Subgroups — C₃₁×#34269 #21491028

Order 7936.

3426 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#535	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2367	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2366	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2358	C₂	1	8	yes	maximal, complemented
7	C₃₁×#2324	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2311	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#535	C₄	1	16	yes	complemented
16	C₂×C₃₁×#127	C₂²	1	2	yes	—
17	C₂×C₃₁×#124	C₂²	1	2	yes	—
18	C₂×C₃₁×#121	C₂²	1	4	yes	—
19	C₂×C₃₁×#120	C₂²	1	4	yes	—
20	C₃₁×#518	—	2	8	no	—
21	C₄×#41×C₃₁	—	2	8	no	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
23	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
28	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
37	C₃₁×#391	C₂²	1	2	yes	complemented
38	C₃₁×#390	C₂²	1	2	yes	—
39	C₃₁×#390	C₂²	1	4	yes	—
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	complemented
42	C₃₁×#390	C₂²	1	8	yes	complemented
43	C₃₁×#389	C₂²	1	2	yes	complemented
44	C₃₁×#389	C₂²	1	2	yes	complemented
45	C₃₁×#388	C₂²	1	8	yes	complemented
46	C₃₁×#388	C₂²	1	8	yes	complemented
47	C₃₁×#386	C₂²	1	4	yes	—
48	C₃₁×#386	C₂²	1	8	yes	complemented
49	C₃₁×#386	C₂²	1	8	yes	complemented
50	C₃₁×#385	C₂²	1	4	yes	—
51	C₃₁×#385	C₂²	1	8	yes	complemented
52	C₃₁×#380	C₂²	1	4	yes	complemented
53	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
54	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
56	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
58	C₂⁵×C₃₁	C₂³	1	2	yes	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	—	2	4	no	—
61	C₂×#41×C₃₁	C₂×C₄	1	8	yes	complemented
62	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	4	no	—
64	C₂²×D₄×C₃₁	—	2	4	no	—
65	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#127	C₂×C₄	1	16	yes	complemented
78	C₃₁×#124	C₂×C₄	1	16	yes	complemented
79	C₃₁×#121	C₂×C₄	1	32	yes	complemented
80	C₃₁×#120	C₂×C₄	1	32	yes	complemented
81	C₄×Q₈×C₃₁	—	2	8	no	—
82	C₄×D₄×C₃₁	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	8	no	—
84	C₄×D₄×C₃₁	—	2	16	no	—
85	C₄×D₄×C₃₁	—	2	16	no	—
86	C₄×D₄×C₃₁	—	4	16	no	—
87	C₃₁×#117	—	2	4	no	—
88	C₃₁×#117	—	2	8	no	—
89	C₃₁×#117	C₂×C₄	1	8	yes	complemented
90	C₃₁×#117	D₄	1	8	yes	complemented
91	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
101	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄²×C₃₁	—	2	4	no	—
112	C₂×C₄²×C₃₁	—	2	8	no	—
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₃₁×#95	C₂³	1	8	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	8	yes	—
124	C₃₁×#95	C₂³	1	4	yes	—
125	C₂⁴×C₃₁	—	2	4	no	—
126	C₂⁴×C₃₁	—	2	4	no	—
127	C₂⁴×C₃₁	—	4	4	no	—
128	C₂⁴×C₃₁	—	4	4	no	—
129	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
131	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
133	#41×C₃₁	—	2	32	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	32	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	4	8	no	—
147	C₂×D₄×C₃₁	—	4	8	no	—
148	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
149	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₃₁	C₂²×C₄	1	32	yes	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	8	no	—
160	C₂²×C₄×C₃₁	—	2	16	no	—
161	C₂²×C₄×C₃₁	—	4	4	no	—
162	C₂²×C₄×C₃₁	—	4	2	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	—	4	8	no	—
165	C₂²×C₄×C₃₁	—	4	8	no	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₃₁	#41	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	8	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	4	4	no	—
185	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	16	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	16	no	—
191	C₂²⋊C₄×C₃₁	—	4	8	no	—
192	C₂²⋊C₄×C₃₁	—	4	8	no	—
193	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
195	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
196	C₄²×C₃₁	—	2	2	no	—
197	C₄²×C₃₁	—	2	2	no	—
198	C₄²×C₃₁	—	2	8	no	—
199	C₄²×C₃₁	—	2	4	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	—	4	8	no	—
202	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
203	C₄²×C₃₁	C₂×D₄	1	8	yes	—
204	#34269	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	4	no	—
209	C₂³×C₃₁	—	2	2	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	2	8	no	—
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	2	4	no	—
214	C₂³×C₃₁	—	2	8	no	—
215	C₂³×C₃₁	—	2	16	no	—
216	C₂³×C₃₁	—	4	16	no	—
217	C₂³×C₃₁	—	8	8	no	—
218	C₂³×C₃₁	—	8	4	no	—
219	C₂³×C₃₁	—	8	4	no	—
220	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
