Subgroups — #29589992

Order 15616.

314 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447618	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	#21443783	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₂²⋊#11758185	—	2	1	no	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	#16839515	—	2	1	no	—
10	#16839515	C₂²	1	2	yes	—
11	C₆₁×#533	—	2	1	no	—
12	C₆₁×#500	C₄	1	1	yes	characteristic
13	C₆₁×#494	C₄	1	1	yes	characteristic
14	C₆₁×#487	—	2	1	no	—
15	C₆₁×#467	—	2	1	no	—
16	C₆₁×#467	C₄	1	2	yes	—
17	C₄×Q₁₆×C₆₁	—	2	1	no	—
18	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	4	1	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	4	1	no	—
24	C₂×Q₁₆×C₆₁	—	2	1	no	—
25	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₆₁	—	2	1	no	—
29	C₄⋊Q₈×C₆₁	—	2	1	no	—
30	#125×C₆₁	—	4	1	no	—
31	C₂²⋊Q₈×C₆₁	—	2	2	no	—
32	C₂²⋊Q₈×C₆₁	—	4	1	no	—
33	C₄×Q₈×C₆₁	—	2	2	no	—
34	C₄×Q₈×C₆₁	—	4	1	no	—
35	#117×C₆₁	—	4	1	no	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	#107×C₆₁	—	2	1	no	—
38	#107×C₆₁	C₂×C₄	1	2	yes	—
39	#103×C₆₁	—	2	2	no	—
40	#103×C₆₁	—	2	2	no	—
41	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₆₁	—	2	2	no	—
46	C₂×Q₈×C₆₁	—	4	1	no	—
47	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
50	Q₁₆×C₆₁	—	2	4	no	—
51	Q₁₆×C₆₁	—	4	2	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	4	1	no	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₆₁	—	2	2	no	—
59	C₄⋊C₄×C₆₁	—	4	1	no	—
60	C₄⋊C₄×C₆₁	—	4	1	no	—
61	C₄⋊C₄×C₆₁	—	4	2	no	—
62	C₄⋊C₄×C₆₁	—	4	1	no	—
63	C₄⋊C₄×C₆₁	—	4	2	no	—
64	C₂²⋊C₄×C₆₁	—	4	2	no	—
65	C₄²×C₆₁	—	4	2	no	—
66	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₁	D₈	1	1	yes	characteristic
68	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂×#92418	—	4	2	no	—
71	C₂×#92418	—	8	2	no	—
72	#11758185	—	4	2	no	—
73	C₂³×C₆₁	#102	1	1	yes	characteristic
74	Q₈×C₆₁	—	4	2	no	—
75	Q₈×C₆₁	—	4	2	no	—
76	Q₈×C₆₁	—	8	2	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	4	1	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	—	4	2	no	—
82	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
86	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	#131	1	2	yes	—
94	#92418	—	8	2	no	—
95	#11646	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	—	2	1	no	—
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	#358	1	2	yes	—
99	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	8	2	no	—
104	C₄×C₆₁	—	8	2	no	—
105	C₄×C₆₁	#358	1	2	yes	—
106	C₄×C₆₁	#408	1	1	yes	characteristic
107	C₄×C₆₁	#421	1	1	yes	characteristic
108	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
109	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
110	#3170	D₆₁	1	1	yes	characteristic
111	#2162	—	61	1	no	—
112	C₂×#335	—	61	1	no	—
113	#1332	—	61	4	no	—
114	C₂×C₆₁	—	2	2	no	—
115	C₂×C₆₁	#1480	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
117	C₂×C₆₁	#2143	1	1	yes	characteristic
118	#533	—	2	1	no	—
119	#500	Dic₆₁	1	1	yes	characteristic
120	#494	Dic₆₁	1	1	yes	characteristic
121	#487	—	2	1	no	—
122	#467	—	2	1	no	—
123	#467	Dic₆₁	1	2	yes	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
126	C₂×C₄⋊C₈	—	122	1	no	—
127	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
128	C₄⋊C₁₆	—	122	1	no	—
129	C₂²⋊C₁₆	—	122	1	no	—
130	C₄×C₁₆	—	61	2	no	—
131	#335	—	61	2	no	—
132	#335	—	122	1	no	—
133	C₆₁	#11646	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×Dic₆₁	1	2	yes	—
136	C₂²×C₈	—	244	1	no	—
137	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	—	2	1	no	—
141	#125	—	4	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	#117	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
148	C₂×C₁₆	—	122	2	no	—
149	#107	—	2	1	no	—
150	#107	C₂×Dic₆₁	1	2	yes	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	—	244	1	no	—
153	#103	—	2	2	no	—
154	#103	—	2	2	no	—
155	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	#11758199	1	1	yes	characteristic
162	C₂²×C₄	#11758200	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	488	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	#11758199	1	1	yes	characteristic
183	C₄²	#11758200	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
185	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
186	C₁₆	—	244	2	no	—
187	C₂³	#14519698	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#14519698	1	1	yes	characteristic
197	C₂×C₄	#14519698	1	1	yes	characteristic
198	C₂×C₄	#14519698	1	1	yes	characteristic
199	C₂×C₄	#14519720	1	2	yes	—
200	C₂×C₄	#14519723	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
202	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
203	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	488	2	no	—
208	C₈	#14519784	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#16839566	1	2	yes	—
212	C₂²	C₂×#14519698	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	#16839566	1	2	yes	—
219	C₄	#16839729	1	1	yes	characteristic
220	C₄	#16839779	1	1	yes	characteristic
221	C₄	C₂×#14519720	1	1	yes	characteristic
222	C₄	C₂×#14519723	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	#21444218	1	1	yes	characteristic
225	C₂	C₂×#16839566	1	1	yes	characteristic
226	C₂	#21447547	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#3170	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	#102
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#3170	D₆₁
hypocenter	C₆₁	#11646
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #21443783	C₂	semidirect
133. C₆₁	#11646	semidirect

Maximal subgroups

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

subgroup	index
#21447618	2
C₂×#16839515	2
#21443783	2
C₆₁×#3170	2
#11646	61