Subgroups — C₁₃₁×#8224 #45694108

Order 33536.

528 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1745×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	C₂×#329×C₁₃₁	C₂	1	2	yes	maximal
4	#1295×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₁	C₄	1	1	yes	characteristic
8	C₂×#97×C₁₃₁	C₂²	1	4	yes	—
9	#391×C₁₃₁	C₄	1	1	yes	characteristic
10	#390×C₁₃₁	C₄	1	1	yes	characteristic
11	#388×C₁₃₁	—	2	2	no	—
12	#382×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
14	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
15	#336×C₁₃₁	—	2	2	no	—
16	#336×C₁₃₁	—	2	2	no	—
17	#329×C₁₃₁	C₄	1	8	yes	complemented
18	C₂²×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₁	—	2	1	no	—
21	C₂²×C₈×C₁₃₁	—	2	4	no	—
22	#124×C₁₃₁	—	2	2	no	—
23	#124×C₁₃₁	C₂×C₄	1	4	yes	—
24	C₄×Q₈×C₁₃₁	—	4	2	no	—
25	C₄×D₄×C₁₃₁	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄²×C₁₃₁	—	2	1	no	—
33	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
38	#97×C₁₃₁	—	2	4	no	—
39	#97×C₁₃₁	C₂×C₄	1	8	yes	complemented
40	#95×C₁₃₁	—	2	2	no	—
41	#95×C₁₃₁	—	2	2	no	—
42	C₂⁴×C₁₃₁	—	2	1	no	—
43	C₂×Q₈×C₁₃₁	—	2	1	no	—
44	C₂×Q₈×C₁₃₁	—	4	1	no	—
45	C₂×Q₈×C₁₃₁	—	4	1	no	—
46	C₂×Q₈×C₁₃₁	C₄²	1	2	yes	—
47	C₂×D₄×C₁₃₁	—	2	1	no	—
48	C₂×D₄×C₁₃₁	—	4	1	no	—
49	C₂×D₄×C₁₃₁	—	4	1	no	—
50	C₂×D₄×C₁₃₁	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₃₁	—	2	1	no	—
52	C₂²×C₄×C₁₃₁	—	2	1	no	—
53	C₂²×C₄×C₁₃₁	—	2	1	no	—
54	C₂²×C₄×C₁₃₁	—	4	4	no	—
55	C₂²×C₄×C₁₃₁	—	4	2	no	—
56	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃₁	—	2	8	no	—
62	C₂×C₈×C₁₃₁	—	4	4	no	—
63	C₂×C₈×C₁₃₁	—	4	4	no	—
64	C₄⋊C₄×C₁₃₁	—	4	1	no	—
65	C₄⋊C₄×C₁₃₁	—	4	1	no	—
66	C₄⋊C₄×C₁₃₁	—	4	2	no	—
67	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
68	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
69	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
70	C₄²×C₁₃₁	—	2	2	no	—
71	C₄²×C₁₃₁	—	4	1	no	—
72	C₄²×C₁₃₁	—	4	2	no	—
73	C₄²×C₁₃₁	—	4	2	no	—
74	C₄²×C₁₃₁	—	4	2	no	—
75	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃₁	#41	1	2	yes	—
77	C₂³×C₁₃₁	—	2	2	no	—
78	C₂³×C₁₃₁	—	4	1	no	—
79	C₂³×C₁₃₁	—	4	1	no	—
80	C₂³×C₁₃₁	—	4	2	no	—
81	C₂³×C₁₃₁	—	4	2	no	—
82	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₁₃₁	—	4	2	no	—
84	Q₈×C₁₃₁	—	4	2	no	—
85	D₄×C₁₃₁	—	4	2	no	—
86	D₄×C₁₃₁	—	4	2	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	—	2	2	no	—
91	C₂×C₄×C₁₃₁	—	2	2	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	1	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	—	2	2	no	—
96	C₂×C₄×C₁₃₁	—	2	2	no	—
97	C₂×C₄×C₁₃₁	—	2	2	no	—
98	C₂×C₄×C₁₃₁	—	4	1	no	—
99	C₂×C₄×C₁₃₁	—	4	1	no	—
100	C₂×C₄×C₁₃₁	—	4	1	no	—
101	C₂×C₄×C₁₃₁	—	4	1	no	—
102	C₂×C₄×C₁₃₁	—	4	2	no	—
103	C₂×C₄×C₁₃₁	—	4	2	no	—
104	C₂×C₄×C₁₃₁	—	8	4	no	—
105	C₂×C₄×C₁₃₁	#100	1	2	yes	—
106	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₄×D₄	1	4	yes	—
109	C₈×C₁₃₁	—	4	8	no	—
110	C₂²×C₁₃₁	—	4	2	no	—
111	C₂²×C₁₃₁	—	4	4	no	—
112	C₂²×C₁₃₁	—	8	1	no	—
113	C₂²×C₁₃₁	—	8	1	no	—
114	C₂²×C₁₃₁	—	8	2	no	—
115	C₂²×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₃₁	C₂×#100	1	1	yes	characteristic
117	C₂²×C₁₃₁	#413	1	1	yes	characteristic
118	C₂²×C₁₃₁	#421	1	2	yes	—
119	C₂²×C₁₃₁	#421	1	2	yes	—
120	C₄×C₁₃₁	—	2	1	no	—
121	C₄×C₁₃₁	—	2	1	no	—
122	C₄×C₁₃₁	—	2	4	no	—
123	C₄×C₁₃₁	—	2	2	no	—
