Subgroups — #29573269

Order 15616.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454753	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	#21444510	C₂	1	2	yes	maximal
4	#21443693	C₂	1	2	yes	maximal
5	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839892	—	2	1	no	—
8	#16839892	C₄	1	2	yes	—
9	C₈×Dic₁₂₂	—	2	1	no	—
10	C₈×Dic₁₂₂	C₄	1	2	yes	—
11	#16839788	—	2	1	no	—
12	#16839786	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#92418	C₂²	1	2	yes	—
14	#16839717	—	2	1	no	—
15	#16839515	—	2	1	no	—
16	#16839515	—	2	1	no	—
17	#16839510	—	2	2	no	—
18	#16839507	—	2	2	no	—
19	C₂×#108×C₆₁	—	2	1	no	—
20	C₂×#108×C₆₁	—	2	1	no	—
21	C₆₁×#423	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#11758194	—	2	1	no	—
24	C₂²×#92418	—	2	2	no	—
25	#14519777	D₄	1	1	yes	characteristic
26	#14519777	D₄	1	1	yes	characteristic
27	#14519769	—	2	1	no	—
28	#14519756	—	2	1	no	—
29	#14519756	Q₈	1	2	yes	—
30	C₄×Dic₁₂₂	—	2	1	no	—
31	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
32	#14519736	—	2	1	no	—
33	C₄⋊#92418	—	2	1	no	—
34	C₄⋊#92418	—	2	2	no	—
35	C₄⋊#92418	—	2	2	no	—
36	C₄⋊#92418	C₂×C₄	1	2	yes	—
37	#14519706	—	2	2	no	—
38	#14519706	—	2	2	no	—
39	#14519687	—	2	2	no	—
40	C₈×Dic₆₁	—	2	2	no	—
41	C₂×M₁₆×C₆₁	—	2	2	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	#108×C₆₁	—	4	2	no	—
46	#108×C₆₁	—	4	2	no	—
47	C₄⋊C₈×C₆₁	—	2	2	no	—
48	C₄⋊C₈×C₆₁	—	2	2	no	—
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
56	C₂×Dic₁₂₂	—	2	1	no	—
57	C₂×Dic₁₂₂	C₄⋊C₄	1	2	yes	—
58	M₁₆×C₆₁	—	4	2	no	—
59	M₁₆×C₆₁	—	4	2	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	—	4	2	no	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
68	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
69	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₁	#41	1	1	yes	characteristic
73	C₂²⋊Dic₆₁	—	2	2	no	—
74	C₄⋊Dic₆₁	—	2	1	no	—
75	C₄⋊Dic₆₁	—	2	1	no	—
76	C₄⋊Dic₆₁	—	2	1	no	—
77	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
78	#11758196	—	2	2	no	—
79	#11758196	—	2	2	no	—
80	C₄×Dic₆₁	—	2	2	no	—
81	#11758194	—	4	1	no	—
82	#11758194	—	4	1	no	—
83	C₂×#92418	—	2	2	no	—
84	C₂×#92418	—	4	2	no	—
85	C₂×#92418	—	8	2	no	—
86	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	4	1	no	—
90	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
100	C₂×Dic₆₁	—	2	2	no	—
101	C₂×Dic₆₁	—	2	2	no	—
102	Dic₁₂₂	—	4	1	no	—
103	Dic₁₂₂	—	4	1	no	—
104	Dic₁₂₂	—	4	2	no	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	1	no	—
107	C₈×C₆₁	—	8	2	no	—
108	C₈×C₆₁	#103	1	2	yes	—
109	C₈×C₆₁	#103	1	2	yes	—
110	#92418	—	4	2	no	—
111	#92418	—	8	2	no	—
112	#8685	—	61	1	no	maximal, Sylow
113	C₂²×C₆₁	—	2	1	no	—
114	C₂²×C₆₁	—	2	1	no	—
115	C₂²×C₆₁	#380	1	1	yes	characteristic
116	C₂²×C₆₁	#475	1	1	yes	characteristic
117	C₂²×C₆₁	#477	1	1	yes	characteristic
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
122	C₄×C₆₁	#417	1	1	yes	characteristic
123	C₄×C₆₁	#460	1	1	yes	characteristic
124	C₄×C₆₁	#462	1	1	yes	characteristic
125	C₄×C₆₁	#472	1	1	yes	characteristic
126	C₄×C₆₁	#473	1	1	yes	characteristic
127	Dic₆₁	—	4	2	no	—
128	Dic₆₁	—	4	2	no	—
129	#2960	—	61	1	no	—
130	#1848	D₆₁	1	1	yes	characteristic
131	C₂×#335	—	61	1	no	—
132	#1550	—	61	2	no	—
133	#1292	—	61	2	no	—
134	C₂×C₆₁	—	2	1	no	—
135	C₂×C₆₁	—	2	1	no	—
136	C₂×C₆₁	#1803	1	1	yes	characteristic
137	C₂×C₆₁	#1813	1	1	yes	characteristic
138	C₂×C₆₁	#1816	1	1	yes	characteristic
139	#533	—	61	1	no	—
140	#446	—	61	2	no	—
141	#446	—	122	1	no	—
142	C₈×Q₈	—	61	2	no	—
143	C₈×Q₈	—	122	1	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	—	2	1	no	—
146	#424	—	122	1	no	—
147	#423	—	61	1	no	—
148	#423	C₂×D₆₁	1	2	yes	—
149	C₂×C₄⋊C₈	—	61	2	no	—
150	#405	—	122	1	no	—
