Subgroups — C₂×#1567×C₁₃₁ #45703675

Order 33536.

854 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₃₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1567×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#335×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₃₁	—	2	2	no	—
6	C₂³×C₈×C₁₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
8	#423×C₁₃₁	—	2	4	no	—
9	#423×C₁₃₁	C₂²	1	8	yes	—
10	C₂×C₄⋊C₈×C₁₃₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₁	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₁₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	4	yes	—
17	#335×C₁₃₁	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₁	Q₈	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₁	—	2	8	no	—
22	C₂×M₁₆×C₁₃₁	—	4	8	no	—
23	C₂²×C₈×C₁₃₁	—	2	8	no	—
24	C₂²×C₈×C₁₃₁	—	2	4	no	—
25	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₁	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₃₁	—	2	8	no	—
37	C₄⋊C₈×C₁₃₁	—	2	4	no	—
38	C₄⋊C₈×C₁₃₁	C₂³	1	8	yes	—
39	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₃₁	C₂³	1	4	yes	—
43	C₂⁴×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	—	2	4	no	—
45	C₂²×C₄×C₁₃₁	—	2	4	no	—
46	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	D₈	1	4	yes	—
53	C₂²×C₄×C₁₃₁	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
60	M₁₆×C₁₃₁	—	4	16	no	—
61	C₂×C₈×C₁₃₁	—	2	4	no	—
62	C₂×C₈×C₁₃₁	—	2	16	no	—
63	C₂×C₈×C₁₃₁	—	2	8	no	—
64	C₂×C₈×C₁₃₁	—	4	8	no	—
65	C₂×C₈×C₁₃₁	M₁₆	1	8	yes	complemented
66	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
68	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₃₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
75	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₁₃₁	C₂⁴	1	1	yes	characteristic
78	C₂³×C₁₃₁	—	2	2	no	—
79	C₂³×C₁₃₁	—	2	2	no	—
80	C₂³×C₁₃₁	#107	1	4	yes	—
81	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	4	no	—
86	C₂×C₄×C₁₃₁	—	2	4	no	—
87	C₂×C₄×C₁₃₁	—	2	4	no	—
88	C₂×C₄×C₁₃₁	—	2	4	no	—
89	C₂×C₄×C₁₃₁	#107	1	8	yes	—
90	C₂×C₄×C₁₃₁	#107	1	4	yes	—
91	C₂×C₄×C₁₃₁	#108	1	4	yes	—
92	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃₁	C₂×M₁₆	1	4	yes	—
99	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	4	yes	—
102	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₁₃₁	—	2	4	no	—
107	C₈×C₁₃₁	—	4	8	no	—
108	C₈×C₁₃₁	C₂×M₁₆	1	8	yes	complemented
109	C₂²×C₁₃₁	—	2	8	no	—
110	C₂²×C₁₃₁	—	2	2	no	—
111	C₂²×C₁₃₁	—	2	2	no	—
112	C₂²×C₁₃₁	#423	1	2	yes	—
113	C₂²×C₁₃₁	C₂×#107	1	4	yes	—
114	C₂²×C₁₃₁	C₂×#107	1	2	yes	—
115	C₂²×C₁₃₁	C₂×#108	1	2	yes	—
116	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₁₃₁	—	2	2	no	—
118	C₄×C₁₃₁	—	2	1	no	—
119	C₄×C₁₃₁	—	2	1	no	—
120	C₄×C₁₃₁	#423	1	2	yes	—
121	C₄×C₁₃₁	C₂×#107	1	4	yes	—
122	C₄×C₁₃₁	C₂×#107	1	2	yes	—
123	C₄×C₁₃₁	C₂×#108	1	4	yes	—
124	C₄×C₁₃₁	C₂²×M₁₆	1	2	yes	—
125	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
126	C₄×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
127	C₂×C₁₃₁	—	2	4	no	—
128	C₂×C₁₃₁	#1567	1	4	yes	direct product factor
129	C₂×C₁₃₁	C₂×#423	1	1	yes	characteristic
130	C₂×C₁₃₁	C₂²×#107	1	1	yes	characteristic
131	C₂×C₁₃₁	C₂²×#108	1	1	yes	characteristic
132	C₂×#1567	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₁₃₁	C₂×#1567	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#423	C₂×C₁₃₁	1	2	yes	—
135	C₂²×C₄×C₈	C₂×C₁₃₁	1	1	yes	characteristic
136	#1567	C₂×C₁₃₁	1	8	yes	direct product factor
137	C₂×#335	C₂×C₁₃₁	1	4	yes	complemented
138	C₂²×M₁₆	—	2	2	no	—
139	C₂³×C₈	C₄×C₁₃₁	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
141	#423	—	2	4	no	—
142	#423	C₂²×C₁₃₁	1	8	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	4	yes	—
145	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₄×C₁₃₁	1	4	yes	—
147	C₂×C₄×C₈	C₄×C₁₃₁	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₁₃₁	1	4	yes	—
