Subgroups — C₃₁×#24830 #21481589

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2563	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2085	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1879	C₂	1	2	yes	maximal
10	C₃₁×#1804	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#474	—	2	4	no	—
18	C₃₁×#442	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₃₁×#393	—	2	1	no	—
29	C₃₁×#393	—	2	1	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#382	C₂²	1	2	yes	—
34	C₃₁×#382	C₂²	1	1	yes	characteristic
35	C₃₁×#380	—	2	2	no	—
36	C₃₁×#380	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#95	—	2	1	no	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	Q₈	1	4	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	4	2	no	—
62	C₄×D₄×C₃₁	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
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	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	4	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	4	2	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	D₄	1	1	yes	characteristic
92	C₃₁×#95	D₄	1	1	yes	characteristic
93	C₂⁴×C₃₁	—	2	1	no	—
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	—	8	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
107	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂×D₄×C₃₁	#41	1	2	yes	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	4	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	8	2	no	—
119	C₂²×C₄×C₃₁	—	8	2	no	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	D₈×C₃₁	—	2	8	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	8	2	no	—
145	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₃₁	—	2	4	no	—
147	C₂²⋊C₄×C₃₁	—	4	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	8	2	no	—
150	C₂²⋊C₄×C₃₁	—	8	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	C₂×D₄	1	2	yes	—
154	C₄²×C₃₁	#41	1	2	yes	—
155	#24830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	1	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₃₁	—	8	4	no	—
168	D₄×C₃₁	—	2	2	no	—
169	D₄×C₃₁	—	2	2	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	4	1	no	—
174	D₄×C₃₁	—	4	1	no	—
175	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	8	4	no	—
195	C₂×C₄×C₃₁	—	8	4	no	—
196	C₂×C₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	#121	1	2	yes	—
198	C₂×C₄×C₃₁	#121	1	2	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	—	4	2	no	—
208	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
210	#2563	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#2085	C₂×C₃₁	1	2	yes	complemented
213	#2067	C₂×C₃₁	1	2	yes	complemented
214	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1879	C₂×C₃₁	1	2	yes	—
217	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1804	C₂×C₃₁	1	2	yes	complemented
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	4	no	—
224	C₂²×C₃₁	—	4	4	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	8	1	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
230	C₂²×C₃₁	#546	1	1	yes	characteristic
231	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
232	C₂²×C₃₁	#575	1	2	yes	—
233	C₂²×C₃₁	#576	1	2	yes	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	8	1	no	—
240	C₄×C₃₁	—	8	2	no	—
241	C₄×C₃₁	—	8	2	no	—
242	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
243	C₄×C₃₁	#537	1	1	yes	characteristic
244	C₄×C₃₁	#552	1	1	yes	characteristic
245	C₄×C₃₁	#554	1	1	yes	characteristic
246	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
250	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
251	#474	—	2	4	no	—
252	#442	—	2	4	no	—
253	C₂×#107	C₂²×C₃₁	1	2	yes	—
254	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#102	C₂²×C₃₁	1	2	yes	—
259	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#102	C₂²×C₃₁	1	2	yes	—
261	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
262	#393	—	2	1	no	—
263	#393	—	2	1	no	—
264	#390	C₂²×C₃₁	1	2	yes	—
265	#389	C₂²×C₃₁	1	2	yes	—
266	#385	—	2	2	no	—
267	#382	C₂²×C₃₁	1	2	yes	—
268	#382	C₂²×C₃₁	1	1	yes	characteristic
269	#380	—	2	2	no	—
270	#380	—	2	1	no	—
271	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#95	—	2	1	no	—
273	#340	C₂²×C₃₁	1	2	yes	—
274	#340	C₂²×C₃₁	1	2	yes	—
275	#340	C₂²×C₃₁	1	2	yes	—
276	#336	C₂²×C₃₁	1	2	yes	—
277	C₂×C₃₁	—	4	4	no	—
278	C₂×C₃₁	—	8	2	no	—
279	C₂×C₃₁	#2417	1	1	yes	characteristic