151	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
152	#335	—	122	1	no	—
153	#335	—	122	1	no	—
154	#333	—	122	2	no	—
155	#332	—	122	2	no	—
156	C₆₁	#8685	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	122	1	no	—
159	C₂²×C₈	—	122	2	no	—
160	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	61	1	no	—
163	C₄⋊Q₈	—	61	1	no	—
164	#125	—	122	1	no	—
165	C₂²⋊Q₈	—	61	2	no	—
166	C₂²⋊Q₈	—	122	1	no	—
167	C₄×Q₈	—	61	2	no	—
168	C₄×Q₈	—	122	1	no	—
169	#117	—	122	1	no	—
170	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
171	#108	—	4	2	no	—
172	#108	—	4	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	61	2	no	—
176	C₄⋊C₈	—	122	2	no	—
177	C₄⋊C₈	—	122	2	no	—
178	C₄⋊C₈	—	122	1	no	—
179	C₄⋊C₈	—	122	2	no	—
180	C₄⋊C₈	—	122	2	no	—
181	#97	—	122	2	no	—
182	C₄×C₈	—	122	2	no	—
183	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
184	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
185	C₄×C₈	D₂₄₄	1	1	yes	characteristic
186	C₄×C₈	D₂₄₄	1	1	yes	characteristic
187	C₂×Q₈	—	61	2	no	—
188	C₂×Q₈	—	122	1	no	—
189	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
190	C₂²×C₄	#11758226	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	244	1	no	—
195	M₁₆	—	244	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	122	2	no	—
200	C₂×C₈	—	488	2	no	—
201	C₂×C₈	—	244	2	no	—
202	C₂×C₈	#11758198	1	1	yes	characteristic
203	C₂×C₈	#11758198	1	1	yes	characteristic
204	C₂×C₈	#11758198	1	1	yes	characteristic
205	C₂×C₈	#11758198	1	1	yes	characteristic
206	C₂×C₈	#11758200	1	2	yes	—
207	C₂×C₈	#11758202	1	2	yes	—
208	C₄⋊C₄	—	61	2	no	—
209	C₄⋊C₄	—	122	2	no	—
210	C₄⋊C₄	—	122	1	no	—
211	C₄⋊C₄	—	122	1	no	—
212	C₄⋊C₄	—	122	1	no	—
213	C₄⋊C₄	—	122	2	no	—
214	C₂²⋊C₄	—	122	2	no	—
215	C₄²	—	122	2	no	—
216	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
217	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
218	C₄²	D₄×D₆₁	1	1	yes	characteristic
219	C₄²	#11758224	1	1	yes	characteristic
220	C₂³	#14519753	1	1	yes	characteristic
221	Q₈	—	244	2	no	—
222	Q₈	—	244	1	no	—
223	Q₈	—	244	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	122	2	no	—
228	C₂×C₄	—	122	2	no	—
229	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
231	C₂×C₄	#14519735	1	1	yes	characteristic
232	C₂×C₄	#14519748	1	1	yes	characteristic
233	C₂×C₄	#14519752	1	1	yes	characteristic
234	C₂×C₄	#14519757	1	1	yes	characteristic
235	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
236	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
237	C₂×C₄	#14519803	1	1	yes	characteristic
238	C₂×C₄	#14519808	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	8	2	no	—
242	C₈	—	488	2	no	—
243	C₈	—	244	2	no	—
244	C₈	#14519700	1	2	yes	—
245	C₈	#14519700	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#16839619	1	1	yes	characteristic
249	C₂²	#16840044	1	1	yes	characteristic
250	C₂²	#16840054	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	244	2	no	—
255	C₄	—	244	2	no	—
256	C₄	C₂×#14519700	1	1	yes	characteristic
257	C₄	#16839761	1	1	yes	characteristic
258	C₄	#16839959	1	1	yes	characteristic
259	C₄	#16839969	1	1	yes	characteristic
260	C₄	#16840033	1	1	yes	characteristic
261	C₄	#16840038	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#21445821	1	1	yes	characteristic
265	C₂	#21445866	1	1	yes	characteristic
266	C₂	#21445880	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839619
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#8685
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
156. C₆₁	#8685	semidirect

Maximal subgroups

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

subgroup	index
#21454753	2
C₂×#16839515	2
#21444510	2
#21443693	2
C₆₁×#1848	2
#8685	61