Subgroups — #29578684

Order 15616.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446583	C₂	1	2	yes	maximal
2	#21445976	C₂	1	1	yes	maximal, characteristic
3	#21445686	C₂	1	1	yes	maximal, characteristic
4	#21443703	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519709	—	2	2	no	—
7	C₂×#14519709	—	2	1	no	—
8	C₂×#14519709	—	2	1	no	—
9	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
10	C₂×#14519687	—	2	1	no	—
11	#16839632	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839525	—	2	2	no	—
14	#16839516	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#335	C₄	1	4	yes	complemented
18	C₂²×#92418	—	2	1	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	D₄	1	2	yes	—
23	C₂×C₄×Dic₆₁	—	2	1	no	—
24	#14519709	—	4	4	no	—
25	#14519709	—	4	2	no	—
26	#14519709	—	4	2	no	—
27	C₄⋊#92418	—	2	1	no	—
28	C₄⋊#92418	Q₈	1	2	yes	—
29	#14519687	—	4	2	no	—
30	#14519684	—	2	1	no	—
31	C₂²×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₁	Q₈	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	2	2	no	—
43	C₂²×Dic₆₁	—	4	2	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	2	4	no	—
47	C₂×C₈×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
52	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
53	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₄⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	8	2	no	—
64	C₄×Dic₆₁	—	4	1	no	—
65	C₄×Dic₆₁	—	4	1	no	—
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	—	4	1	no	—
68	C₂×#92418	—	4	1	no	—
69	C₂³×C₆₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#137	1	2	yes	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	8	2	no	—
87	C₂×Dic₆₁	—	8	2	no	—
88	C₂×Dic₆₁	—	8	2	no	—
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	—	4	4	no	—
91	C₈×C₆₁	#106	1	4	yes	complemented
92	#92418	—	4	2	no	—
93	#9618	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#389	1	1	yes	characteristic
95	C₂²×C₆₁	#460	1	1	yes	characteristic
96	C₂²×C₆₁	#461	1	1	yes	characteristic
97	C₂²×C₆₁	#468	1	2	yes	—
98	C₂²×C₆₁	#471	1	2	yes	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
102	C₄×C₆₁	#428	1	1	yes	characteristic
103	C₄×C₆₁	#441	1	1	yes	characteristic
104	C₄×C₆₁	#442	1	1	yes	characteristic
105	C₄×C₆₁	#475	1	1	yes	characteristic
106	C₄×C₆₁	#476	1	1	yes	characteristic
107	C₄×C₆₁	#501	1	2	yes	—
108	Dic₆₁	—	8	2	no	—
109	Dic₆₁	—	16	2	no	—
110	#1948	—	61	2	no	—
111	#1844	—	61	1	no	—
112	#1770	—	61	1	no	—
113	C₂×#335	D₆₁	1	1	yes	characteristic
114	#1296	—	61	2	no	—
115	C₂×C₆₁	#1686	1	2	yes	—
116	C₂×C₆₁	#1691	1	2	yes	—
117	C₂×C₆₁	#1943	1	1	yes	characteristic
118	C₂×C₆₁	#1948	1	1	yes	characteristic
119	C₂×C₆₁	#1951	1	1	yes	characteristic
120	C₂×#106	—	122	2	no	—
121	C₂×#106	—	122	1	no	—
122	C₂×#106	—	122	1	no	—
123	C₂×C₄⋊C₈	—	61	1	no	—
124	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
125	C₂×#97	—	122	1	no	—
126	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
127	#384	—	61	2	no	—
128	C₄×C₄⋊C₄	—	61	1	no	—
129	#340	—	122	2	no	—
130	#336	—	122	1	no	—
131	#335	Dic₆₁	1	4	yes	complemented
132	C₆₁	#9618	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	122	1	no	—
135	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
136	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	61	2	no	—
138	C₂×C₄⋊C₄	—	122	2	no	—
139	C₂×C₄⋊C₄	—	122	2	no	—
140	C₂×C₄⋊C₄	—	122	2	no	—
141	C₂×C₄²	—	122	1	no	—
142	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
143	#106	—	244	4	no	—
144	#106	—	244	2	no	—
145	#106	—	244	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	61	2	no	—
148	C₄⋊C₈	—	122	1	no	—
149	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
150	#97	—	244	2	no	—
151	C₄×C₈	Dic₁₂₂	1	2	yes	—
152	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
153	#95	—	122	1	no	—
154	C₂²×C₄	—	122	2	no	—
155	C₂²×C₄	—	244	2	no	—
156	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	#11758226	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	244	1	no	—
166	C₂×C₈	—	244	1	no	—
167	C₂×C₈	#11758190	1	2	yes	—
168	C₂×C₈	#11758190	1	2	yes	—
169	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
170	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
171	C₄⋊C₄	—	122	2	no	—
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	488	2	no	—
174	C₄⋊C₄	—	244	2	no	—
175	C₄⋊C₄	—	244	2	no	—
176	C₄⋊C₄	—	244	2	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₄²	—	244	1	no	—
179	C₄²	—	244	1	no	—
180	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
181	C₄²	#11758224	1	1	yes	characteristic
182	C₄²	Q₈×D₆₁	1	1	yes	characteristic
183	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
184	C₂³	#14519752	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	488	2	no	—
187	C₂×C₄	—	488	2	no	—
188	C₂×C₄	—	488	2	no	—
189	C₂×C₄	—	244	2	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
195	C₂×C₄	#14519757	1	1	yes	characteristic
196	C₂×C₄	#14519768	1	1	yes	characteristic
197	C₂×C₄	#14519776	1	1	yes	characteristic
198	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
199	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
200	C₂×C₄	#14519801	1	2	yes	—
201	C₂×C₄	#14519805	1	1	yes	characteristic
202	C₂×C₄	#14519809	1	2	yes	—
203	C₂×C₄	#14519810	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	244	2	no	—
207	C₈	#14519709	1	4	yes	complemented
208	C₂²	#16839661	1	1	yes	characteristic
209	C₂²	#16839963	1	1	yes	characteristic
210	C₂²	#16839968	1	1	yes	characteristic
211	C₂²	#16840009	1	2	yes	—
212	C₂²	#16840026	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	976	2	no	—
216	C₄	—	488	2	no	—
217	C₄	C₂×#14519709	1	1	yes	characteristic
218	C₄	#16839804	1	1	yes	characteristic
219	C₄	#16839869	1	1	yes	characteristic
220	C₄	#16839874	1	1	yes	characteristic
221	C₄	#16840048	1	1	yes	characteristic
222	C₄	#16840053	1	1	yes	characteristic
223	C₄	#16840170	1	2	yes	—
224	C₂	#21445292	1	2	yes	—
225	C₂	#21445319	1	2	yes	—
226	C₂	#21446558	1	1	yes	characteristic
227	C₂	#21446583	1	1	yes	characteristic
228	C₂	#21446600	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519752
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#9618
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₆₁×#335	C₄	semidirect
91. C₈×C₆₁	#106	semidirect
131. #335	Dic₆₁	semidirect
132. C₆₁	#9618	semidirect
207. C₈	#14519709	semidirect

Maximal subgroups

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

subgroup	index
#21446583	2
#21445976	2
#21445686	2
#21443703	2
C₂×C₆₁×#335	2
#9618	61