Subgroups — C₃₁×#19484 #21476243

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#486	C₂²	1	2	yes	—
18	C₃₁×#482	—	2	1	no	—
19	C₃₁×#477	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#476	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#460	—	2	1	no	—
22	C₃₁×#460	C₂²	1	2	yes	—
23	C₃₁×#460	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	—	2	1	no	—
26	C₃₁×#459	—	2	1	no	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#449	—	2	1	no	—
30	C₃₁×#447	—	2	2	no	—
31	C₃₁×#446	C₂²	1	2	yes	complemented
32	C₄×SD₁₆×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	—	2	1	no	—
34	C₃₁×#435	—	2	2	no	—
35	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
36	C₃₁×#432	—	2	1	no	—
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#431	C₂²	1	1	yes	characteristic
39	C₃₁×#423	C₂²	1	1	yes	characteristic
40	C₃₁×#423	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×#418	—	2	1	no	—
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#334	C₂²	1	2	yes	complemented
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	2	yes	—
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂×M₁₆×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	C₂³	1	2	yes	—
69	C₃₁×#127	C₂³	1	1	yes	characteristic
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#121	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
80	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#117	—	2	1	no	—
82	C₃₁×#117	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	1	no	—
86	C₃₁×#106	—	2	1	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
92	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
93	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
96	C₃₁×#103	—	2	1	no	—
97	C₃₁×#102	—	2	1	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#102	—	2	2	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	1	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	2	no	—
106	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
107	C₃₁×#97	—	2	2	no	—
108	C₄×C₈×C₃₁	—	2	1	no	—
109	C₄×C₈×C₃₁	C₂³	1	2	yes	—
110	C₂⁴×C₃₁	—	4	1	no	—
111	#41×C₃₁	—	2	2	no	—
112	#41×C₃₁	—	4	1	no	—
113	#41×C₃₁	—	8	2	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	4	2	no	—
124	C₂×D₄×C₃₁	—	4	2	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	4	1	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	—	8	1	no	—
130	C₂×D₄×C₃₁	—	8	1	no	—
131	C₂×D₄×C₃₁	—	8	2	no	—
132	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	4	1	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
140	SD₁₆×C₃₁	—	4	2	no	—
141	SD₁₆×C₃₁	—	4	4	no	—
142	D₈×C₃₁	—	4	2	no	—
143	D₈×C₃₁	—	4	4	no	—
144	D₈×C₃₁	—	8	2	no	—
145	D₈×C₃₁	—	8	2	no	—
146	M₁₆×C₃₁	—	4	2	no	—
147	M₁₆×C₃₁	—	8	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	8	2	no	—
153	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₃₁	—	2	1	no	—
162	C₂²⋊C₄×C₃₁	—	2	1	no	—
163	C₂²⋊C₄×C₃₁	—	2	1	no	—
164	C₂²⋊C₄×C₃₁	—	4	1	no	—
165	C₂²⋊C₄×C₃₁	—	4	1	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₃₁	#41	1	1	yes	characteristic
172	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
173	#19484	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	8	2	no	—
178	C₂³×C₃₁	—	8	2	no	—
179	C₂³×C₃₁	—	8	2	no	—
180	C₂³×C₃₁	#121	1	1	yes	characteristic
181	Q₈×C₃₁	—	2	1	no	—
182	Q₈×C₃₁	—	2	1	no	—
183	Q₈×C₃₁	—	8	1	no	—
184	D₄×C₃₁	—	2	1	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	1	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	1	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	1	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	8	2	no	—
195	D₄×C₃₁	—	8	1	no	—
196	D₄×C₃₁	—	8	2	no	—
197	D₄×C₃₁	—	8	2	no	—
198	D₄×C₃₁	—	8	2	no	—
199	D₄×C₃₁	—	8	2	no	—
200	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
