Subgroups — C₁₃₁×#23216 #45709100

Order 33536.

962 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2366×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2365×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#1988×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1982×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1934×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1925×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1862×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₁₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
17	#486×C₁₃₁	C₄	1	8	yes	complemented
18	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
23	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
24	C₂×#102×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
27	#391×C₁₃₁	—	2	1	no	—
28	#390×C₁₃₁	—	2	1	no	—
29	#390×C₁₃₁	C₂²	1	2	yes	—
30	#389×C₁₃₁	C₂²	1	2	yes	—
31	#387×C₁₃₁	—	2	2	no	—
32	#386×C₁₃₁	—	2	2	no	—
33	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×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	1	yes	characteristic
40	#340×C₁₃₁	C₂²	1	1	yes	characteristic
41	#336×C₁₃₁	C₂²	1	2	yes	—
42	C₂²×Q₈×C₁₃₁	—	2	1	no	—
43	C₂²×D₄×C₁₃₁	—	2	1	no	—
44	C₂²×D₄×C₁₃₁	—	2	1	no	—
45	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₁₃₁	—	2	2	no	—
47	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
51	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
52	#127×C₁₃₁	—	2	2	no	—
53	#127×C₁₃₁	C₂×C₄	1	4	yes	—
54	C₄×Q₈×C₁₃₁	—	4	4	no	—
55	C₄×D₄×C₁₃₁	—	2	4	no	—
56	C₄×D₄×C₁₃₁	—	4	2	no	—
57	C₄×D₄×C₁₃₁	—	4	4	no	—
58	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄²×C₁₃₁	—	2	1	no	—
71	C₂×C₄²×C₁₃₁	—	2	1	no	—
72	C₂×C₄²×C₁₃₁	—	2	2	no	—
73	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
75	#107×C₁₃₁	—	2	2	no	—
76	#107×C₁₃₁	—	2	2	no	—
77	C₄⋊C₈×C₁₃₁	—	2	2	no	—
78	#102×C₁₃₁	—	2	4	no	—
79	#102×C₁₃₁	—	4	2	no	—
80	#102×C₁₃₁	C₂×C₄	1	8	yes	complemented
81	#102×C₁₃₁	C₂×C₄	1	8	yes	complemented
82	C₄×C₈×C₁₃₁	—	4	2	no	—
83	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
84	#95×C₁₃₁	—	2	2	no	—
85	#95×C₁₃₁	—	2	2	no	—
86	#95×C₁₃₁	—	2	1	no	—
87	#95×C₁₃₁	—	2	1	no	—
88	#95×C₁₃₁	D₄	1	2	yes	—
89	C₂⁴×C₁₃₁	—	2	2	no	—
90	C₂×Q₈×C₁₃₁	—	2	4	no	—
91	C₂×Q₈×C₁₃₁	—	4	2	no	—
92	C₂×Q₈×C₁₃₁	—	4	2	no	—
93	C₂×D₄×C₁₃₁	—	2	4	no	—
94	C₂×D₄×C₁₃₁	—	2	4	no	—
95	C₂×D₄×C₁₃₁	—	2	2	no	—
96	C₂×D₄×C₁₃₁	—	2	2	no	—
97	C₂×D₄×C₁₃₁	—	2	2	no	—
98	C₂×D₄×C₁₃₁	—	2	2	no	—
99	C₂×D₄×C₁₃₁	—	2	2	no	—
100	C₂×D₄×C₁₃₁	—	4	2	no	—
101	C₂×D₄×C₁₃₁	—	4	2	no	—
102	C₂×D₄×C₁₃₁	—	8	1	no	—
103	C₂×D₄×C₁₃₁	—	8	1	no	—
104	C₂×D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₁₃₁	—	2	1	no	—
106	C₂²×C₄×C₁₃₁	—	2	2	no	—
107	C₂²×C₄×C₁₃₁	—	2	1	no	—
108	C₂²×C₄×C₁₃₁	—	2	1	no	—
109	C₂²×C₄×C₁₃₁	—	2	1	no	—
110	C₂²×C₄×C₁₃₁	—	4	4	no	—
111	C₂²×C₄×C₁₃₁	—	4	4	no	—
112	C₂²×C₄×C₁₃₁	—	8	2	no	—
113	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₁₃₁	—	2	1	no	—
119	C₂×C₈×C₁₃₁	—	2	1	no	—
120	C₂×C₈×C₁₃₁	—	2	1	no	—
121	C₂×C₈×C₁₃₁	—	2	1	no	—
122	C₂×C₈×C₁₃₁	—	2	2	no	—
123	C₂×C₈×C₁₃₁	—	2	2	no	—
124	C₂×C₈×C₁₃₁	—	2	2	no	—
125	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₁₃₁	—	2	4	no	—
128	C₄⋊C₄×C₁₃₁	—	2	2	no	—
129	C₄⋊C₄×C₁₃₁	—	2	2	no	—
130	C₄⋊C₄×C₁₃₁	—	2	1	no	—
131	C₄⋊C₄×C₁₃₁	—	2	1	no	—
