Subgroups — C₃₁×#22230 #21478989

Order 7936.

846 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#486	C₄	1	8	yes	complemented
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#391	—	2	2	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#387	—	2	2	no	—
28	C₃₁×#386	—	2	2	no	—
29	C₃₁×#386	—	2	2	no	—
30	C₃₁×#382	C₂²	1	1	yes	characteristic
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#376	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₂²×Q₈×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
51	C₃₁×#127	—	2	2	no	—
52	C₃₁×#127	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₃₁	—	4	2	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#107	—	4	2	no	—
68	C₃₁×#106	—	4	2	no	—
69	C₃₁×#102	—	4	2	no	—
70	C₃₁×#102	—	4	2	no	—
71	C₃₁×#102	C₂×C₄	1	8	yes	complemented
72	C₃₁×#102	C₂×C₄	1	8	yes	complemented
73	C₃₁×#97	—	2	2	no	—
74	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	1	no	—
77	C₃₁×#95	—	2	1	no	—
78	C₃₁×#95	—	2	1	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	D₄	1	2	yes	—
82	C₃₁×#95	D₄	1	2	yes	—
83	C₂⁴×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	—	8	2	no	—
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	—	2	4	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	8	2	no	—
94	C₂×D₄×C₃₁	—	8	2	no	—
95	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	4	4	no	—
100	C₂²×C₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	8	2	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	8	2	no	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₃₁	—	4	4	no	—
128	C₂²⋊C₄×C₃₁	—	8	2	no	—
129	C₂²⋊C₄×C₃₁	—	8	2	no	—
130	C₄²×C₃₁	—	2	1	no	—
131	C₄²×C₃₁	—	2	1	no	—
132	C₄²×C₃₁	—	2	1	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	8	2	no	—
135	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
136	C₄²×C₃₁	#41	1	2	yes	—
137	#22230	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂³×C₃₁	—	2	4	no	—
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₃₁	—	8	4	no	—
147	D₄×C₃₁	—	2	4	no	—
148	D₄×C₃₁	—	2	4	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	8	4	no	—
152	D₄×C₃₁	—	8	4	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	8	8	no	—
171	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
178	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
179	C₈×C₃₁	—	2	2	no	—
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
184	#2331	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2328	C₂×C₃₁	1	1	yes	characteristic
186	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1961	C₂×C₃₁	1	1	yes	characteristic
188	#1926	C₂×C₃₁	1	2	yes	complemented
189	#1925	C₂×C₃₁	1	2	yes	complemented
190	#1864	C₂×C₃₁	1	2	yes	complemented
191	#1862	C₂×C₃₁	1	2	yes	complemented
192	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	4	4	no	—
195	C₂²×C₃₁	—	4	4	no	—
196	C₂²×C₃₁	—	4	4	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₃₁	#535	1	1	yes	characteristic
203	C₂²×C₃₁	#536	1	1	yes	characteristic
204	C₂²×C₃₁	#576	1	2	yes	—
205	C₂²×C₃₁	#577	1	2	yes	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	4	4	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	#520	1	1	yes	characteristic
214	C₄×C₃₁	#520	1	1	yes	characteristic
215	C₄×C₃₁	#542	1	1	yes	characteristic
216	C₄×C₃₁	#542	1	1	yes	characteristic
217	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
220	#486	C₄×C₃₁	1	8	yes	complemented
221	C₂×#107	C₂²×C₃₁	1	2	yes	—
222	C₂×#106	C₂²×C₃₁	1	2	yes	—
223	C₂×#102	C₂²×C₃₁	1	2	yes	—
224	C₂×#102	C₂²×C₃₁	1	2	yes	—
225	C₂×#102	C₂²×C₃₁	1	2	yes	—
226	C₂×#102	C₂²×C₃₁	1	2	yes	—
227	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	#391	—	2	2	no	—
230	#390	C₂²×C₃₁	1	2	yes	—
231	#389	C₂²×C₃₁	1	2	yes	—
232	#387	—	2	2	no	—
233	#386	—	2	2	no	—
234	#386	—	2	2	no	—
235	#382	C₂²×C₃₁	1	1	yes	characteristic
236	#382	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
238	#376	C₂²×C₃₁	1	1	yes	characteristic
239	#340	C₂²×C₃₁	1	2	yes	—
240	#340	C₂²×C₃₁	1	2	yes	—
241	#336	C₂²×C₃₁	1	1	yes	characteristic
242	#336	C₂²×C₃₁	1	1	yes	characteristic
243	#336	C₂²×C₃₁	1	1	yes	characteristic
244	#336	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₃₁	—	8	4	no	—
