Subgroups — C₂×#2343×C₁₃₁ #45719988

Order 33536.

3006 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2343×C₁₃₁	C₂	1	16	yes	maximal, direct product factor
3	C₂×#387×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	C₂×#387×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₂×#385×C₁₃₁	C₂	1	2	yes	maximal
6	C₂×#383×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#382×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#381×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊C₄×C₁₃₁	C₂	1	2	yes	maximal
11	C₂×#376×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#95×C₁₃₁	C₂	1	1	yes	maximal, characteristic
13	C₂⁴×C₄×C₁₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₁	C₄	1	16	yes	complemented
16	C₂×#117×C₁₃₁	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
28	#387×C₁₃₁	C₂²	1	16	yes	complemented
29	#387×C₁₃₁	C₂²	1	16	yes	complemented
30	#385×C₁₃₁	C₂²	1	16	yes	—
31	#383×C₁₃₁	C₂²	1	8	yes	complemented
32	#382×C₁₃₁	C₂²	1	8	yes	complemented
33	#381×C₁₃₁	C₂²	1	8	yes	—
34	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	8	yes	complemented
35	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	16	yes	—
36	#376×C₁₃₁	C₂²	1	8	yes	complemented
37	C₂×#95×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂×#95×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
40	C₂×#95×C₁₃₁	C₂²	1	8	yes	—
41	C₂×#95×C₁₃₁	C₂²	1	1	yes	characteristic
42	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
43	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
44	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
45	C₂×#95×C₁₃₁	C₂²	1	1	yes	characteristic
46	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
47	C₂×#95×C₁₃₁	C₂²	1	2	yes	—
48	C₂⁵×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₁₃₁	—	2	8	no	—
50	C₂³×C₄×C₁₃₁	—	2	8	no	—
51	C₂³×C₄×C₁₃₁	—	2	8	no	—
52	C₂³×C₄×C₁₃₁	—	4	4	no	—
53	C₂³×C₄×C₁₃₁	C₂×C₄	1	8	yes	—
54	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₁₃₁	C₂³	1	8	yes	—
58	C₂³×C₄×C₁₃₁	C₂³	1	4	yes	—
59	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
66	#117×C₁₃₁	C₂×C₄	1	64	yes	complemented
67	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
68	C₂×C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	complemented
70	C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	complemented
71	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₁₃₁	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	—
77	C₂×C₂²⋊C₄×C₁₃₁	C₂×C₄	1	8	yes	—
78	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	4	yes	—
81	C₂×C₄²×C₁₃₁	—	2	8	no	—
82	C₂×C₄²×C₁₃₁	—	4	4	no	—
83	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
84	C₂×C₄²×C₁₃₁	C₂×C₄	1	8	yes	complemented
85	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
86	C₂×C₄²×C₁₃₁	C₂³	1	8	yes	—
87	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
88	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
89	#95×C₁₃₁	C₂³	1	4	yes	—
90	#95×C₁₃₁	C₂³	1	4	yes	—
