Subgroups — C₁₃₁×#33464 #45719348

Order 33536.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3414×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2957×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2179×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2178×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2161×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#359×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
9	#568×C₁₃₁	—	2	2	no	—
10	#510×C₁₃₁	C₄	1	4	yes	complemented
11	#509×C₁₃₁	C₄	1	4	yes	complemented
12	#508×C₁₃₁	C₂²	1	4	yes	—
13	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	#444×C₁₃₁	—	2	4	no	—
16	#443×C₁₃₁	C₂²	1	8	yes	complemented
17	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
18	#421×C₁₃₁	—	2	2	no	—
19	#405×C₁₃₁	C₂²	1	1	yes	characteristic
20	#368×C₁₃₁	C₂²	1	4	yes	complemented
21	#359×C₁₃₁	C₂²	1	4	yes	complemented
22	#359×C₁₃₁	C₂²	1	4	yes	complemented
23	#346×C₁₃₁	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
25	C₂×#41×C₁₃₁	—	2	2	no	—
26	#135×C₁₃₁	—	2	2	no	—
27	#135×C₁₃₁	—	2	2	no	—
28	#135×C₁₃₁	C₂×C₄	1	4	yes	—
29	#135×C₁₃₁	C₂³	1	4	yes	—
30	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
32	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
33	#131×C₁₃₁	—	2	8	no	—
34	#131×C₁₃₁	—	4	4	no	—
35	C₂×M₁₆×C₁₃₁	—	2	2	no	—
36	C₂×M₁₆×C₁₃₁	—	4	4	no	—
37	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
39	#117×C₁₃₁	D₄	1	1	yes	characteristic
40	Q₃₂×C₁₃₁	C₂×C₄	1	4	yes	—
41	SD₃₂×C₁₃₁	C₂×C₄	1	8	yes	—
42	D₁₆×C₁₃₁	C₂×C₄	1	4	yes	—
43	M₃₂×C₁₃₁	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₁₃₁	—	2	2	no	—
45	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
48	#108×C₁₃₁	—	2	2	no	—
49	#108×C₁₃₁	C₂³	1	4	yes	—
50	#104×C₁₃₁	—	2	8	no	—
51	#97×C₁₃₁	D₄	1	2	yes	—
52	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
53	#41×C₁₃₁	—	2	4	no	—
54	#41×C₁₃₁	—	2	4	no	—
55	#41×C₁₃₁	—	4	2	no	—
56	#41×C₁₃₁	—	4	2	no	—
57	C₂×Q₈×C₁₃₁	—	2	2	no	—
58	C₂×D₄×C₁₃₁	—	2	2	no	—
59	C₂×D₄×C₁₃₁	—	4	2	no	—
60	C₂²×C₄×C₁₃₁	—	4	2	no	—
61	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₁₃₁	—	2	2	no	—
63	Q₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
64	SD₁₆×C₁₃₁	—	2	4	no	—
65	SD₁₆×C₁₃₁	—	2	4	no	—
66	D₈×C₁₃₁	—	2	2	no	—
67	D₈×C₁₃₁	C₂²×C₄	1	4	yes	—
68	M₁₆×C₁₃₁	—	2	4	no	—
69	M₁₆×C₁₃₁	—	4	2	no	—
70	M₁₆×C₁₃₁	—	4	8	no	—
71	M₁₆×C₁₃₁	D₈	1	4	yes	—
72	C₂×C₈×C₁₃₁	—	4	2	no	—
73	C₂×C₈×C₁₃₁	—	4	8	no	—
74	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃₁	#41	1	2	yes	—
78	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₁₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
81	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
82	C₁₆×C₁₃₁	—	2	1	no	—
83	C₁₆×C₁₃₁	—	2	1	no	—
84	C₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
85	C₂³×C₁₃₁	—	4	2	no	—
86	C₂³×C₁₃₁	#135	1	1	yes	characteristic
87	Q₈×C₁₃₁	—	2	4	no	—
88	Q₈×C₁₃₁	—	4	2	no	—
89	D₄×C₁₃₁	—	2	4	no	—
90	D₄×C₁₃₁	—	4	4	no	—
91	D₄×C₁₃₁	—	4	4	no	—
92	D₄×C₁₃₁	—	4	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	4	2	no	—
95	C₂×C₄×C₁₃₁	—	4	4	no	—
96	C₂×C₄×C₁₃₁	—	4	4	no	—
97	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₃₁	#135	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₁₃₁	—	4	4	no	—
102	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
103	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
104	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
106	C₂²×C₁₃₁	—	4	1	no	—
107	C₂²×C₁₃₁	—	4	4	no	—
108	C₂²×C₁₃₁	—	16	2	no	—
109	C₂²×C₁₃₁	C₄×D₈	1	2	yes	—
110	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
111	C₄×C₁₃₁	—	4	2	no	—
112	C₄×C₁₃₁	—	4	4	no	—
113	C₄×C₁₃₁	C₄×D₈	1	2	yes	—
114	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
116	C₂×C₁₃₁	—	2	1	no	—
117	C₂×C₁₃₁	—	2	2	no	—
118	C₂×C₁₃₁	—	8	4	no	—
119	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
120	#33464	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₃₁	#33464	1	1	yes	characteristic, direct product factor, Sylow
