Subgroups — C₃₁×#19972 #21476731

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3284	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1674	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1641	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1625	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	2	no	—
13	C₃₁×#550	C₂²	1	2	yes	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#493	—	2	1	no	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#492	C₂²	1	2	yes	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#480	—	2	2	no	—
21	C₃₁×#474	C₂²	1	2	yes	—
22	C₃₁×#466	—	2	1	no	—
23	C₃₁×#465	—	2	2	no	—
24	C₃₁×#462	C₂²	1	2	yes	—
25	C₃₁×#459	—	2	2	no	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#452	—	2	2	no	—
29	C₃₁×#449	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₃₁×#107	—	2	1	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#334	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	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
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	C₂×#41×C₃₁	—	2	2	no	—
47	C₃₁×#135	—	4	8	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	—	2	2	no	—
61	C₃₁×#127	C₂³	1	2	yes	—
62	C₃₁×#123	—	4	2	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	2	2	no	—
66	C₃₁×#121	—	4	2	no	—
67	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	C₂³	1	2	yes	—
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×#106	D₄	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	—	2	1	no	—
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	D₄	1	1	yes	characteristic
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	4	2	no	—
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	2	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	4	4	no	—
97	#41×C₃₁	—	8	2	no	—
98	#41×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
105	C₂×D₄×C₃₁	—	4	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	Q₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	4	no	—
115	SD₁₆×C₃₁	—	8	2	no	—
116	SD₁₆×C₃₁	D₈	1	4	yes	complemented
117	D₈×C₃₁	—	2	2	no	—
118	D₈×C₃₁	—	4	4	no	—
119	D₈×C₃₁	—	4	2	no	—
120	D₈×C₃₁	—	8	1	no	—
121	D₈×C₃₁	—	8	2	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	—	4	2	no	—
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₃₁	—	2	2	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
145	#19972	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	2	1	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	C₂×D₈	1	2	yes	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	2	1	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	8	4	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	8	4	no	—
163	D₄×C₃₁	—	16	2	no	—
164	D₄×C₃₁	C₂×D₈	1	2	yes	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	4	no	—
175	C₂×C₄×C₃₁	—	8	4	no	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	—	4	1	no	—
185	C₈×C₃₁	C₂×D₈	1	2	yes	—
186	C₈×C₃₁	#135	1	2	yes	complemented
187	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#3334	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3284	C₂×C₃₁	1	2	yes	complemented
190	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1674	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1665	C₂×C₃₁	1	2	yes	complemented
194	#1641	C₂×C₃₁	1	2	yes	complemented
195	#1625	C₂×C₃₁	1	2	yes	complemented
196	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	8	2	no	—
205	C₂²×C₃₁	—	8	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	D₄²	1	1	yes	characteristic
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
215	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
216	C₄×C₃₁	#576	1	1	yes	characteristic
217	C₄×C₃₁	#577	1	1	yes	characteristic
218	C₂×#135	—	2	2	no	—
219	#550	C₂²×C₃₁	1	2	yes	—
220	#547	C₂²×C₃₁	1	2	yes	—
221	#500	C₂²×C₃₁	1	1	yes	characteristic
222	#493	—	2	1	no	—
223	#493	C₂²×C₃₁	1	1	yes	characteristic
224	#492	C₂²×C₃₁	1	2	yes	—
225	#486	C₂²×C₃₁	1	2	yes	—
226	#480	—	2	2	no	—
227	#474	C₂²×C₃₁	1	2	yes	—
228	#466	—	2	1	no	—
229	#465	—	2	2	no	—
230	#462	C₂²×C₃₁	1	2	yes	—
231	#459	—	2	2	no	—
232	#459	C₂²×C₃₁	1	2	yes	—
233	#459	C₂²×C₃₁	1	2	yes	—
234	#452	—	2	2	no	—
235	#449	—	2	2	no	—
236	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
238	C₄×D₈	—	2	1	no	—
239	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₈×D₄	C₂²×C₃₁	1	2	yes	—
242	C₂×#107	—	2	1	no	—
