₄⋊C₈	—	61	2	no	—
128	C₂×C₄⋊C₈	—	61	1	no	—
129	C₂×C₄⋊C₈	—	122	1	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	#384	C₂×D₆₁	1	2	yes	—
132	C₄×C₄⋊C₄	—	61	1	no	—
133	#340	—	122	2	no	—
134	#336	—	122	1	no	—
135	C₆₁	#10650	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	122	2	no	—
137	C₂²×C₈	—	122	1	no	—
138	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	61	1	no	—
144	C₂×C₄⋊C₄	—	61	1	no	—
145	C₂×C₄²	—	122	1	no	—
146	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
147	#107	—	4	2	no	—
148	#107	—	4	2	no	—
149	#107	—	244	4	no	—
150	C₄⋊C₈	—	61	4	no	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	—	122	2	no	—
153	C₄⋊C₈	—	244	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	122	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	122	1	no	—
158	C₂²×C₄	—	122	1	no	—
159	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
160	C₂²×C₄	#11758224	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	122	4	no	—
167	C₂×C₈	—	244	1	no	—
168	C₂×C₈	—	244	1	no	—
169	C₂×C₈	—	244	1	no	—
170	C₂×C₈	—	244	2	no	—
171	C₂×C₈	—	244	2	no	—
172	C₂×C₈	#11758199	1	2	yes	—
173	C₂×C₈	#11758202	1	2	yes	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	8	2	no	—
179	C₄⋊C₄	—	61	2	no	—
180	C₄⋊C₄	—	122	1	no	—
181	C₄⋊C₄	—	122	2	no	—
182	C₄²	—	244	1	no	—
183	C₄²	—	244	1	no	—
184	C₄²	#11758222	1	2	yes	—
185	C₄²	Q₈×D₆₁	1	2	yes	—
186	C₂³	#14519748	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	122	2	no	—
193	C₂×C₄	—	244	1	no	—
194	C₂×C₄	—	244	1	no	—
195	C₂×C₄	—	244	1	no	—
196	C₂×C₄	—	244	1	no	—
197	C₂×C₄	—	244	1	no	—
198	C₂×C₄	#14519755	1	2	yes	—
199	C₂×C₄	#14519760	1	1	yes	characteristic
200	C₂×C₄	#14519767	1	2	yes	—
201	C₂×C₄	#14519775	1	1	yes	characteristic
202	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
204	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
205	C₂×C₄	#14519800	1	1	yes	characteristic
206	C₂×C₄	#14519801	1	2	yes	—
207	C₂×C₄	#14519803	1	1	yes	characteristic
208	C₂×C₄	#14519808	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	488	1	no	—
212	C₈	—	244	4	no	—
213	C₂²	#16839693	1	1	yes	characteristic
214	C₂²	#16839954	1	1	yes	characteristic
215	C₂²	#16839969	1	1	yes	characteristic
216	C₂²	#16840000	1	1	yes	characteristic
217	C₂²	#16840012	1	1	yes	characteristic
218	C₂²	#16840029	1	1	yes	characteristic
219	C₂²	#16840034	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	16	2	no	—
222	C₄	—	488	1	no	—
223	C₄	—	244	2	no	—
224	C₄	#16840056	1	2	yes	—
225	C₄	#16840065	1	2	yes	—
226	C₄	#16840075	1	1	yes	characteristic
227	C₄	#16840097	1	1	yes	characteristic
228	C₄	#16840165	1	1	yes	characteristic
229	C₄	#16840168	1	1	yes	characteristic
230	C₂	#21445181	1	1	yes	characteristic
231	C₂	#21445220	1	1	yes	characteristic
232	C₂	#21445295	1	1	yes	characteristic
233	C₂	#21445334	1	1	yes	characteristic
234	C₂	#21447291	1	1	yes	characteristic
235	C₂	#21447313	1	1	yes	characteristic
236	C₂	#21447325	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1845	C₂
socle, prime radical	C₂³×C₆₁	#121
hypercenter	#1845	D₆₁
hypocenter	C₆₁	#10650
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
135. C₆₁	#10650	semidirect

Maximal subgroups

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

subgroup	index
#21446575	2
#21446537	2
#21443702	2
#21443654	2
C₆₁×#1845	2
#10650	61