122	#3414	C₂×C₁₃₁	1	1	yes	characteristic
123	#2957	C₂×C₁₃₁	1	2	yes	complemented
124	#2179	C₂×C₁₃₁	1	4	yes	complemented
125	#2178	C₂×C₁₃₁	1	4	yes	complemented
126	#2161	C₂×C₁₃₁	1	1	yes	characteristic, complemented
127	C₂×#359	C₂×C₁₃₁	1	2	yes	complemented
128	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
129	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
130	#568	—	2	2	no	—
131	#510	C₄×C₁₃₁	1	4	yes	complemented
132	#509	C₄×C₁₃₁	1	4	yes	complemented
133	#508	C₂²×C₁₃₁	1	4	yes	—
134	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
136	#444	—	2	4	no	—
137	#443	C₂²×C₁₃₁	1	8	yes	complemented
138	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
139	#421	—	2	2	no	—
140	#405	C₂²×C₁₃₁	1	1	yes	characteristic
141	#368	C₂²×C₁₃₁	1	4	yes	complemented
142	#359	C₂²×C₁₃₁	1	4	yes	complemented
143	#359	C₂²×C₁₃₁	1	4	yes	complemented
144	#346	C₂²×C₁₃₁	1	2	yes	complemented
145	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
146	C₂×#41	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	C₂×C₄×C₁₃₁	1	4	yes	—
150	#135	C₂³×C₁₃₁	1	4	yes	—
151	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
154	#131	—	2	8	no	—
155	#131	—	4	4	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	4	4	no	—
158	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
160	#117	D₄×C₁₃₁	1	1	yes	characteristic
161	Q₃₂	C₂×C₄×C₁₃₁	1	4	yes	—
162	SD₃₂	C₂×C₄×C₁₃₁	1	8	yes	—
163	D₁₆	C₂×C₄×C₁₃₁	1	4	yes	—
164	M₃₂	C₂×C₄×C₁₃₁	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
169	#108	—	2	2	no	—
170	#108	C₂³×C₁₃₁	1	4	yes	—
171	#104	—	2	8	no	—
172	#97	D₄×C₁₃₁	1	2	yes	—
173	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×C₄×C₁₃₁	1	4	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	8	no	—
192	M₁₆	D₈×C₁₃₁	1	4	yes	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	8	no	—
195	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
198	C₂×C₈	#41×C₁₃₁	1	2	yes	—
199	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	#135×C₁₃₁	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
219	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
220	C₂×C₄	#135×C₁₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
222	C₈	—	4	4	no	—
223	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
224	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
225	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
226	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	C₄×D₈×C₁₃₁	1	2	yes	—
231	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₄×D₈×C₁₃₁	1	2	yes	—
235	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
236	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂×C₁₃₁	C₂×C₄×D₈
prime radical	#3414×C₁₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂×M₁₆×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2957×C₁₃₁	C₂	semidirect
3. #2179×C₁₃₁	C₂	semidirect
4. #2178×C₁₃₁	C₂	semidirect
5. #2161×C₁₃₁	C₂	semidirect
6. C₂×#359×C₁₃₁	C₂	semidirect
7. #2108×C₁₃₁	C₂	semidirect
10. #510×C₁₃₁	C₄	semidirect
11. #509×C₁₃₁	C₄	semidirect
16. #443×C₁₃₁	C₂²	semidirect
20. #368×C₁₃₁	C₂²	semidirect
21. #359×C₁₃₁	C₂²	semidirect
22. #359×C₁₃₁	C₂²	semidirect
23. #346×C₁₃₁	C₂²	semidirect
24. C₄×C₁₆×C₁₃₁	C₂²	semidirect
120. #33464	C₁₃₁	direct (N × H)
121. C₁₃₁	#33464	direct (N × H)
123. #2957	C₂×C₁₃₁	semidirect
124. #2179	C₂×C₁₃₁	semidirect
125. #2178	C₂×C₁₃₁	semidirect
126. #2161	C₂×C₁₃₁	semidirect
127. C₂×#359	C₂×C₁₃₁	semidirect
128. #2108	C₂×C₁₃₁	semidirect
131. #510	C₄×C₁₃₁	semidirect
132. #509	C₄×C₁₃₁	semidirect
137. #443	C₂²×C₁₃₁	semidirect
141. #368	C₂²×C₁₃₁	semidirect
142. #359	C₂²×C₁₃₁	semidirect
143. #359	C₂²×C₁₃₁	semidirect
144. #346	C₂²×C₁₃₁	semidirect
145. C₄×C₁₆	C₂²×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3414×C₁₃₁	2
#2957×C₁₃₁	2
#2179×C₁₃₁	2
#2178×C₁₃₁	2
#2161×C₁₃₁	2
C₂×#359×C₁₃₁	2
#2108×C₁₃₁	2
#33464	131

Subgroups — C131×#33464 #45719348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33464 #45719348