Subgroups — #30597959

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519844	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840603	C₂	1	2	yes	maximal
4	#21460641	C₂	1	2	yes	maximal
5	#21445667	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519844	C₄	1	2	yes	—
7	C₂×C₄×#92420	C₄	1	2	yes	—
8	C₂×C₄×#92420	C₄	1	2	yes	—
9	C₂×C₄×#92420	C₂²	1	2	yes	—
10	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
12	#16840628	C₄	1	2	yes	—
13	#16840603	—	2	4	no	—
14	C₂×#14519706	C₄	1	2	yes	—
15	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
16	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
17	#16839517	C₄	1	1	yes	characteristic
18	C₆₁×#336	C₄	1	1	yes	characteristic
19	C₂²×#92420	C₂×C₄	1	4	yes	—
20	C₂²×#92420	C₂×C₄	1	2	yes	—
21	C₂²×#92420	C₂×C₄	1	2	yes	—
22	#14519844	—	2	4	no	—
23	C₄×#92420	—	2	4	no	—
24	C₄×#92420	—	2	4	no	—
25	C₄×#92420	—	2	2	no	—
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	—	2	2	no	—
28	C₄×#92420	—	2	1	no	—
29	C₄×#92420	D₄	1	2	yes	—
30	C₄×#92420	Q₈	1	2	yes	—
31	C₂²×#92418	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
38	#14519706	C₈	1	8	yes	complemented
39	C₈×Dic₆₁	—	2	2	no	—
40	C₈×Dic₆₁	—	2	2	no	—
41	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
47	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
48	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×#92420	—	2	2	no	—
51	C₂×#92420	—	2	2	no	—
52	C₂×#92420	—	2	2	no	—
53	C₂×#92420	—	2	2	no	—
54	C₂×#92420	—	2	4	no	—
55	C₂×#92420	—	2	4	no	—
56	C₂×#92420	—	2	4	no	—
57	C₂×#92420	C₄⋊C₄	1	4	yes	—
58	C₂×#92420	M₁₆	1	4	yes	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
62	C₄²×C₆₁	—	2	2	no	—
63	C₄×Dic₆₁	—	2	2	no	—
64	C₄×Dic₆₁	—	2	2	no	—
65	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
66	C₄×Dic₆₁	M₁₆	1	4	yes	—
67	C₄×Dic₆₁	M₁₆	1	4	yes	—
68	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
70	C₄×Dic₆₁	#41	1	1	yes	characteristic
71	C₄×Dic₆₁	#41	1	1	yes	characteristic
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	—	2	2	no	—
74	C₂×#92418	C₂×C₈	1	4	yes	—
75	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	#97	1	2	yes	—
81	C₂×C₄×C₆₁	#97	1	2	yes	—
82	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₆₁	#131	1	2	yes	—
86	C₂×Dic₆₁	—	2	1	no	—
87	C₂×Dic₆₁	—	2	1	no	—
88	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
89	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
90	C₂×Dic₆₁	#97	1	2	yes	—
91	C₂×Dic₆₁	#97	1	2	yes	—
92	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₆₁	#117	1	1	yes	characteristic
94	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
96	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
97	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
98	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
99	C₂×Dic₆₁	#131	1	2	yes	—
100	#92420	—	2	4	no	—
101	#92420	—	2	4	no	—
102	#92420	—	4	4	no	—
103	C₈×C₆₁	—	2	4	no	—
104	#92418	—	2	4	no	—
105	#8261	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
109	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
110	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
111	C₂²×C₆₁	#405	1	1	yes	characteristic
112	C₂²×C₆₁	#405	1	1	yes	characteristic
113	C₄×C₆₁	—	2	2	no	—
114	C₄×C₆₁	—	2	2	no	—
115	C₄×C₆₁	C₈×D₄	1	2	yes	—
116	C₄×C₆₁	C₈×Q₈	1	2	yes	—
117	Dic₆₁	—	2	2	no	—
118	Dic₆₁	#322	1	4	yes	complemented
119	Dic₆₁	#423	1	2	yes	—
120	Dic₆₁	#432	1	2	yes	—
121	Dic₆₁	#436	1	2	yes	—
122	Dic₆₁	#446	1	2	yes	—
123	C₄×C₄⋊C₈	—	61	1	no	—
124	C₄×C₄⋊C₈	—	61	1	no	—
125	C₄²×C₈	—	61	1	no	—
126	C₂×#322	—	61	2	no	—
127	#1273	—	61	2	no	—
128	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
129	C₂×C₆₁	C₈×M₁₆	1	1	yes	characteristic
130	C₂×C₆₁	#1451	1	1	yes	characteristic
131	C₂×C₆₁	#1452	1	1	yes	characteristic
132	C₂×C₆₁	#1767	1	1	yes	characteristic
