Subgroups — C₂×#1328×C₁₃₁ #45697341

Order 33536.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#424×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#349×C₁₃₁	C₂	1	2	yes	maximal
3	#1328×C₁₃₁	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₁₃₁	C₄	1	1	yes	characteristic
6	C₂×#117×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₃₁	—	2	2	no	—
8	#424×C₁₃₁	—	2	2	no	—
9	#424×C₁₃₁	C₄	1	2	yes	—
10	#424×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₃₁	—	2	2	no	—
12	#349×C₁₃₁	—	2	2	no	—
13	#349×C₁₃₁	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₁₃₁	—	4	1	no	—
16	C₂×M₁₆×C₁₃₁	—	4	1	no	—
17	C₂×M₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₃₁	D₄	1	2	yes	—
19	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₁	—	2	1	no	—
21	C₂²×C₈×C₁₃₁	—	2	1	no	—
22	C₂²×C₈×C₁₃₁	—	4	1	no	—
23	C₂²×C₈×C₁₃₁	—	4	1	no	—
24	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃₁	D₄	1	2	yes	—
26	#117×C₁₃₁	—	2	1	no	—
27	#117×C₁₃₁	—	2	1	no	—
28	#117×C₁₃₁	C₈	1	2	yes	—
29	#117×C₁₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
32	C₂×C₄²×C₁₃₁	—	2	1	no	—
33	M₃₂×C₁₃₁	—	2	4	no	—
34	M₃₂×C₁₃₁	—	4	2	no	—
35	C₂×C₁₆×C₁₃₁	—	4	2	no	—
36	C₄⋊C₈×C₁₃₁	—	2	4	no	—
37	C₄⋊C₈×C₁₃₁	—	2	4	no	—
38	C₂⁴×C₁₃₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₁	—	2	1	no	—
40	C₂²×C₄×C₁₃₁	—	2	1	no	—
41	C₂²×C₄×C₁₃₁	—	2	1	no	—
42	C₂²×C₄×C₁₃₁	—	2	1	no	—
43	C₂²×C₄×C₁₃₁	—	2	2	no	—
44	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₁₃₁	—	4	1	no	—
51	M₁₆×C₁₃₁	—	4	2	no	—
52	M₁₆×C₁₃₁	—	4	1	no	—
53	C₂×C₈×C₁₃₁	—	2	2	no	—
54	C₂×C₈×C₁₃₁	—	2	2	no	—
55	C₂×C₈×C₁₃₁	—	2	2	no	—
56	C₂×C₈×C₁₃₁	—	4	2	no	—
57	C₂×C₈×C₁₃₁	—	4	1	no	—
58	C₂×C₈×C₁₃₁	—	4	1	no	—
59	C₂×C₈×C₁₃₁	—	4	2	no	—
60	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₄⋊C₄×C₁₃₁	—	2	2	no	—
63	C₄⋊C₄×C₁₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	4	yes	—
66	C₄²×C₁₃₁	—	2	2	no	—
67	C₄²×C₁₃₁	—	2	2	no	—
68	C₁₆×C₁₃₁	—	4	4	no	—
69	C₂³×C₁₃₁	—	2	1	no	—
70	C₂³×C₁₃₁	—	2	1	no	—
71	C₂³×C₁₃₁	—	4	2	no	—
72	C₂³×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
73	C₂³×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
74	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃₁	—	2	1	no	—
76	C₂×C₄×C₁₃₁	—	2	1	no	—
77	C₂×C₄×C₁₃₁	—	2	1	no	—
78	C₂×C₄×C₁₃₁	—	2	1	no	—
79	C₂×C₄×C₁₃₁	—	2	1	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
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	4	no	—
86	C₂×C₄×C₁₃₁	—	4	4	no	—
87	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	#99	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	#99	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	#101	1	1	yes	characteristic
92	C₂×C₄×C₁₃₁	#101	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₁₃₁	—	2	2	no	—
96	C₈×C₁₃₁	—	2	2	no	—
97	C₈×C₁₃₁	—	4	2	no	—
98	C₂²×C₁₃₁	—	2	2	no	—
99	C₂²×C₁₃₁	—	2	1	no	—
100	C₂²×C₁₃₁	—	2	1	no	—
101	C₂²×C₁₃₁	—	4	1	no	—
102	C₂²×C₁₃₁	—	4	1	no	—
103	C₂²×C₁₃₁	—	4	2	no	—
104	C₂²×C₁₃₁	—	4	2	no	—
105	C₂²×C₁₃₁	#324	1	1	yes	characteristic
106	C₂²×C₁₃₁	#324	1	1	yes	characteristic
107	C₂²×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₁₃₁	—	2	1	no	—
109	C₄×C₁₃₁	—	2	1	no	—
110	C₄×C₁₃₁	—	4	4	no	—
111	C₄×C₁₃₁	#324	1	1	yes	characteristic
112	C₄×C₁₃₁	#324	1	1	yes	characteristic
113	C₄×C₁₃₁	C₂×#99	1	1	yes	characteristic
114	C₄×C₁₃₁	C₂×#101	1	1	yes	characteristic
115	C₂×C₁₃₁	—	2	1	no	—
