Subgroups — C₃₁×#19110 #21475869

Order 7936.

716 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1679	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	—	2	1	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#488	—	2	1	no	—
17	C₃₁×#488	—	2	1	no	—
18	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#483	—	2	1	no	—
23	C₃₁×#480	—	2	1	no	—
24	C₃₁×#476	C₂²	1	2	yes	—
25	C₃₁×#474	C₂²	1	2	yes	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#450	—	2	1	no	—
29	C₃₁×#447	—	2	1	no	—
30	C₄×SD₁₆×C₃₁	—	2	2	no	—
31	C₄×D₈×C₃₁	—	2	2	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₈×D₄×C₃₁	—	2	1	no	—
34	C₈×D₄×C₃₁	C₂²	1	2	yes	—
35	C₈×D₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#416	—	2	1	no	—
40	C₃₁×#416	—	2	1	no	—
41	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
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×#332	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	4	4	no	—
58	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#127	C₂³	1	1	yes	characteristic
64	C₃₁×#125	—	2	1	no	—
65	C₃₁×#124	—	2	1	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₄×Q₈×C₃₁	—	2	1	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	D₄	1	2	yes	—
77	C₄×D₄×C₃₁	C₂³	1	2	yes	—
78	C₃₁×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#107	—	2	2	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	2	1	no	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
88	C₃₁×#103	—	2	1	no	—
89	C₃₁×#103	—	2	1	no	—
90	C₃₁×#103	—	2	1	no	—
91	C₃₁×#103	—	2	1	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	1	no	—
96	C₃₁×#102	—	2	1	no	—
97	C₃₁×#102	—	2	1	no	—
98	C₃₁×#102	—	2	1	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×C₂²⋊C₈	—	2	1	no	—
101	C₃₁×C₂²⋊C₈	—	2	1	no	—
102	C₃₁×C₂²⋊C₈	—	2	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	2	no	—
104	C₄×C₈×C₃₁	—	2	1	no	—
105	C₄×C₈×C₃₁	—	2	1	no	—
106	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
107	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
110	C₂⁴×C₃₁	—	2	1	no	—
111	#41×C₃₁	—	4	2	no	—
112	#41×C₃₁	—	4	2	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	4	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	4	1	no	—
122	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
132	SD₁₆×C₃₁	—	4	4	no	—
133	D₈×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	4	4	no	—
140	C₂×C₈×C₃₁	—	4	4	no	—
141	C₂×C₈×C₃₁	—	4	2	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	4	1	no	—
154	C₄⋊C₄×C₃₁	—	4	1	no	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²⋊C₄×C₃₁	—	2	2	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	1	no	—
159	C₂²⋊C₄×C₃₁	—	4	1	no	—
160	C₂²⋊C₄×C₃₁	—	4	1	no	—
161	C₄²×C₃₁	—	4	1	no	—
162	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₃₁	#41	1	1	yes	characteristic
165	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
166	#19110	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	4	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	#121	1	1	yes	characteristic
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	4	1	no	—
175	Q₈×C₃₁	—	4	1	no	—
176	Q₈×C₃₁	—	8	1	no	—
177	D₄×C₃₁	—	2	4	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	1	no	—
181	D₄×C₃₁	—	4	1	no	—
182	D₄×C₃₁	—	8	1	no	—
183	D₄×C₃₁	—	8	2	no	—
184	D₄×C₃₁	#135	1	4	yes	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	1	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	4	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	8	2	no	—
196	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
197	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
198	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
199	C₂×C₄×C₃₁	#135	1	2	yes	—
200	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₈×C₃₁	—	2	1	no	—
