Subgroups — C₃₁×#19973 #21476732

Order 7936.

712 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1641	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1627	C₂	1	2	yes	maximal, complemented
10	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂²×Q₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#550	C₂²	1	2	yes	—
15	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#486	—	2	2	no	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#486	C₂²	1	2	yes	—
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#474	C₂²	1	2	yes	—
23	C₃₁×#467	—	2	1	no	—
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#462	C₂²	1	2	yes	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₃₁×#459	—	2	2	no	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#451	—	2	2	no	—
30	C₃₁×#449	—	2	2	no	—
31	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	—	2	2	no	—
34	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#332	C₂²	1	2	yes	—
40	C₃₁×#331	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₃₁×#135	—	4	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	4	4	no	—
52	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×#127	—	2	2	no	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₃₁×C₂²⋊Q₈	—	2	2	no	—
65	C₃₁×C₂²⋊Q₈	—	4	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	4	2	no	—
68	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	D₄	1	2	yes	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×#102	D₄	1	1	yes	characteristic
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	#41×C₃₁	—	4	4	no	—
97	#41×C₃₁	—	8	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	—	4	4	no	—
101	C₂×Q₈×C₃₁	—	8	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	—	4	2	no	—
114	Q₁₆×C₃₁	—	4	4	no	—
115	Q₁₆×C₃₁	—	4	2	no	—
116	Q₁₆×C₃₁	—	8	1	no	—
117	Q₁₆×C₃₁	D₈	1	4	yes	complemented
118	SD₁₆×C₃₁	—	4	4	no	—
119	D₈×C₃₁	—	2	2	no	—
120	D₈×C₃₁	—	4	4	no	—
121	D₈×C₃₁	—	4	2	no	—
122	D₈×C₃₁	—	8	1	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	4	2	no	—
128	C₂×C₈×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
146	#19973	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	2	2	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	2	no	—
153	Q₈×C₃₁	—	8	4	no	—
154	Q₈×C₃₁	C₂×D₈	1	4	yes	—
155	D₄×C₃₁	—	2	1	no	—
156	D₄×C₃₁	—	2	1	no	—
157	D₄×C₃₁	—	2	1	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	8	4	no	—
163	D₄×C₃₁	—	8	2	no	—
164	D₄×C₃₁	—	16	2	no	—
165	D₄×C₃₁	#135	1	2	yes	complemented
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	8	4	no	—
176	C₂×C₄×C₃₁	—	8	4	no	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	—	4	1	no	—
185	C₈×C₃₁	—	4	1	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₈×C₃₁	C₂×D₈	1	2	yes	—
188	#3358	C₂×C₃₁	1	2	yes	complemented
189	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
190	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1682	C₂×C₃₁	1	2	yes	complemented
193	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1641	C₂×C₃₁	1	2	yes	complemented
196	#1627	C₂×C₃₁	1	2	yes	complemented
197	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	—	4	1	no	—
204	C₂²×C₃₁	—	4	1	no	—
205	C₂²×C₃₁	—	8	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	D₄²	1	1	yes	characteristic
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
216	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
217	C₄×C₃₁	#576	1	1	yes	characteristic
218	C₄×C₃₁	#578	1	1	yes	characteristic
219	C₂×#135	—	2	1	no	—
220	C₂²×Q₁₆	—	2	1	no	—
221	#550	C₂²×C₃₁	1	2	yes	—
222	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
223	#494	C₂²×C₃₁	1	1	yes	characteristic
224	#493	C₂²×C₃₁	1	1	yes	characteristic
225	#486	—	2	2	no	—
226	#486	C₂²×C₃₁	1	2	yes	—
227	#486	C₂²×C₃₁	1	2	yes	—
228	#474	—	2	2	no	—
229	#474	C₂²×C₃₁	1	2	yes	—
230	#467	—	2	1	no	—
231	#466	—	2	1	no	—
232	#462	C₂²×C₃₁	1	2	yes	—
233	#460	C₂²×C₃₁	1	2	yes	—
234	#459	—	2	2	no	—
235	#459	C₂²×C₃₁	1	2	yes	—
236	#451	—	2	2	no	—
237	#449	—	2	2	no	—
238	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
239	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
240	C₄×D₈	—	2	2	no	—
241	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
244	#335	C₂²×C₃₁	1	1	yes	characteristic
245	#335	C₂²×C₃₁	1	1	yes	characteristic
246	#332	C₂²×C₃₁	1	2	yes	—
247	#331	C₂²×C₃₁	1	2	yes	—
248	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
