Subgroups — #30597473

Order 15616.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461050	C₂	1	2	yes	maximal
2	#21461022	C₂	1	2	yes	maximal, complemented
3	#21460659	C₂	1	2	yes	maximal, complemented
4	C₂×#16839717	C₂	1	1	yes	maximal, characteristic
5	#16840778	C₂²	1	4	yes	—
6	#16840607	C₄	1	4	yes	—
7	#16840605	C₄	1	4	yes	—
8	#16840603	—	2	2	no	—
9	C₈×#92420	—	2	2	no	—
10	C₂²×#11758194	C₄	1	1	yes	characteristic
11	C₂×#14519736	C₂²	1	1	yes	characteristic
12	#16839717	—	2	2	no	—
13	#16839717	—	2	2	no	—
14	C₂×#14519687	C₄	1	1	yes	characteristic
15	C₂×#14519687	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
19	C₄⋊#92420	—	2	4	no	—
20	C₂²⋊#92420	C₂×C₄	1	8	yes	—
21	C₄×#92420	—	2	4	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	—	2	2	no	—
24	C₂×#11758194	C₈	1	4	yes	complemented
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	#14519736	—	2	2	no	—
28	#14519736	—	2	2	no	—
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
33	#14519687	—	2	2	no	—
34	#14519687	—	2	2	no	—
35	C₈×Dic₆₁	—	2	1	no	—
36	C₈×Dic₆₁	—	2	1	no	—
37	C₈×Dic₆₁	D₄	1	2	yes	—
38	C₈×Dic₆₁	D₄	1	2	yes	—
39	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	—	2	1	no	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂²×C₈×C₆₁	—	2	2	no	—
43	C₂²×C₈×C₆₁	C₈	1	4	yes	complemented
44	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	—	2	1	no	—
50	C₂²×C₄×C₆₁	—	2	1	no	—
51	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₆₁	C₄²	1	2	yes	—
56	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
57	C₂×#92420	—	2	8	no	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	2	4	no	—
62	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
67	C₂²⋊Dic₆₁	—	2	4	no	—
68	C₄⋊Dic₆₁	—	2	4	no	—
69	C₄×Dic₆₁	—	2	1	no	—
70	C₄×Dic₆₁	—	2	1	no	—
71	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
72	C₄×Dic₆₁	#41	1	2	yes	—
73	#11758194	—	2	8	no	—
74	C₂×#92418	—	2	1	no	—
75	C₂×#92418	—	2	1	no	—
76	C₂×#92418	C₂²⋊C₄	1	2	yes	—
77	C₂×#92418	C₂×C₈	1	2	yes	—
78	C₂×#92418	C₂×C₈	1	2	yes	—
79	C₂×#92418	M₁₆	1	2	yes	—
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	—	4	2	no	—
83	C₂³×C₆₁	C₄×C₈	1	2	yes	—
84	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	2	no	—
90	C₂×C₄×C₆₁	—	2	2	no	—
91	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₆₁	—	2	2	no	—
99	C₂×Dic₆₁	—	4	2	no	—
100	C₂×Dic₆₁	—	4	2	no	—
101	C₂×Dic₆₁	C₄×D₄	1	4	yes	—
102	#92420	—	4	8	no	—
103	C₈×C₆₁	—	2	2	no	—
104	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
105	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
106	#92418	—	2	2	no	—
107	#92418	C₂²⋊C₈	1	4	yes	—
108	#8156	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	2	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	2	no	—
112	C₂²×C₆₁	—	4	2	no	—
113	C₂²×C₆₁	—	4	2	no	—
114	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₆₁	#431	1	1	yes	characteristic
121	C₄×C₆₁	#431	1	1	yes	characteristic
122	Dic₆₁	—	2	4	no	—
123	Dic₆₁	—	4	2	no	—
124	C₂×#405	—	61	1	no	—
125	#1458	—	61	2	no	—
126	#1450	—	61	2	no	—
127	#1281	—	61	2	no	—
128	C₂×C₆₁	—	2	1	no	—
129	C₂×C₆₁	—	2	1	no	—
130	C₂×C₆₁	—	4	2	no	—
131	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₆₁	#1757	1	2	yes	—
133	C₂²×M₁₆	—	61	1	no	—
134	C₂³×C₈	#6923	1	1	yes	characteristic
135	C₂×#117	—	61	1	no	—
136	#431	—	61	4	no	—
137	#405	—	122	2	no	—
