Subgroups — C₃₁×#29367 #21486126

Order 7936.

914 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	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2041	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2014	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1877	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#391	C₂²	1	1	yes	characteristic
24	C₃₁×#391	C₂²	1	1	yes	characteristic
25	C₃₁×#390	—	2	1	no	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#389	C₂²	1	1	yes	characteristic
29	C₃₁×#389	C₂²	1	1	yes	characteristic
30	C₃₁×#387	—	2	1	no	—
31	C₃₁×#387	—	2	1	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#376	—	2	1	no	—
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	4	no	—
49	C₂×SD₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	4	2	no	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×#127	—	2	4	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₄×D₄×C₃₁	—	4	2	no	—
58	C₄×D₄×C₃₁	—	4	4	no	—
59	C₄×D₄×C₃₁	—	4	4	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	4	2	no	—
82	C₃₁×#103	—	4	2	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	4	2	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₂⁴×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	#41	1	2	yes	—
98	C₂×D₄×C₃₁	—	2	4	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	4	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	#41	1	4	yes	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	4	4	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	SD₁₆×C₃₁	—	4	8	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	4	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	#41	1	2	yes	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₂²⋊C₄×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	—	2	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	4	2	no	—
156	#29367	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	4	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₃₁	—	2	4	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	2	4	no	—
168	D₄×C₃₁	—	2	4	no	—
169	D₄×C₃₁	—	4	8	no	—
170	D₄×C₃₁	—	4	8	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	8	8	no	—
203	C₂×C₄×C₃₁	#135	1	2	yes	—
204	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
206	C₂×C₄×C₃₁	#137	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
211	C₈×C₃₁	—	4	2	no	—
212	C₈×C₃₁	—	4	2	no	—
213	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
214	#2295	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#2100	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#2041	C₂×C₃₁	1	1	yes	characteristic
218	#2014	C₂×C₃₁	1	1	yes	characteristic
219	#1925	C₂×C₃₁	1	2	yes	complemented
220	#1880	C₂×C₃₁	1	2	yes	complemented
221	#1877	C₂×C₃₁	1	2	yes	complemented
222	#1862	C₂×C₃₁	1	2	yes	complemented
223	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂²×C₃₁	—	4	4	no	—
225	C₂²×C₃₁	—	4	4	no	—
226	C₂²×C₃₁	—	4	4	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	—	8	2	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	#525	1	1	yes	characteristic
232	C₂²×C₃₁	#525	1	1	yes	characteristic
233	C₂²×C₃₁	#535	1	1	yes	characteristic
234	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	1	no	—
246	C₄×C₃₁	—	8	1	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	#551	1	2	yes	—
250	C₄×C₃₁	#553	1	2	yes	—
251	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
252	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄×D₄	—	2	1	no	—
255	C₂×C₄×D₄	—	2	1	no	—
256	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
257	C₂×#107	C₂²×C₃₁	1	2	yes	—
258	C₂×#103	C₂²×C₃₁	1	2	yes	—
259	C₂×#102	C₂²×C₃₁	1	2	yes	—
260	C₂×#102	C₂²×C₃₁	1	2	yes	—
261	C₂×#102	C₂²×C₃₁	1	2	yes	—
262	#391	C₂²×C₃₁	1	1	yes	characteristic
263	#391	C₂²×C₃₁	1	1	yes	characteristic
264	#390	—	2	1	no	—
265	#390	—	2	1	no	—
266	#390	C₂²×C₃₁	1	2	yes	—
267	#389	C₂²×C₃₁	1	1	yes	characteristic
268	#389	C₂²×C₃₁	1	1	yes	characteristic
269	#387	—	2	1	no	—
270	#387	—	2	1	no	—
271	#384	C₂²×C₃₁	1	1	yes	characteristic
272	C₄×C₄⋊C₄	—	2	1	no	—