221	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
222	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
223	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
224	C₂³×C₃₁	C₂×#41	1	2	yes	—
225	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
226	Q₈×C₃₁	—	2	8	no	—
227	D₄×C₃₁	—	2	16	no	—
228	D₄×C₃₁	—	2	16	no	—
229	D₄×C₃₁	—	2	16	no	—
230	D₄×C₃₁	—	2	8	no	—
231	D₄×C₃₁	—	4	32	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	16	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	8	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	16	no	—
250	C₂×C₄×C₃₁	—	4	8	no	—
251	C₂×C₄×C₃₁	—	8	16	no	—
252	C₂×C₄×C₃₁	—	8	4	no	—
253	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	complemented
254	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
255	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
256	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
262	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
263	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
264	C₂×#535	C₂×C₃₁	1	1	yes	characteristic
265	C₂×#518	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2367	C₂×C₃₁	1	4	yes	complemented
268	#2366	C₂×C₃₁	1	4	yes	complemented
269	#2358	C₂×C₃₁	1	8	yes	complemented
270	#2324	C₂×C₃₁	1	4	yes	complemented
271	#2311	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₄×#121	C₂×C₃₁	1	4	yes	complemented
273	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂×#390	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	4	no	—
281	C₂²×C₃₁	—	2	8	no	—
282	C₂²×C₃₁	—	2	8	no	—
283	C₂²×C₃₁	—	2	16	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	8	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	2	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	8	16	no	—
291	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
292	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
293	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
294	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
295	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
296	C₂²×C₃₁	#585	1	2	yes	—
297	C₄×C₃₁	—	2	4	no	—
298	C₄×C₃₁	—	2	8	no	—
299	C₄×C₃₁	—	2	4	no	—
300	C₄×C₃₁	—	2	8	no	—
301	C₄×C₃₁	—	4	2	no	—
302	C₄×C₃₁	—	4	4	no	—
303	C₄×C₃₁	—	4	4	no	—
304	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
305	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
306	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
307	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
308	#535	C₄×C₃₁	1	16	yes	complemented
309	C₂×#127	C₂²×C₃₁	1	2	yes	—
310	C₂×#124	C₂²×C₃₁	1	2	yes	—
311	C₂×#121	C₂²×C₃₁	1	4	yes	—
312	C₂×#120	C₂²×C₃₁	1	4	yes	—
313	#518	—	2	8	no	—
314	C₄×#41	—	2	8	no	—
315	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
316	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
317	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
318	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
320	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
321	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
325	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
326	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
327	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
328	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
329	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
330	#391	C₂²×C₃₁	1	2	yes	complemented
331	#390	C₂²×C₃₁	1	2	yes	—
332	#390	C₂²×C₃₁	1	4	yes	—
333	#390	C₂²×C₃₁	1	2	yes	—
334	#390	C₂²×C₃₁	1	2	yes	complemented
335	#390	C₂²×C₃₁	1	8	yes	complemented
336	#389	C₂²×C₃₁	1	2	yes	complemented
337	#389	C₂²×C₃₁	1	2	yes	complemented
338	#388	C₂²×C₃₁	1	8	yes	complemented
339	#388	C₂²×C₃₁	1	8	yes	complemented
340	#386	C₂²×C₃₁	1	4	yes	—
341	#386	C₂²×C₃₁	1	8	yes	complemented
342	#386	C₂²×C₃₁	1	8	yes	complemented
343	#385	C₂²×C₃₁	1	4	yes	—
344	#385	C₂²×C₃₁	1	8	yes	complemented
345	#380	C₂²×C₃₁	1	4	yes	complemented
346	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
347	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
349	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
350	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
351	C₂×C₃₁	—	2	4	no	—
352	C₂×C₃₁	—	4	4	no	—
353	C₂×C₃₁	—	4	8	no	—
354	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
355	C₂×C₃₁	C₄×H(5,𝔽₂)	1	2	yes	—
356	C₂×C₃₁	#3488	1	2	yes	—
357	C₂×C₃₁	#3508	1	1	yes	characteristic
358	C₂×C₃₁	#3510	1	1	yes	characteristic
359	C₂⁵	C₂³×C₃₁	1	2	yes	—
360	C₂×#41	—	2	4	no	—
361	C₂×#41	—	2	4	no	—
362	C₂×#41	C₂×C₄×C₃₁	1	8	yes	complemented
