Subgroups — #29573683

Order 15616.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445971	C₂	1	1	yes	maximal, characteristic
2	#21445565	C₂	1	1	yes	maximal, characteristic
3	#21443653	C₂	1	4	yes	maximal
4	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	1	no	—
6	C₂×C₄⋊#92418	—	2	1	no	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	2	yes	—
9	#16839517	C₄	1	2	yes	—
10	#16839517	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
12	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
14	C₄³×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×#92418	—	2	2	no	—
16	C₂²×#92418	C₂×C₄	1	4	yes	—
17	C₄⋊#92418	—	2	4	no	—
18	C₄⋊#92418	—	4	2	no	—
19	C₄⋊#92418	—	4	2	no	—
20	C₄×#92418	—	2	2	no	—
21	C₄×#92418	—	2	2	no	—
22	C₂²×C₈×C₆₁	—	2	2	no	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	4	no	—
31	C₄⋊C₈×C₆₁	—	4	2	no	—
32	C₄⋊C₈×C₆₁	—	4	2	no	—
33	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₄²×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	2	2	no	—
45	C₄²×C₆₁	—	4	1	no	—
46	C₄²×C₆₁	—	4	1	no	—
47	C₄²×C₆₁	D₈	1	2	yes	—
48	C₄²×C₆₁	SD₁₆	1	2	yes	—
49	C₄²×C₆₁	SD₁₆	1	2	yes	—
50	C₄²×C₆₁	Q₁₆	1	2	yes	—
51	C₄²×C₆₁	#41	1	2	yes	—
52	C₄²×C₆₁	#41	1	2	yes	—
53	C₂×#92418	—	2	4	no	—
54	C₂×#92418	—	2	4	no	—
55	C₂×#92418	—	2	4	no	—
56	C₂×#92418	—	4	2	no	—
57	C₂×#92418	—	4	2	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	1	no	—
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	#106	1	2	yes	—
71	C₂×C₄×C₆₁	#106	1	2	yes	—
72	C₂×C₄×C₆₁	#107	1	2	yes	—
73	C₂×C₄×C₆₁	#107	1	2	yes	—
74	C₂×C₄×C₆₁	#117	1	2	yes	—
75	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#125	1	2	yes	—
78	C₂×C₄×C₆₁	#131	1	2	yes	—
79	C₂×C₄×C₆₁	#131	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₆₁	—	8	2	no	—
85	#92418	—	4	4	no	—
86	#92418	—	8	2	no	—
87	#8787	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	#382	1	1	yes	characteristic
89	C₂²×C₆₁	#424	1	1	yes	characteristic
90	C₂²×C₆₁	#424	1	1	yes	characteristic
91	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
92	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
93	C₂²×C₆₁	#430	1	1	yes	characteristic
94	C₂²×C₆₁	#430	1	1	yes	characteristic
95	C₄×C₆₁	—	2	2	no	—
96	C₄×C₆₁	—	2	2	no	—
97	C₄×C₆₁	—	2	2	no	—
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	#433	1	4	yes	—
101	C₄×C₆₁	#486	1	1	yes	characteristic
102	C₄×C₆₁	#486	1	1	yes	characteristic
103	C₄×C₆₁	#487	1	1	yes	characteristic
104	C₄×C₆₁	#487	1	1	yes	characteristic
105	#1842	—	61	1	no	—
106	#1842	D₆₁	1	1	yes	characteristic
107	#1744	—	61	1	no	—
108	#1277	—	61	4	no	—
109	C₂×C₆₁	#1570	1	1	yes	characteristic
110	C₂×C₆₁	#1570	1	1	yes	characteristic
111	C₂×C₆₁	#1571	1	1	yes	characteristic
112	C₂×C₆₁	#1571	1	1	yes	characteristic
113	C₂×C₆₁	#1832	1	1	yes	characteristic
114	C₂×C₆₁	#1834	1	1	yes	characteristic
115	C₂×C₆₁	#1838	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	61	2	no	—
119	C₂×C₄⋊C₈	—	122	1	no	—
120	C₂×C₄⋊C₈	—	122	1	no	—
121	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
122	C₂×C₄×C₈	—	61	2	no	—
123	C₄³	C₂×D₆₁	1	1	yes	characteristic
124	#336	—	61	2	no	—
125	#336	—	61	2	no	—
126	C₆₁	#8787	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	61	4	no	—
129	C₂²×C₈	—	122	2	no	—
130	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
131	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	—	122	4	no	—
141	C₄⋊C₈	—	244	2	no	—
142	C₄⋊C₈	—	244	2	no	—
143	C₄×C₈	—	122	2	no	—
144	C₄×C₈	—	122	2	no	—
145	C₂²×C₄	#11758196	1	1	yes	characteristic
146	C₂²×C₄	#11758196	1	1	yes	characteristic
147	C₂²×C₄	#11758196	1	1	yes	characteristic
148	C₂²×C₄	#11758196	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	122	4	no	—
156	C₂×C₈	—	122	4	no	—
157	C₂×C₈	—	122	4	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	—	244	2	no	—
160	C₂×C₈	—	244	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#11758199	1	2	yes	—
166	C₄²	#11758200	1	2	yes	—
167	C₄²	#11758201	1	2	yes	—
168	C₄²	#11758202	1	2	yes	—
169	C₄²	#11758224	1	2	yes	—
170	C₄²	#11758226	1	2	yes	—
171	C₂³	C₂×#11758196	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	#14519708	1	2	yes	—
183	C₂×C₄	#14519708	1	2	yes	—
184	C₂×C₄	#14519710	1	2	yes	—
185	C₂×C₄	#14519710	1	2	yes	—
186	C₂×C₄	#14519735	1	2	yes	—
187	C₂×C₄	#14519764	1	1	yes	characteristic
188	C₂×C₄	#14519766	1	1	yes	characteristic
189	C₂×C₄	#14519768	1	2	yes	—
190	C₂×C₄	#14519783	1	2	yes	—
191	C₂×C₄	#14519785	1	2	yes	—
192	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
193	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
194	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
195	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	488	2	no	—
198	C₈	—	244	4	no	—
199	C₂²	#16839627	1	1	yes	characteristic
200	C₂²	#16839788	1	1	yes	characteristic
201	C₂²	#16839790	1	1	yes	characteristic
202	C₂²	C₂×#14519708	1	1	yes	characteristic
203	C₂²	C₂×#14519710	1	1	yes	characteristic
204	C₂²	#16839810	1	1	yes	characteristic
205	C₂²	#16839810	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#16839822	1	4	yes	—
212	C₄	#16840108	1	1	yes	characteristic
213	C₄	#16840110	1	1	yes	characteristic
214	C₄	#16840112	1	1	yes	characteristic
215	C₄	#16840114	1	1	yes	characteristic
216	C₂	#21444589	1	1	yes	characteristic
217	C₂	#21444591	1	1	yes	characteristic
218	C₂	#21444593	1	1	yes	characteristic
219	C₂	#21444595	1	1	yes	characteristic
220	C₂	#21445938	1	1	yes	characteristic
221	C₂	#21445946	1	1	yes	characteristic
222	C₂	#21445959	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1842	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1842	D₆₁
hypocenter	C₆₁	#8787
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
126. C₆₁	#8787	semidirect

Maximal subgroups

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

subgroup	index
#21445971	2
#21445565	2
#21443653	2
C₆₁×#1842	2
#8787	61