Subgroups — #30670361

Order 16768.

412 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃₁⋊#401	C₂	1	2	yes	maximal, complemented
2	#22553240	C₂	1	2	yes	maximal
3	#22553210	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553210	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553206	C₂	1	2	yes	maximal
6	#22553194	C₂	1	2	yes	maximal, complemented
7	#22553178	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553172	C₂	1	1	yes	maximal, characteristic
9	#22553172	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₃₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
21	C₁₃₁⋊#95	C₂²	1	2	yes	—
22	C₁₃₁⋊#95	C₂²	1	2	yes	—
23	C₁₃₁⋊#95	C₂²	1	2	yes	—
24	C₁₃₁⋊#95	C₂²	1	2	yes	—
25	C₁₃₁⋊#95	C₂²	1	2	yes	—
26	C₁₃₁⋊#95	C₂²	1	2	yes	—
27	C₁₃₁⋊#95	C₂²	1	2	yes	—
28	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
33	#95×C₁₃₁	C₂²	1	1	yes	characteristic
34	#95×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
36	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
37	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
39	C₄⋊Dic₁₃₁	—	2	2	no	—
40	C₄⋊Dic₁₃₁	—	2	2	no	—
41	C₄⋊Dic₁₃₁	—	2	2	no	—
42	C₄⋊Dic₁₃₁	—	2	2	no	—
43	Dic₁₃₁⋊C₄	—	2	4	no	—
44	Dic₁₃₁⋊C₄	—	4	2	no	—
45	C₂²⋊Dic₁₃₁	—	2	4	no	—
46	C₂²⋊Dic₁₃₁	—	4	2	no	—
47	C₄×Dic₁₃₁	—	4	2	no	—
48	C₂⁴×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	—	2	2	no	—
50	C₂²×C₄×C₁₃₁	—	2	2	no	—
51	C₂²×C₄×C₁₃₁	—	2	2	no	—
52	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
58	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
60	C₄²×C₁₃₁	—	2	2	no	—
61	C₄²×C₁₃₁	—	2	2	no	—
62	C₂×Dic₁₃₁	—	2	2	no	—
63	C₂×Dic₁₃₁	—	2	2	no	—
64	C₂×Dic₁₃₁	—	2	2	no	—
65	C₂×Dic₁₃₁	—	2	2	no	—
66	C₂×Dic₁₃₁	—	2	2	no	—
67	C₂×Dic₁₃₁	—	2	2	no	—
68	C₂×Dic₁₃₁	—	2	2	no	—
69	C₂×Dic₁₃₁	—	2	2	no	—
70	C₂×Dic₁₃₁	—	2	2	no	—
71	C₂×Dic₁₃₁	—	2	2	no	—
72	C₂×Dic₁₃₁	—	2	2	no	—
73	C₂×Dic₁₃₁	—	2	2	no	—
74	C₂³×C₁₃₁	—	2	1	no	—
75	C₂³×C₁₃₁	—	2	1	no	—
76	C₂³×C₁₃₁	—	2	1	no	—
77	C₂³×C₁₃₁	—	2	1	no	—
78	C₂³×C₁₃₁	—	2	1	no	—
79	C₂³×C₁₃₁	—	2	1	no	—
80	C₂³×C₁₃₁	#41	1	2	yes	—
81	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₁₃₁	—	2	1	no	—
83	C₂×C₄×C₁₃₁	—	2	1	no	—
84	C₂×C₄×C₁₃₁	—	2	1	no	—
85	C₂×C₄×C₁₃₁	—	2	1	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	1	no	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	1	no	—
92	C₂×C₄×C₁₃₁	—	2	1	no	—
93	C₂×C₄×C₁₃₁	—	2	1	no	—
94	C₂×C₄×C₁₃₁	—	2	1	no	—
95	C₂×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₄×C₁₃₁	—	4	4	no	—
97	C₂×C₄×C₁₃₁	#41	1	2	yes	—
98	C₂×C₄×C₁₃₁	#41	1	2	yes	—
99	C₂×C₄×C₁₃₁	#41	1	2	yes	—
100	C₂×C₄×C₁₃₁	#41	1	2	yes	—
101	C₂²×C₁₃₁	—	2	2	no	—
102	C₂²×C₁₃₁	—	2	2	no	—
103	C₂²×C₁₃₁	—	2	2	no	—
104	C₂²×C₁₃₁	—	4	1	no	—
105	C₂²×C₁₃₁	—	4	1	no	—
106	C₂²×C₁₃₁	—	4	1	no	—
107	C₂²×C₁₃₁	—	4	1	no	—
108	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₁₃₁	—	2	2	no	—
116	C₄×C₁₃₁	—	4	1	no	—
117	C₄×C₁₃₁	—	4	1	no	—
118	C₄×C₁₃₁	—	4	1	no	—
119	C₄×C₁₃₁	—	4	1	no	—
120	C₄×C₁₃₁	#126	1	4	yes	—
121	Dic₁₃₁	—	4	2	no	—
122	Dic₁₃₁	—	4	2	no	—
123	Dic₁₃₁	—	4	2	no	—
124	Dic₁₃₁	—	4	2	no	—
125	C₂×C₁₃₁	—	4	2	no	—
126	C₂×C₁₃₁	C₂×#126	1	1	yes	characteristic
127	C₂×C₁₃₁	#529	1	1	yes	characteristic
128	C₂×C₁₃₁	#542	1	1	yes	characteristic
129	C₂×C₁₃₁	#552	1	1	yes	characteristic
130	C₂×C₁₃₁	#553	1	1	yes	characteristic
131	C₂×C₁₃₁	#553	1	1	yes	characteristic
132	C₂×C₁₃₁	#556	1	1	yes	characteristic
133	C₁₃₁	#2494	1	1	yes	characteristic, complemented, Sylow
134	#2494	—	131	1	no	maximal, Sylow
135	#401	—	131	2	no	—