205	C₈×C₃₁	—	2	1	no	—
206	C₈×C₃₁	—	2	1	no	—
207	C₈×C₃₁	—	2	1	no	—
208	C₈×C₃₁	—	4	2	no	—
209	C₈×C₃₁	—	4	2	no	—
210	C₈×C₃₁	—	4	2	no	—
211	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#3100	C₂×C₃₁	1	1	yes	characteristic
214	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1683	C₂×C₃₁	1	2	yes	complemented
216	#1681	C₂×C₃₁	1	2	yes	complemented
217	#1679	C₂×C₃₁	1	2	yes	complemented
218	#1677	C₂×C₃₁	1	2	yes	complemented
219	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₃₁	—	2	1	no	—
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	2	4	no	—
225	C₂²×C₃₁	—	4	8	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	8	1	no	—
231	C₂²×C₃₁	#463	1	2	yes	—
232	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	2	1	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	8	1	no	—
238	C₄×C₃₁	—	8	1	no	—
239	C₄×C₃₁	—	8	1	no	—
240	C₄×C₃₁	#463	1	2	yes	—
241	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
242	C₄×C₃₁	C₂×#135	1	2	yes	—
243	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂³×C₈	—	2	1	no	—
245	#533	C₂²×C₃₁	1	1	yes	characteristic
246	#532	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
248	#488	—	2	1	no	—
249	#488	—	2	1	no	—
250	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
251	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
253	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	#483	—	2	1	no	—
255	#480	—	2	1	no	—
256	#476	C₂²×C₃₁	1	2	yes	—
257	#474	C₂²×C₃₁	1	2	yes	—
258	#461	C₂²×C₃₁	1	2	yes	—
259	#459	C₂²×C₃₁	1	2	yes	—
260	#450	—	2	1	no	—
261	#447	—	2	1	no	—
262	C₄×SD₁₆	—	2	2	no	—
263	C₄×D₈	—	2	2	no	—
264	C₈×D₄	—	2	1	no	—
265	C₈×D₄	—	2	1	no	—
266	C₈×D₄	C₂²×C₃₁	1	2	yes	—
267	C₈×D₄	C₂²×C₃₁	1	2	yes	—
268	#431	C₂²×C₃₁	1	1	yes	characteristic
269	#431	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	#416	—	2	1	no	—
272	#416	—	2	1	no	—
273	C₂×C₂²⋊C₈	—	2	1	no	—
274	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	#335	C₂²×C₃₁	1	2	yes	complemented
276	#334	C₂²×C₃₁	1	2	yes	complemented
277	#333	C₂²×C₃₁	1	2	yes	complemented
278	#332	C₂²×C₃₁	1	2	yes	complemented
279	#332	C₂²×C₃₁	1	2	yes	complemented
280	#331	C₂²×C₃₁	1	2	yes	complemented
281	C₂×C₃₁	—	2	2	no	—
282	C₂×C₃₁	—	4	4	no	—
283	C₂×C₃₁	—	16	1	no	—
284	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
285	C₂×C₃₁	#3050	1	1	yes	characteristic
286	C₂×C₃₁	#3067	1	1	yes	characteristic
287	C₂×#41	—	2	1	no	—
288	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
289	C₂³×C₄	—	2	1	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	—	4	4	no	—
296	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
298	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
301	#127	C₂³×C₃₁	1	1	yes	characteristic
302	#125	—	2	1	no	—
303	#124	—	2	1	no	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	#121	—	2	1	no	—
307	#121	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	D₄×C₃₁	1	2	yes	—
315	C₄×D₄	C₂³×C₃₁	1	2	yes	—
316	#117	—	2	1	no	—
317	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#107	—	2	2	no	—
321	#106	—	2	2	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
324	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
325	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
326	#103	—	2	1	no	—
327	#103	—	2	1	no	—
328	#103	—	2	1	no	—
329	#103	—	2	1	no	—
330	#103	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	—	2	1	no	—
334	#102	—	2	1	no	—
335	#102	—	2	1	no	—
336	#102	—	2	1	no	—
337	#102	—	2	2	no	—
338	C₂²⋊C₈	—	2	1	no	—
339	C₂²⋊C₈	—	2	1	no	—
340	C₂²⋊C₈	—	2	2	no	—
341	C₂²⋊C₈	—	2	2	no	—
342	C₄×C₈	—	2	1	no	—
343	C₄×C₈	—	2	1	no	—
344	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
345	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
