Subgroups — C₁₃₁×#34226 #45720110

Order 33536.

2706 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2367×C₁₃₁	C₂	1	4	yes	maximal, complemented
3	#2363×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2357×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2349×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2341×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2339×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#2324×C₁₃₁	C₂	1	4	yes	maximal, complemented
9	#2319×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	2	yes	maximal, complemented
11	C₄×#127×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	#2285×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#2275×C₁₃₁	C₂	1	2	yes	maximal, complemented
14	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₁₃₁	C₂²	1	2	yes	—
16	#532×C₁₃₁	C₄	1	16	yes	complemented
17	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#127×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#121×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#121×C₁₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₁₃₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
28	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
29	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
31	#391×C₁₃₁	C₂²	1	4	yes	—
32	#390×C₁₃₁	C₂²	1	4	yes	—
33	#390×C₁₃₁	C₂²	1	4	yes	—
34	#390×C₁₃₁	C₂²	1	4	yes	—
35	#390×C₁₃₁	C₂²	1	8	yes	—
36	#389×C₁₃₁	C₂²	1	4	yes	—
37	#389×C₁₃₁	C₂²	1	4	yes	—
38	#388×C₁₃₁	C₂²	1	8	yes	—
39	#388×C₁₃₁	C₂²	1	8	yes	—
40	#386×C₁₃₁	C₂²	1	8	yes	—
41	#385×C₁₃₁	C₂²	1	8	yes	—
42	#384×C₁₃₁	C₂²	1	2	yes	—
43	#382×C₁₃₁	C₂²	1	2	yes	—
44	#381×C₁₃₁	C₂²	1	4	yes	—
45	#380×C₁₃₁	C₂²	1	2	yes	—
46	#380×C₁₃₁	C₂²	1	2	yes	—
47	#380×C₁₃₁	C₂²	1	4	yes	—
48	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
55	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
57	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
58	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	8	yes	—
60	#376×C₁₃₁	C₂²	1	1	yes	characteristic
61	#376×C₁₃₁	C₂²	1	1	yes	characteristic
62	C₂×#95×C₁₃₁	C₂²	1	4	yes	—
63	C₄³×C₁₃₁	C₂²	1	2	yes	—
64	C₂×#41×C₁₃₁	—	4	8	no	—
65	C₂×#41×C₁₃₁	C₂×C₄	1	16	yes	complemented
66	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₁₃₁	C₂³	1	8	yes	—
75	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
76	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
77	#127×C₁₃₁	C₂×C₄	1	8	yes	—
78	#124×C₁₃₁	C₂×C₄	1	16	yes	complemented
79	#121×C₁₃₁	C₂×C₄	1	16	yes	complemented
80	#121×C₁₃₁	C₂×C₄	1	16	yes	complemented
81	C₄×Q₈×C₁₃₁	—	4	8	no	—
82	C₄×D₄×C₁₃₁	—	2	16	no	—
83	C₄×D₄×C₁₃₁	—	4	8	no	—
84	C₄×D₄×C₁₃₁	—	4	8	no	—
85	C₄×D₄×C₁₃₁	C₂×C₄	1	16	yes	complemented
86	C₄×D₄×C₁₃₁	C₂×C₄	1	16	yes	complemented
87	#117×C₁₃₁	—	2	8	no	—
88	#117×C₁₃₁	—	4	4	no	—
89	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
105	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
106	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
107	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