150	#335	C₂²×C₁₃₁	1	16	yes	complemented
151	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
153	C₂³×C₄	Q₈×C₁₃₁	1	1	yes	characteristic
154	C₂×M₁₆	—	2	8	no	—
155	C₂×M₁₆	—	4	8	no	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₁₃₁	1	8	yes	—
160	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
165	C₂×C₄²	Q₈×C₁₃₁	1	1	yes	characteristic
166	C₂×C₄²	Q₈×C₁₃₁	1	2	yes	—
167	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
169	C₄⋊C₈	—	2	8	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	C₂³×C₁₃₁	1	8	yes	—
172	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
173	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
174	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
175	C₄×C₈	C₂³×C₁₃₁	1	4	yes	—
176	C₂⁴	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
180	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
183	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₄	D₈×C₁₃₁	1	4	yes	—
186	C₂²×C₄	Q₁₆×C₁₃₁	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
193	M₁₆	—	4	16	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	16	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	4	8	no	—
198	C₂×C₈	M₁₆×C₁₃₁	1	8	yes	complemented
199	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
201	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
202	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
203	C₄²	C₄⋊C₄×C₁₃₁	1	2	yes	—
204	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
205	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
206	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
208	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
209	C₄²	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
210	C₄²	C₂⁴×C₁₃₁	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#107×C₁₃₁	1	4	yes	—
214	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
215	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	#107×C₁₃₁	1	8	yes	—
223	C₂×C₄	#107×C₁₃₁	1	4	yes	—
224	C₂×C₄	#108×C₁₃₁	1	4	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	2	yes	—
231	C₂×C₄	C₂×M₁₆×C₁₃₁	1	4	yes	—
232	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
234	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	4	yes	—
235	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	4	8	no	—
241	C₈	C₂×M₁₆×C₁₃₁	1	8	yes	complemented
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	#423×C₁₃₁	1	2	yes	—
246	C₂²	C₂×#107×C₁₃₁	1	4	yes	—
247	C₂²	C₂×#107×C₁₃₁	1	2	yes	—
248	C₂²	C₂×#108×C₁₃₁	1	2	yes	—
249	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	#423×C₁₃₁	1	2	yes	—
254	C₄	C₂×#107×C₁₃₁	1	2	yes	—
255	C₄	C₂×#107×C₁₃₁	1	4	yes	—
256	C₄	C₂×#108×C₁₃₁	1	4	yes	—
257	C₄	C₂²×M₁₆×C₁₃₁	1	2	yes	—
258	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
259	C₄	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
260	C₂	—	2	4	no	—
261	C₂	#1567×C₁₃₁	1	4	yes	direct product factor
262	C₂	C₂×#423×C₁₃₁	1	1	yes	characteristic
263	C₂	C₂²×#107×C₁₃₁	1	1	yes	characteristic
264	C₂	C₂²×#108×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₄²	C₂⁴×C₁₃₁
socle	C₂³×C₁₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₄²×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1567×C₁₃₁	C₂	direct (N × H)
4. C₂×#335×C₁₃₁	C₂	semidirect
17. #335×C₁₃₁	C₂²	semidirect
65. C₂×C₈×C₁₃₁	M₁₆	semidirect
108. C₈×C₁₃₁	C₂×M₁₆	semidirect
128. C₂×C₁₃₁	#1567	direct (N × H)
132. C₂×#1567	C₁₃₁	direct (N × H)
133. C₁₃₁	C₂×#1567	direct (N × H)
136. #1567	C₂×C₁₃₁	direct (N × H)
137. C₂×#335	C₂×C₁₃₁	semidirect
150. #335	C₂²×C₁₃₁	semidirect
198. C₂×C₈	M₁₆×C₁₃₁	semidirect
241. C₈	C₂×M₁₆×C₁₃₁	semidirect
261. C₂	#1567×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₃₁	2
C₂²×C₄×C₈×C₁₃₁	2
#1567×C₁₃₁	2
C₂×#335×C₁₃₁	2
C₂×#1567	131

Subgroups — C2×#1567×C131 #45703675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1567×C₁₃₁ #45703675