Subgroups — C₁₃₁×#22504 #45708388

Order 33536.

1182 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#436×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1990×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1988×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#1894×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1872×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1825×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#1803×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1736×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
13	#556×C₁₃₁	—	2	4	no	—
14	#556×C₁₃₁	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×#126×C₁₃₁	—	2	2	no	—
17	C₂×#124×C₁₃₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₁₃₁	—	2	1	no	—
20	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
25	#436×C₁₃₁	—	2	4	no	—
26	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×#103×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂×#103×C₁₃₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
32	#389×C₁₃₁	—	2	1	no	—
33	#385×C₁₃₁	—	2	2	no	—
34	#384×C₁₃₁	C₂²	1	1	yes	characteristic
35	#380×C₁₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
37	#340×C₁₃₁	C₂²	1	2	yes	—
38	#340×C₁₃₁	C₂²	1	2	yes	—
39	#340×C₁₃₁	C₂²	1	2	yes	—
40	#340×C₁₃₁	C₂²	1	2	yes	—
41	#340×C₁₃₁	C₂²	1	1	yes	characteristic
42	#340×C₁₃₁	C₂²	1	2	yes	—
43	#340×C₁₃₁	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₁₃₁	—	2	1	no	—
45	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₃₁	—	2	2	no	—
48	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
49	C₂×M₁₆×C₁₃₁	—	2	4	no	—
50	C₂×M₁₆×C₁₃₁	—	2	4	no	—
51	C₂×M₁₆×C₁₃₁	—	4	2	no	—
52	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
57	#126×C₁₃₁	—	2	16	no	—
58	#124×C₁₃₁	—	2	8	no	—
59	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
60	C₂²⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
61	C₄×Q₈×C₁₃₁	—	2	8	no	—
62	C₄×Q₈×C₁₃₁	—	2	2	no	—
63	C₄×Q₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
64	C₄×D₄×C₁₃₁	—	4	4	no	—
65	C₄×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
66	C₄×D₄×C₁₃₁	D₄	1	4	yes	—
67	#117×C₁₃₁	—	2	4	no	—
68	#117×C₁₃₁	C₂×C₄	1	8	yes	complemented
69	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
77	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
87	C₂×C₄²×C₁₃₁	—	2	2	no	—
88	C₂×C₄²×C₁₃₁	—	2	1	no	—
89	C₂×C₄²×C₁₃₁	—	2	1	no	—
90	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₁₃₁	—	2	2	no	—
92	#103×C₁₃₁	—	2	2	no	—
93	#103×C₁₃₁	—	2	2	no	—
94	#103×C₁₃₁	—	4	2	no	—
95	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
96	C₄×C₈×C₁₃₁	—	2	2	no	—
97	#95×C₁₃₁	—	2	2	no	—
98	#95×C₁₃₁	—	2	1	no	—
99	C₂⁴×C₁₃₁	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₁₃₁	—	2	4	no	—
101	C₂×Q₈×C₁₃₁	—	2	1	no	—
102	C₂×Q₈×C₁₃₁	—	2	1	no	—
103	C₂×Q₈×C₁₃₁	—	2	1	no	—
104	C₂×Q₈×C₁₃₁	—	2	1	no	—