109	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
111	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
112	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
113	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
118	#95×C₁₃₁	C₂³	1	2	yes	—
119	#95×C₁₃₁	C₂³	1	8	yes	—
120	#95×C₁₃₁	C₂³	1	4	yes	—
121	#95×C₁₃₁	C₂³	1	8	yes	—
122	#95×C₁₃₁	C₂³	1	1	yes	characteristic
123	#95×C₁₃₁	C₂³	1	4	yes	—
124	#95×C₁₃₁	C₂³	1	1	yes	characteristic
125	#95×C₁₃₁	C₂³	1	2	yes	—
126	#95×C₁₃₁	C₂³	1	1	yes	characteristic
127	#95×C₁₃₁	C₂³	1	1	yes	characteristic
128	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
129	C₂⁴×C₁₃₁	C₂⁴	1	4	yes	—
130	#41×C₁₃₁	—	4	32	no	—
131	C₂×Q₈×C₁₃₁	—	2	8	no	—
132	C₂×Q₈×C₁₃₁	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₁₃₁	—	2	8	no	—
134	C₂×D₄×C₁₃₁	—	2	4	no	—
135	C₂×D₄×C₁₃₁	—	2	8	no	—
136	C₂×D₄×C₁₃₁	—	2	4	no	—
137	C₂×D₄×C₁₃₁	—	4	8	no	—
138	C₂×D₄×C₁₃₁	C₂²×C₄	1	8	yes	—
139	C₂×D₄×C₁₃₁	C₂²×C₄	1	8	yes	—
140	C₂×D₄×C₁₃₁	C₂²×C₄	1	8	yes	—
141	C₂×D₄×C₁₃₁	C₂²×C₄	1	16	yes	—
142	C₂²×C₄×C₁₃₁	—	2	4	no	—
143	C₂²×C₄×C₁₃₁	—	2	2	no	—
144	C₂²×C₄×C₁₃₁	—	2	4	no	—
145	C₂²×C₄×C₁₃₁	—	2	2	no	—
146	C₂²×C₄×C₁₃₁	—	2	16	no	—
147	C₂²×C₄×C₁₃₁	—	4	8	no	—
148	C₂²×C₄×C₁₃₁	—	4	8	no	—
149	C₂²×C₄×C₁₃₁	—	4	8	no	—
150	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
152	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
154	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₁₃₁	—	2	4	no	—
167	C₄⋊C₄×C₁₃₁	—	2	4	no	—
168	C₄⋊C₄×C₁₃₁	—	2	8	no	—
169	C₄⋊C₄×C₁₃₁	—	2	8	no	—
170	C₄⋊C₄×C₁₃₁	—	2	8	no	—
171	C₄⋊C₄×C₁₃₁	—	4	8	no	—
172	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
173	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
177	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
178	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
179	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
180	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
181	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
182	C₄²×C₁₃₁	—	2	4	no	—
183	C₄²×C₁₃₁	—	2	4	no	—
184	C₄²×C₁₃₁	—	2	4	no	—
185	C₄²×C₁₃₁	—	2	4	no	—
186	C₄²×C₁₃₁	—	2	4	no	—
187	C₄²×C₁₃₁	—	4	4	no	—
188	C₄²×C₁₃₁	—	4	8	no	—
189	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
190	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
191	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
192	C₄²×C₁₃₁	C₂²×C₄	1	4	yes	—
193	C₄²×C₁₃₁	#41	1	8	yes	complemented
194	C₂³×C₁₃₁	—	2	2	no	—
195	C₂³×C₁₃₁	—	2	8	no	—
196	C₂³×C₁₃₁	—	2	1	no	—
197	C₂³×C₁₃₁	—	2	4	no	—
198	C₂³×C₁₃₁	—	2	8	no	—
199	C₂³×C₁₃₁	—	2	2	no	—
200	C₂³×C₁₃₁	—	2	1	no	—
201	C₂³×C₁₃₁	—	2	4	no	—
202	C₂³×C₁₃₁	C₂³×C₄	1	8	yes	—
203	C₂³×C₁₃₁	C₂³×C₄	1	2	yes	—
204	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
205	Q₈×C₁₃₁	—	2	16	no	—
206	D₄×C₁₃₁	—	2	16	no	—
207	D₄×C₁₃₁	—	2	16	no	—
208	D₄×C₁₃₁	—	2	16	no	—
209	D₄×C₁₃₁	—	4	16	no	—
210	C₂×C₄×C₁₃₁	—	2	2	no	—
211	C₂×C₄×C₁₃₁	—	2	2	no	—
212	C₂×C₄×C₁₃₁	—	2	2	no	—
213	C₂×C₄×C₁₃₁	—	2	2	no	—
214	C₂×C₄×C₁₃₁	—	2	1	no	—
215	C₂×C₄×C₁₃₁	—	2	8	no	—