346	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₃₁	#19110	1	1	yes	characteristic, direct product factor, Sylow
349	C₂⁴	—	2	1	no	—
350	#41	—	4	2	no	—
351	#41	—	4	2	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×D₄	—	2	4	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	1	no	—
368	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
371	SD₁₆	—	4	4	no	—
372	D₈	—	4	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	4	4	no	—
379	C₂×C₈	—	4	4	no	—
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	4	1	no	—
393	C₄⋊C₄	—	4	1	no	—
394	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	4	1	no	—
398	C₂²⋊C₄	—	4	1	no	—
399	C₂²⋊C₄	—	4	1	no	—
400	C₄²	—	4	1	no	—
401	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₄²	#41×C₃₁	1	1	yes	characteristic
404	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂³	—	2	4	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	1	no	—
410	C₂³	C₃₁×#121	1	1	yes	characteristic
411	Q₈	—	4	2	no	—
412	Q₈	—	4	1	no	—
413	Q₈	—	4	1	no	—
414	Q₈	—	8	1	no	—
415	D₄	—	2	4	no	—
416	D₄	—	2	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	1	no	—
419	D₄	—	4	1	no	—
420	D₄	—	8	1	no	—
421	D₄	—	8	2	no	—
422	D₄	C₃₁×#135	1	4	yes	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	8	2	no	—
434	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
435	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
436	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
437	C₂×C₄	C₃₁×#135	1	2	yes	—
438	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
442	C₈	—	2	1	no	—
443	C₈	—	2	1	no	—
444	C₈	—	2	1	no	—
445	C₈	—	2	1	no	—
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	8	1	no	—
458	C₂²	C₃₁×#463	1	2	yes	—
459	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
460	C₄	—	2	1	no	—
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	8	1	no	—
465	C₄	—	8	1	no	—
466	C₄	—	8	1	no	—
467	C₄	C₃₁×#463	1	2	yes	—
468	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
469	C₄	C₂×C₃₁×#135	1	2	yes	—
470	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
471	C₂	—	2	2	no	—
472	C₂	—	4	4	no	—
473	C₂	—	16	1	no	—
474	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
475	C₂	C₃₁×#3050	1	1	yes	characteristic
476	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₃₁×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3131	C₂	semidirect
2. C₃₁×#3101	C₂	semidirect
4. C₂×C₈×D₄×C₃₁	C₂	semidirect
5. C₃₁×#1683	C₂	semidirect
6. C₃₁×#1681	C₂	semidirect
7. C₃₁×#1679	C₂	semidirect
8. C₃₁×#1677	C₂	semidirect
9. C₃₁×#1564	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₃₁×#1540	C₂	semidirect
18. C₃₁×#487	C₂²	semidirect
19. C₃₁×#487	C₂²	semidirect
20. C₃₁×#486	C₂²	semidirect
21. C₃₁×#486	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
166. #19110	C₃₁	direct (N × H)
211. #3131	C₂×C₃₁	semidirect
212. #3101	C₂×C₃₁	semidirect
214. C₂×C₈×D₄	C₂×C₃₁	semidirect
215. #1683	C₂×C₃₁	semidirect
216. #1681	C₂×C₃₁	semidirect
217. #1679	C₂×C₃₁	semidirect
218. #1677	C₂×C₃₁	semidirect
219. #1564	C₂×C₃₁	semidirect
220. #1542	C₂×C₃₁	semidirect
221. #1540	C₂×C₃₁	semidirect
250. #487	C₂²×C₃₁	semidirect
251. #487	C₂²×C₃₁	semidirect
252. #486	C₂²×C₃₁	semidirect
253. #486	C₂²×C₃₁	semidirect
275. #335	C₂²×C₃₁	semidirect
276. #334	C₂²×C₃₁	semidirect
277. #333	C₂²×C₃₁	semidirect
278. #332	C₂²×C₃₁	semidirect
279. #332	C₂²×C₃₁	semidirect
280. #331	C₂²×C₃₁	semidirect
348. C₃₁	#19110	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×#3131	2
C₃₁×#3101	2
C₃₁×#3100	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1683	2
C₃₁×#1681	2
C₃₁×#1679	2
C₃₁×#1677	2
C₃₁×#1564	2
C₃₁×#1542	2
C₃₁×#1540	2
#19110	31

Subgroups — C31×#19110 #21475869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19110 #21475869