363	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×D₄	—	2	4	no	—
365	C₂²×D₄	—	2	4	no	—
366	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
367	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
368	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
369	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
372	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
373	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
374	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
375	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
376	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
377	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
378	#127	C₂×C₄×C₃₁	1	16	yes	complemented
379	#124	C₂×C₄×C₃₁	1	16	yes	complemented
380	#121	C₂×C₄×C₃₁	1	32	yes	complemented
381	#120	C₂×C₄×C₃₁	1	32	yes	complemented
382	C₄×Q₈	—	2	8	no	—
383	C₄×D₄	—	2	16	no	—
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	—	2	16	no	—
386	C₄×D₄	—	2	16	no	—
387	C₄×D₄	—	4	16	no	—
388	#117	—	2	4	no	—
389	#117	—	2	8	no	—
390	#117	C₂×C₄×C₃₁	1	8	yes	complemented
391	#117	D₄×C₃₁	1	8	yes	complemented
392	C₂×C₄⋊C₄	—	2	4	no	—
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₂²⋊C₄	—	2	16	no	—
402	C₂×C₂²⋊C₄	—	4	8	no	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
412	C₂×C₄²	—	2	4	no	—
413	C₂×C₄²	—	2	8	no	—
414	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
415	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
417	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
420	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
421	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
422	#95	C₂³×C₃₁	1	8	yes	—
423	#95	C₂³×C₃₁	1	4	yes	—
424	#95	C₂³×C₃₁	1	8	yes	—
425	#95	C₂³×C₃₁	1	4	yes	—
426	C₃₁	#34269	1	1	yes	characteristic, direct product factor, Sylow
427	C₂⁴	—	2	4	no	—
428	C₂⁴	—	2	4	no	—
429	C₂⁴	—	4	4	no	—
430	C₂⁴	—	4	4	no	—
431	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
432	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
433	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
434	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
435	#41	—	2	32	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	32	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	4	8	no	—
450	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
451	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
452	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
453	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
454	C₂×D₄	C₂²×C₄×C₃₁	1	32	yes	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	8	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	16	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	8	no	—
467	C₂²×C₄	—	4	8	no	—
468	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
469	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
470	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₂²×C₄	#41×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	4	4	no	—
487	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
488	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	16	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	16	no	—
493	C₂²⋊C₄	—	4	8	no	—
494	C₂²⋊C₄	—	4	8	no	—
495	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
496	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
497	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	8	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	4	no	—
503	C₄²	—	4	8	no	—
504	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
505	C₄²	C₂×D₄×C₃₁	1	8	yes	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	8	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	8	no	—
516	C₂³	—	2	16	no	—
517	C₂³	—	4	16	no	—
518	C₂³	—	8	8	no	—
519	C₂³	—	8	4	no	—
520	C₂³	—	8	4	no	—
521	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
522	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
523	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
524	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
525	C₂³	C₂×#41×C₃₁	1	2	yes	—
526	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
527	Q₈	—	2	8	no	—
528	D₄	—	2	16	no	—
529	D₄	—	2	16	no	—
530	D₄	—	2	16	no	—
531	D₄	—	2	8	no	—
532	D₄	—	4	32	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	16	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	4	16	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	—	8	16	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	complemented
555	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
