Subgroups — #29573059

Order 15616.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	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×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₂₂	—	2	2	no	—
8	C₈×Dic₁₂₂	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×C₈×Dic₆₁	—	2	1	no	—
14	#16839524	—	2	2	no	—
15	#16839513	—	2	2	no	—
16	C₂×#106×C₆₁	—	2	1	no	—
17	C₂×#106×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	C₈×Dic₆₁	—	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	#106×C₆₁	—	4	2	no	—
44	#106×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	1	no	—
54	C₂×Dic₁₂₂	—	2	2	no	—
55	C₂×Dic₁₂₂	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₁₂₂	SD₁₆	1	4	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	2	no	—
86	C₂×C₄×C₆₁	—	8	2	no	—
87	C₂×C₄×C₆₁	—	8	2	no	—
88	C₂×C₄×C₆₁	—	8	2	no	—
89	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#135	1	2	yes	—
99	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
101	C₂×Dic₆₁	—	2	2	no	—
102	C₂×Dic₆₁	—	2	4	no	—
103	C₂×Dic₆₁	—	2	2	no	—
104	C₂×Dic₆₁	—	4	2	no	—
105	Dic₁₂₂	—	2	2	no	—
106	Dic₁₂₂	—	2	4	no	—
107	Dic₁₂₂	#106	1	4	yes	complemented
108	C₈×C₆₁	—	2	2	no	—
109	C₈×C₆₁	#103	1	4	yes	complemented
110	#92418	—	4	2	no	—
111	#92418	—	8	2	no	—
112	#8643	—	61	1	no	maximal, Sylow
113	C₂²×C₆₁	#380	1	1	yes	characteristic
114	C₂²×C₆₁	#460	1	1	yes	characteristic
115	C₂²×C₆₁	#462	1	1	yes	characteristic
116	C₂²×C₆₁	#465	1	2	yes	—
117	C₂²×C₆₁	#471	1	2	yes	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	2	1	no	—
120	C₄×C₆₁	—	16	2	no	—
121	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
122	C₄×C₆₁	#417	1	1	yes	characteristic
123	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
124	C₄×C₆₁	#427	1	1	yes	characteristic
125	C₄×C₆₁	#465	1	2	yes	—
126	C₄×C₆₁	#475	1	1	yes	characteristic
127	C₄×C₆₁	#477	1	1	yes	characteristic
128	Dic₆₁	—	2	4	no	—
129	Dic₆₁	—	4	2	no	—
130	C₂×C₈×Q₈	—	61	1	no	—
131	#1844	D₆₁	1	1	yes	characteristic
132	#1803	—	61	2	no	—
133	C₂×#334	—	61	1	no	—
134	#1296	—	61	2	no	—
135	C₂×C₆₁	#1666	1	2	yes	—
136	C₂×C₆₁	#1672	1	2	yes	—
137	C₂×C₆₁	#1803	1	1	yes	characteristic
138	C₂×C₆₁	#1807	1	1	yes	characteristic
139	C₂×C₆₁	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	61	1	no	—
141	C₈×Q₈	—	61	4	no	—
142	C₈×Q₈	—	122	2	no	—
143	C₂×#106	—	2	1	no	—
144	C₂×#106	—	2	1	no	—
145	C₂×C₄⋊C₈	—	61	1	no	—
146	C₂×C₄⋊C₈	—	61	2	no	—
147	C₂×C₄⋊C₈	—	122	1	no	—
148	C₂×#103	—	122	2	no	—
149	C₂×C₄×C₈	—	122	1	no	—
150	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
151	#384	C₂×D₆₁	1	2	yes	—
152	#340	—	122	2	no	—
153	#334	—	122	2	no	—
154	C₆₁	#8643	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	61	1	no	—
156	C₂²×C₈	—	122	1	no	—
157	C₂²×C₈	—	122	2	no	—
158	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄×Q₈	—	61	2	no	—
161	C₄×Q₈	—	61	2	no	—
162	C₄×Q₈	—	122	1	no	—
163	C₄×Q₈	—	122	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	61	1	no	—
168	C₂×C₄⋊C₄	—	122	1	no	—
169	C₂×C₄²	—	122	1	no	—
170	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
171	#106	—	4	2	no	—
172	#106	—	4	2	no	—
173	C₄⋊C₈	—	61	2	no	—
174	C₄⋊C₈	—	122	4	no	—
175	C₄⋊C₈	—	122	1	no	—
