Subgroups — C₁₃₁×#26082 #45711966

Order 33536.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3164×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#3084×C₁₃₁	C₂	1	2	yes	maximal
3	#2913×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#2090×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2061×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#1952×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1945×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#1860×C₁₃₁	C₂	1	2	yes	maximal
9	#1846×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#1816×C₁₃₁	C₂	1	1	yes	maximal, characteristic
11	#1764×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#131×C₁₃₁	C₂²	1	1	yes	characteristic
13	#544×C₁₃₁	C₂²	1	2	yes	—
14	#520×C₁₃₁	C₂²	1	2	yes	—
15	#501×C₁₃₁	—	2	2	no	—
16	#500×C₁₃₁	—	2	1	no	—
17	#499×C₁₃₁	—	2	1	no	—
18	#479×C₁₃₁	—	2	2	no	—
19	#478×C₁₃₁	—	2	2	no	—
20	#476×C₁₃₁	—	2	2	no	—
21	#475×C₁₃₁	—	2	2	no	—
22	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
23	#428×C₁₃₁	—	2	2	no	—
24	#428×C₁₃₁	C₂²	1	2	yes	—
25	#428×C₁₃₁	C₂²	1	2	yes	—
26	#428×C₁₃₁	C₂²	1	2	yes	—
27	#427×C₁₃₁	—	2	1	no	—
28	#427×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂×#107×C₁₃₁	—	2	1	no	—
30	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
32	C₂×#106×C₁₃₁	C₂²	1	2	yes	—
33	#424×C₁₃₁	C₂²	1	2	yes	—
34	C₂×#104×C₁₃₁	C₂²	1	1	yes	characteristic
35	#417×C₁₃₁	C₂²	1	2	yes	—
36	#416×C₁₃₁	C₂²	1	1	yes	characteristic
37	#405×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
39	#382×C₁₃₁	C₂²	1	2	yes	—
40	#343×C₁₃₁	C₂²	1	2	yes	—
41	#343×C₁₃₁	C₂²	1	2	yes	—
42	#341×C₁₃₁	C₂²	1	1	yes	characteristic
43	#339×C₁₃₁	C₂²	1	1	yes	characteristic
44	#339×C₁₃₁	C₂²	1	2	yes	—
45	C₂×#41×C₁₃₁	C₂³	1	1	yes	characteristic
46	#131×C₁₃₁	—	2	2	no	—
47	#131×C₁₃₁	Q₈	1	4	yes	complemented
48	C₂×M₁₆×C₁₃₁	—	2	1	no	—
49	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₁	—	2	1	no	—
53	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
57	#125×C₁₃₁	—	2	2	no	—
58	#125×C₁₃₁	—	2	2	no	—
59	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
60	C₄×Q₈×C₁₃₁	—	2	2	no	—
61	C₄×D₄×C₁₃₁	—	2	2	no	—
62	C₄×D₄×C₁₃₁	—	4	2	no	—
63	#117×C₁₃₁	—	2	2	no	—
64	#117×C₁₃₁	C₂³	1	1	yes	characteristic
65	#117×C₁₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
69	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
70	#108×C₁₃₁	—	2	2	no	—
71	#107×C₁₃₁	—	2	1	no	—
72	#107×C₁₃₁	—	2	1	no	—
73	#107×C₁₃₁	—	2	1	no	—
74	#107×C₁₃₁	—	2	1	no	—
75	#107×C₁₃₁	—	2	2	no	—
76	#107×C₁₃₁	—	2	2	no	—
77	#107×C₁₃₁	—	2	2	no	—
78	#107×C₁₃₁	D₄	1	1	yes	characteristic
79	#107×C₁₃₁	D₄	1	1	yes	characteristic
80	#106×C₁₃₁	—	2	2	no	—
81	#106×C₁₃₁	—	2	2	no	—
82	#106×C₁₃₁	—	2	2	no	—
83	#106×C₁₃₁	—	2	2	no	—
84	#106×C₁₃₁	—	2	2	no	—
85	#106×C₁₃₁	D₄	1	2	yes	—
86	C₄⋊C₈×C₁₃₁	—	2	2	no	—
87	C₄⋊C₈×C₁₃₁	—	2	2	no	—
88	#104×C₁₃₁	Q₈	1	4	yes	complemented
89	#103×C₁₃₁	—	2	2	no	—
90	#103×C₁₃₁	D₄	1	1	yes	characteristic
91	#103×C₁₃₁	D₄	1	1	yes	characteristic
92	#102×C₁₃₁	—	2	2	no	—
93	#102×C₁₃₁	D₄	1	1	yes	characteristic
94	#102×C₁₃₁	D₄	1	1	yes	characteristic
