Subgroups — C₁₃₁×#12820 #45698704

Order 33536.

414 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	C₂×#443×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2139×C₁₃₁	C₂	1	2	yes	maximal
4	#1846×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1387×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×#131×C₁₃₁	—	2	1	no	—
8	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
9	#443×C₁₃₁	—	2	2	no	—
10	#443×C₁₃₁	C₄	1	4	yes	complemented
11	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#106×C₁₃₁	—	2	1	no	—
14	C₂×#104×C₁₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	#382×C₁₃₁	—	2	1	no	—
17	#364×C₁₃₁	—	2	2	no	—
18	#358×C₁₃₁	—	2	2	no	—
19	#357×C₁₃₁	—	2	2	no	—
20	C₂×#41×C₁₃₁	—	2	1	no	—
21	#135×C₁₃₁	—	2	1	no	—
22	#135×C₁₃₁	—	2	1	no	—
23	#135×C₁₃₁	C₂×C₄	1	2	yes	—
24	#135×C₁₃₁	Q₈	1	2	yes	—
25	C₂×Q₁₆×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₁₃₁	—	2	1	no	—
27	C₂×D₈×C₁₃₁	D₄	1	1	yes	characteristic
28	#131×C₁₃₁	—	2	4	no	—
29	#131×C₁₃₁	—	4	2	no	—
30	C₂×M₁₆×C₁₃₁	—	2	1	no	—
31	C₂×M₁₆×C₁₃₁	—	4	1	no	—
32	C₂²×C₈×C₁₃₁	—	4	1	no	—
33	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	M₃₂×C₁₃₁	—	2	2	no	—
38	M₃₂×C₁₃₁	—	2	2	no	—
39	C₂×C₁₆×C₁₃₁	—	2	2	no	—
40	#108×C₁₃₁	—	2	1	no	—
41	#108×C₁₃₁	C₂×C₄	1	2	yes	—
42	#107×C₁₃₁	—	2	2	no	—
43	#107×C₁₃₁	—	2	2	no	—
44	#106×C₁₃₁	—	4	1	no	—
45	#106×C₁₃₁	—	4	1	no	—
46	#104×C₁₃₁	—	2	4	no	—
47	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
49	#95×C₁₃₁	—	4	2	no	—
50	#41×C₁₃₁	—	2	2	no	—
51	#41×C₁₃₁	—	2	2	no	—
52	#41×C₁₃₁	—	4	1	no	—
53	#41×C₁₃₁	—	4	1	no	—
54	C₂×Q₈×C₁₃₁	—	2	1	no	—
55	C₂×D₄×C₁₃₁	—	2	1	no	—
56	C₂×D₄×C₁₃₁	—	4	1	no	—
57	C₂²×C₄×C₁₃₁	—	2	1	no	—
58	C₂²×C₄×C₁₃₁	—	4	1	no	—
59	C₂²×C₄×C₁₃₁	—	4	2	no	—
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₁₃₁	—	2	1	no	—
62	Q₁₆×C₁₃₁	C₄⋊C₄	1	2	yes	—
63	SD₁₆×C₁₃₁	—	2	2	no	—
64	SD₁₆×C₁₃₁	—	2	2	no	—
65	D₈×C₁₃₁	—	2	1	no	—
66	D₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
67	M₁₆×C₁₃₁	—	2	2	no	—
68	M₁₆×C₁₃₁	—	4	1	no	—
69	M₁₆×C₁₃₁	—	4	4	no	—
70	C₂×C₈×C₁₃₁	—	4	1	no	—
71	C₂×C₈×C₁₃₁	—	4	4	no	—
72	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
74	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
75	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
76	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃₁	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₁₃₁	—	4	2	no	—
83	C₄⋊C₄×C₁₃₁	—	4	2	no	—
84	C₄²×C₁₃₁	—	2	1	no	—
85	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
86	C₁₆×C₁₃₁	—	4	2	no	—
87	C₂³×C₁₃₁	—	4	1	no	—
88	C₂³×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
89	Q₈×C₁₃₁	—	2	2	no	—
90	Q₈×C₁₃₁	—	4	1	no	—
91	D₄×C₁₃₁	—	2	2	no	—
92	D₄×C₁₃₁	—	4	2	no	—
93	D₄×C₁₃₁	—	4	2	no	—
94	D₄×C₁₃₁	—	4	1	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	4	2	no	—
97	C₂×C₄×C₁₃₁	—	4	1	no	—
98	C₂×C₄×C₁₃₁	—	4	2	no	—
99	C₂×C₄×C₁₃₁	—	4	2	no	—
100	C₂×C₄×C₁₃₁	—	8	4	no	—
101	C₂×C₄×C₁₃₁	—	8	2	no	—
102	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
108	C₈×C₁₃₁	—	4	2	no	—
109	C₈×C₁₃₁	#102	1	2	yes	—
110	C₈×C₁₃₁	#102	1	2	yes	—
111	C₈×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₁₃₁	#117	1	1	yes	characteristic
113	C₈×C₁₃₁	#121	1	1	yes	characteristic
114	C₈×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₁₃₁	—	2	2	no	—
116	C₂²×C₁₃₁	—	4	2	no	—
117	C₂²×C₁₃₁	—	8	2	no	—
118	C₂²×C₁₃₁	#417	1	1	yes	characteristic
119	C₂²×C₁₃₁	#459	1	1	yes	characteristic
120	C₂²×C₁₃₁	#476	1	1	yes	characteristic
121	C₄×C₁₃₁	—	4	2	no	—
122	C₄×C₁₃₁	—	4	2	no	—
123	C₄×C₁₃₁	—	16	2	no	—
124	C₄×C₁₃₁	#380	1	1	yes	characteristic
125	C₄×C₁₃₁	C₂×#102	1	1	yes	characteristic
126	C₄×C₁₃₁	#461	1	1	yes	characteristic
127	C₄×C₁₃₁	#474	1	1	yes	characteristic
128	C₂×C₁₃₁	—	2	1	no	—
129	C₂×C₁₃₁	—	2	1	no	—
130	C₂×C₁₃₁	—	8	2	no	—
131	C₂×C₁₃₁	#1802	1	1	yes	characteristic
132	C₂×C₁₃₁	#2212	1	2	yes	—