91	#95×C₁₃₁	C₂³	1	8	yes	—
92	#95×C₁₃₁	C₂³	1	4	yes	—
93	#95×C₁₃₁	C₂³	1	8	yes	—
94	#95×C₁₃₁	C₂³	1	8	yes	—
95	#95×C₁₃₁	C₂³	1	8	yes	—
96	#95×C₁₃₁	C₂³	1	4	yes	—
97	#95×C₁₃₁	C₂³	1	8	yes	—
98	#95×C₁₃₁	C₂³	1	8	yes	—
99	C₂⁴×C₁₃₁	—	2	4	no	—
100	C₂⁴×C₁₃₁	—	2	2	no	—
101	C₂⁴×C₁₃₁	—	2	2	no	—
102	C₂⁴×C₁₃₁	C₂²×C₄	1	4	yes	—
103	C₂⁴×C₁₃₁	#41	1	8	yes	—
104	C₂⁴×C₁₃₁	C₂⁴	1	2	yes	—
105	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₁	—	2	32	no	—
107	C₂²×C₄×C₁₃₁	—	2	4	no	—
108	C₂²×C₄×C₁₃₁	—	2	4	no	—
109	C₂²×C₄×C₁₃₁	—	2	16	no	—
110	C₂²×C₄×C₁₃₁	—	2	4	no	—
111	C₂²×C₄×C₁₃₁	—	2	4	no	—
112	C₂²×C₄×C₁₃₁	—	2	4	no	—
113	C₂²×C₄×C₁₃₁	—	2	2	no	—
114	C₂²×C₄×C₁₃₁	—	2	4	no	—
115	C₂²×C₄×C₁₃₁	—	2	2	no	—
116	C₂²×C₄×C₁₃₁	—	2	2	no	—
117	C₂²×C₄×C₁₃₁	—	2	2	no	—
118	C₂²×C₄×C₁₃₁	—	2	4	no	—
119	C₂²×C₄×C₁₃₁	—	2	4	no	—
120	C₂²×C₄×C₁₃₁	—	2	4	no	—
121	C₂²×C₄×C₁₃₁	—	2	4	no	—
122	C₂²×C₄×C₁₃₁	—	2	16	no	—
123	C₂²×C₄×C₁₃₁	—	2	4	no	—
124	C₂²×C₄×C₁₃₁	—	2	4	no	—
125	C₂²×C₄×C₁₃₁	—	2	4	no	—
126	C₂²×C₄×C₁₃₁	—	2	4	no	—
127	C₂²×C₄×C₁₃₁	—	2	4	no	—
128	C₂²×C₄×C₁₃₁	—	2	16	no	—
129	C₂²×C₄×C₁₃₁	—	4	8	no	—
130	C₂²×C₄×C₁₃₁	—	4	32	no	—
131	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
138	C₂²×C₄×C₁₃₁	#41	1	8	yes	—
139	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
149	C₄⋊C₄×C₁₃₁	—	2	16	no	—
150	C₄⋊C₄×C₁₃₁	—	4	8	no	—
151	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	complemented
152	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	complemented
153	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
154	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
155	C₂²⋊C₄×C₁₃₁	C₂²×C₄	1	16	yes	—
156	C₄²×C₁₃₁	—	2	16	no	—
157	C₄²×C₁₃₁	—	4	8	no	—
158	C₄²×C₁₃₁	C₂²×C₄	1	16	yes	complemented
159	C₄²×C₁₃₁	C₂²×C₄	1	16	yes	complemented
160	C₂³×C₁₃₁	—	2	8	no	—
161	C₂³×C₁₃₁	—	2	8	no	—
162	C₂³×C₁₃₁	—	2	16	no	—
163	C₂³×C₁₃₁	—	2	16	no	—
164	C₂³×C₁₃₁	—	2	4	no	—
165	C₂³×C₁₃₁	—	2	2	no	—
166	C₂³×C₁₃₁	—	2	2	no	—
167	C₂³×C₁₃₁	#117	1	16	yes	—
168	C₂³×C₁₃₁	C₂³×C₄	1	4	yes	—
169	C₂³×C₁₃₁	C₂³×C₄	1	2	yes	—
170	C₂³×C₁₃₁	C₂×#41	1	8	yes	—
171	C₂³×C₁₃₁	C₂×#41	1	4	yes	—
172	C₂³×C₁₃₁	C₂×#41	1	4	yes	—
173	C₂³×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
174	C₂³×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
175	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
176	C₂×C₄×C₁₃₁	—	2	32	no	—
177	C₂×C₄×C₁₃₁	—	2	8	no	—
178	C₂×C₄×C₁₃₁	—	2	8	no	—
179	C₂×C₄×C₁₃₁	—	2	8	no	—
180	C₂×C₄×C₁₃₁	—	2	8	no	—
181	C₂×C₄×C₁₃₁	—	2	8	no	—
182	C₂×C₄×C₁₃₁	—	2	8	no	—
183	C₂×C₄×C₁₃₁	—	2	16	no	—
184	C₂×C₄×C₁₃₁	—	2	4	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	4	no	—
191	C₂×C₄×C₁₃₁	—	2	2	no	—
192	C₂×C₄×C₁₃₁	—	2	2	no	—
193	C₂×C₄×C₁₃₁	—	2	2	no	—
194	C₂×C₄×C₁₃₁	—	2	4	no	—
195	C₂×C₄×C₁₃₁	—	2	4	no	—
196	C₂×C₄×C₁₃₁	—	2	4	no	—
197	C₂×C₄×C₁₃₁	—	2	4	no	—
198	C₂×C₄×C₁₃₁	—	2	4	no	—
199	C₂×C₄×C₁₃₁	—	2	4	no	—
200	C₂×C₄×C₁₃₁	—	2	4	no	—
201	C₂×C₄×C₁₃₁	—	2	4	no	—
202	C₂×C₄×C₁₃₁	—	2	4	no	—
203	C₂×C₄×C₁₃₁	—	4	32	no	—
