Subgroups — #30597378

Order 15616.

478 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	#21461534	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519832	C₂	1	1	yes	maximal, characteristic
3	#21460641	C₂	1	2	yes	maximal
4	#21460641	C₂	1	2	yes	maximal
5	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519832	C₄	1	2	yes	—
7	C₂×#14519832	C₄	1	2	yes	—
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	C₂×C₄×#92420	C₂²	1	2	yes	—
10	#16840628	C₄	1	2	yes	—
11	#16840628	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
19	C₂²×#92420	C₂×C₄	1	4	yes	—
20	C₂²×#92420	C₂×C₄	1	4	yes	—
21	#14519832	—	2	4	no	—
22	#14519832	C₈	1	8	yes	complemented
23	C₄×#92420	—	2	2	no	—
24	C₄×#92420	—	2	2	no	—
25	C₄×#92420	—	2	2	no	—
26	C₄×#92420	—	2	2	no	—
27	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
35	C₄×#92418	—	2	2	no	—
36	C₈×Dic₆₁	—	2	2	no	—
37	C₈×Dic₆₁	—	2	2	no	—
38	C₈×Dic₆₁	—	2	2	no	—
39	C₈×Dic₆₁	—	2	2	no	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
48	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
49	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×#92420	—	2	4	no	—
52	C₂×#92420	—	2	4	no	—
53	C₂×#92420	—	2	4	no	—
54	C₂×#92420	—	2	4	no	—
55	C₂×#92420	—	2	4	no	—
56	C₂×#92420	C₂×C₈	1	8	yes	—
57	C₂×C₈×C₆₁	—	2	1	no	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
62	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
63	C₄²×C₆₁	—	2	1	no	—
64	C₄²×C₆₁	—	2	1	no	—
65	C₄×Dic₆₁	—	2	1	no	—
66	C₄×Dic₆₁	—	2	1	no	—
67	C₄×Dic₆₁	—	2	2	no	—
68	C₄×Dic₆₁	—	2	2	no	—
69	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
70	C₄×Dic₆₁	M₁₆	1	4	yes	—
71	C₄×Dic₆₁	#41	1	2	yes	—
72	C₄×Dic₆₁	#41	1	2	yes	—
73	C₂×#92418	—	2	1	no	—
74	C₂×#92418	—	2	1	no	—
75	C₂×#92418	—	2	1	no	—
76	C₂×#92418	—	2	1	no	—
77	C₂×#92418	M₁₆	1	2	yes	—
78	C₂×#92418	M₁₆	1	2	yes	—
79	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	1	no	—
84	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₆₁	#97	1	2	yes	—
86	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
87	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
92	C₂×Dic₆₁	—	2	1	no	—
93	C₂×Dic₆₁	—	2	1	no	—
94	C₂×Dic₆₁	—	2	1	no	—
95	C₂×Dic₆₁	—	2	1	no	—
96	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
97	C₂×Dic₆₁	#97	1	2	yes	—
98	C₂×Dic₆₁	#117	1	2	yes	—
99	C₂×Dic₆₁	#117	1	2	yes	—
100	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
101	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
102	C₂×Dic₆₁	#131	1	2	yes	—
103	C₂×Dic₆₁	#131	1	2	yes	—
104	#92420	—	2	8	no	—
105	#92420	—	4	4	no	—
106	C₈×C₆₁	—	2	2	no	—
107	C₈×C₆₁	—	2	2	no	—
108	#92418	—	2	2	no	—
109	#92418	—	2	2	no	—
110	#8132	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	#376	1	1	yes	characteristic
112	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
114	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
115	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
116	C₂²×C₆₁	#405	1	1	yes	characteristic
117	C₂²×C₆₁	#405	1	1	yes	characteristic
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	#432	1	1	yes	characteristic
121	C₄×C₆₁	#432	1	1	yes	characteristic
122	C₄×C₆₁	#432	1	1	yes	characteristic
123	C₄×C₆₁	#432	1	1	yes	characteristic
124	Dic₆₁	—	2	2	no	—
125	Dic₆₁	—	2	2	no	—
126	Dic₆₁	#431	1	4	yes	—
127	Dic₆₁	#433	1	4	yes	—
128	#1744	—	61	1	no	—
129	C₄×#97	—	61	1	no	—
130	C₄²×C₈	—	61	1	no	—
131	#1273	—	61	2	no	—
132	#1273	—	61	2	no	—
133	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
134	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
135	C₂×C₆₁	#1451	1	1	yes	characteristic
136	C₂×C₆₁	#1451	1	1	yes	characteristic
137	C₂×C₆₁	#1744	1	1	yes	characteristic
138	C₂×C₆₁	#1745	1	1	yes	characteristic
139	C₂×C₆₁	#1746	1	1	yes	characteristic
140	C₂×#97	—	61	2	no	—
141	C₂×#97	—	61	2	no	—
142	C₂×C₄×C₈	—	61	2	no	—
143	C₂×C₄×C₈	—	61	2	no	—