280	C₂×C₃₁	#3040	1	1	yes	characteristic
281	C₂×C₃₁	#3041	1	1	yes	characteristic
282	C₂×C₃₁	#3319	1	2	yes	—
283	C₂×C₃₁	#3389	1	2	yes	—
284	C₂²×Q₈	—	2	1	no	—
285	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	—	2	2	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂×D₈	Q₈×C₃₁	1	4	yes	complemented
293	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
294	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	—	2	2	no	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	2	2	no	—
300	C₄×D₄	—	2	2	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	—	4	2	no	—
303	C₄×D₄	D₄×C₃₁	1	4	yes	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
317	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
318	#107	—	2	4	no	—
319	#106	—	2	2	no	—
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	4	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	4	no	—
324	#102	—	2	2	no	—
325	#102	—	4	2	no	—
326	#97	—	2	2	no	—
327	#95	—	2	1	no	—
328	#95	—	2	2	no	—
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	—	2	1	no	—
332	#95	D₄×C₃₁	1	1	yes	characteristic
333	#95	D₄×C₃₁	1	1	yes	characteristic
334	C₃₁	#24830	1	1	yes	characteristic, direct product factor, Sylow
335	C₂⁴	—	2	1	no	—
336	C₂⁴	—	2	1	no	—
337	C₂×Q₈	—	4	2	no	—
338	C₂×Q₈	—	8	2	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
349	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
350	C₂×D₄	#41×C₃₁	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	4	no	—
357	C₂²×C₄	—	4	4	no	—
358	C₂²×C₄	—	4	4	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	—	8	2	no	—
361	C₂²×C₄	—	8	2	no	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
367	D₈	—	2	8	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	8	2	no	—
387	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	4	2	no	—
391	C₂²⋊C₄	—	8	2	no	—
392	C₂²⋊C₄	—	8	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	C₂×D₄×C₃₁	1	2	yes	—
396	C₄²	#41×C₃₁	1	2	yes	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	1	no	—
404	C₂³	—	4	1	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	Q₈	—	8	4	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	2	2	no	—
413	D₄	—	2	2	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	1	no	—
416	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	8	4	no	—
436	C₂×C₄	—	8	4	no	—
437	C₂×C₄	—	8	8	no	—
438	C₂×C₄	C₃₁×#121	1	2	yes	—
439	C₂×C₄	C₃₁×#121	1	2	yes	—
440	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
445	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
446	C₈	—	2	2	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	4	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	4	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	8	1	no	—
458	C₂²	—	8	2	no	—
459	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
460	C₂²	C₃₁×#546	1	1	yes	characteristic
461	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
462	C₂²	C₃₁×#575	1	2	yes	—
463	C₂²	C₃₁×#576	1	2	yes	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	8	1	no	—
470	C₄	—	8	2	no	—
471	C₄	—	8	2	no	—
472	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
473	C₄	C₃₁×#537	1	1	yes	characteristic
474	C₄	C₃₁×#552	1	1	yes	characteristic
475	C₄	C₃₁×#554	1	1	yes	characteristic
476	C₂	—	4	4	no	—
477	C₂	—	8	2	no	—
478	C₂	C₃₁×#2417	1	1	yes	characteristic
479	C₂	C₃₁×#3040	1	1	yes	characteristic
480	C₂	C₃₁×#3041	1	1	yes	characteristic
481	C₂	C₃₁×#3319	1	2	yes	—
482	C₂	C₃₁×#3389	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#474	C₂	semidirect
3. C₃₁×#2563	C₂	semidirect
4. C₃₁×#2349	C₂	semidirect
5. C₃₁×#2085	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1804	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
51. C₂×D₈×C₃₁	Q₈	semidirect
155. #24830	C₃₁	direct (N × H)
208. C₂×#474	C₂×C₃₁	semidirect
210. #2563	C₂×C₃₁	semidirect
211. #2349	C₂×C₃₁	semidirect
212. #2085	C₂×C₃₁	semidirect
213. #2067	C₂×C₃₁	semidirect
214. #1993	C₂×C₃₁	semidirect
215. #1940	C₂×C₃₁	semidirect
217. #1804	C₂×C₃₁	semidirect
218. #1804	C₂×C₃₁	semidirect
292. C₂×D₈	Q₈×C₃₁	semidirect
334. C₃₁	#24830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#474	2
C₂×C₃₁×#442	2
C₃₁×#2563	2
C₃₁×#2349	2
C₃₁×#2085	2
C₃₁×#2067	2
C₃₁×#1993	2
C₃₁×#1940	2
C₃₁×#1879	2
C₃₁×#1804	2
#24830	31

Subgroups — C31×#24830 #21481589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24830 #21481589