133	#12820	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₁	#12820	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#443	C₂×C₁₃₁	1	1	yes	characteristic
136	C₂×#364	C₂×C₁₃₁	1	1	yes	characteristic, complemented
137	#2139	C₂×C₁₃₁	1	2	yes	—
138	#1846	C₂×C₁₃₁	1	1	yes	characteristic, complemented
139	#1387	C₂×C₁₃₁	1	2	yes	complemented
140	C₂×#135	C₂²×C₁₃₁	1	1	yes	characteristic
141	C₂×#131	—	2	1	no	—
142	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
143	#443	—	2	2	no	—
144	#443	C₄×C₁₃₁	1	4	yes	complemented
145	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
146	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
147	C₂×#106	—	2	1	no	—
148	C₂×#104	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
150	#382	—	2	1	no	—
151	#364	—	2	2	no	—
152	#358	—	2	2	no	—
153	#357	—	2	2	no	—
154	C₂×#41	—	2	1	no	—
155	#135	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	C₂×C₄×C₁₃₁	1	2	yes	—
158	#135	Q₈×C₁₃₁	1	2	yes	—
159	C₂×Q₁₆	D₄×C₁₃₁	1	1	yes	characteristic
160	C₂×SD₁₆	—	2	1	no	—
161	C₂×D₈	D₄×C₁₃₁	1	1	yes	characteristic
162	#131	—	2	4	no	—
163	#131	—	4	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	—	4	1	no	—
166	C₂²×C₈	—	4	1	no	—
167	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	#108	—	2	1	no	—
175	#108	C₂×C₄×C₁₃₁	1	2	yes	—
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#106	—	4	1	no	—
179	#106	—	4	1	no	—
180	#104	—	2	4	no	—
181	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
182	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
183	#95	—	4	2	no	—
184	#41	—	2	2	no	—
185	#41	—	2	2	no	—
186	#41	—	4	1	no	—
187	#41	—	4	1	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
195	Q₁₆	—	2	1	no	—
196	Q₁₆	C₄⋊C₄×C₁₃₁	1	2	yes	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	—	2	2	no	—
199	D₈	—	2	1	no	—
200	D₈	C₄⋊C₄×C₁₃₁	1	2	yes	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	4	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
207	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
208	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
209	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
213	C₂×C₈	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
214	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
215	C₂×C₈	#41×C₁₃₁	1	1	yes	characteristic
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
220	C₁₆	—	4	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
237	C₂×C₄	#121×C₁₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
241	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
242	C₈	—	4	2	no	—
243	C₈	#102×C₁₃₁	1	2	yes	—
244	C₈	#102×C₁₃₁	1	2	yes	—
245	C₈	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
246	C₈	#117×C₁₃₁	1	1	yes	characteristic
247	C₈	#121×C₁₃₁	1	1	yes	characteristic
248	C₈	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	#417×C₁₃₁	1	1	yes	characteristic
253	C₂²	#459×C₁₃₁	1	1	yes	characteristic
254	C₂²	#476×C₁₃₁	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	16	2	no	—
258	C₄	#380×C₁₃₁	1	1	yes	characteristic
259	C₄	C₂×#102×C₁₃₁	1	1	yes	characteristic
260	C₄	#461×C₁₃₁	1	1	yes	characteristic
261	C₄	#474×C₁₃₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	8	2	no	—
265	C₂	#1802×C₁₃₁	1	1	yes	characteristic
266	C₂	#2212×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
prime radical	C₂×#135×C₁₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₃₁	C₂	semidirect
4. #1846×C₁₃₁	C₂	semidirect
5. #1387×C₁₃₁	C₂	semidirect
10. #443×C₁₃₁	C₄	semidirect
133. #12820	C₁₃₁	direct (N × H)
134. C₁₃₁	#12820	direct (N × H)
136. C₂×#364	C₂×C₁₃₁	semidirect
138. #1846	C₂×C₁₃₁	semidirect
139. #1387	C₂×C₁₃₁	semidirect
144. #443	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443×C₁₃₁	2
C₂×#364×C₁₃₁	2
#2139×C₁₃₁	2
#1846×C₁₃₁	2
#1387×C₁₃₁	2
#12820	131

Subgroups — C131×#12820 #45698704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#12820 #45698704