Subgroups — #29579848

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446781	C₂	1	2	yes	maximal
2	#21446622	C₂	1	2	yes	maximal
3	#21445976	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519709	—	2	2	no	—
7	C₂×#14519701	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839632	C₂²	1	2	yes	—
11	#16839524	—	2	4	no	—
12	#16839483	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#500	—	2	2	no	—
15	C₆₁×#500	C₄	1	4	yes	complemented
16	C₂×#107×C₆₁	—	2	2	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×#92418	D₄	1	2	yes	—
19	C₂²×#92418	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	#14519709	—	4	4	no	—
24	#14519701	—	4	8	no	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₁	—	2	2	no	—
32	C₄⋊Q₈×C₆₁	—	2	2	no	—
33	C₄⋊Q₈×C₆₁	—	2	2	no	—
34	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	#107×C₆₁	—	2	8	no	—
39	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₁	D₄	1	2	yes	—
41	C₂×Q₈×C₆₁	—	2	4	no	—
42	C₂×Q₈×C₆₁	—	4	2	no	—
43	C₂×Q₈×C₆₁	—	4	4	no	—
44	C₂×Q₈×C₆₁	—	4	4	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
52	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₆₁	—	2	4	no	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₄⋊C₄×C₆₁	—	4	2	no	—
56	C₄⋊C₄×C₆₁	—	4	2	no	—
57	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	C₄²×C₆₁	#41	1	1	yes	characteristic
61	C₄⋊Dic₆₁	—	4	4	no	—
62	C₄⋊Dic₆₁	—	4	4	no	—
63	C₄⋊Dic₆₁	—	4	4	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	—	2	4	no	—
67	C₂×#92418	—	2	4	no	—
68	C₂³×C₆₁	#127	1	1	yes	characteristic
69	Q₈×C₆₁	—	4	8	no	—
70	Q₈×C₆₁	—	8	4	no	—
71	C₂×C₄×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	—	8	2	no	—
73	C₂×C₄×C₆₁	—	8	2	no	—
74	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₆₁	#137	1	4	yes	—
84	C₂×Dic₆₁	—	8	2	no	—
85	C₂×Dic₆₁	—	8	2	no	—
86	C₂×Dic₆₁	—	8	2	no	—
87	C₈×C₆₁	#103	1	8	yes	complemented
88	#92418	—	4	2	no	—
89	#92418	—	4	2	no	—
90	#9836	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	#390	1	1	yes	characteristic
92	C₂²×C₆₁	#492	1	1	yes	characteristic
93	C₂²×C₆₁	#494	1	1	yes	characteristic
94	C₂²×C₆₁	#496	1	2	yes	—
95	C₂²×C₆₁	#497	1	2	yes	—
96	C₄×C₆₁	—	8	4	no	—
97	C₄×C₆₁	C₂×#103	1	2	yes	—
98	C₄×C₆₁	#441	1	2	yes	—
99	C₄×C₆₁	#442	1	2	yes	—
100	C₄×C₆₁	#469	1	2	yes	—
101	C₄×C₆₁	#470	1	2	yes	—
102	C₄×C₆₁	#487	1	2	yes	—
103	Dic₆₁	—	16	2	no	—
104	C₂×#500	D₆₁	1	1	yes	characteristic
105	#1990	—	61	2	no	—
106	#1956	—	61	2	no	—
107	#1844	—	61	1	no	—
108	C₂×#322	—	61	1	no	—
109	C₂×C₆₁	#1714	1	2	yes	—
110	C₂×C₆₁	#1723	1	2	yes	—
111	C₂×C₆₁	#1956	1	1	yes	characteristic
112	C₂×C₆₁	#1971	1	1	yes	characteristic
113	C₂×C₆₁	#1972	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
115	#500	—	2	2	no	—
116	#500	Dic₆₁	1	4	yes	complemented
117	C₂×#107	—	2	2	no	—
118	C₂×#106	—	122	2	no	—
119	C₂×#103	—	122	4	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
123	#384	—	61	2	no	—
124	#340	—	122	4	no	—
125	#322	—	122	2	no	—
126	C₆₁	#9836	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	—	61	2	no	—
129	C₂²×C₈	—	61	2	no	—
130	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	122	2	no	—
138	C₂×C₄⋊C₄	—	122	2	no	—
139	C₂×C₄⋊C₄	—	122	2	no	—
140	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
141	#107	—	2	8	no	—
142	#106	—	244	4	no	—
143	#103	—	244	8	no	—
144	C₄×C₈	—	122	1	no	—
145	C₄×C₈	—	122	1	no	—
146	C₄×C₈	—	122	1	no	—
147	C₄×C₈	—	122	1	no	—
148	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
149	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	244	2	no	—
156	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
159	C₂×C₈	—	122	4	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	—	122	4	no	—
163	C₂×C₈	#11758201	1	4	yes	—
164	C₂×C₈	#11758202	1	4	yes	—
165	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	244	4	no	—
171	C₄⋊C₄	—	244	4	no	—
172	C₄⋊C₄	—	244	4	no	—
173	C₄²	#11758224	1	1	yes	characteristic
174	C₄²	#11758224	1	1	yes	characteristic
175	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂³	#14519773	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	488	2	no	—
184	C₂×C₄	—	488	2	no	—
185	C₂×C₄	—	488	2	no	—
186	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
189	C₂×C₄	#14519748	1	1	yes	characteristic
190	C₂×C₄	#14519748	1	1	yes	characteristic
191	C₂×C₄	#14519764	1	1	yes	characteristic
192	C₂×C₄	C₂×#11758201	1	2	yes	—
193	C₂×C₄	C₂×#11758202	1	2	yes	—
194	C₂×C₄	#14519801	1	4	yes	—
195	C₂×C₄	#14519809	1	4	yes	—
196	C₈	—	244	2	no	—
197	C₈	—	244	2	no	—
198	C₈	#14519701	1	8	yes	complemented
199	C₂²	#16839665	1	1	yes	characteristic
200	C₂²	#16840136	1	1	yes	characteristic
201	C₂²	#16840141	1	1	yes	characteristic
202	C₂²	#16840150	1	2	yes	—
203	C₂²	#16840155	1	2	yes	—
204	C₄	—	8	4	no	—
205	C₄	—	976	2	no	—
206	C₄	C₂×#14519701	1	2	yes	—
207	C₄	#16839869	1	2	yes	—
208	C₄	#16839874	1	2	yes	—
209	C₄	#16840012	1	2	yes	—
210	C₄	#16840017	1	2	yes	—
211	C₄	#16840112	1	2	yes	—
212	C₂	#21445442	1	2	yes	—
213	C₂	#21445497	1	2	yes	—
214	C₂	#21446622	1	1	yes	characteristic
215	C₂	#21446691	1	1	yes	characteristic
216	C₂	#21446696	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#9836
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. C₆₁×#500	C₄	semidirect
87. C₈×C₆₁	#103	semidirect
116. #500	Dic₆₁	semidirect
126. C₆₁	#9836	semidirect
198. C₈	#14519701	semidirect

Maximal subgroups

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

subgroup	index
#21446781	2
#21446622	2
#21445976	2
C₂×#16839483	2
C₂×C₆₁×#500	2
#9836	61