124	C₄×C₁₃₁	—	4	1	no	—
125	C₄×C₁₃₁	—	4	2	no	—
126	C₄×C₁₃₁	—	4	2	no	—
127	C₄×C₁₃₁	—	8	1	no	—
128	C₄×C₁₃₁	—	8	1	no	—
129	C₂×C₁₃₁	—	8	2	no	—
130	C₂×C₁₃₁	#1533	1	1	yes	characteristic
131	C₂×C₁₃₁	#1535	1	2	yes	—
132	C₂×C₁₃₁	#1536	1	1	yes	characteristic
133	C₂×C₁₃₁	C₄×#100	1	1	yes	characteristic
134	C₂×C₁₃₁	#1759	1	2	yes	—
135	#8224	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₁	#8224	1	1	yes	characteristic, direct product factor, Sylow
137	C₄×#124	C₂×C₁₃₁	1	1	yes	characteristic
138	#1745	C₂×C₁₃₁	1	2	yes	complemented
139	C₂×#329	C₂×C₁₃₁	1	2	yes	—
140	#1295	C₂×C₁₃₁	1	2	yes	complemented
141	C₂×#124	C₂²×C₁₃₁	1	1	yes	characteristic
142	C₂×C₄×Q₈	C₄×C₁₃₁	1	1	yes	characteristic
143	C₂×C₄×D₄	C₄×C₁₃₁	1	1	yes	characteristic
144	C₂×#97	C₂²×C₁₃₁	1	4	yes	—
145	#391	C₄×C₁₃₁	1	1	yes	characteristic
146	#390	C₄×C₁₃₁	1	1	yes	characteristic
147	#388	—	2	2	no	—
148	#382	C₂²×C₁₃₁	1	1	yes	characteristic
149	C₄×C₂²⋊C₄	—	2	2	no	—
150	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
151	#336	—	2	2	no	—
152	#336	—	2	2	no	—
153	#329	C₄×C₁₃₁	1	8	yes	complemented
154	C₂²×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
155	C₂²×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
156	C₂³×C₄	—	2	1	no	—
157	C₂²×C₈	—	2	4	no	—
158	#124	—	2	2	no	—
159	#124	C₂×C₄×C₁₃₁	1	4	yes	—
160	C₄×Q₈	—	4	2	no	—
161	C₄×D₄	—	4	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
172	C₂×C₄²	D₄×C₁₃₁	1	2	yes	—
173	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
174	#97	—	2	4	no	—
175	#97	C₂×C₄×C₁₃₁	1	8	yes	complemented
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	C₂⁴	—	2	1	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	C₄²×C₁₃₁	1	2	yes	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	C₄²×C₁₃₁	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	2	8	no	—
204	C₂²⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	2	no	—
211	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
212	C₄²	#41×C₁₃₁	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	#100×C₁₃₁	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
243	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
244	C₂×C₄	C₄×D₄×C₁₃₁	1	4	yes	—
245	C₈	—	4	8	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	1	no	—
251	C₂²	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
252	C₂²	C₂×#100×C₁₃₁	1	1	yes	characteristic
253	C₂²	#413×C₁₃₁	1	1	yes	characteristic
254	C₂²	#421×C₁₃₁	1	2	yes	—
255	C₂²	#421×C₁₃₁	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	8	1	no	—
264	C₄	—	8	1	no	—
265	C₂	—	8	2	no	—
266	C₂	#1533×C₁₃₁	1	1	yes	characteristic
267	C₂	#1535×C₁₃₁	1	2	yes	—
268	C₂	#1536×C₁₃₁	1	1	yes	characteristic
269	C₂	C₄×#100×C₁₃₁	1	1	yes	characteristic
270	C₂	#1759×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1745×C₁₃₁	C₂	semidirect
4. #1295×C₁₃₁	C₂	semidirect
17. #329×C₁₃₁	C₄	semidirect
39. #97×C₁₃₁	C₂×C₄	semidirect
135. #8224	C₁₃₁	direct (N × H)
136. C₁₃₁	#8224	direct (N × H)
138. #1745	C₂×C₁₃₁	semidirect
140. #1295	C₂×C₁₃₁	semidirect
153. #329	C₄×C₁₃₁	semidirect
175. #97	C₂×C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₁₃₁	2
#1745×C₁₃₁	2
C₂×#329×C₁₃₁	2
#1295×C₁₃₁	2
#8224	131

Subgroups — C131×#8224 #45694108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8224 #45694108