138	#405	—	122	2	no	—
139	C₂×#97	—	61	1	no	—
140	C₂×#97	—	61	1	no	—
141	C₂×C₄×C₈	—	61	1	no	—
142	C₂×C₄×C₈	—	61	1	no	—
143	#324	—	61	4	no	—
144	#323	—	61	4	no	—
145	#322	—	122	2	no	—
146	C₈²	—	122	2	no	—
147	C₆₁	#8156	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
149	C₂×M₁₆	—	61	4	no	—
150	C₂×M₁₆	—	122	2	no	—
151	C₂×M₁₆	—	122	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	61	1	no	—
156	C₂²×C₈	—	61	1	no	—
157	C₂²×C₈	#92420	1	4	yes	complemented
158	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
159	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
160	#117	—	122	2	no	—
161	#117	—	122	2	no	—
162	C₂×C₄⋊C₄	—	61	2	no	—
163	C₂×C₂²⋊C₄	—	61	2	no	—
164	C₂×C₄²	—	61	1	no	—
165	C₂×C₄²	—	61	1	no	—
166	C₄⋊C₈	—	122	4	no	—
167	C₂²⋊C₈	—	61	8	no	—
168	#97	—	122	2	no	—
169	#97	—	122	2	no	—
170	C₄×C₈	—	61	2	no	—
171	C₄×C₈	—	61	2	no	—
172	C₄×C₈	—	122	4	no	—
173	C₄×C₈	—	122	1	no	—
174	C₄×C₈	—	122	1	no	—
175	C₂⁴	C₄×#6923	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	61	2	no	—
181	C₂²×C₄	—	61	2	no	—
182	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
183	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
184	C₂²×C₄	C₂×#92420	1	2	yes	—
185	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
186	M₁₆	—	122	8	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	—	61	2	no	—
193	C₂×C₈	—	61	2	no	—
194	C₂×C₈	—	61	2	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	—	122	1	no	—
197	C₂×C₈	—	122	8	no	—
198	C₂×C₈	C₂×#92420	1	2	yes	—
199	C₂×C₈	C₂×#92420	1	2	yes	—
200	C₂×C₈	#11758217	1	2	yes	—
201	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
203	C₄⋊C₄	—	122	4	no	—
204	C₂²⋊C₄	—	122	4	no	—
205	C₄²	—	61	2	no	—
206	C₄²	—	61	2	no	—
207	C₄²	—	122	1	no	—
208	C₄²	—	122	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₄×#92420	1	2	yes	—
213	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	61	4	no	—
221	C₂×C₄	—	122	2	no	—
222	C₂×C₄	—	244	2	no	—
223	C₂×C₄	—	244	2	no	—
224	C₂×C₄	C₄×#92420	1	2	yes	—
225	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
226	C₂×C₄	#14519856	1	1	yes	characteristic
227	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
228	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
230	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	61	4	no	—
233	C₈	—	122	2	no	—
234	C₈	—	244	8	no	—
235	C₈	C₂²⋊#92420	1	2	yes	—
236	C₈	C₂²⋊#92420	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	C₄×C₂²⋊#6923	1	1	yes	characteristic
243	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
244	C₂²	C₄×#11758217	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	122	4	no	—
247	C₄	—	244	2	no	—
248	C₄	C₂×C₂²⋊#92420	1	1	yes	characteristic
249	C₄	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
250	C₄	#16840778	1	1	yes	characteristic
251	C₄	#16840779	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	4	2	no	—
255	C₂	C₄×C₂²⋊#92420	1	1	yes	characteristic
256	C₂	#21461574	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	#8156
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #21461022	C₂	semidirect
3. #21460659	C₂	semidirect
24. C₂×#11758194	C₈	semidirect
43. C₂²×C₈×C₆₁	C₈	semidirect
147. C₆₁	#8156	semidirect
157. C₂²×C₈	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461050	2
#21461022	2
#21460659	2
C₂×#16839717	2
#8156	61