95	#97×C₁₃₁	—	2	1	no	—
96	C₄×C₈×C₁₃₁	—	2	1	no	—
97	C₄×C₈×C₁₃₁	—	2	2	no	—
98	#95×C₁₃₁	—	4	2	no	—
99	#41×C₁₃₁	—	2	1	no	—
100	#41×C₁₃₁	—	2	1	no	—
101	#41×C₁₃₁	C₂×Q₈	1	2	yes	—
102	#41×C₁₃₁	C₂×Q₈	1	2	yes	—
103	C₂×Q₈×C₁₃₁	—	4	2	no	—
104	C₂×Q₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₁₃₁	—	2	1	no	—
106	C₂×D₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₃₁	—	2	1	no	—
108	C₂²×C₄×C₁₃₁	—	2	1	no	—
109	C₂²×C₄×C₁₃₁	—	2	2	no	—
110	C₂²×C₄×C₁₃₁	—	2	2	no	—
111	C₂²×C₄×C₁₃₁	—	4	2	no	—
112	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₁₃₁	—	2	1	no	—
114	M₁₆×C₁₃₁	—	2	1	no	—
115	M₁₆×C₁₃₁	—	2	4	no	—
116	M₁₆×C₁₃₁	—	2	2	no	—
117	M₁₆×C₁₃₁	—	2	2	no	—
118	M₁₆×C₁₃₁	C₂×Q₈	1	2	yes	—
119	M₁₆×C₁₃₁	C₂×Q₈	1	2	yes	—
120	C₂×C₈×C₁₃₁	—	2	1	no	—
121	C₂×C₈×C₁₃₁	—	2	1	no	—
122	C₂×C₈×C₁₃₁	—	2	1	no	—
123	C₂×C₈×C₁₃₁	—	2	1	no	—
124	C₂×C₈×C₁₃₁	—	2	4	no	—
125	C₂×C₈×C₁₃₁	—	2	2	no	—
126	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₁₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₁₃₁	—	2	2	no	—
133	C₄⋊C₄×C₁₃₁	—	2	2	no	—
134	C₄⋊C₄×C₁₃₁	—	2	2	no	—
135	C₄⋊C₄×C₁₃₁	—	2	2	no	—
136	C₄⋊C₄×C₁₃₁	—	4	2	no	—
137	C₄⋊C₄×C₁₃₁	—	4	2	no	—
138	C₄⋊C₄×C₁₃₁	—	4	2	no	—
139	C₄⋊C₄×C₁₃₁	—	4	2	no	—
140	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₁₃₁	C₂×D₄	1	2	yes	—
144	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
147	C₄²×C₁₃₁	—	2	1	no	—
148	C₄²×C₁₃₁	—	2	2	no	—
149	C₄²×C₁₃₁	—	2	1	no	—
150	C₄²×C₁₃₁	—	4	2	no	—
151	C₄²×C₁₃₁	C₂×Q₈	1	2	yes	—
152	C₂³×C₁₃₁	—	2	1	no	—
153	C₂³×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
154	Q₈×C₁₃₁	—	2	1	no	—
155	Q₈×C₁₃₁	—	16	2	no	—
156	Q₈×C₁₃₁	C₄⋊Q₈	1	2	yes	—
157	D₄×C₁₃₁	—	2	2	no	—
158	D₄×C₁₃₁	—	2	2	no	—
159	D₄×C₁₃₁	—	2	1	no	—
160	D₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
161	C₂×C₄×C₁₃₁	—	2	1	no	—
162	C₂×C₄×C₁₃₁	—	2	2	no	—
163	C₂×C₄×C₁₃₁	—	2	2	no	—
164	C₂×C₄×C₁₃₁	—	2	2	no	—
165	C₂×C₄×C₁₃₁	—	2	1	no	—
166	C₂×C₄×C₁₃₁	—	2	2	no	—
167	C₂×C₄×C₁₃₁	—	2	1	no	—
168	C₂×C₄×C₁₃₁	—	2	2	no	—
169	C₂×C₄×C₁₃₁	—	4	1	no	—
170	C₂×C₄×C₁₃₁	—	4	1	no	—
171	C₂×C₄×C₁₃₁	—	4	2	no	—
172	C₂×C₄×C₁₃₁	—	4	2	no	—
173	C₂×C₄×C₁₃₁	—	8	2	no	—
174	C₂×C₄×C₁₃₁	—	8	2	no	—
175	C₂×C₄×C₁₃₁	—	16	2	no	—
176	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
182	C₈×C₁₃₁	—	2	2	no	—
183	C₈×C₁₃₁	—	2	1	no	—
184	C₈×C₁₃₁	—	2	1	no	—
185	C₈×C₁₃₁	—	2	2	no	—
186	C₈×C₁₃₁	—	4	2	no	—
187	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
188	C₈×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
189	C₂²×C₁₃₁	—	2	1	no	—
190	C₂²×C₁₃₁	—	2	1	no	—
191	C₂²×C₁₃₁	—	2	2	no	—
192	C₂²×C₁₃₁	—	4	1	no	—
193	C₂²×C₁₃₁	—	4	1	no	—
194	C₂²×C₁₃₁	#544	1	1	yes	characteristic
195	C₂²×C₁₃₁	D₄²	1	1	yes	characteristic
196	C₂²×C₁₃₁	#554	1	1	yes	characteristic
197	C₄×C₁₃₁	—	2	2	no	—
198	C₄×C₁₃₁	—	4	2	no	—
199	C₄×C₁₃₁	—	4	1	no	—
200	C₄×C₁₃₁	—	4	2	no	—
201	C₄×C₁₃₁	—	4	2	no	—
202	C₄×C₁₃₁	—	4	1	no	—
203	C₄×C₁₃₁	—	8	2	no	—
204	C₄×C₁₃₁	—	8	2	no	—
205	C₄×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
207	C₄×C₁₃₁	#547	1	1	yes	characteristic
208	C₄×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