216	C₂×C₄×C₁₃₁	—	2	2	no	—
217	C₂×C₄×C₁₃₁	—	2	2	no	—
218	C₂×C₄×C₁₃₁	—	2	1	no	—
219	C₂×C₄×C₁₃₁	—	2	8	no	—
220	C₂×C₄×C₁₃₁	—	2	8	no	—
221	C₂×C₄×C₁₃₁	—	2	8	no	—
222	C₂×C₄×C₁₃₁	—	2	2	no	—
223	C₂×C₄×C₁₃₁	—	2	2	no	—
224	C₂×C₄×C₁₃₁	—	2	8	no	—
225	C₂×C₄×C₁₃₁	—	2	2	no	—
226	C₂×C₄×C₁₃₁	—	2	2	no	—
227	C₂×C₄×C₁₃₁	—	2	2	no	—
228	C₂×C₄×C₁₃₁	—	2	8	no	—
229	C₂×C₄×C₁₃₁	—	2	8	no	—
230	C₂×C₄×C₁₃₁	—	2	2	no	—
231	C₂×C₄×C₁₃₁	—	4	16	no	—
232	C₂×C₄×C₁₃₁	—	8	16	no	—
233	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
234	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₁₃₁	C₂³×C₄	1	8	yes	—
237	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₁₃₁	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
241	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
242	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
243	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	4	yes	—
244	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
245	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
246	C₂²×C₁₃₁	—	2	2	no	—
247	C₂²×C₁₃₁	—	2	8	no	—
248	C₂²×C₁₃₁	—	2	2	no	—
249	C₂²×C₁₃₁	—	2	8	no	—
250	C₂²×C₁₃₁	—	2	2	no	—
251	C₂²×C₁₃₁	—	2	8	no	—
252	C₂²×C₁₃₁	—	4	8	no	—
253	C₂²×C₁₃₁	—	4	2	no	—
254	C₂²×C₁₃₁	—	4	2	no	—
255	C₂²×C₁₃₁	—	4	8	no	—
256	C₂²×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
257	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
258	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
259	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²×C₁₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
261	C₄×C₁₃₁	—	2	2	no	—
262	C₄×C₁₃₁	—	2	2	no	—
263	C₄×C₁₃₁	—	2	8	no	—
264	C₄×C₁₃₁	—	2	2	no	—
265	C₄×C₁₃₁	—	2	4	no	—
266	C₄×C₁₃₁	—	2	4	no	—
267	C₄×C₁₃₁	—	4	2	no	—
268	C₄×C₁₃₁	—	4	8	no	—
269	C₄×C₁₃₁	—	4	2	no	—
270	C₄×C₁₃₁	C₄×#41	1	4	yes	—
271	C₄×C₁₃₁	#518	1	4	yes	—
272	C₄×C₁₃₁	#520	1	4	yes	—
273	C₂×C₁₃₁	—	4	2	no	—
274	C₂×C₁₃₁	—	4	8	no	—
275	C₂×C₁₃₁	C₂×C₄×#41	1	1	yes	characteristic
276	C₂×C₁₃₁	C₂×#518	1	1	yes	characteristic
277	C₂×C₁₃₁	C₂×#520	1	1	yes	characteristic
278	C₂×C₁₃₁	#3508	1	2	yes	—
279	C₂×C₁₃₁	#3511	1	2	yes	—
280	#34226	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₁₃₁	#34226	1	1	yes	characteristic, direct product factor, Sylow
282	C₂×#532	C₂×C₁₃₁	1	1	yes	characteristic
283	#2367	C₂×C₁₃₁	1	4	yes	complemented
284	#2363	C₂×C₁₃₁	1	4	yes	complemented
285	#2357	C₂×C₁₃₁	1	2	yes	complemented
286	#2349	C₂×C₁₃₁	1	2	yes	complemented
287	#2341	C₂×C₁₃₁	1	2	yes	complemented
288	#2339	C₂×C₁₃₁	1	2	yes	complemented
289	#2324	C₂×C₁₃₁	1	4	yes	complemented
290	#2319	C₂×C₁₃₁	1	2	yes	complemented
291	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	2	yes	complemented
292	C₄×#127	C₂×C₁₃₁	1	2	yes	complemented
293	#2285	C₂×C₁₃₁	1	1	yes	characteristic, complemented
294	#2275	C₂×C₁₃₁	1	2	yes	complemented
295	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic, complemented
296	C₂²×#41	C₂²×C₁₃₁	1	2	yes	—
297	#532	C₄×C₁₃₁	1	16	yes	complemented
298	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
