Subgroups — #29578755

Order 15616.

310 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	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21445686	C₂	1	1	yes	maximal, characteristic
3	#21444590	C₂	1	2	yes	maximal
4	#21443721	C₂	1	2	yes	maximal
5	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	—	2	1	no	—
7	C₂×#14519713	—	2	1	no	—
8	#16839786	C₂²	1	2	yes	—
9	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
10	C₂×#14519687	—	2	1	no	—
11	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	—	2	1	no	—
13	#16839513	—	2	4	no	—
14	C₆₁×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#335	C₄	1	2	yes	complemented
17	C₆₁×#334	C₄	1	2	yes	complemented
18	C₂×#11758194	—	2	2	no	—
19	C₂²×#92418	—	2	1	no	—
20	C₂×C₄⋊Dic₆₁	—	2	1	no	—
21	C₂×C₄×Dic₆₁	—	2	1	no	—
22	#14519713	—	4	2	no	—
23	#14519713	—	4	2	no	—
24	C₄⋊#92418	—	2	1	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	D₄	1	2	yes	—
28	#14519687	—	4	2	no	—
29	#14519684	Q₈	1	2	yes	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	2	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	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₄⋊Dic₆₁	—	4	2	no	—
58	C₄×Dic₆₁	—	4	1	no	—
59	C₄×Dic₆₁	—	4	1	no	—
60	#11758194	—	4	2	no	—
61	#11758194	—	4	2	no	—
62	C₂×#92418	—	2	2	no	—
63	C₂×#92418	—	4	1	no	—
64	C₂×#92418	—	4	1	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	4	1	no	—
69	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	—	4	2	no	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	4	4	no	—
85	C₈×C₆₁	#106	1	2	yes	—
86	C₈×C₆₁	#106	1	2	yes	—
87	#92418	—	4	2	no	—
88	#92418	—	8	2	no	—
89	#9631	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	#389	1	1	yes	characteristic
93	C₂²×C₆₁	#475	1	1	yes	characteristic
94	C₂²×C₆₁	#476	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
99	C₄×C₆₁	#428	1	1	yes	characteristic
100	C₄×C₆₁	#444	1	2	yes	—
101	C₄×C₆₁	#460	1	1	yes	characteristic
102	C₄×C₆₁	#461	1	1	yes	characteristic
103	Dic₆₁	—	8	2	no	—
104	#1848	—	61	1	no	—
105	#1770	—	61	1	no	—
106	#1570	—	61	2	no	—
107	#1564	D₆₁	1	1	yes	characteristic
108	#1303	—	61	2	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	—	2	1	no	—
111	C₂×C₆₁	#1946	1	1	yes	characteristic
112	C₂×C₆₁	#1948	1	1	yes	characteristic
113	C₂×C₆₁	#1953	1	1	yes	characteristic
114	#431	C₂×D₆₁	1	2	yes	—
115	C₂×#108	—	122	1	no	—
116	C₂×#108	—	122	1	no	—
117	#423	—	61	2	no	—
118	C₂×C₄⋊C₈	—	61	1	no	—
119	C₂×#97	—	122	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	61	1	no	—
122	#336	—	122	1	no	—
123	#335	Dic₆₁	1	2	yes	complemented
124	#334	—	122	4	no	—
125	#334	Dic₆₁	1	2	yes	complemented
126	C₆₁	#9631	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	122	2	no	—
128	C₂²×C₈	—	122	1	no	—
129	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
130	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	122	1	no	—
132	C₂×C₄²	—	122	1	no	—
133	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
134	#108	—	244	2	no	—
135	#108	—	244	2	no	—
136	C₄⋊C₈	—	61	2	no	—
137	C₄⋊C₈	—	122	2	no	—
138	C₄⋊C₈	—	122	1	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	244	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
145	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
148	#95	—	61	2	no	—
149	C₂²×C₄	—	122	2	no	—
150	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
151	C₂²×C₄	#11758224	1	1	yes	characteristic
152	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
153	M₁₆	—	244	2	no	—
154	M₁₆	—	244	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	122	2	no	—
159	C₂×C₈	—	244	2	no	—
160	C₂×C₈	—	244	1	no	—
161	C₂×C₈	—	244	1	no	—
162	C₂×C₈	#11758190	1	2	yes	—
163	C₂×C₈	#11758190	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
168	C₄⋊C₄	—	244	2	no	—
169	C₄²	—	244	1	no	—
170	C₄²	—	244	1	no	—
171	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
172	C₄²	D₄×D₆₁	1	1	yes	characteristic
173	C₄²	#11758226	1	1	yes	characteristic
174	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
175	C₂³	#14519757	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	244	2	no	—
179	C₂×C₄	—	244	2	no	—
180	C₂×C₄	—	244	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
184	C₂×C₄	#14519752	1	1	yes	characteristic
185	C₂×C₄	#14519768	1	1	yes	characteristic
186	C₂×C₄	#14519776	1	1	yes	characteristic
187	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
188	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
189	C₂×C₄	#14519805	1	1	yes	characteristic
190	C₂×C₄	#14519810	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	488	2	no	—
195	C₈	—	244	2	no	—
196	C₈	#14519709	1	2	yes	—
197	C₈	#14519709	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16839661	1	1	yes	characteristic
201	C₂²	#16840048	1	1	yes	characteristic
202	C₂²	#16840053	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	488	2	no	—
207	C₄	C₂×#14519709	1	1	yes	characteristic
208	C₄	#16839804	1	1	yes	characteristic
209	C₄	#16839884	1	2	yes	—
210	C₄	#16839963	1	1	yes	characteristic
211	C₄	#16839968	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#21446573	1	1	yes	characteristic
215	C₂	#21446583	1	1	yes	characteristic
216	C₂	#21446610	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519752
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1564	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1564	D₆₁
hypocenter	C₆₁	#9631
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. C₆₁×#335	C₄	semidirect
17. C₆₁×#334	C₄	semidirect
123. #335	Dic₆₁	semidirect
125. #334	Dic₆₁	semidirect
126. C₆₁	#9631	semidirect

Maximal subgroups

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

subgroup	index
#21445991	2
#21445686	2
#21444590	2
#21443721	2
C₆₁×#1564	2
#9631	61