217	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₈×C₃₁	—	2	2	no	—
222	C₈×C₃₁	—	2	2	no	—
223	C₈×C₃₁	—	4	2	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	2	no	—
227	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#3106	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#2990	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1698	C₂×C₃₁	1	2	yes	complemented
232	#1688	C₂×C₃₁	1	2	yes	complemented
233	#1677	C₂×C₃₁	1	2	yes	complemented
234	#1667	C₂×C₃₁	1	2	yes	complemented
235	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1560	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	1	no	—
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	—	8	1	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	—	8	1	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	16	4	no	—
251	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	4	1	no	—
258	C₄×C₃₁	—	8	1	no	—
259	C₄×C₃₁	#472	1	2	yes	—
260	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₄×C₃₁	#575	1	1	yes	characteristic
262	C₄×C₃₁	#576	1	1	yes	characteristic
263	C₄×C₃₁	#577	1	1	yes	characteristic
264	C₂×AGL(1,ℤ/8)	—	2	1	no	—
265	#532	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
267	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
268	#493	C₂²×C₃₁	1	2	yes	—
269	#486	C₂²×C₃₁	1	2	yes	—
270	#482	—	2	1	no	—
271	#477	C₂²×C₃₁	1	1	yes	characteristic, complemented
272	#476	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#460	—	2	1	no	—
274	#460	C₂²×C₃₁	1	2	yes	—
275	#460	C₂²×C₃₁	1	1	yes	characteristic, complemented
276	#459	—	2	1	no	—
277	#459	—	2	1	no	—
278	#459	—	2	1	no	—
279	#459	C₂²×C₃₁	1	2	yes	—
280	#459	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#449	—	2	1	no	—
282	#447	—	2	2	no	—
283	#446	C₂²×C₃₁	1	2	yes	complemented
284	C₄×SD₁₆	—	2	1	no	—
285	C₄×D₈	—	2	1	no	—
286	#435	—	2	2	no	—
287	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
288	#432	—	2	1	no	—
289	#431	C₂²×C₃₁	1	1	yes	characteristic
290	#431	C₂²×C₃₁	1	1	yes	characteristic
291	#423	C₂²×C₃₁	1	1	yes	characteristic
292	#423	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₈	—	2	1	no	—
294	#418	—	2	1	no	—
295	#335	C₂²×C₃₁	1	2	yes	complemented
296	#334	C₂²×C₃₁	1	2	yes	complemented
297	#333	C₂²×C₃₁	1	2	yes	complemented
298	#331	C₂²×C₃₁	1	2	yes	—
299	#331	C₂²×C₃₁	1	2	yes	—
300	#331	C₂²×C₃₁	1	2	yes	complemented
301	C₂×C₃₁	—	4	1	no	—
302	C₂×C₃₁	—	4	2	no	—
303	C₂×C₃₁	—	16	1	no	—
304	C₂×C₃₁	—	16	2	no	—
305	C₂×C₃₁	#3045	1	1	yes	characteristic
306	C₂×C₃₁	#3055	1	1	yes	characteristic
307	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
308	C₂×#41	—	2	1	no	—
309	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
310	C₂²×D₄	—	2	1	no	—
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	—	2	1	no	—
313	AGL(1,ℤ/8)	—	4	4	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	1	no	—
316	C₂×D₈	—	2	2	no	—
317	C₂×D₈	—	2	1	no	—
318	C₂×D₈	—	2	2	no	—
319	C₂×D₈	—	2	2	no	—
320	C₂×M₁₆	—	2	1	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
324	#127	—	2	1	no	—
325	#127	—	2	1	no	—
326	#127	—	2	1	no	—
327	#127	C₂³×C₃₁	1	2	yes	—
328	#127	C₂³×C₃₁	1	1	yes	characteristic
329	#124	—	2	1	no	—
330	#121	—	2	1	no	—
331	#121	—	2	1	no	—
332	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
339	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
340	#117	—	2	1	no	—
341	#117	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#107	—	2	1	no	—
345	#106	—	2	1	no	—
346	C₄⋊C₈	—	2	2	no	—
347	C₄⋊C₈	—	2	1	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
350	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
351	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
352	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
355	#103	—	2	1	no	—
356	#102	—	2	1	no	—
357	#102	—	2	2	no	—
358	#102	—	2	2	no	—
359	#102	—	2	2	no	—