299	C₂×#127	C₂²×C₁₃₁	1	1	yes	characteristic
300	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
301	C₂×#121	C₂²×C₁₃₁	1	2	yes	—
302	C₂×#121	C₂²×C₁₃₁	1	2	yes	—
303	C₂×C₄×Q₈	C₂²×C₁₃₁	1	4	yes	—
304	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
305	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
306	C₂×C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
307	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
308	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
309	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
310	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
311	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
312	#391	C₂²×C₁₃₁	1	4	yes	—
313	#390	C₂²×C₁₃₁	1	4	yes	—
314	#390	C₂²×C₁₃₁	1	4	yes	—
315	#390	C₂²×C₁₃₁	1	4	yes	—
316	#390	C₂²×C₁₃₁	1	8	yes	—
317	#389	C₂²×C₁₃₁	1	4	yes	—
318	#389	C₂²×C₁₃₁	1	4	yes	—
319	#388	C₂²×C₁₃₁	1	8	yes	—
320	#388	C₂²×C₁₃₁	1	8	yes	—
321	#386	C₂²×C₁₃₁	1	8	yes	—
322	#385	C₂²×C₁₃₁	1	8	yes	—
323	#384	C₂²×C₁₃₁	1	2	yes	—
324	#382	C₂²×C₁₃₁	1	2	yes	—
325	#381	C₂²×C₁₃₁	1	4	yes	—
326	#380	C₂²×C₁₃₁	1	4	yes	—
327	#380	C₂²×C₁₃₁	1	2	yes	—
328	#380	C₂²×C₁₃₁	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
331	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
333	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
334	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
335	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
336	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
337	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
338	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
339	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	8	yes	—
341	#376	C₂²×C₁₃₁	1	1	yes	characteristic
342	#376	C₂²×C₁₃₁	1	1	yes	characteristic
343	C₂×#95	C₂²×C₁₃₁	1	4	yes	—
344	C₄³	C₂²×C₁₃₁	1	2	yes	—
345	C₂×#41	—	4	8	no	—
346	C₂×#41	C₂×C₄×C₁₃₁	1	16	yes	complemented
347	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
348	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
349	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
350	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
351	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
352	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₁₃₁	1	8	yes	—
356	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
357	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
358	#127	C₂×C₄×C₁₃₁	1	8	yes	—
359	#124	C₂×C₄×C₁₃₁	1	16	yes	complemented
360	#121	C₂×C₄×C₁₃₁	1	16	yes	complemented
361	#121	C₂×C₄×C₁₃₁	1	16	yes	complemented
362	C₄×Q₈	—	4	8	no	—
363	C₄×D₄	—	2	16	no	—
364	C₄×D₄	—	4	8	no	—
365	C₄×D₄	—	4	8	no	—
366	C₄×D₄	C₂×C₄×C₁₃₁	1	16	yes	complemented
367	C₄×D₄	C₂×C₄×C₁₃₁	1	16	yes	complemented
368	#117	—	2	8	no	—
369	#117	—	4	4	no	—
370	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
376	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
386	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
387	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