556	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
557	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
558	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
560	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
563	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
564	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	8	no	—
572	C₂²	—	2	16	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	8	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	16	no	—
580	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
581	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
582	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
583	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
584	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
585	C₂²	C₃₁×#585	1	2	yes	—
586	C₄	—	2	4	no	—
587	C₄	—	2	8	no	—
588	C₄	—	2	4	no	—
589	C₄	—	2	8	no	—
590	C₄	—	4	2	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	4	no	—
593	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
594	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
595	C₂	—	2	4	no	—
596	C₂	—	4	8	no	—
597	C₂	—	4	4	no	—
598	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
599	C₂	C₄×C₃₁×H(5,𝔽₂)	1	2	yes	—
600	C₂	C₃₁×#3488	1	2	yes	—
601	C₂	C₃₁×#3508	1	1	yes	characteristic
602	C₂	C₃₁×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#518	C₂	semidirect
3. C₂×C₄×#41×C₃₁	C₂	semidirect
4. C₃₁×#2367	C₂	semidirect
5. C₃₁×#2366	C₂	semidirect
6. C₃₁×#2358	C₂	semidirect
7. C₃₁×#2324	C₂	semidirect
8. C₃₁×#2311	C₂	semidirect
9. C₄×C₃₁×#121	C₂	semidirect
10. C₃₁×#2296	C₂	semidirect
11. C₂×C₃₁×#390	C₂	semidirect
12. C₃₁×#2274	C₂	semidirect
15. C₃₁×#535	C₄	semidirect
22. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
26. C₂×C₄×D₄×C₃₁	C₂²	semidirect
27. C₂×C₄×D₄×C₃₁	C₂²	semidirect
31. C₂×C₃₁×#117	C₂²	semidirect
36. C₂²×C₄²×C₃₁	C₂²	semidirect
37. C₃₁×#391	C₂²	semidirect
41. C₃₁×#390	C₂²	semidirect
42. C₃₁×#390	C₂²	semidirect
43. C₃₁×#389	C₂²	semidirect
44. C₃₁×#389	C₂²	semidirect
45. C₃₁×#388	C₂²	semidirect
46. C₃₁×#388	C₂²	semidirect
48. C₃₁×#386	C₂²	semidirect
49. C₃₁×#386	C₂²	semidirect
51. C₃₁×#385	C₂²	semidirect
52. C₃₁×#380	C₂²	semidirect
53. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
54. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
55. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
57. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
61. C₂×#41×C₃₁	C₂×C₄	semidirect
65. C₂²×D₄×C₃₁	C₂×C₄	semidirect
77. C₃₁×#127	C₂×C₄	semidirect
78. C₃₁×#124	C₂×C₄	semidirect
79. C₃₁×#121	C₂×C₄	semidirect
80. C₃₁×#120	C₂×C₄	semidirect
89. C₃₁×#117	C₂×C₄	semidirect
90. C₃₁×#117	D₄	semidirect
204. #34269	C₃₁	direct (N × H)
253. C₂×C₄×C₃₁	C₄×D₄	semidirect
265. C₂×#518	C₂×C₃₁	semidirect
266. C₂×C₄×#41	C₂×C₃₁	semidirect
267. #2367	C₂×C₃₁	semidirect
268. #2366	C₂×C₃₁	semidirect
269. #2358	C₂×C₃₁	semidirect
270. #2324	C₂×C₃₁	semidirect
271. #2311	C₂×C₃₁	semidirect
272. C₄×#121	C₂×C₃₁	semidirect
273. #2296	C₂×C₃₁	semidirect
274. C₂×#390	C₂×C₃₁	semidirect
275. #2274	C₂×C₃₁	semidirect
308. #535	C₄×C₃₁	semidirect
315. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
319. C₂×C₄×D₄	C₂²×C₃₁	semidirect
320. C₂×C₄×D₄	C₂²×C₃₁	semidirect
324. C₂×#117	C₂²×C₃₁	semidirect
329. C₂²×C₄²	C₂²×C₃₁	semidirect
330. #391	C₂²×C₃₁	semidirect
334. #390	C₂²×C₃₁	semidirect
335. #390	C₂²×C₃₁	semidirect
336. #389	C₂²×C₃₁	semidirect
337. #389	C₂²×C₃₁	semidirect
338. #388	C₂²×C₃₁	semidirect
339. #388	C₂²×C₃₁	semidirect
341. #386	C₂²×C₃₁	semidirect
342. #386	C₂²×C₃₁	semidirect
344. #385	C₂²×C₃₁	semidirect
345. #380	C₂²×C₃₁	semidirect
346. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
347. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
348. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
350. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
362. C₂×#41	C₂×C₄×C₃₁	semidirect
366. C₂²×D₄	C₂×C₄×C₃₁	semidirect
378. #127	C₂×C₄×C₃₁	semidirect
379. #124	C₂×C₄×C₃₁	semidirect
380. #121	C₂×C₄×C₃₁	semidirect
381. #120	C₂×C₄×C₃₁	semidirect
390. #117	C₂×C₄×C₃₁	semidirect
391. #117	D₄×C₃₁	semidirect
426. C₃₁	#34269	direct (N × H)
554. C₂×C₄	C₄×D₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#535	2
C₂×C₃₁×#518	2
C₂×C₄×#41×C₃₁	2
C₃₁×#2367	2
C₃₁×#2366	2
C₃₁×#2358	2
C₃₁×#2324	2
C₃₁×#2311	2
C₄×C₃₁×#121	2
C₃₁×#2296	2
C₂×C₃₁×#390	2
C₃₁×#2274	2
#34269	31

Subgroups — C31×#34269 #21491028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34269 #21491028