209	C₂×C₁₃₁	—	2	1	no	—
210	C₂×C₁₃₁	—	2	1	no	—
211	C₂×C₁₃₁	—	4	2	no	—
212	C₂×C₁₃₁	#2440	1	1	yes	characteristic
213	C₂×C₁₃₁	#3290	1	1	yes	characteristic
214	C₂×C₁₃₁	#3290	1	1	yes	characteristic
215	#26082	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₁₃₁	#26082	1	1	yes	characteristic, direct product factor, Sylow
217	#3164	C₂×C₁₃₁	1	1	yes	characteristic, complemented
218	#3084	C₂×C₁₃₁	1	2	yes	—
219	#2913	C₂×C₁₃₁	1	1	yes	characteristic
220	#2090	C₂×C₁₃₁	1	2	yes	complemented
221	#2061	C₂×C₁₃₁	1	2	yes	complemented
222	#1952	C₂×C₁₃₁	1	1	yes	characteristic, complemented
223	#1945	C₂×C₁₃₁	1	1	yes	characteristic, complemented
224	#1860	C₂×C₁₃₁	1	2	yes	—
225	#1846	C₂×C₁₃₁	1	1	yes	characteristic, complemented
226	#1816	C₂×C₁₃₁	1	1	yes	characteristic
227	#1764	C₂×C₁₃₁	1	1	yes	characteristic
228	C₂×#131	C₂²×C₁₃₁	1	1	yes	characteristic
229	#544	C₂²×C₁₃₁	1	2	yes	—
230	#520	C₂²×C₁₃₁	1	2	yes	—
231	#501	—	2	2	no	—
232	#500	—	2	1	no	—
233	#499	—	2	1	no	—
234	#479	—	2	2	no	—
235	#478	—	2	2	no	—
236	#476	—	2	2	no	—
237	#475	—	2	2	no	—
238	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
239	#428	—	2	2	no	—
240	#428	C₂²×C₁₃₁	1	2	yes	—
241	#428	C₂²×C₁₃₁	1	2	yes	—
242	#428	C₂²×C₁₃₁	1	2	yes	—
243	#427	—	2	1	no	—
244	#427	C₂²×C₁₃₁	1	1	yes	characteristic
245	C₂×#107	—	2	1	no	—
246	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
247	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
248	C₂×#106	C₂²×C₁₃₁	1	2	yes	—
249	#424	C₂²×C₁₃₁	1	2	yes	—
250	C₂×#104	C₂²×C₁₃₁	1	1	yes	characteristic
251	#417	C₂²×C₁₃₁	1	2	yes	—
252	#416	C₂²×C₁₃₁	1	1	yes	characteristic
253	#405	C₂²×C₁₃₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
255	#382	C₂²×C₁₃₁	1	2	yes	—
256	#343	C₂²×C₁₃₁	1	2	yes	—
257	#343	C₂²×C₁₃₁	1	2	yes	—
258	#341	C₂²×C₁₃₁	1	1	yes	characteristic
259	#339	C₂²×C₁₃₁	1	1	yes	characteristic
260	#339	C₂²×C₁₃₁	1	2	yes	—
261	C₂×#41	C₂³×C₁₃₁	1	1	yes	characteristic
262	#131	—	2	2	no	—
263	#131	Q₈×C₁₃₁	1	4	yes	complemented
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
266	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
267	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	2	no	—
273	#125	—	2	2	no	—
274	#125	—	2	2	no	—
275	C₂²⋊Q₈	—	4	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	—	4	2	no	—
279	#117	—	2	2	no	—
280	#117	C₂³×C₁₃₁	1	2	yes	—
281	#117	C₂³×C₁₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
285	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
286	#108	—	2	2	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	1	no	—
291	#107	—	2	2	no	—
292	#107	—	2	2	no	—
293	#107	—	2	2	no	—
294	#107	D₄×C₁₃₁	1	1	yes	characteristic
295	#107	D₄×C₁₃₁	1	1	yes	characteristic
296	#106	—	2	2	no	—
297	#106	—	2	2	no	—
298	#106	—	2	2	no	—
299	#106	—	2	2	no	—
300	#106	—	2	2	no	—
301	#106	D₄×C₁₃₁	1	2	yes	—
302	C₄⋊C₈	—	2	2	no	—
303	C₄⋊C₈	—	2	2	no	—
304	#104	Q₈×C₁₃₁	1	4	yes	complemented
305	#103	—	2	2	no	—
306	#103	D₄×C₁₃₁	1	1	yes	characteristic
307	#103	D₄×C₁₃₁	1	1	yes	characteristic
308	#102	—	2	2	no	—