246	C₂×C₃₁	#2326	1	1	yes	characteristic
247	C₂×C₃₁	#2948	1	1	yes	characteristic
248	C₂×C₃₁	#2949	1	1	yes	characteristic
249	C₂×C₃₁	#3224	1	2	yes	—
250	C₂×C₃₁	#3241	1	2	yes	—
251	C₂²×Q₈	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
255	C₂³×C₄	—	2	2	no	—
256	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
262	#127	—	2	2	no	—
263	#127	C₂×C₄×C₃₁	1	4	yes	—
264	C₄×D₄	—	4	2	no	—
265	C₄×D₄	—	4	2	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₄²	—	2	2	no	—
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
278	#107	—	4	2	no	—
279	#106	—	4	2	no	—
280	#102	—	4	2	no	—
281	#102	—	4	2	no	—
282	#102	C₂×C₄×C₃₁	1	8	yes	complemented
283	#102	C₂×C₄×C₃₁	1	8	yes	complemented
284	#97	—	2	2	no	—
285	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
286	#95	—	2	2	no	—
287	#95	—	2	1	no	—
288	#95	—	2	1	no	—
289	#95	—	2	1	no	—
290	#95	—	2	1	no	—
291	#95	—	2	2	no	—
292	#95	D₄×C₃₁	1	2	yes	—
293	#95	D₄×C₃₁	1	2	yes	—
294	C₃₁	#22230	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	—	2	2	no	—
296	C₂×Q₈	—	2	4	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	8	2	no	—
306	C₂×D₄	—	8	2	no	—
307	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	—	8	2	no	—
313	C₂²×C₄	—	8	2	no	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
330	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	8	2	no	—
338	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
339	C₂²⋊C₄	—	4	4	no	—
340	C₂²⋊C₄	—	8	2	no	—
341	C₂²⋊C₄	—	8	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	8	2	no	—
347	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
348	C₄²	#41×C₃₁	1	2	yes	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	Q₈	—	8	4	no	—
358	D₄	—	2	4	no	—
359	D₄	—	2	4	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	2	no	—
362	D₄	—	8	4	no	—
363	D₄	—	8	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	8	no	—
382	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
384	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
387	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
388	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
389	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	2	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
402	C₂²	C₃₁×#535	1	1	yes	characteristic
403	C₂²	C₃₁×#536	1	1	yes	characteristic
404	C₂²	C₃₁×#576	1	2	yes	—
405	C₂²	C₃₁×#577	1	2	yes	—
406	C₄	—	2	2	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	4	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	2	no	—
412	C₄	—	8	2	no	—
413	C₄	C₃₁×#520	1	1	yes	characteristic
414	C₄	C₃₁×#520	1	1	yes	characteristic
415	C₄	C₃₁×#542	1	1	yes	characteristic
416	C₄	C₃₁×#542	1	1	yes	characteristic
417	C₂	—	8	4	no	—
418	C₂	C₃₁×#2326	1	1	yes	characteristic
419	C₂	C₃₁×#2948	1	1	yes	characteristic
420	C₂	C₃₁×#2949	1	1	yes	characteristic
421	C₂	C₃₁×#3224	1	2	yes	—
422	C₂	C₃₁×#3241	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2331	C₂	semidirect
4. C₃₁×#1993	C₂	semidirect
6. C₃₁×#1926	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1864	C₂	semidirect
9. C₃₁×#1862	C₂	semidirect
10. C₃₁×#1846	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
15. C₃₁×#486	C₄	semidirect
50. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
71. C₃₁×#102	C₂×C₄	semidirect
72. C₃₁×#102	C₂×C₄	semidirect
74. C₄×C₈×C₃₁	C₂×C₄	semidirect
137. #22230	C₃₁	direct (N × H)
184. #2331	C₂×C₃₁	semidirect
186. #1993	C₂×C₃₁	semidirect
188. #1926	C₂×C₃₁	semidirect
189. #1925	C₂×C₃₁	semidirect
190. #1864	C₂×C₃₁	semidirect
191. #1862	C₂×C₃₁	semidirect
192. #1846	C₂×C₃₁	semidirect
193. #1746	C₂×C₃₁	semidirect
220. #486	C₄×C₃₁	semidirect
261. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
282. #102	C₂×C₄×C₃₁	semidirect
283. #102	C₂×C₄×C₃₁	semidirect
285. C₄×C₈	C₂×C₄×C₃₁	semidirect
294. C₃₁	#22230	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×#486	2
C₃₁×#2331	2
C₃₁×#2328	2
C₃₁×#1993	2
C₃₁×#1961	2
C₃₁×#1926	2
C₃₁×#1925	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1846	2
C₃₁×#1746	2
#22230	31

Subgroups — C31×#22230 #21478989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22230 #21478989