136	#397	—	131	2	no	—
137	#389	—	131	1	no	—
138	#389	—	131	1	no	—
139	#388	—	131	2	no	—
140	#387	—	131	2	no	—
141	#382	—	131	1	no	—
142	#380	—	131	1	no	—
143	#380	—	131	1	no	—
144	C₄×C₄⋊C₄	—	131	1	no	—
145	C₄×C₂²⋊C₄	D₁₃₁	1	1	yes	characteristic
146	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	131	1	no	—
148	C₂×C₄⋊C₄	—	131	1	no	—
149	C₂×C₄⋊C₄	—	131	1	no	—
150	C₂×C₄⋊C₄	—	131	1	no	—
151	C₂×C₄⋊C₄	—	131	2	no	—
152	C₂×C₄⋊C₄	—	131	2	no	—
153	C₂×C₂²⋊C₄	—	131	2	no	—
154	C₂×C₂²⋊C₄	—	131	2	no	—
155	C₂×C₂²⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
157	C₂×C₄²	—	131	2	no	—
158	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
159	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
160	#95	—	131	2	no	—
161	#95	—	131	2	no	—
162	#95	—	131	2	no	—
163	#95	—	131	2	no	—
164	#95	—	131	2	no	—
165	#95	—	131	2	no	—
166	#95	—	131	2	no	—
167	#95	C₂×D₁₃₁	1	1	yes	characteristic
168	#95	C₂×D₁₃₁	1	1	yes	characteristic
169	C₂⁴	C₂²×D₁₃₁	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	131	2	no	—
174	C₂²×C₄	—	131	2	no	—
175	C₂²×C₄	—	131	2	no	—
176	C₂²×C₄	—	131	2	no	—
177	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
183	C₄⋊C₄	—	262	2	no	—
184	C₄⋊C₄	—	262	2	no	—
185	C₄⋊C₄	—	262	4	no	—
186	C₄⋊C₄	—	262	2	no	—
187	C₄⋊C₄	—	262	2	no	—
188	C₄⋊C₄	—	524	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	262	4	no	—
192	C₂²⋊C₄	—	524	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	524	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	#14523513	1	2	yes	—
203	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	262	2	no	—
220	C₂×C₄	—	262	2	no	—
221	C₂×C₄	—	262	2	no	—
222	C₂×C₄	—	262	2	no	—
223	C₂×C₄	—	262	2	no	—
224	C₂×C₄	—	262	2	no	—
225	C₂×C₄	—	262	2	no	—
226	C₂×C₄	—	262	2	no	—
227	C₂×C₄	—	262	2	no	—
228	C₂×C₄	—	262	2	no	—
229	C₂×C₄	—	262	2	no	—
230	C₂×C₄	—	262	2	no	—
231	C₂×C₄	#14523512	1	2	yes	—
232	C₂×C₄	#14523512	1	2	yes	—
233	C₂×C₄	#14523513	1	2	yes	—
234	C₂×C₄	#14523513	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂×#14523512	1	1	yes	characteristic
243	C₂²	C₂×#14523512	1	1	yes	characteristic
244	C₂²	C₂×#14523513	1	1	yes	characteristic
245	C₂²	C₂×#14523513	1	1	yes	characteristic
246	C₂²	C₂×#14523513	1	1	yes	characteristic
247	C₂²	#16862292	1	1	yes	characteristic
248	C₂²	#16862293	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	524	2	no	—
255	C₄	—	524	2	no	—
256	C₄	—	524	2	no	—
257	C₄	—	524	2	no	—
258	C₄	#16862239	1	4	yes	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#16862239	1	1	yes	characteristic
261	C₂	#22553846	1	1	yes	characteristic
262	C₂	#22553933	1	1	yes	characteristic
263	C₂	#22554002	1	1	yes	characteristic
264	C₂	#22554008	1	1	yes	characteristic
265	C₂	#22554008	1	1	yes	characteristic
266	C₂	#22554032	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₁
commutator, socle, cosocle	C₂³×C₁₃₁	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₁₃₁	C₂
prime radical	C₂⁴×C₁₃₁	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₃₁
hypocenter	C₁₃₁	#2494
LCS	G ▷ C₂³×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₁₃₁⋊#401	C₂	semidirect
3. #22553210	C₂	semidirect
4. #22553210	C₂	semidirect
6. #22553194	C₂	semidirect
7. #22553178	C₂	semidirect
10. C₄×C₄⋊Dic₁₃₁	C₂	semidirect
133. C₁₃₁	#2494	semidirect

Maximal subgroups

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

subgroup	index
C₁₃₁⋊#401	2
#22553240	2
#22553210	2
#22553206	2
#22553194	2
#22553178	2
#22553172	2
C₄×C₄⋊Dic₁₃₁	2
C₄×C₂²⋊C₄×C₁₃₁	2
#2494	131