309	#102	D₄×C₁₃₁	1	1	yes	characteristic
310	#102	D₄×C₁₃₁	1	1	yes	characteristic
311	#97	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	2	2	no	—
314	#95	—	4	2	no	—
315	#41	—	2	1	no	—
316	#41	—	2	1	no	—
317	#41	C₂×Q₈×C₁₃₁	1	2	yes	—
318	#41	C₂×Q₈×C₁₃₁	1	2	yes	—
319	C₂×Q₈	—	4	2	no	—
320	C₂×Q₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
329	M₁₆	—	2	1	no	—
330	M₁₆	—	2	1	no	—
331	M₁₆	—	2	4	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	C₂×Q₈×C₁₃₁	1	2	yes	—
335	M₁₆	C₂×Q₈×C₁₃₁	1	2	yes	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	4	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
346	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
347	C₂×C₈	#41×C₁₃₁	1	2	yes	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
357	C₄⋊C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
358	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
359	C₄⋊C₄	C₂×D₄×C₁₃₁	1	2	yes	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	4	2	no	—
367	C₄²	C₂×Q₈×C₁₃₁	1	2	yes	—
368	C₂³	—	2	1	no	—
369	C₂³	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
370	Q₈	—	2	1	no	—
371	Q₈	—	16	2	no	—
372	Q₈	C₄⋊Q₈×C₁₃₁	1	2	yes	—
373	D₄	—	2	2	no	—
374	D₄	—	2	2	no	—
375	D₄	—	2	1	no	—
376	D₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	16	2	no	—
392	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
397	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	2	2	no	—
402	C₈	—	4	2	no	—
403	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
404	C₈	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	#544×C₁₃₁	1	1	yes	characteristic
411	C₂²	D₄²×C₁₃₁	1	1	yes	characteristic
412	C₂²	#554×C₁₃₁	1	1	yes	characteristic
413	C₄	—	2	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	2	no	—
421	C₄	C₂×C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
422	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
423	C₄	#547×C₁₃₁	1	1	yes	characteristic
424	C₄	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
425	C₂	—	2	1	no	—
426	C₂	—	2	1	no	—
427	C₂	—	4	2	no	—
428	C₂	#2440×C₁₃₁	1	1	yes	characteristic
429	C₂	#3290×C₁₃₁	1	1	yes	characteristic
430	C₂	#3290×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂×#41×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3164×C₁₃₁	C₂	semidirect
4. #2090×C₁₃₁	C₂	semidirect
5. #2061×C₁₃₁	C₂	semidirect
6. #1952×C₁₃₁	C₂	semidirect
7. #1945×C₁₃₁	C₂	semidirect
9. #1846×C₁₃₁	C₂	semidirect
47. #131×C₁₃₁	Q₈	semidirect
88. #104×C₁₃₁	Q₈	semidirect
215. #26082	C₁₃₁	direct (N × H)
216. C₁₃₁	#26082	direct (N × H)
217. #3164	C₂×C₁₃₁	semidirect
220. #2090	C₂×C₁₃₁	semidirect
221. #2061	C₂×C₁₃₁	semidirect
222. #1952	C₂×C₁₃₁	semidirect
223. #1945	C₂×C₁₃₁	semidirect
225. #1846	C₂×C₁₃₁	semidirect
263. #131	Q₈×C₁₃₁	semidirect
304. #104	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3164×C₁₃₁	2
#3084×C₁₃₁	2
#2913×C₁₃₁	2
#2090×C₁₃₁	2
#2061×C₁₃₁	2
#1952×C₁₃₁	2
#1945×C₁₃₁	2
#1860×C₁₃₁	2
#1846×C₁₃₁	2
#1816×C₁₃₁	2
#1764×C₁₃₁	2
#26082	131

Subgroups — C131×#26082 #45711966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#26082 #45711966