360	#102	—	2	2	no	—
361	#102	—	2	1	no	—
362	#102	—	2	1	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	2	no	—
365	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
366	#97	—	2	2	no	—
367	C₄×C₈	—	2	1	no	—
368	C₄×C₈	C₂³×C₃₁	1	2	yes	—
369	C₃₁	#19484	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	—	4	1	no	—
371	#41	—	2	2	no	—
372	#41	—	4	1	no	—
373	#41	—	8	2	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	8	1	no	—
390	C₂×D₄	—	8	1	no	—
391	C₂×D₄	—	8	2	no	—
392	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
400	SD₁₆	—	4	2	no	—
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	2	no	—
403	D₈	—	4	4	no	—
404	D₈	—	8	2	no	—
405	D₈	—	8	2	no	—
406	M₁₆	—	4	2	no	—
407	M₁₆	—	8	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	8	2	no	—
413	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₄²	#41×C₃₁	1	1	yes	characteristic
432	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	—	8	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	C₃₁×#121	1	1	yes	characteristic
440	Q₈	—	2	1	no	—
441	Q₈	—	2	1	no	—
442	Q₈	—	8	1	no	—
443	D₄	—	2	1	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	1	no	—
447	D₄	—	4	1	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	1	no	—
452	D₄	—	4	2	no	—
453	D₄	—	8	2	no	—
454	D₄	—	8	1	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	—	8	2	no	—
459	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	8	2	no	—
474	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
475	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
476	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	2	no	—
482	C₈	—	4	2	no	—
483	C₈	—	4	1	no	—
484	C₈	—	4	1	no	—
485	C₈	—	4	2	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	1	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	16	4	no	—
499	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	1	no	—
506	C₄	—	8	1	no	—
507	C₄	C₃₁×#472	1	2	yes	—
508	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
509	C₄	C₃₁×#575	1	1	yes	characteristic
510	C₄	C₃₁×#576	1	1	yes	characteristic
511	C₄	C₃₁×#577	1	1	yes	characteristic
512	C₂	—	4	1	no	—
513	C₂	—	4	2	no	—
514	C₂	—	16	2	no	—
515	C₂	—	16	1	no	—
516	C₂	C₃₁×#3045	1	1	yes	characteristic
517	C₂	C₃₁×#3055	1	1	yes	characteristic
518	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3205	C₂	semidirect
2. C₃₁×#3106	C₂	semidirect
3. C₃₁×#3038	C₂	semidirect
4. C₃₁×#2990	C₂	semidirect
5. C₃₁×#1698	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1677	C₂	semidirect
8. C₃₁×#1667	C₂	semidirect
9. C₃₁×#1574	C₂	semidirect
10. C₃₁×#1560	C₂	semidirect
11. C₃₁×#1545	C₂	semidirect
19. C₃₁×#477	C₂²	semidirect
20. C₃₁×#476	C₂²	semidirect
23. C₃₁×#460	C₂²	semidirect
28. C₃₁×#459	C₂²	semidirect
31. C₃₁×#446	C₂²	semidirect
35. C₈×D₄×C₃₁	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
173. #19484	C₃₁	direct (N × H)
227. #3205	C₂×C₃₁	semidirect
228. #3106	C₂×C₃₁	semidirect
229. #3038	C₂×C₃₁	semidirect
230. #2990	C₂×C₃₁	semidirect
231. #1698	C₂×C₃₁	semidirect
232. #1688	C₂×C₃₁	semidirect
233. #1677	C₂×C₃₁	semidirect
234. #1667	C₂×C₃₁	semidirect
235. #1574	C₂×C₃₁	semidirect
236. #1560	C₂×C₃₁	semidirect
237. #1545	C₂×C₃₁	semidirect
271. #477	C₂²×C₃₁	semidirect
272. #476	C₂²×C₃₁	semidirect
275. #460	C₂²×C₃₁	semidirect
280. #459	C₂²×C₃₁	semidirect
283. #446	C₂²×C₃₁	semidirect
287. C₈×D₄	C₂²×C₃₁	semidirect
295. #335	C₂²×C₃₁	semidirect
296. #334	C₂²×C₃₁	semidirect
297. #333	C₂²×C₃₁	semidirect
300. #331	C₂²×C₃₁	semidirect
369. C₃₁	#19484	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3205	2
C₃₁×#3106	2
C₃₁×#3038	2
C₃₁×#2990	2
C₃₁×#1698	2
C₃₁×#1688	2
C₃₁×#1677	2
C₃₁×#1667	2
C₃₁×#1574	2
C₃₁×#1560	2
C₃₁×#1545	2
#19484	31

Subgroups — C31×#19484 #21476243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19484 #21476243