105	C₂×Q₈×C₁₃₁	—	2	1	no	—
106	C₂×Q₈×C₁₃₁	—	4	2	no	—
107	C₂×Q₈×C₁₃₁	—	4	2	no	—
108	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
109	C₂×D₄×C₁₃₁	—	2	2	no	—
110	C₂×D₄×C₁₃₁	—	2	2	no	—
111	C₂×D₄×C₁₃₁	C₂²⋊C₄	1	4	yes	—
112	C₂²×C₄×C₁₃₁	—	2	2	no	—
113	C₂²×C₄×C₁₃₁	—	2	2	no	—
114	C₂²×C₄×C₁₃₁	—	2	2	no	—
115	C₂²×C₄×C₁₃₁	—	2	2	no	—
116	C₂²×C₄×C₁₃₁	—	2	1	no	—
117	C₂²×C₄×C₁₃₁	—	2	1	no	—
118	C₂²×C₄×C₁₃₁	—	2	1	no	—
119	C₂²×C₄×C₁₃₁	—	2	2	no	—
120	C₂²×C₄×C₁₃₁	—	2	1	no	—
121	C₂²×C₄×C₁₃₁	—	2	2	no	—
122	C₂²×C₄×C₁₃₁	—	4	4	no	—
123	C₂²×C₄×C₁₃₁	—	4	4	no	—
124	C₂²×C₄×C₁₃₁	—	8	2	no	—
125	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
126	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₁₃₁	—	4	8	no	—
132	C₂×C₈×C₁₃₁	—	2	1	no	—
133	C₂×C₈×C₁₃₁	—	2	1	no	—
134	C₂×C₈×C₁₃₁	—	2	1	no	—
135	C₂×C₈×C₁₃₁	—	2	2	no	—
136	C₂×C₈×C₁₃₁	—	2	1	no	—
137	C₂×C₈×C₁₃₁	—	2	2	no	—
138	C₂×C₈×C₁₃₁	—	2	2	no	—
139	C₂×C₈×C₁₃₁	—	2	1	no	—
140	C₂×C₈×C₁₃₁	#41	1	2	yes	—
141	C₄⋊C₄×C₁₃₁	—	2	8	no	—
142	C₄⋊C₄×C₁₃₁	—	2	4	no	—
143	C₄⋊C₄×C₁₃₁	—	2	4	no	—
144	C₄⋊C₄×C₁₃₁	—	2	1	no	—
145	C₄⋊C₄×C₁₃₁	—	2	2	no	—
146	C₄⋊C₄×C₁₃₁	—	2	1	no	—
147	C₄⋊C₄×C₁₃₁	—	2	1	no	—
148	C₄⋊C₄×C₁₃₁	—	2	2	no	—
149	C₄⋊C₄×C₁₃₁	—	2	2	no	—
150	C₄⋊C₄×C₁₃₁	—	2	1	no	—
151	C₄⋊C₄×C₁₃₁	—	4	4	no	—
152	C₄⋊C₄×C₁₃₁	—	4	2	no	—
153	C₄⋊C₄×C₁₃₁	—	8	1	no	—
154	C₄⋊C₄×C₁₃₁	C₂²⋊C₄	1	4	yes	—
155	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
157	C₄⋊C₄×C₁₃₁	#41	1	2	yes	—
158	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
159	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
160	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₁₃₁	C₂²⋊C₄	1	8	yes	—
162	C₄²×C₁₃₁	—	2	8	no	—
163	C₄²×C₁₃₁	—	2	4	no	—
164	C₄²×C₁₃₁	—	4	2	no	—
165	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
166	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
167	C₂³×C₁₃₁	—	2	2	no	—
168	C₂³×C₁₃₁	—	2	2	no	—
169	C₂³×C₁₃₁	—	2	2	no	—
170	C₂³×C₁₃₁	—	2	2	no	—
171	C₂³×C₁₃₁	—	2	2	no	—
172	C₂³×C₁₃₁	—	2	2	no	—
173	C₂³×C₁₃₁	C₂×C₂²⋊C₄	1	4	yes	—
174	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₁₃₁	—	2	2	no	—
176	Q₈×C₁₃₁	—	2	2	no	—
177	Q₈×C₁₃₁	—	4	1	no	—
178	Q₈×C₁₃₁	—	4	1	no	—
179	Q₈×C₁₃₁	—	4	8	no	—
180	D₄×C₁₃₁	—	2	8	no	—
181	C₂×C₄×C₁₃₁	—	2	2	no	—
182	C₂×C₄×C₁₃₁	—	2	2	no	—
183	C₂×C₄×C₁₃₁	—	2	1	no	—
184	C₂×C₄×C₁₃₁	—	2	1	no	—
185	C₂×C₄×C₁₃₁	—	2	4	no	—
186	C₂×C₄×C₁₃₁	—	2	4	no	—
187	C₂×C₄×C₁₃₁	—	2	4	no	—
188	C₂×C₄×C₁₃₁	—	2	4	no	—
189	C₂×C₄×C₁₃₁	—	2	2	no	—
190	C₂×C₄×C₁₃₁	—	2	2	no	—
191	C₂×C₄×C₁₃₁	—	2	4	no	—
192	C₂×C₄×C₁₃₁	—	2	2	no	—
193	C₂×C₄×C₁₃₁	—	4	1	no	—
194	C₂×C₄×C₁₃₁	—	4	1	no	—
195	C₂×C₄×C₁₃₁	—	4	1	no	—
196	C₂×C₄×C₁₃₁	—	4	2	no	—
197	C₂×C₄×C₁₃₁	—	4	2	no	—
198	C₂×C₄×C₁₃₁	—	4	1	no	—
199	C₂×C₄×C₁₃₁	—	4	1	no	—
200	C₂×C₄×C₁₃₁	—	4	1	no	—
201	C₂×C₄×C₁₃₁	—	4	2	no	—
202	C₂×C₄×C₁₃₁	—	4	1	no	—
203	C₂×C₄×C₁₃₁	—	4	1	no	—
204	C₂×C₄×C₁₃₁	—	4	2	no	—
205	C₂×C₄×C₁₃₁	—	4	2	no	—