249	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₈²	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₃₁	—	4	2	no	—
253	C₂×C₃₁	—	8	1	no	—
254	C₂×C₃₁	—	16	2	no	—
255	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
256	C₂×C₃₁	#3285	1	1	yes	characteristic
257	C₂×C₃₁	#3290	1	1	yes	characteristic
258	C₂×#41	—	2	2	no	—
259	C₂²×Q₈	—	2	2	no	—
260	#135	—	4	4	no	—
261	C₂×Q₁₆	—	2	2	no	—
262	C₂×Q₁₆	—	2	1	no	—
263	C₂×Q₁₆	—	2	1	no	—
264	C₂×Q₁₆	—	4	4	no	—
265	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×D₈	—	2	2	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	—	2	1	no	—
270	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	2	no	—
274	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
275	#127	—	2	2	no	—
276	#127	C₂³×C₃₁	1	2	yes	—
277	C₂²⋊Q₈	—	2	2	no	—
278	C₂²⋊Q₈	—	4	2	no	—
279	#121	—	2	2	no	—
280	#121	—	4	2	no	—
281	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
282	C₄×D₄	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	2	no	—
287	#107	D₄×C₃₁	1	1	yes	characteristic
288	#107	D₄×C₃₁	1	1	yes	characteristic
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
293	#103	—	2	2	no	—
294	#103	D₄×C₃₁	1	2	yes	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	4	2	no	—
301	#102	D₄×C₃₁	1	1	yes	characteristic
302	#102	D₄×C₃₁	1	1	yes	characteristic
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₄×C₈	—	2	2	no	—
306	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₃₁	#19973	1	1	yes	characteristic, direct product factor, Sylow
310	#41	—	4	4	no	—
311	#41	—	8	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	4	2	no	—
314	C₂×Q₈	—	4	4	no	—
315	C₂×Q₈	—	8	2	no	—
316	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	4	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	Q₁₆	—	4	4	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	8	1	no	—
331	Q₁₆	D₈×C₃₁	1	4	yes	complemented
332	SD₁₆	—	4	4	no	—
333	D₈	—	2	2	no	—
334	D₈	—	4	4	no	—
335	D₈	—	4	2	no	—
336	D₈	—	8	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	1	no	—
356	C₂²⋊C₄	—	4	2	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
360	C₂³	—	2	1	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	4	2	no	—
363	Q₈	—	2	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	8	4	no	—
367	Q₈	C₂×D₈×C₃₁	1	4	yes	—
368	D₄	—	2	1	no	—
369	D₄	—	2	1	no	—
370	D₄	—	2	1	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	8	4	no	—
376	D₄	—	8	2	no	—
377	D₄	—	16	2	no	—
378	D₄	C₃₁×#135	1	2	yes	complemented
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	—	8	4	no	—
390	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	4	1	no	—
398	C₈	—	4	1	no	—
399	C₈	—	4	2	no	—
400	C₈	C₂×D₈×C₃₁	1	2	yes	—
401	C₂²	—	2	2	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	4	1	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	8	2	no	—
410	C₂²	D₄²×C₃₁	1	1	yes	characteristic
411	C₄	—	2	4	no	—
412	C₄	—	2	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
418	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
419	C₄	C₃₁×#576	1	1	yes	characteristic
420	C₄	C₃₁×#578	1	1	yes	characteristic
421	C₂	—	4	2	no	—
422	C₂	—	8	1	no	—
423	C₂	—	16	2	no	—
424	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
425	C₂	C₃₁×#3285	1	1	yes	characteristic
426	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3358	C₂	semidirect
2. C₃₁×#3287	C₂	semidirect
3. D₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1709	C₂	semidirect
5. C₃₁×#1682	C₂	semidirect
6. C₃₁×#1669	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1641	C₂	semidirect
9. C₃₁×#1627	C₂	semidirect
10. C₈×Q₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
117. Q₁₆×C₃₁	D₈	semidirect
146. #19973	C₃₁	direct (N × H)
165. D₄×C₃₁	#135	semidirect
188. #3358	C₂×C₃₁	semidirect
189. #3287	C₂×C₃₁	semidirect
190. D₄×Q₁₆	C₂×C₃₁	semidirect
191. #1709	C₂×C₃₁	semidirect
192. #1682	C₂×C₃₁	semidirect
193. #1669	C₂×C₃₁	semidirect
194. #1667	C₂×C₃₁	semidirect
195. #1641	C₂×C₃₁	semidirect
196. #1627	C₂×C₃₁	semidirect
197. C₈×Q₁₆	C₂×C₃₁	semidirect
198. C₈×D₈	C₂×C₃₁	semidirect
309. C₃₁	#19973	direct (N × H)
331. Q₁₆	D₈×C₃₁	semidirect
378. D₄	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3358	2
C₃₁×#3287	2
D₄×Q₁₆×C₃₁	2
C₃₁×#1709	2
C₃₁×#1682	2
C₃₁×#1669	2
C₃₁×#1667	2
C₃₁×#1641	2
C₃₁×#1627	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#19973	31

Subgroups — C31×#19973 #21476732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19973 #21476732