388	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
389	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
392	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
393	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
394	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
399	#95	C₂³×C₁₃₁	1	1	yes	characteristic
400	#95	C₂³×C₁₃₁	1	4	yes	—
401	#95	C₂³×C₁₃₁	1	8	yes	—
402	#95	C₂³×C₁₃₁	1	1	yes	characteristic
403	#95	C₂³×C₁₃₁	1	2	yes	—
404	#95	C₂³×C₁₃₁	1	1	yes	characteristic
405	#95	C₂³×C₁₃₁	1	1	yes	characteristic
406	#95	C₂³×C₁₃₁	1	2	yes	—
407	#95	C₂³×C₁₃₁	1	4	yes	—
408	#95	C₂³×C₁₃₁	1	8	yes	—
409	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
410	C₂⁴	C₂⁴×C₁₃₁	1	4	yes	—
411	#41	—	4	32	no	—
412	C₂×Q₈	—	2	8	no	—
413	C₂×Q₈	C₂²×C₄×C₁₃₁	1	8	yes	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	4	8	no	—
419	C₂×D₄	C₂²×C₄×C₁₃₁	1	8	yes	—
420	C₂×D₄	C₂²×C₄×C₁₃₁	1	8	yes	—
421	C₂×D₄	C₂²×C₄×C₁₃₁	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₁₃₁	1	16	yes	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	16	no	—
428	C₂²×C₄	—	4	8	no	—
429	C₂²×C₄	—	4	8	no	—
430	C₂²×C₄	—	4	8	no	—
431	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
432	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
433	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
434	C₂²×C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
435	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
437	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	4	8	no	—
453	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
454	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	8	yes	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	16	no	—
458	C₂²⋊C₄	—	4	4	no	—
459	C₂²⋊C₄	—	4	8	no	—
460	C₂²⋊C₄	—	4	8	no	—
461	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
462	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
463	C₄²	—	2	4	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	4	no	—
467	C₄²	—	2	4	no	—
468	C₄²	—	4	4	no	—
469	C₄²	—	4	8	no	—
470	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
471	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
472	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
473	C₄²	C₂²×C₄×C₁₃₁	1	4	yes	—
474	C₄²	#41×C₁₃₁	1	8	yes	complemented
475	C₂³	—	2	2	no	—
476	C₂³	—	2	8	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	4	no	—
483	C₂³	C₂³×C₄×C₁₃₁	1	8	yes	—
484	C₂³	C₂³×C₄×C₁₃₁	1	2	yes	—
485	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
486	Q₈	—	2	16	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	16	no	—
489	D₄	—	2	16	no	—
490	D₄	—	4	16	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	8	no	—
501	C₂×C₄	—	2	8	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	16	no	—
513	C₂×C₄	—	8	16	no	—
514	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₁₃₁	1	8	yes	—
518	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
520	C₂×C₄	C₂³×C₄×C₁₃₁	1	2	yes	—
521	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
522	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