204	C₂×C₄×C₁₃₁	—	4	8	no	—
205	C₂×C₄×C₁₃₁	—	4	8	no	—
206	C₂×C₄×C₁₃₁	—	4	8	no	—
207	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
208	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
209	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
210	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
211	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
212	C₂×C₄×C₁₃₁	C₂³×C₄	1	4	yes	—
213	C₂×C₄×C₁₃₁	C₂×#41	1	8	yes	—
214	C₂²×C₁₃₁	—	2	32	no	—
215	C₂²×C₁₃₁	—	2	4	no	—
216	C₂²×C₁₃₁	—	2	4	no	—
217	C₂²×C₁₃₁	—	2	8	no	—
218	C₂²×C₁₃₁	—	2	8	no	—
219	C₂²×C₁₃₁	C₂×#117	1	8	yes	—
220	C₂²×C₁₃₁	C₂×#117	1	4	yes	—
221	C₂²×C₁₃₁	C₄×#41	1	4	yes	—
222	C₂²×C₁₃₁	#520	1	4	yes	—
223	C₂²×C₁₃₁	#551	1	8	yes	—
224	C₂²×C₁₃₁	#552	1	8	yes	—
225	C₂²×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
226	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
227	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
228	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂²×C₁₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
230	C₄×C₁₃₁	—	2	4	no	—
231	C₄×C₁₃₁	—	2	4	no	—
232	C₄×C₁₃₁	—	2	4	no	—
233	C₄×C₁₃₁	—	2	4	no	—
234	C₄×C₁₃₁	—	2	4	no	—
235	C₄×C₁₃₁	—	2	4	no	—
236	C₄×C₁₃₁	—	2	8	no	—
237	C₄×C₁₃₁	—	4	4	no	—
238	C₄×C₁₃₁	—	4	4	no	—
239	C₄×C₁₃₁	—	4	4	no	—
240	C₂×C₁₃₁	—	2	8	no	—
241	C₂×C₁₃₁	#2343	1	8	yes	direct product factor
242	C₂×C₁₃₁	C₂²×#117	1	1	yes	characteristic
243	C₂×C₁₃₁	C₂×C₄×#41	1	1	yes	characteristic
244	C₂×C₁₃₁	C₂×#520	1	1	yes	characteristic
245	C₂×C₁₃₁	C₂×#551	1	2	yes	—
246	C₂×C₁₃₁	C₂×#552	1	2	yes	—
247	C₂×#2343	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₁₃₁	C₂×#2343	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²×#117	C₂×C₁₃₁	1	1	yes	characteristic
250	#2343	C₂×C₁₃₁	1	16	yes	direct product factor
251	C₂×#387	C₂×C₁₃₁	1	2	yes	complemented
252	C₂×#387	C₂×C₁₃₁	1	2	yes	complemented
253	C₂×#385	C₂×C₁₃₁	1	2	yes	—
254	C₂×#383	C₂×C₁₃₁	1	1	yes	characteristic, complemented
255	C₂×#382	C₂×C₁₃₁	1	1	yes	characteristic, complemented
256	C₂×#381	C₂×C₁₃₁	1	1	yes	characteristic
257	C₂×C₄×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
258	C₂×C₄×C₂²⋊C₄	C₂×C₁₃₁	1	2	yes	—
259	C₂×#376	C₂×C₁₃₁	1	1	yes	characteristic, complemented
260	C₂²×#95	C₂×C₁₃₁	1	1	yes	characteristic
261	C₂⁴×C₄	C₂²×C₁₃₁	1	2	yes	—
262	C₂⁴×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
263	C₂×#117	C₄×C₁₃₁	1	16	yes	complemented
264	C₂×#117	C₂²×C₁₃₁	1	8	yes	—
265	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
266	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
267	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
269	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
270	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
271	C₂²×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
272	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
273	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
274	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