206	C₂×C₄×C₁₃₁	—	4	1	no	—
207	C₂×C₄×C₁₃₁	—	8	8	no	—
208	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	4	yes	—
209	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
212	C₂×C₄×C₁₃₁	#137	1	2	yes	—
213	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
219	C₈×C₁₃₁	—	2	2	no	—
220	C₈×C₁₃₁	—	4	2	no	—
221	C₈×C₁₃₁	—	4	1	no	—
222	C₂²×C₁₃₁	—	2	4	no	—
223	C₂²×C₁₃₁	—	2	4	no	—
224	C₂²×C₁₃₁	—	2	4	no	—
225	C₂²×C₁₃₁	—	4	2	no	—
226	C₂²×C₁₃₁	—	4	2	no	—
227	C₂²×C₁₃₁	—	4	2	no	—
228	C₂²×C₁₃₁	—	4	2	no	—
229	C₂²×C₁₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₁₃₁	#546	1	1	yes	characteristic
231	C₂²×C₁₃₁	#547	1	1	yes	characteristic
232	C₂²×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂²×C₁₃₁	C₂×#137	1	1	yes	characteristic
234	C₂²×C₁₃₁	#577	1	1	yes	characteristic
235	C₂²×C₁₃₁	#578	1	1	yes	characteristic
236	C₄×C₁₃₁	—	2	4	no	—
237	C₄×C₁₃₁	—	2	2	no	—
238	C₄×C₁₃₁	—	4	2	no	—
239	C₄×C₁₃₁	—	4	4	no	—
240	C₄×C₁₃₁	—	4	2	no	—
241	C₄×C₁₃₁	—	4	2	no	—
242	C₄×C₁₃₁	—	8	1	no	—
243	C₄×C₁₃₁	—	8	2	no	—
244	C₄×C₁₃₁	#417	1	4	yes	—
245	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
246	C₄×C₁₃₁	#520	1	1	yes	characteristic
247	C₄×C₁₃₁	#546	1	1	yes	characteristic
248	C₄×C₁₃₁	#547	1	1	yes	characteristic
249	C₂×C₁₃₁	—	4	4	no	—
250	C₂×C₁₃₁	#2339	1	1	yes	characteristic
251	C₂×C₁₃₁	C₂×#417	1	1	yes	characteristic
252	C₂×C₁₃₁	#2898	1	1	yes	characteristic
253	C₂×C₁₃₁	#3324	1	1	yes	characteristic
254	C₂×C₁₃₁	#3328	1	1	yes	characteristic
255	C₂×C₁₃₁	#3353	1	1	yes	characteristic
256	C₂×C₁₃₁	#3356	1	1	yes	characteristic
257	#22504	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₁₃₁	#22504	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×#556	C₂×C₁₃₁	1	1	yes	characteristic
260	C₂×#436	C₂×C₁₃₁	1	1	yes	characteristic
261	D₄×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic
262	#1990	C₂×C₁₃₁	1	1	yes	characteristic, complemented
263	#1988	C₂×C₁₃₁	1	1	yes	characteristic, complemented
264	#1894	C₂×C₁₃₁	1	2	yes	complemented
265	#1872	C₂×C₁₃₁	1	2	yes	complemented
266	#1825	C₂×C₁₃₁	1	2	yes	complemented
267	#1803	C₂×C₁₃₁	1	1	yes	characteristic, complemented
268	C₄×#103	C₂×C₁₃₁	1	1	yes	characteristic, complemented
269	#1736	C₂×C₁₃₁	1	2	yes	complemented
270	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
271	#556	—	2	4	no	—
272	#556	C₄×C₁₃₁	1	8	yes	complemented
273	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
274	C₂×#126	—	2	2	no	—
275	C₂×#124	—	2	1	no	—
276	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
277	C₂×C₄×Q₈	—	2	1	no	—
278	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
279	C₂×C₄×D₄	—	2	1	no	—
280	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
281	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
282	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
283	#436	—	2	4	no	—
284	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