243	#335	C₂²×C₃₁	1	1	yes	characteristic
244	#334	C₂²×C₃₁	1	1	yes	characteristic
245	#332	C₂²×C₃₁	1	2	yes	—
246	#331	C₂²×C₃₁	1	2	yes	—
247	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
248	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
249	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₈²	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₃₁	—	4	2	no	—
252	C₂×C₃₁	—	8	2	no	—
253	C₂×C₃₁	—	16	2	no	—
254	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
255	C₂×C₃₁	#3284	1	1	yes	characteristic
256	C₂×C₃₁	#3290	1	1	yes	characteristic
257	C₂×#41	—	2	2	no	—
258	C₂×#41	—	2	2	no	—
259	#135	—	4	8	no	—
260	C₂×Q₁₆	—	2	2	no	—
261	C₂×SD₁₆	—	2	2	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
272	#127	—	2	2	no	—
273	#127	C₂³×C₃₁	1	2	yes	—
274	#123	—	4	2	no	—
275	C₂²⋊Q₈	—	2	2	no	—
276	#121	—	2	2	no	—
277	#121	—	2	2	no	—
278	#121	—	4	2	no	—
279	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₄×D₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	#107	—	2	1	no	—
285	#107	—	2	1	no	—
286	#107	—	2	2	no	—
287	#107	D₄×C₃₁	1	1	yes	characteristic
288	#106	D₄×C₃₁	1	1	yes	characteristic
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
291	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
293	#103	—	2	2	no	—
294	#103	D₄×C₃₁	1	1	yes	characteristic
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	4	2	no	—
299	#102	D₄×C₃₁	1	2	yes	—
300	#102	D₄×C₃₁	1	1	yes	characteristic
301	C₂²⋊C₈	—	2	2	no	—
302	C₂²⋊C₈	—	2	1	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₃₁	#19972	1	1	yes	characteristic, direct product factor, Sylow
308	#41	—	4	4	no	—
309	#41	—	4	4	no	—
310	#41	—	8	2	no	—
311	#41	—	8	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	—	4	2	no	—
326	Q₁₆	—	4	4	no	—
327	SD₁₆	—	4	4	no	—
328	SD₁₆	—	8	2	no	—
329	SD₁₆	D₈×C₃₁	1	4	yes	complemented
330	D₈	—	2	2	no	—
331	D₈	—	4	4	no	—
332	D₈	—	4	2	no	—
333	D₈	—	8	1	no	—
334	D₈	—	8	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	4	4	no	—
340	C₂×C₈	—	4	2	no	—
341	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	2	1	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
358	C₂³	—	2	1	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	4	2	no	—
361	Q₈	—	2	1	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	C₂×D₈×C₃₁	1	2	yes	—
365	D₄	—	2	2	no	—
366	D₄	—	2	1	no	—
367	D₄	—	2	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	8	4	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	4	no	—
375	D₄	—	16	2	no	—
376	D₄	C₂×D₈×C₃₁	1	2	yes	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	8	4	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	C₂×D₈×C₃₁	1	2	yes	—
398	C₈	C₃₁×#135	1	2	yes	complemented
399	C₂²	—	2	2	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	D₄²×C₃₁	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	2	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
416	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
417	C₄	C₃₁×#576	1	1	yes	characteristic
418	C₄	C₃₁×#577	1	1	yes	characteristic
419	C₂	—	4	2	no	—
420	C₂	—	8	2	no	—
421	C₂	—	16	2	no	—
422	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
423	C₂	C₃₁×#3284	1	1	yes	characteristic
424	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×#3391	C₂	semidirect
2. C₃₁×#3334	C₂	semidirect
3. C₃₁×#3284	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1674	C₂	semidirect
6. C₃₁×#1667	C₂	semidirect
7. C₃₁×#1665	C₂	semidirect
8. C₃₁×#1641	C₂	semidirect
9. C₃₁×#1625	C₂	semidirect
10. C₈×SD₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
116. SD₁₆×C₃₁	D₈	semidirect
145. #19972	C₃₁	direct (N × H)
186. C₈×C₃₁	#135	semidirect
187. #3391	C₂×C₃₁	semidirect
188. #3334	C₂×C₃₁	semidirect
189. #3284	C₂×C₃₁	semidirect
190. #1706	C₂×C₃₁	semidirect
191. #1674	C₂×C₃₁	semidirect
192. #1667	C₂×C₃₁	semidirect
193. #1665	C₂×C₃₁	semidirect
194. #1641	C₂×C₃₁	semidirect
195. #1625	C₂×C₃₁	semidirect
196. C₈×SD₁₆	C₂×C₃₁	semidirect
197. C₈×D₈	C₂×C₃₁	semidirect
307. C₃₁	#19972	direct (N × H)
329. SD₁₆	D₈×C₃₁	semidirect
398. C₈	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×#3391	2
C₃₁×#3334	2
C₃₁×#3284	2
C₃₁×#1706	2
C₃₁×#1674	2
C₃₁×#1667	2
C₃₁×#1665	2
C₃₁×#1641	2
C₃₁×#1625	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#19972	31

Subgroups — C31×#19972 #21476731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19972 #21476731