273	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	#376	—	2	1	no	—
275	C₄³	C₂²×C₃₁	1	1	yes	characteristic
276	#340	C₂²×C₃₁	1	2	yes	—
277	#340	C₂²×C₃₁	1	2	yes	—
278	#336	C₂²×C₃₁	1	1	yes	characteristic
279	#336	C₂²×C₃₁	1	1	yes	characteristic
280	#336	C₂²×C₃₁	1	2	yes	—
281	#336	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	—	8	4	no	—
283	C₂×C₃₁	#2598	1	1	yes	characteristic
284	C₂×C₃₁	#3105	1	1	yes	characteristic
285	C₂×C₃₁	#3106	1	1	yes	characteristic
286	C₂×C₃₁	#3110	1	1	yes	characteristic
287	C₂×C₃₁	#3112	1	1	yes	characteristic
288	C₂×C₃₁	#3210	1	1	yes	characteristic
289	C₂×C₃₁	#3217	1	1	yes	characteristic
290	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	—	2	1	no	—
293	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
294	C₂³×C₄	—	2	2	no	—
295	C₂×SD₁₆	—	2	4	no	—
296	C₂×SD₁₆	—	2	4	no	—
297	C₂×SD₁₆	—	4	2	no	—
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
300	#127	—	2	4	no	—
301	C₄×Q₈	—	2	2	no	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×D₄	—	4	2	no	—
304	C₄×D₄	—	4	2	no	—
305	C₄×D₄	—	4	4	no	—
306	C₄×D₄	—	4	4	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	#107	—	4	2	no	—
329	#103	—	4	2	no	—
330	#102	—	2	4	no	—
331	#102	—	2	4	no	—
332	#102	—	4	2	no	—
333	#95	—	2	1	no	—
334	#95	—	2	1	no	—
335	#95	—	2	1	no	—
336	#95	—	2	1	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	C₃₁	#29367	1	1	yes	characteristic, direct product factor, Sylow
341	C₂⁴	—	2	2	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	#41×C₃₁	1	2	yes	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	—	4	2	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	#41×C₃₁	1	4	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
373	SD₁₆	—	4	8	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	#41×C₃₁	1	2	yes	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	4	4	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	4	2	no	—
404	C₂³	—	2	4	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	Q₈	—	2	4	no	—
413	Q₈	—	4	2	no	—
414	D₄	—	2	4	no	—
415	D₄	—	2	4	no	—
416	D₄	—	4	8	no	—
417	D₄	—	4	8	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	8	8	no	—
450	C₂×C₄	C₃₁×#135	1	2	yes	—
451	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
452	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
453	C₂×C₄	C₃₁×#137	1	2	yes	—
454	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
456	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	8	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	C₃₁×#525	1	1	yes	characteristic
468	C₂²	C₃₁×#525	1	1	yes	characteristic
469	C₂²	C₃₁×#535	1	1	yes	characteristic
470	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
471	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
472	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
473	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	8	1	no	—
483	C₄	—	8	2	no	—
484	C₄	—	8	1	no	—
485	C₄	C₃₁×#551	1	2	yes	—
486	C₄	C₃₁×#553	1	2	yes	—
487	C₂	—	8	4	no	—
488	C₂	C₃₁×#2598	1	1	yes	characteristic
489	C₂	C₃₁×#3105	1	1	yes	characteristic
490	C₂	C₃₁×#3106	1	1	yes	characteristic
491	C₂	C₃₁×#3110	1	1	yes	characteristic
492	C₂	C₃₁×#3112	1	1	yes	characteristic
493	C₂	C₃₁×#3210	1	1	yes	characteristic
494	C₂	C₃₁×#3217	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2295	C₂	semidirect
3. C₃₁×#2100	C₂	semidirect
4. C₃₁×#2095	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1880	C₂	semidirect
9. C₃₁×#1877	C₂	semidirect
10. C₃₁×#1862	C₂	semidirect
11. C₃₁×#1839	C₂	semidirect
156. #29367	C₃₁	direct (N × H)
214. #2295	C₂×C₃₁	semidirect
215. #2100	C₂×C₃₁	semidirect
216. #2095	C₂×C₃₁	semidirect
219. #1925	C₂×C₃₁	semidirect
220. #1880	C₂×C₃₁	semidirect
221. #1877	C₂×C₃₁	semidirect
222. #1862	C₂×C₃₁	semidirect
223. #1839	C₂×C₃₁	semidirect
340. C₃₁	#29367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#127	2
C₃₁×#2295	2
C₃₁×#2100	2
C₃₁×#2095	2
C₃₁×#2041	2
C₃₁×#2014	2
C₃₁×#1925	2
C₃₁×#1880	2
C₃₁×#1877	2
C₃₁×#1862	2
C₃₁×#1839	2
#29367	31

Subgroups — C31×#29367 #21486126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29367 #21486126