523	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
524	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	4	yes	—
525	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
526	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	8	no	—
537	C₂²	C₂⁴×C₄×C₁₃₁	1	1	yes	characteristic
538	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
539	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
540	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	2	yes	—
541	C₂²	C₂×2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	8	no	—
547	C₄	—	2	2	no	—
548	C₄	—	4	8	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	2	no	—
551	C₄	C₄×#41×C₁₃₁	1	4	yes	—
552	C₄	#518×C₁₃₁	1	4	yes	—
553	C₄	#520×C₁₃₁	1	4	yes	—
554	C₂	—	4	2	no	—
555	C₂	—	4	8	no	—
556	C₂	C₂×C₄×#41×C₁₃₁	1	1	yes	characteristic
557	C₂	C₂×#518×C₁₃₁	1	1	yes	characteristic
558	C₂	C₂×#520×C₁₃₁	1	1	yes	characteristic
559	C₂	#3508×C₁₃₁	1	2	yes	—
560	C₂	#3511×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₃₁
Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
prime radical	C₂×#532×C₁₃₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2367×C₁₃₁	C₂	semidirect
3. #2363×C₁₃₁	C₂	semidirect
4. #2357×C₁₃₁	C₂	semidirect
5. #2349×C₁₃₁	C₂	semidirect
6. #2341×C₁₃₁	C₂	semidirect
7. #2339×C₁₃₁	C₂	semidirect
8. #2324×C₁₃₁	C₂	semidirect
9. #2319×C₁₃₁	C₂	semidirect
10. C₄×C₄⋊Q₈×C₁₃₁	C₂	semidirect
11. C₄×#127×C₁₃₁	C₂	semidirect
12. #2285×C₁₃₁	C₂	semidirect
13. #2275×C₁₃₁	C₂	semidirect
14. C₄×#117×C₁₃₁	C₂	semidirect
16. #532×C₁₃₁	C₄	semidirect
65. C₂×#41×C₁₃₁	C₂×C₄	semidirect
76. C₄⋊Q₈×C₁₃₁	C₂×C₄	semidirect
78. #124×C₁₃₁	C₂×C₄	semidirect
79. #121×C₁₃₁	C₂×C₄	semidirect
80. #121×C₁₃₁	C₂×C₄	semidirect
85. C₄×D₄×C₁₃₁	C₂×C₄	semidirect
86. C₄×D₄×C₁₃₁	C₂×C₄	semidirect
105. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
193. C₄²×C₁₃₁	#41	semidirect
280. #34226	C₁₃₁	direct (N × H)
281. C₁₃₁	#34226	direct (N × H)
283. #2367	C₂×C₁₃₁	semidirect
284. #2363	C₂×C₁₃₁	semidirect
285. #2357	C₂×C₁₃₁	semidirect
286. #2349	C₂×C₁₃₁	semidirect
287. #2341	C₂×C₁₃₁	semidirect
288. #2339	C₂×C₁₃₁	semidirect
289. #2324	C₂×C₁₃₁	semidirect
290. #2319	C₂×C₁₃₁	semidirect
291. C₄×C₄⋊Q₈	C₂×C₁₃₁	semidirect
292. C₄×#127	C₂×C₁₃₁	semidirect
293. #2285	C₂×C₁₃₁	semidirect
294. #2275	C₂×C₁₃₁	semidirect
295. C₄×#117	C₂×C₁₃₁	semidirect
297. #532	C₄×C₁₃₁	semidirect
346. C₂×#41	C₂×C₄×C₁₃₁	semidirect
357. C₄⋊Q₈	C₂×C₄×C₁₃₁	semidirect
359. #124	C₂×C₄×C₁₃₁	semidirect
360. #121	C₂×C₄×C₁₃₁	semidirect
361. #121	C₂×C₄×C₁₃₁	semidirect
366. C₄×D₄	C₂×C₄×C₁₃₁	semidirect
367. C₄×D₄	C₂×C₄×C₁₃₁	semidirect
386. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
474. C₄²	#41×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₁₃₁	2
#2367×C₁₃₁	2
#2363×C₁₃₁	2
#2357×C₁₃₁	2
#2349×C₁₃₁	2
#2341×C₁₃₁	2
#2339×C₁₃₁	2
#2324×C₁₃₁	2
#2319×C₁₃₁	2
C₄×C₄⋊Q₈×C₁₃₁	2
C₄×#127×C₁₃₁	2
#2285×C₁₃₁	2
#2275×C₁₃₁	2
C₄×#117×C₁₃₁	2
#34226	131

Subgroups — C131×#34226 #45720110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34226 #45720110