Subgroups — #29573109

Order 15616.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839892	C₂	1	1	yes	maximal, characteristic
2	#21445821	C₂	1	2	yes	maximal
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	#21443702	C₂	1	2	yes	maximal
5	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	#16839892	—	2	2	no	—
8	#16839892	C₄	1	4	yes	complemented
9	C₂×C₄⋊#92418	C₂²	1	2	yes	—
10	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
11	C₂×#14519706	—	2	1	no	—
12	C₂×#14519700	—	2	2	no	—
13	C₂×#14519687	—	2	1	no	—
14	#16839524	—	2	2	no	—
15	#16839513	—	2	2	no	—
16	C₂×#107×C₆₁	—	2	1	no	—
17	C₂×#107×C₆₁	—	2	1	no	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#384	C₂²	1	2	yes	—
20	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
21	C₂²×#92418	—	2	1	no	—
22	C₂²×#92418	—	2	2	no	—
23	C₄×Dic₁₂₂	—	2	1	no	—
24	C₄×Dic₁₂₂	—	2	1	no	—
25	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₂₂	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
28	C₂×#11758196	—	2	1	no	—
29	C₂×C₄×Dic₆₁	—	2	1	no	—
30	C₄⋊#92418	—	2	1	no	—
31	C₄⋊#92418	—	2	2	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	C₂×C₄	1	2	yes	—
34	#14519706	—	2	4	no	—
35	#14519700	—	4	4	no	—
36	#14519687	—	2	4	no	—
37	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	#107×C₆₁	—	4	2	no	—
44	#107×C₆₁	—	4	2	no	—
45	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₁	D₄	1	2	yes	—
47	C₂²×C₄×C₆₁	—	4	2	no	—
48	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
51	C₂²×Dic₆₁	—	2	1	no	—
52	C₂²×Dic₆₁	—	2	1	no	—
53	C₂×Dic₁₂₂	—	2	2	no	—
54	C₂×Dic₁₂₂	—	2	2	no	—
55	C₂×Dic₁₂₂	—	2	1	no	—
56	C₂×Dic₁₂₂	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₁	—	2	1	no	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₆₁	—	4	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	4	2	no	—
66	C₄⋊C₄×C₆₁	—	4	2	no	—
67	C₄⋊C₄×C₆₁	—	8	2	no	—
68	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₆₁	#41	1	1	yes	characteristic
72	C₄⋊Dic₆₁	—	2	1	no	—
73	C₄⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
74	#11758196	—	2	2	no	—
75	#11758196	—	2	2	no	—
76	C₄×Dic₆₁	—	2	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₂×#92418	—	2	2	no	—
79	C₂×#92418	—	4	1	no	—
80	C₂×#92418	—	4	1	no	—
81	C₂×#92418	—	4	2	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂×#92418	—	4	2	no	—
84	C₂³×C₆₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	8	2	no	—
88	C₂×C₄×C₆₁	—	8	2	no	—
89	C₂×C₄×C₆₁	—	8	2	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	C₂²⋊Q₈	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	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
101	C₂×C₄×C₆₁	#137	1	2	yes	—
102	C₂×Dic₆₁	—	2	2	no	—
103	C₂×Dic₆₁	—	2	2	no	—
104	C₂×Dic₆₁	—	4	1	no	—
105	C₂×Dic₆₁	—	4	1	no	—
106	C₂×Dic₆₁	—	4	1	no	—
107	C₂×Dic₆₁	—	4	1	no	—
108	Dic₁₂₂	—	2	4	no	—
109	Dic₁₂₂	—	4	2	no	—
110	C₈×C₆₁	—	2	2	no	—
111	C₈×C₆₁	#103	1	4	yes	complemented
112	#92418	—	4	2	no	—
113	#92418	—	8	2	no	—
114	#8653	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	#380	1	1	yes	characteristic
116	C₂²×C₆₁	#460	1	1	yes	characteristic
117	C₂²×C₆₁	#462	1	1	yes	characteristic
118	C₂²×C₆₁	#469	1	1	yes	characteristic
119	C₂²×C₆₁	#470	1	1	yes	characteristic
120	C₂²×C₆₁	#472	1	1	yes	characteristic
121	C₂²×C₆₁	#473	1	1	yes	characteristic
122	C₄×C₆₁	—	2	1	no	—
123	C₄×C₆₁	—	2	1	no	—
124	C₄×C₆₁	—	16	2	no	—
125	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
126	C₄×C₆₁	#417	1	1	yes	characteristic
127	C₄×C₆₁	#428	1	2	yes	—
128	C₄×C₆₁	#468	1	2	yes	—
129	C₄×C₆₁	#475	1	1	yes	characteristic
130	C₄×C₆₁	#477	1	1	yes	characteristic
131	Dic₆₁	—	4	2	no	—
132	Dic₆₁	—	4	2	no	—
133	C₂×#446	—	61	1	no	—
134	#1845	D₆₁	1	1	yes	characteristic
135	#1803	—	61	2	no	—
136	C₂×#334	—	61	1	no	—
137	#1296	—	61	2	no	—
138	C₂×C₆₁	#1688	1	1	yes	characteristic
139	C₂×C₆₁	#1690	1	1	yes	characteristic
140	C₂×C₆₁	#1694	1	1	yes	characteristic
141	C₂×C₆₁	#1696	1	1	yes	characteristic
142	C₂×C₆₁	#1803	1	1	yes	characteristic
143	C₂×C₆₁	#1809	1	1	yes	characteristic
144	C₂×C₆₁	#1816	1	1	yes	characteristic
145	C₂×C₄×Q₈	—	61	1	no	—
146	#446	—	61	4	no	—
147	#446	—	122	2	no	—