275	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
276	#387	C₂²×C₁₃₁	1	16	yes	complemented
277	#387	C₂²×C₁₃₁	1	16	yes	complemented
278	#385	C₂²×C₁₃₁	1	16	yes	—
279	#383	C₂²×C₁₃₁	1	8	yes	complemented
280	#382	C₂²×C₁₃₁	1	8	yes	complemented
281	#381	C₂²×C₁₃₁	1	8	yes	—
282	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	8	yes	complemented
283	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	16	yes	—
284	#376	C₂²×C₁₃₁	1	8	yes	complemented
285	C₂×#95	C₂²×C₁₃₁	1	8	yes	—
286	C₂×#95	C₂²×C₁₃₁	1	1	yes	characteristic
287	C₂×#95	C₂²×C₁₃₁	1	1	yes	characteristic
288	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
289	C₂×#95	C₂²×C₁₃₁	1	1	yes	characteristic
290	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
291	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
292	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
293	C₂×#95	C₂²×C₁₃₁	1	1	yes	characteristic
294	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
295	C₂×#95	C₂²×C₁₃₁	1	2	yes	—
296	C₂⁵	C₂³×C₁₃₁	1	1	yes	characteristic
297	C₂³×C₄	—	2	8	no	—
298	C₂³×C₄	—	2	8	no	—
299	C₂³×C₄	—	2	8	no	—
300	C₂³×C₄	—	4	4	no	—
301	C₂³×C₄	C₂×C₄×C₁₃₁	1	8	yes	—
302	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
303	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
304	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
305	C₂³×C₄	C₂³×C₁₃₁	1	8	yes	—
306	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₁₃₁	1	4	yes	—
312	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
313	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
314	#117	C₂×C₄×C₁₃₁	1	64	yes	complemented
315	C₂×C₄⋊C₄	—	2	8	no	—
316	C₂×C₄⋊C₄	—	4	4	no	—
317	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	complemented
318	C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	complemented
319	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
320	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
321	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
323	C₂×C₂²⋊C₄	—	2	8	no	—
324	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₁	1	8	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	4	yes	—
329	C₂×C₄²	—	2	8	no	—
330	C₂×C₄²	—	4	4	no	—
331	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
332	C₂×C₄²	C₂×C₄×C₁₃₁	1	8	yes	complemented
333	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
334	C₂×C₄²	C₂³×C₁₃₁	1	8	yes	—
335	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
336	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
337	#95	C₂³×C₁₃₁	1	4	yes	—
338	#95	C₂³×C₁₃₁	1	4	yes	—
339	#95	C₂³×C₁₃₁	1	8	yes	—
340	#95	C₂³×C₁₃₁	1	4	yes	—
341	#95	C₂³×C₁₃₁	1	8	yes	—
342	#95	C₂³×C₁₃₁	1	8	yes	—
343	#95	C₂³×C₁₃₁	1	8	yes	—
344	#95	C₂³×C₁₃₁	1	4	yes	—
345	#95	C₂³×C₁₃₁	1	8	yes	—
346	#95	C₂³×C₁₃₁	1	8	yes	—
347	C₂⁴	—	2	4	no	—
348	C₂⁴	—	2	2	no	—
349	C₂⁴	—	2	2	no	—
350	C₂⁴	C₂²×C₄×C₁₃₁	1	4	yes	—
351	C₂⁴	#41×C₁₃₁	1	8	yes	—
352	C₂⁴	C₂⁴×C₁₃₁	1	2	yes	—
353	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