132	C₄⋊C₄×C₁₃₁	—	2	1	no	—
133	C₄⋊C₄×C₁₃₁	—	2	1	no	—
134	C₄⋊C₄×C₁₃₁	—	2	1	no	—
135	C₄⋊C₄×C₁₃₁	—	2	1	no	—
136	C₄⋊C₄×C₁₃₁	—	4	4	no	—
137	C₄⋊C₄×C₁₃₁	D₈	1	4	yes	—
138	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
139	C₄⋊C₄×C₁₃₁	#41	1	2	yes	—
140	C₄⋊C₄×C₁₃₁	#41	1	2	yes	—
141	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
143	C₂²⋊C₄×C₁₃₁	—	8	2	no	—
144	C₄²×C₁₃₁	—	2	2	no	—
145	C₄²×C₁₃₁	—	2	2	no	—
146	C₄²×C₁₃₁	—	4	2	no	—
147	C₄²×C₁₃₁	—	4	4	no	—
148	C₄²×C₁₃₁	—	4	2	no	—
149	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
150	C₄²×C₁₃₁	#41	1	2	yes	—
151	C₂³×C₁₃₁	—	2	4	no	—
152	C₂³×C₁₃₁	—	4	2	no	—
153	C₂³×C₁₃₁	—	4	2	no	—
154	C₂³×C₁₃₁	—	4	2	no	—
155	C₂³×C₁₃₁	—	4	2	no	—
156	C₂³×C₁₃₁	—	4	2	no	—
157	C₂³×C₁₃₁	—	4	2	no	—
158	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₁₃₁	—	4	8	no	—
160	D₄×C₁₃₁	—	2	4	no	—
161	D₄×C₁₃₁	—	2	4	no	—
162	D₄×C₁₃₁	—	4	2	no	—
163	D₄×C₁₃₁	—	4	2	no	—
164	D₄×C₁₃₁	—	4	8	no	—
165	D₄×C₁₃₁	—	8	4	no	—
166	C₂×C₄×C₁₃₁	—	2	2	no	—
167	C₂×C₄×C₁₃₁	—	2	2	no	—
168	C₂×C₄×C₁₃₁	—	2	1	no	—
169	C₂×C₄×C₁₃₁	—	2	1	no	—
170	C₂×C₄×C₁₃₁	—	2	4	no	—
171	C₂×C₄×C₁₃₁	—	2	2	no	—
172	C₂×C₄×C₁₃₁	—	2	2	no	—
173	C₂×C₄×C₁₃₁	—	2	2	no	—
174	C₂×C₄×C₁₃₁	—	2	2	no	—
175	C₂×C₄×C₁₃₁	—	2	2	no	—
176	C₂×C₄×C₁₃₁	—	4	1	no	—
177	C₂×C₄×C₁₃₁	—	4	2	no	—
178	C₂×C₄×C₁₃₁	—	4	1	no	—
179	C₂×C₄×C₁₃₁	—	4	2	no	—
180	C₂×C₄×C₁₃₁	—	4	1	no	—
181	C₂×C₄×C₁₃₁	—	4	1	no	—
182	C₂×C₄×C₁₃₁	—	4	1	no	—
183	C₂×C₄×C₁₃₁	—	4	1	no	—
184	C₂×C₄×C₁₃₁	—	4	1	no	—
185	C₂×C₄×C₁₃₁	—	4	1	no	—
186	C₂×C₄×C₁₃₁	—	4	1	no	—
187	C₂×C₄×C₁₃₁	—	4	1	no	—
188	C₂×C₄×C₁₃₁	—	8	8	no	—
189	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
192	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
193	C₂×C₄×C₁₃₁	#135	1	2	yes	—
194	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
200	C₈×C₁₃₁	—	2	2	no	—
201	C₈×C₁₃₁	—	2	2	no	—
202	C₈×C₁₃₁	—	4	2	no	—
203	C₂²×C₁₃₁	—	4	4	no	—
204	C₂²×C₁₃₁	—	4	4	no	—
205	C₂²×C₁₃₁	—	4	4	no	—
206	C₂²×C₁₃₁	—	8	2	no	—
207	C₂²×C₁₃₁	—	8	2	no	—
208	C₂²×C₁₃₁	—	8	2	no	—
209	C₂²×C₁₃₁	—	8	2	no	—
210	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₁₃₁	#546	1	1	yes	characteristic
212	C₂²×C₁₃₁	#548	1	1	yes	characteristic
213	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
214	C₂²×C₁₃₁	C₂×#135	1	1	yes	characteristic
215	C₂²×C₁₃₁	#577	1	1	yes	characteristic
216	C₂²×C₁₃₁	#577	1	1	yes	characteristic
217	C₄×C₁₃₁	—	2	2	no	—
218	C₄×C₁₃₁	—	2	4	no	—
219	C₄×C₁₃₁	—	4	2	no	—
220	C₄×C₁₃₁	—	4	4	no	—
221	C₄×C₁₃₁	—	4	2	no	—
222	C₄×C₁₃₁	—	4	2	no	—
223	C₄×C₁₃₁	—	8	1	no	—
224	C₄×C₁₃₁	—	8	1	no	—
225	C₄×C₁₃₁	C₄×D₈	1	4	yes	—
226	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
227	C₄×C₁₃₁	#520	1	1	yes	characteristic
228	C₄×C₁₃₁	#556	1	1	yes	characteristic
229	C₄×C₁₃₁	#559	1	1	yes	characteristic
230	C₂×C₁₃₁	—	8	4	no	—
231	C₂×C₁₃₁	#2367	1	1	yes	characteristic
232	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
233	C₂×C₁₃₁	#2949	1	1	yes	characteristic
234	C₂×C₁₃₁	#3294	1	1	yes	characteristic
235	C₂×C₁₃₁	#3304	1	1	yes	characteristic
236	C₂×C₁₃₁	#3358	1	1	yes	characteristic
237	C₂×C₁₃₁	#3365	1	1	yes	characteristic
238	#23216	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₁₃₁	#23216	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×#486	C₂×C₁₃₁	1	1	yes	characteristic