176	C₄⋊C₈	—	122	2	no	—
177	C₄⋊C₈	—	122	2	no	—
178	#103	—	244	4	no	—
179	C₄×C₈	—	122	4	no	—
180	C₄×C₈	C₄×D₆₁	1	2	yes	—
181	C₄×C₈	D₂₄₄	1	2	yes	—
182	C₂×Q₈	—	61	2	no	—
183	C₂×Q₈	—	61	4	no	—
184	C₂×Q₈	—	122	1	no	—
185	C₂×Q₈	—	122	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	122	1	no	—
188	C₂²×C₄	—	122	1	no	—
189	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
190	C₂²×C₄	#11758224	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	122	2	no	—
195	C₂×C₈	—	244	2	no	—
196	C₂×C₈	—	244	2	no	—
197	C₂×C₈	—	244	2	no	—
198	C₂×C₈	—	244	1	no	—
199	C₂×C₈	—	244	1	no	—
200	C₂×C₈	#11758198	1	2	yes	—
201	C₂×C₈	#11758198	1	2	yes	—
202	C₂×C₈	#11758200	1	2	yes	—
203	C₂×C₈	#11758202	1	2	yes	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	—	61	2	no	—
210	C₄⋊C₄	—	122	2	no	—
211	C₄⋊C₄	—	122	1	no	—
212	C₄⋊C₄	—	122	2	no	—
213	C₄²	—	122	2	no	—
214	C₄²	—	122	2	no	—
215	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
217	C₄²	Q₈×D₆₁	1	1	yes	characteristic
218	C₄²	#11758226	1	1	yes	characteristic
219	C₂³	#14519748	1	1	yes	characteristic
220	Q₈	—	61	4	no	—
221	Q₈	—	122	4	no	—
222	Q₈	—	122	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	122	4	no	—
228	C₂×C₄	—	122	2	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	244	2	no	—
231	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
233	C₂×C₄	#14519735	1	1	yes	characteristic
234	C₂×C₄	#14519752	1	1	yes	characteristic
235	C₂×C₄	#14519753	1	1	yes	characteristic
236	C₂×C₄	#14519757	1	1	yes	characteristic
237	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
238	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
239	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
240	C₂×C₄	#14519798	1	2	yes	—
241	C₂×C₄	#14519803	1	1	yes	characteristic
242	C₂×C₄	#14519808	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	488	2	no	—
245	C₈	—	244	2	no	—
246	C₈	#14519700	1	4	yes	complemented
247	C₂²	#16839619	1	1	yes	characteristic
248	C₂²	#16839959	1	1	yes	characteristic
249	C₂²	#16839969	1	1	yes	characteristic
250	C₂²	#16839988	1	2	yes	—
251	C₂²	#16840022	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	16	2	no	—
255	C₄	—	122	4	no	—
256	C₄	—	244	2	no	—
257	C₄	C₂×#14519700	1	1	yes	characteristic
258	C₄	#16839761	1	1	yes	characteristic
259	C₄	#106×D₆₁	1	1	yes	characteristic
260	C₄	#16839797	1	1	yes	characteristic
261	C₄	#16839992	1	2	yes	—
262	C₄	#16840044	1	1	yes	characteristic
263	C₄	#16840054	1	1	yes	characteristic
264	C₂	#21445164	1	2	yes	—
265	C₂	#21445198	1	2	yes	—
266	C₂	#21445821	1	1	yes	characteristic
267	C₂	#21445841	1	1	yes	characteristic
268	C₂	#21445879	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519748
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1844	C₂
socle, prime radical	C₂³×C₆₁	#121
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1844	D₆₁
hypocenter	C₆₁	#8643
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₂₂	C₄	semidirect
107. Dic₁₂₂	#106	semidirect
109. C₈×C₆₁	#103	semidirect
154. C₆₁	#8643	semidirect
246. 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×C₈×Dic₁₂₂	2
#21445821	2
C₂×#16839513	2
#21443702	2
C₆₁×#1844	2
#8643	61