354	C₂²×C₄	—	2	32	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	16	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	16	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	16	no	—
377	C₂²×C₄	—	4	8	no	—
378	C₂²×C₄	—	4	32	no	—
379	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
380	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
381	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
382	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
383	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
384	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
385	C₂²×C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
386	C₂²×C₄	#41×C₁₃₁	1	8	yes	—
387	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
388	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
389	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
390	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
391	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
392	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
393	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
397	C₄⋊C₄	—	2	16	no	—
398	C₄⋊C₄	—	4	8	no	—
399	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	complemented
400	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	complemented
401	C₂²⋊C₄	—	2	16	no	—
402	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
403	C₂²⋊C₄	C₂²×C₄×C₁₃₁	1	16	yes	—
404	C₄²	—	2	16	no	—
405	C₄²	—	4	8	no	—
406	C₄²	C₂²×C₄×C₁₃₁	1	16	yes	complemented
407	C₄²	C₂²×C₄×C₁₃₁	1	16	yes	complemented
408	C₂³	—	2	8	no	—
409	C₂³	—	2	8	no	—
410	C₂³	—	2	16	no	—
411	C₂³	—	2	16	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	#117×C₁₃₁	1	16	yes	—
416	C₂³	C₂³×C₄×C₁₃₁	1	4	yes	—
417	C₂³	C₂³×C₄×C₁₃₁	1	2	yes	—
418	C₂³	C₂×#41×C₁₃₁	1	8	yes	—
419	C₂³	C₂×#41×C₁₃₁	1	4	yes	—
420	C₂³	C₂×#41×C₁₃₁	1	4	yes	—
421	C₂³	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
422	C₂³	2₋¹⁺⁴×C₁₃₁	1	2	yes	—
423	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
424	C₂×C₄	—	2	32	no	—
425	C₂×C₄	—	2	8	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	8	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	2	16	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	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	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	32	no	—
452	C₂×C₄	—	4	8	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
456	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
457	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
458	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
459	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
460	C₂×C₄	C₂³×C₄×C₁₃₁	1	4	yes	—
461	C₂×C₄	C₂×#41×C₁₃₁	1	8	yes	—
462	C₂²	—	2	32	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	2	8	no	—
467	C₂²	C₂×#117×C₁₃₁	1	8	yes	—
468	C₂²	C₂×#117×C₁₃₁	1	4	yes	—
469	C₂²	C₄×#41×C₁₃₁	1	4	yes	—
470	C₂²	#520×C₁₃₁	1	4	yes	—
471	C₂²	#551×C₁₃₁	1	8	yes	—