241	#2366	C₂×C₁₃₁	1	1	yes	characteristic
242	#2365	C₂×C₁₃₁	1	1	yes	characteristic, complemented
243	#1988	C₂×C₁₃₁	1	1	yes	characteristic, complemented
244	#1982	C₂×C₁₃₁	1	1	yes	characteristic
245	#1934	C₂×C₁₃₁	1	2	yes	complemented
246	#1925	C₂×C₁₃₁	1	2	yes	complemented
247	#1862	C₂×C₁₃₁	1	2	yes	complemented
248	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
249	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
250	C₄×#102	C₂×C₁₃₁	1	2	yes	complemented
251	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
252	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
253	C₂×C₄×Q₈	—	2	1	no	—
254	C₂×C₄×D₄	—	2	1	no	—
255	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
256	#486	C₄×C₁₃₁	1	8	yes	complemented
257	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
258	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
260	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
261	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
262	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
263	C₂×#102	C₂²×C₁₃₁	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
266	#391	—	2	1	no	—
267	#390	—	2	1	no	—
268	#390	C₂²×C₁₃₁	1	2	yes	—
269	#389	C₂²×C₁₃₁	1	2	yes	—
270	#387	—	2	2	no	—
271	#386	—	2	2	no	—
272	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
273	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
274	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
275	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
276	#340	C₂²×C₁₃₁	1	2	yes	—
277	#340	C₂²×C₁₃₁	1	2	yes	—
278	#340	C₂²×C₁₃₁	1	1	yes	characteristic
279	#340	C₂²×C₁₃₁	1	1	yes	characteristic
280	#336	C₂²×C₁₃₁	1	2	yes	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
285	C₂³×C₄	—	2	2	no	—
286	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
287	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
291	#127	—	2	2	no	—
292	#127	C₂×C₄×C₁₃₁	1	4	yes	—
293	C₄×Q₈	—	4	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	4	2	no	—
296	C₄×D₄	—	4	4	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	D₄×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
313	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
314	#107	—	2	2	no	—
315	#107	—	2	2	no	—
316	C₄⋊C₈	—	2	2	no	—
317	#102	—	2	4	no	—
318	#102	—	4	2	no	—
319	#102	C₂×C₄×C₁₃₁	1	8	yes	complemented
320	#102	C₂×C₄×C₁₃₁	1	8	yes	complemented
321	C₄×C₈	—	4	2	no	—
322	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
323	#95	—	2	1	no	—
324	#95	—	2	1	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	D₄×C₁₃₁	1	2	yes	—
328	C₂⁴	—	2	2	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	8	1	no	—
342	C₂×D₄	—	8	1	no	—
343	C₂×D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	8	2	no	—
352	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
365	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	D₈×C₁₃₁	1	4	yes	—
377	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
378	C₄⋊C₄	#41×C₁₃₁	1	2	yes	—
379	C₄⋊C₄	#41×C₁₃₁	1	2	yes	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	8	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	4	2	no	—
386	C₄²	—	4	4	no	—
387	C₄²	—	4	2	no	—
388	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
389	C₄²	#41×C₁₃₁	1	2	yes	—