285	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
286	C₂×#103	C₂²×C₁₃₁	1	1	yes	characteristic
287	C₂×#103	C₂²×C₁₃₁	1	2	yes	—
288	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
290	#389	—	2	1	no	—
291	#385	—	2	2	no	—
292	#384	C₂²×C₁₃₁	1	1	yes	characteristic
293	#380	C₂²×C₁₃₁	1	2	yes	—
294	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
295	#340	C₂²×C₁₃₁	1	2	yes	—
296	#340	C₂²×C₁₃₁	1	1	yes	characteristic
297	#340	C₂²×C₁₃₁	1	2	yes	—
298	#340	C₂²×C₁₃₁	1	2	yes	—
299	#340	C₂²×C₁₃₁	1	2	yes	—
300	#340	C₂²×C₁₃₁	1	2	yes	—
301	#340	C₂²×C₁₃₁	1	1	yes	characteristic
302	C₂²×Q₈	—	2	1	no	—
303	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
304	C₂²×D₄	D₄×C₁₃₁	1	1	yes	characteristic
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
307	C₂×M₁₆	—	2	4	no	—
308	C₂×M₁₆	—	2	4	no	—
309	C₂×M₁₆	—	4	2	no	—
310	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
311	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
315	#126	—	2	16	no	—
316	#124	—	2	8	no	—
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
319	C₄×Q₈	—	2	8	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
322	C₄×D₄	—	4	4	no	—
323	C₄×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
324	C₄×D₄	D₄×C₁₃₁	1	4	yes	—
325	#117	—	2	4	no	—
326	#117	C₂×C₄×C₁₃₁	1	8	yes	complemented
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	4	2	no	—
335	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
349	C₄⋊C₈	—	2	2	no	—
350	#103	—	2	2	no	—
351	#103	—	2	2	no	—
352	#103	—	4	2	no	—
353	C₂²⋊C₈	—	2	4	no	—
354	C₄×C₈	—	2	2	no	—
355	#95	—	2	1	no	—
356	#95	—	2	2	no	—
357	C₂⁴	C₂×D₄×C₁₃₁	1	2	yes	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	4	2	no	—
365	C₂×Q₈	—	4	2	no	—
366	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	C₂²⋊C₄×C₁₃₁	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	4	4	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	8	2	no	—
383	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
384	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
387	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
389	M₁₆	—	4	8	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	#41×C₁₃₁	1	2	yes	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	4	4	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	8	1	no	—
412	C₄⋊C₄	C₂²⋊C₄×C₁₃₁	1	4	yes	—
413	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
414	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
415	C₄⋊C₄	#41×C₁₃₁	1	2	yes	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	C₂²⋊C₄×C₁₃₁	1	8	yes	—
420	C₄²	—	2	8	no	—
421	C₄²	—	2	4	no	—
422	C₄²	—	4	2	no	—
423	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
424	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	C₂×C₂²⋊C₄×C₁₃₁	1	4	yes	—
432	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
433	Q₈	—	2	2	no	—