144	C₂×C₄×C₈	—	61	1	no	—
145	C₂×C₄×C₈	—	61	1	no	—
146	C₂×C₄×C₈	—	61	1	no	—
147	C₂×C₄×C₈	—	61	1	no	—
148	C₂×C₄×C₈	—	61	1	no	—
149	C₂×C₄×C₈	#6923	1	1	yes	characteristic
150	C₄³	—	61	1	no	—
151	#336	—	61	2	no	—
152	#336	—	61	2	no	—
153	C₆₁	#8132	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₈	—	61	4	no	—
155	C₂²×C₈	—	61	1	no	—
156	C₂²×C₈	—	61	1	no	—
157	C₂²×C₈	—	61	4	no	—
158	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
159	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
160	C₂×C₄²	—	61	1	no	—
161	C₂×C₄²	—	61	1	no	—
162	C₂×C₄²	—	61	1	no	—
163	C₂×C₄²	—	61	1	no	—
164	C₂×C₄²	—	61	1	no	—
165	C₂×C₄²	—	61	1	no	—
166	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
167	#97	—	61	8	no	—
168	#97	—	122	4	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	122	2	no	—
171	C₄×C₈	—	122	2	no	—
172	C₄×C₈	—	122	2	no	—
173	C₄×C₈	—	122	2	no	—
174	C₄×C₈	—	122	2	no	—
175	C₄×C₈	—	122	2	no	—
176	C₄×C₈	—	122	2	no	—
177	C₄×C₈	—	122	2	no	—
178	C₄×C₈	—	122	2	no	—
179	C₂²×C₄	—	61	1	no	—
180	C₂²×C₄	—	61	1	no	—
181	C₂²×C₄	—	61	1	no	—
182	C₂²×C₄	—	61	1	no	—
183	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
184	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
185	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	61	8	no	—
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	—	61	2	no	—
193	C₂×C₈	—	122	4	no	—
194	C₂×C₈	—	122	4	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	—	122	1	no	—
197	C₂×C₈	—	122	1	no	—
198	C₂×C₈	—	122	1	no	—
199	C₂×C₈	—	122	4	no	—
200	C₂×C₈	—	122	4	no	—
201	C₂×C₈	—	122	4	no	—
202	C₂×C₈	#11758213	1	2	yes	—
203	C₂×C₈	#11758217	1	2	yes	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	61	4	no	—
207	C₄²	—	61	4	no	—
208	C₄²	—	61	2	no	—
209	C₄²	—	61	2	no	—
210	C₄²	—	122	2	no	—
211	C₄²	—	122	2	no	—
212	C₄²	—	122	1	no	—
213	C₄²	—	122	1	no	—
214	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	61	2	no	—
220	C₂×C₄	—	61	2	no	—
221	C₂×C₄	—	61	2	no	—
222	C₂×C₄	—	61	2	no	—
223	C₂×C₄	—	61	2	no	—
224	C₂×C₄	—	61	2	no	—
225	C₂×C₄	—	61	2	no	—
226	C₂×C₄	—	61	2	no	—
227	C₂×C₄	—	122	1	no	—
228	C₂×C₄	—	122	1	no	—
229	C₂×C₄	—	122	1	no	—
230	C₂×C₄	—	122	1	no	—
231	C₂×C₄	C₈×#6923	1	2	yes	—
232	C₂×C₄	#14519831	1	2	yes	—
233	C₂×C₄	#14519856	1	1	yes	characteristic
234	C₂×C₄	#14519856	1	1	yes	characteristic
235	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
236	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
237	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
238	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	122	8	no	—
242	C₈	—	122	2	no	—
243	C₈	—	122	2	no	—
244	C₈	—	244	4	no	—
245	C₂²	#16840698	1	1	yes	characteristic
246	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
247	C₂²	C₂×#14519831	1	1	yes	characteristic
248	C₂²	C₄×#11758217	1	1	yes	characteristic
249	C₂²	C₄×#11758213	1	1	yes	characteristic
250	C₂²	#16840725	1	1	yes	characteristic
251	C₂²	#16840725	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	61	4	no	—
255	C₄	—	61	4	no	—
256	C₄	—	122	2	no	—
257	C₄	—	122	2	no	—
258	C₄	#16840782	1	1	yes	characteristic
259	C₄	#16840783	1	1	yes	characteristic
260	C₄	#16840784	1	1	yes	characteristic
261	C₄	#16840785	1	1	yes	characteristic
262	C₂	C₈×#11758217	1	1	yes	characteristic
263	C₂	C₈×#11758213	1	1	yes	characteristic
264	C₂	#21461028	1	1	yes	characteristic
265	C₂	#21461029	1	1	yes	characteristic
266	C₂	#21461532	1	1	yes	characteristic
267	C₂	#21461536	1	1	yes	characteristic
268	C₂	#21461537	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8132
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
22. #14519832	C₈	semidirect
153. C₆₁	#8132	semidirect

Maximal subgroups

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

subgroup	index
#21461534	2
C₄×#14519832	2
#21460641	2
C₄×C₈×Dic₆₁	2
#8132	61