390	C₂³	—	2	4	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
398	Q₈	—	4	8	no	—
399	D₄	—	2	4	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	8	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	8	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	8	8	no	—
428	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
429	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
430	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
431	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
432	C₂×C₄	#135×C₁₃₁	1	2	yes	—
433	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
437	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
438	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
439	C₈	—	2	2	no	—
440	C₈	—	2	2	no	—
441	C₈	—	4	2	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
450	C₂²	#546×C₁₃₁	1	1	yes	characteristic
451	C₂²	#548×C₁₃₁	1	1	yes	characteristic
452	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
453	C₂²	C₂×#135×C₁₃₁	1	1	yes	characteristic
454	C₂²	#577×C₁₃₁	1	1	yes	characteristic
455	C₂²	#577×C₁₃₁	1	1	yes	characteristic
456	C₄	—	2	4	no	—
457	C₄	—	2	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	4	no	—
462	C₄	—	8	1	no	—
463	C₄	—	8	1	no	—
464	C₄	C₄×D₈×C₁₃₁	1	4	yes	—
465	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
466	C₄	#520×C₁₃₁	1	1	yes	characteristic
467	C₄	#556×C₁₃₁	1	1	yes	characteristic
468	C₄	#559×C₁₃₁	1	1	yes	characteristic
469	C₂	—	8	4	no	—
470	C₂	#2367×C₁₃₁	1	1	yes	characteristic
471	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic
472	C₂	#2949×C₁₃₁	1	1	yes	characteristic
473	C₂	#3294×C₁₃₁	1	1	yes	characteristic
474	C₂	#3304×C₁₃₁	1	1	yes	characteristic
475	C₂	#3358×C₁₃₁	1	1	yes	characteristic
476	C₂	#3365×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	C₂×#127×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₄×C₄⋊C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2365×C₁₃₁	C₂	semidirect
4. #1988×C₁₃₁	C₂	semidirect
6. #1934×C₁₃₁	C₂	semidirect
7. #1925×C₁₃₁	C₂	semidirect
8. #1862×C₁₃₁	C₂	semidirect
9. C₄×#107×C₁₃₁	C₂	semidirect
10. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
11. C₄×#102×C₁₃₁	C₂	semidirect
17. #486×C₁₃₁	C₄	semidirect
51. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
80. #102×C₁₃₁	C₂×C₄	semidirect
81. #102×C₁₃₁	C₂×C₄	semidirect
83. C₄×C₈×C₁₃₁	C₂×C₄	semidirect
238. #23216	C₁₃₁	direct (N × H)
239. C₁₃₁	#23216	direct (N × H)
242. #2365	C₂×C₁₃₁	semidirect
243. #1988	C₂×C₁₃₁	semidirect
245. #1934	C₂×C₁₃₁	semidirect
246. #1925	C₂×C₁₃₁	semidirect
247. #1862	C₂×C₁₃₁	semidirect
248. C₄×#107	C₂×C₁₃₁	semidirect
249. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
250. C₄×#102	C₂×C₁₃₁	semidirect
256. #486	C₄×C₁₃₁	semidirect
290. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
319. #102	C₂×C₄×C₁₃₁	semidirect
320. #102	C₂×C₄×C₁₃₁	semidirect
322. C₄×C₈	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×#486×C₁₃₁	2
#2366×C₁₃₁	2
#2365×C₁₃₁	2
#1988×C₁₃₁	2
#1982×C₁₃₁	2
#1934×C₁₃₁	2
#1925×C₁₃₁	2
#1862×C₁₃₁	2
C₄×#107×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
C₄×#102×C₁₃₁	2
#23216	131

Subgroups — C131×#23216 #45709100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23216 #45709100