133	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₆₁	#1770	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	61	2	no	—
136	C₂×C₄⋊C₈	—	61	2	no	—
137	C₂×C₄×C₈	—	61	1	no	—
138	C₂×C₄×C₈	—	61	2	no	—
139	C₂×C₄×C₈	—	61	2	no	—
140	C₂×C₄×C₈	—	61	2	no	—
141	C₂×C₄×C₈	—	61	1	no	—
142	C₂×C₄×C₈	—	61	2	no	—
143	C₄³	—	61	1	no	—
144	#336	—	61	2	no	—
145	#336	—	61	1	no	—
146	#336	#6923	1	1	yes	characteristic
147	#322	—	122	4	no	—
148	C₆₁	#8261	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₈	—	61	2	no	—
150	C₂²×C₈	—	61	2	no	—
151	C₂²×C₈	—	61	2	no	—
152	C₂²×C₈	—	61	4	no	—
153	C₂²×C₈	C₂×#6923	1	2	yes	—
154	C₂×C₄²	—	61	1	no	—
155	C₂×C₄²	—	61	1	no	—
156	C₂×C₄²	—	61	1	no	—
157	C₂×C₄²	—	61	1	no	—
158	C₂×C₄²	—	61	1	no	—
159	C₂×C₄²	—	61	1	no	—
160	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
161	C₄⋊C₈	—	61	8	no	—
162	C₄⋊C₈	—	122	4	no	—
163	C₄×C₈	—	61	2	no	—
164	C₄×C₈	—	61	2	no	—
165	C₄×C₈	—	122	2	no	—
166	C₄×C₈	—	122	2	no	—
167	C₄×C₈	—	122	2	no	—
168	C₄×C₈	—	122	4	no	—
169	C₄×C₈	—	122	1	no	—
170	C₄×C₈	—	122	1	no	—
171	C₄×C₈	—	122	2	no	—
172	C₄×C₈	—	122	4	no	—
173	C₂²×C₄	—	61	1	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	61	1	no	—
177	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
178	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
179	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	61	4	no	—
183	C₂×C₈	—	61	4	no	—
184	C₂×C₈	—	61	4	no	—
185	C₂×C₈	—	122	2	no	—
186	C₂×C₈	—	122	2	no	—
187	C₂×C₈	—	122	2	no	—
188	C₂×C₈	—	122	2	no	—
189	C₂×C₈	—	122	2	no	—
190	C₂×C₈	—	122	2	no	—
191	C₂×C₈	—	122	4	no	—
192	C₂×C₈	—	122	4	no	—
193	C₂×C₈	—	122	4	no	—
194	C₂×C₈	D₆₁⋊C₈	1	4	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	61	4	no	—
197	C₄²	—	61	4	no	—
198	C₄²	—	61	1	no	—
199	C₄²	—	61	1	no	—
200	C₄²	—	61	1	no	—
201	C₄²	—	61	1	no	—
202	C₄²	—	61	4	no	—
203	C₄²	—	122	2	no	—
204	C₄²	—	122	2	no	—
205	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	61	2	no	—
211	C₂×C₄	—	61	2	no	—
212	C₂×C₄	—	61	2	no	—
213	C₂×C₄	—	61	2	no	—
214	C₂×C₄	—	61	2	no	—
215	C₂×C₄	—	61	2	no	—
216	C₂×C₄	—	61	2	no	—
217	C₂×C₄	—	61	1	no	—
218	C₂×C₄	—	61	1	no	—
219	C₂×C₄	—	61	1	no	—
220	C₂×C₄	—	61	1	no	—
221	C₂×C₄	—	61	2	no	—
222	C₂×C₄	—	122	1	no	—
223	C₂×C₄	—	122	1	no	—
224	C₂×C₄	#14519831	1	2	yes	—
225	C₂×C₄	#14519831	1	2	yes	—
226	C₂×C₄	D₄×#6923	1	1	yes	characteristic
227	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
228	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
229	C₂×C₄	#14519864	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	122	4	no	—
232	C₈	—	122	4	no	—
233	C₈	—	122	4	no	—
234	C₈	—	244	4	no	—
235	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
236	C₂²	C₄×D₆₁⋊C₈	1	1	yes	characteristic
237	C₂²	C₂×#14519831	1	1	yes	characteristic
238	C₂²	C₂×#14519831	1	1	yes	characteristic
239	C₂²	M₁₆×#6923	1	1	yes	characteristic
240	C₂²	#16840726	1	1	yes	characteristic
241	C₂²	#16840728	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	61	4	no	—
245	C₄	—	61	2	no	—
246	C₄	—	61	2	no	—
247	C₄	—	61	2	no	—
248	C₄	—	61	2	no	—
249	C₄	—	122	2	no	—
250	C₄	#16840789	1	2	yes	—
251	C₄	#16840811	1	2	yes	—
252	C₂	#21461013	1	1	yes	characteristic
253	C₂	#21461017	1	1	yes	characteristic
254	C₂	#21461030	1	1	yes	characteristic
255	C₂	#21461034	1	1	yes	characteristic
256	C₂	#21461606	1	1	yes	characteristic
257	C₂	#21461611	1	1	yes	characteristic
258	C₂	#21461616	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#336	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#336	#6923
hypocenter	C₆₁	#8261
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
38. #14519706	C₈	semidirect
118. Dic₆₁	#322	semidirect
148. C₆₁	#8261	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519844	2
C₄²×#92420	2
C₂×#16840603	2
#21460641	2
#21445667	2
#8261	61