116	C₂×C₁₃₁	—	2	1	no	—
117	C₂×C₁₃₁	—	4	1	no	—
118	C₂×C₁₃₁	—	4	1	no	—
119	C₂×C₁₃₁	#1328	1	2	yes	direct product factor
120	C₂×C₁₃₁	C₂×#324	1	1	yes	characteristic
121	C₂×#1328	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₁	C₂×#1328	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#424	C₂×C₁₃₁	1	1	yes	characteristic
124	C₂×#349	C₂×C₁₃₁	1	2	yes	—
125	#1328	C₂×C₁₃₁	1	4	yes	direct product factor
126	C₂²×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
127	C₂³×C₈	C₄×C₁₃₁	1	1	yes	characteristic
128	C₂×#117	C₄×C₁₃₁	1	1	yes	characteristic
129	C₂×M₃₂	—	2	2	no	—
130	#424	—	2	2	no	—
131	#424	C₄×C₁₃₁	1	2	yes	—
132	#424	C₂²×C₁₃₁	1	2	yes	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	#349	—	2	2	no	—
135	#349	C₂²×C₁₃₁	1	4	yes	—
136	C₂³×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
137	C₂×M₁₆	—	4	1	no	—
138	C₂×M₁₆	—	4	1	no	—
139	C₂×M₁₆	C₂×C₄×C₁₃₁	1	1	yes	characteristic
140	C₂×M₁₆	D₄×C₁₃₁	1	2	yes	—
141	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	4	1	no	—
145	C₂²×C₈	—	4	1	no	—
146	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
147	C₂²×C₈	D₄×C₁₃₁	1	2	yes	—
148	#117	—	2	1	no	—
149	#117	—	2	1	no	—
150	#117	C₈×C₁₃₁	1	2	yes	—
151	#117	C₂×C₄×C₁₃₁	1	2	yes	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	C₈×C₁₃₁	1	2	yes	—
154	C₂×C₄²	—	2	1	no	—
155	M₃₂	—	2	4	no	—
156	M₃₂	—	4	2	no	—
157	C₂×C₁₆	—	4	2	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₂⁴	C₂×C₈×C₁₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
169	C₂²×C₄	M₁₆×C₁₃₁	1	1	yes	characteristic
170	C₂²×C₄	M₁₆×C₁₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	—	4	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
183	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	4	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₁₆	—	4	4	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
195	C₂³	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
196	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
211	C₂×C₄	#99×C₁₃₁	1	1	yes	characteristic
212	C₂×C₄	#99×C₁₃₁	1	1	yes	characteristic
213	C₂×C₄	#101×C₁₃₁	1	1	yes	characteristic
214	C₂×C₄	#101×C₁₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂×M₁₆×C₁₃₁	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	#324×C₁₃₁	1	1	yes	characteristic
228	C₂²	#324×C₁₃₁	1	1	yes	characteristic
229	C₂²	C₂×C₂²⋊C₈×C₁₃₁	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	4	no	—
233	C₄	#324×C₁₃₁	1	1	yes	characteristic
234	C₄	#324×C₁₃₁	1	1	yes	characteristic
235	C₄	C₂×#99×C₁₃₁	1	1	yes	characteristic
236	C₄	C₂×#101×C₁₃₁	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	#1328×C₁₃₁	1	2	yes	direct product factor
242	C₂	C₂×#324×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#99
commutator	C₂³	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#324
prime radical	C₂⁴×C₁₃₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×M₁₆×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1328×C₁₃₁	C₂	direct (N × H)
119. C₂×C₁₃₁	#1328	direct (N × H)
121. C₂×#1328	C₁₃₁	direct (N × H)
122. C₁₃₁	C₂×#1328	direct (N × H)
125. #1328	C₂×C₁₃₁	direct (N × H)
241. C₂	#1328×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×#424×C₁₃₁	2
C₂×#349×C₁₃₁	2
#1328×C₁₃₁	2
C₂×#1328	131

Subgroups — C2×#1328×C131 #45697341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1328×C₁₃₁ #45697341