148	C₂×#107	—	2	1	no	—
149	C₂×#107	—	2	1	no	—
150	C₂×C₄⋊C₈	—	61	1	no	—
151	C₂×C₄⋊C₈	—	61	2	no	—
152	C₂×C₄⋊C₈	—	122	1	no	—
153	C₂×#103	—	122	2	no	—
154	C₂×#97	—	122	1	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	#384	C₂×D₆₁	1	2	yes	—
157	#340	—	122	2	no	—
158	#334	—	122	2	no	—
159	C₆₁	#8653	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	61	1	no	—
161	C₂²×C₈	—	122	1	no	—
162	C₂²×C₈	—	122	2	no	—
163	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₄×Q₈	—	61	2	no	—
166	C₄×Q₈	—	61	2	no	—
167	C₄×Q₈	—	122	1	no	—
168	C₄×Q₈	—	122	1	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	61	1	no	—
173	C₂×C₄⋊C₄	—	122	1	no	—
174	C₂×C₄²	—	122	1	no	—
175	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
176	#107	—	4	2	no	—
177	#107	—	4	2	no	—
178	C₄⋊C₈	—	61	2	no	—
179	C₄⋊C₈	—	122	4	no	—
180	C₄⋊C₈	—	122	1	no	—
181	C₄⋊C₈	—	122	2	no	—
182	C₄⋊C₈	—	122	2	no	—
183	#103	—	244	4	no	—
184	#97	—	122	4	no	—
185	C₄×C₈	C₄×D₆₁	1	2	yes	—
186	C₄×C₈	D₂₄₄	1	2	yes	—
187	C₂×Q₈	—	61	2	no	—
188	C₂×Q₈	—	122	1	no	—
189	C₂×Q₈	—	122	2	no	—
190	C₂×Q₈	—	122	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	122	1	no	—
193	C₂²×C₄	—	122	1	no	—
194	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
195	C₂²×C₄	#11758224	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	122	2	no	—
200	C₂×C₈	—	244	2	no	—
201	C₂×C₈	—	244	2	no	—
202	C₂×C₈	—	244	2	no	—
203	C₂×C₈	—	244	1	no	—
204	C₂×C₈	—	244	1	no	—
205	C₂×C₈	#11758198	1	2	yes	—
206	C₂×C₈	#11758198	1	2	yes	—
207	C₂×C₈	#11758200	1	2	yes	—
208	C₂×C₈	#11758202	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	8	2	no	—
214	C₄⋊C₄	—	61	2	no	—
215	C₄⋊C₄	—	122	2	no	—
216	C₄⋊C₄	—	122	1	no	—
217	C₄⋊C₄	—	122	2	no	—
218	C₄²	—	122	2	no	—
219	C₄²	—	122	2	no	—
220	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
221	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
222	C₄²	Q₈×D₆₁	1	1	yes	characteristic
223	C₄²	#11758226	1	1	yes	characteristic
224	C₂³	#14519748	1	1	yes	characteristic
225	Q₈	—	122	4	no	—
226	Q₈	—	244	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	122	2	no	—
233	C₂×C₄	—	122	2	no	—
234	C₂×C₄	—	244	1	no	—
235	C₂×C₄	—	244	1	no	—
236	C₂×C₄	—	244	1	no	—
237	C₂×C₄	—	244	1	no	—
238	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
240	C₂×C₄	#14519735	1	1	yes	characteristic
241	C₂×C₄	#14519752	1	1	yes	characteristic
242	C₂×C₄	#14519753	1	1	yes	characteristic
243	C₂×C₄	#14519757	1	1	yes	characteristic
244	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
245	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
246	C₂×C₄	#14519801	1	2	yes	—
247	C₂×C₄	#14519803	1	1	yes	characteristic
248	C₂×C₄	#14519808	1	1	yes	characteristic
249	C₂×C₄	#14519809	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	488	2	no	—
252	C₈	—	244	2	no	—
253	C₈	#14519700	1	4	yes	complemented
254	C₂²	#16839619	1	1	yes	characteristic
255	C₂²	#16839959	1	1	yes	characteristic
256	C₂²	#16839969	1	1	yes	characteristic
257	C₂²	#16840012	1	1	yes	characteristic
258	C₂²	#16840017	1	1	yes	characteristic
259	C₂²	#16840029	1	1	yes	characteristic
260	C₂²	#16840034	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	16	2	no	—
264	C₄	—	244	2	no	—
265	C₄	—	244	2	no	—
266	C₄	C₂×#14519700	1	1	yes	characteristic
267	C₄	#16839761	1	1	yes	characteristic
268	C₄	#16839803	1	2	yes	—
269	C₄	#16840009	1	2	yes	—
270	C₄	#16840044	1	1	yes	characteristic
271	C₄	#16840054	1	1	yes	characteristic
272	C₂	#21445300	1	1	yes	characteristic
273	C₂	#21445310	1	1	yes	characteristic
274	C₂	#21445334	1	1	yes	characteristic
275	C₂	#21445344	1	1	yes	characteristic
276	C₂	#21445821	1	1	yes	characteristic
277	C₂	#21445853	1	1	yes	characteristic
278	C₂	#21445884	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1845	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1845	D₆₁
hypocenter	C₆₁	#8653
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
8. #16839892	C₄	semidirect
111. C₈×C₆₁	#103	semidirect
159. C₆₁	#8653	semidirect
253. C₈	#14519700	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839892	2
#21445821	2
C₂×#16839513	2
#21443702	2
C₆₁×#1845	2
#8653	61