472	C₂²	#552×C₁₃₁	1	8	yes	—
473	C₂²	C₂⁴×C₄×C₁₃₁	1	1	yes	characteristic
474	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
475	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
476	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
477	C₂²	C₂×2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
478	C₄	—	2	4	no	—
479	C₄	—	2	4	no	—
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	4	no	—
483	C₄	—	2	4	no	—
484	C₄	—	2	8	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	4	no	—
488	C₂	—	2	8	no	—
489	C₂	#2343×C₁₃₁	1	8	yes	direct product factor
490	C₂	C₂²×#117×C₁₃₁	1	1	yes	characteristic
491	C₂	C₂×C₄×#41×C₁₃₁	1	1	yes	characteristic
492	C₂	C₂×#520×C₁₃₁	1	1	yes	characteristic
493	C₂	C₂×#551×C₁₃₁	1	2	yes	—
494	C₂	C₂×#552×C₁₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #2343×C₁₃₁	C₂	direct (N × H)
3. C₂×#387×C₁₃₁	C₂	semidirect
4. C₂×#387×C₁₃₁	C₂	semidirect
6. C₂×#383×C₁₃₁	C₂	semidirect
7. C₂×#382×C₁₃₁	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₁₃₁	C₂	semidirect
11. C₂×#376×C₁₃₁	C₂	semidirect
15. C₂×#117×C₁₃₁	C₄	semidirect
28. #387×C₁₃₁	C₂²	semidirect
29. #387×C₁₃₁	C₂²	semidirect
31. #383×C₁₃₁	C₂²	semidirect
32. #382×C₁₃₁	C₂²	semidirect
34. C₄×C₄⋊C₄×C₁₃₁	C₂²	semidirect
36. #376×C₁₃₁	C₂²	semidirect
66. #117×C₁₃₁	C₂×C₄	semidirect
69. C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	semidirect
70. C₂×C₄⋊C₄×C₁₃₁	C₂×C₄	semidirect
83. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
84. C₂×C₄²×C₁₃₁	C₂×C₄	semidirect
151. C₄⋊C₄×C₁₃₁	C₂²×C₄	semidirect
152. C₄⋊C₄×C₁₃₁	C₂²×C₄	semidirect
158. C₄²×C₁₃₁	C₂²×C₄	semidirect
159. C₄²×C₁₃₁	C₂²×C₄	semidirect
241. C₂×C₁₃₁	#2343	direct (N × H)
247. C₂×#2343	C₁₃₁	direct (N × H)
248. C₁₃₁	C₂×#2343	direct (N × H)
250. #2343	C₂×C₁₃₁	direct (N × H)
251. C₂×#387	C₂×C₁₃₁	semidirect
252. C₂×#387	C₂×C₁₃₁	semidirect
254. C₂×#383	C₂×C₁₃₁	semidirect
255. C₂×#382	C₂×C₁₃₁	semidirect
257. C₂×C₄×C₄⋊C₄	C₂×C₁₃₁	semidirect
259. C₂×#376	C₂×C₁₃₁	semidirect
263. C₂×#117	C₄×C₁₃₁	semidirect
276. #387	C₂²×C₁₃₁	semidirect
277. #387	C₂²×C₁₃₁	semidirect
279. #383	C₂²×C₁₃₁	semidirect
280. #382	C₂²×C₁₃₁	semidirect
282. C₄×C₄⋊C₄	C₂²×C₁₃₁	semidirect
284. #376	C₂²×C₁₃₁	semidirect
314. #117	C₂×C₄×C₁₃₁	semidirect
317. C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	semidirect
318. C₂×C₄⋊C₄	C₂×C₄×C₁₃₁	semidirect
331. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
332. C₂×C₄²	C₂×C₄×C₁₃₁	semidirect
399. C₄⋊C₄	C₂²×C₄×C₁₃₁	semidirect
400. C₄⋊C₄	C₂²×C₄×C₁₃₁	semidirect
406. C₄²	C₂²×C₄×C₁₃₁	semidirect
407. C₄²	C₂²×C₄×C₁₃₁	semidirect
489. C₂	#2343×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#117×C₁₃₁	2
#2343×C₁₃₁	2
C₂×#387×C₁₃₁	2
C₂×#385×C₁₃₁	2
C₂×#383×C₁₃₁	2
C₂×#382×C₁₃₁	2
C₂×#381×C₁₃₁	2
C₂×C₄×C₄⋊C₄×C₁₃₁	2
C₂×C₄×C₂²⋊C₄×C₁₃₁	2
C₂×#376×C₁₃₁	2
C₂²×#95×C₁₃₁	2
C₂×#2343	131

Subgroups — C2×#2343×C131 #45719988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2343×C₁₃₁ #45719988