434	Q₈	—	2	2	no	—
435	Q₈	—	4	8	no	—
436	Q₈	—	4	1	no	—
437	Q₈	—	4	1	no	—
438	D₄	—	2	8	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	8	8	no	—
466	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	4	yes	—
467	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
468	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
469	C₂×C₄	AGL(1,ℤ/8)×C₁₃₁	1	2	yes	—
470	C₂×C₄	#137×C₁₃₁	1	2	yes	—
471	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
475	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
476	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	1	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂²×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
488	C₂²	#546×C₁₃₁	1	1	yes	characteristic
489	C₂²	#547×C₁₃₁	1	1	yes	characteristic
490	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
491	C₂²	C₂×#137×C₁₃₁	1	1	yes	characteristic
492	C₂²	#577×C₁₃₁	1	1	yes	characteristic
493	C₂²	#578×C₁₃₁	1	1	yes	characteristic
494	C₄	—	2	4	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	2	no	—
500	C₄	—	8	2	no	—
501	C₄	—	8	1	no	—
502	C₄	#417×C₁₃₁	1	4	yes	—
503	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
504	C₄	#520×C₁₃₁	1	1	yes	characteristic
505	C₄	#546×C₁₃₁	1	1	yes	characteristic
506	C₄	#547×C₁₃₁	1	1	yes	characteristic
507	C₂	—	4	4	no	—
508	C₂	#2339×C₁₃₁	1	1	yes	characteristic
509	C₂	C₂×#417×C₁₃₁	1	1	yes	characteristic
510	C₂	#2898×C₁₃₁	1	1	yes	characteristic
511	C₂	#3324×C₁₃₁	1	1	yes	characteristic
512	C₂	#3328×C₁₃₁	1	1	yes	characteristic
513	C₂	#3353×C₁₃₁	1	1	yes	characteristic
514	C₂	#3356×C₁₃₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #1990×C₁₃₁	C₂	semidirect
5. #1988×C₁₃₁	C₂	semidirect
6. #1894×C₁₃₁	C₂	semidirect
7. #1872×C₁₃₁	C₂	semidirect
8. #1825×C₁₃₁	C₂	semidirect
9. #1803×C₁₃₁	C₂	semidirect
10. C₄×#103×C₁₃₁	C₂	semidirect
11. #1736×C₁₃₁	C₂	semidirect
14. #556×C₁₃₁	C₄	semidirect
56. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
60. C₂²⋊Q₈×C₁₃₁	C₂×C₄	semidirect
63. C₄×Q₈×C₁₃₁	C₂×C₄	semidirect
68. #117×C₁₃₁	C₂×C₄	semidirect
257. #22504	C₁₃₁	direct (N × H)
258. C₁₃₁	#22504	direct (N × H)
262. #1990	C₂×C₁₃₁	semidirect
263. #1988	C₂×C₁₃₁	semidirect
264. #1894	C₂×C₁₃₁	semidirect
265. #1872	C₂×C₁₃₁	semidirect
266. #1825	C₂×C₁₃₁	semidirect
267. #1803	C₂×C₁₃₁	semidirect
268. C₄×#103	C₂×C₁₃₁	semidirect
269. #1736	C₂×C₁₃₁	semidirect
272. #556	C₄×C₁₃₁	semidirect
314. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
318. C₂²⋊Q₈	C₂×C₄×C₁₃₁	semidirect
321. C₄×Q₈	C₂×C₄×C₁₃₁	semidirect
326. #117	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
C₂×#556×C₁₃₁	2
C₂×#436×C₁₃₁	2
D₄×C₄⋊C₄×C₁₃₁	2
#1990×C₁₃₁	2
#1988×C₁₃₁	2
#1894×C₁₃₁	2
#1872×C₁₃₁	2
#1825×C₁₃₁	2
#1803×C₁₃₁	2
C₄×#103×C₁₃₁	2
#1736×C₁₃₁	2
#22504	131

Subgroups — C131×#22504 #45708388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22504 #45708388