Subgroups — C₃₁×#17507 #21474266

Order 7936.

1284 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1492	C₂	1	3	yes	maximal, complemented
5	C₃₁×#1486	C₂	1	3	yes	maximal, complemented
6	C₃₁×#1480	C₂	1	3	yes	maximal, complemented
7	C₂×C₃₁×D₄⋊C₈	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₃₁	—	2	1	no	—
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₃₁×#550	C₄	1	1	yes	characteristic
11	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
12	C₃₁×#548	—	2	1	no	—
13	C₃₁×#547	C₄	1	3	yes	—
14	C₃₁×#546	—	2	3	no	—
15	D₄²×C₃₁	C₄	1	3	yes	—
16	C₃₁×#532	C₂²	1	3	yes	—
17	C₃₁×#525	—	2	3	no	—
18	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
20	C₈×Q₈×C₃₁	—	2	1	no	—
21	C₈×D₄×C₃₁	—	2	3	no	—
22	C₃₁×#431	C₂²	1	3	yes	complemented
23	C₃₁×#424	—	2	3	no	—
24	C₃₁×#423	C₂²	1	3	yes	complemented
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
26	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	complemented
27	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×D₄⋊C₈	C₂²	1	3	yes	—
29	C₃₁×D₄⋊C₈	C₂²	1	3	yes	complemented
30	C₃₁×D₄⋊C₈	C₂²	1	3	yes	complemented
31	C₃₁×D₄⋊C₈	C₂²	1	6	yes	complemented
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂×#41×C₃₁	—	2	3	no	—
34	C₂×#41×C₃₁	—	2	3	no	—
35	C₂×#41×C₃₁	—	4	4	no	—
36	C₂×#41×C₃₁	—	4	4	no	—
37	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×Q₈×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	3	no	—
40	C₂²×D₄×C₃₁	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₃₁	—	2	3	no	—
42	C₃₁×#131	—	4	4	no	—
43	C₂×M₁₆×C₃₁	—	2	3	no	—
44	C₂²×C₈×C₃₁	—	2	3	no	—
45	C₂²×C₈×C₃₁	—	2	3	no	—
46	C₃₁×C₄⋊Q₈	C₂×C₄	1	3	yes	—
47	C₃₁×#127	C₂×C₄	1	3	yes	—
48	C₃₁×#124	—	2	3	no	—
49	C₃₁×#123	—	2	6	no	—
50	C₃₁×C₂²⋊Q₈	—	2	3	no	—
51	C₃₁×C₂²⋊Q₈	—	2	3	no	—
52	C₃₁×#121	—	2	3	no	—
53	C₃₁×#121	—	2	3	no	—
54	C₃₁×#121	—	2	3	no	—
55	C₃₁×#121	C₂×C₄	1	6	yes	—
56	C₃₁×#120	—	2	6	no	—
57	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×D₄×C₃₁	—	2	3	no	—
60	C₄×D₄×C₃₁	—	2	3	no	—
61	C₄×D₄×C₃₁	—	2	3	no	—
62	C₄×D₄×C₃₁	C₂×C₄	1	3	yes	—
63	C₄×D₄×C₃₁	C₂×C₄	1	3	yes	—
64	C₄×D₄×C₃₁	C₂×C₄	1	3	yes	—
65	C₄×D₄×C₃₁	D₄	1	3	yes	—
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	3	yes	—
68	C₃₁×#117	D₄	1	3	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
71	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
72	C₃₁×C₄⋊C₈	—	2	3	no	—
73	C₃₁×C₄⋊C₈	—	2	3	no	—
74	C₃₁×C₄⋊C₈	—	2	3	no	—
75	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
77	C₃₁×C₂²⋊C₈	—	2	3	no	—
78	C₄×C₈×C₃₁	—	2	3	no	—
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	3	yes	—
81	C₂⁴×C₃₁	—	2	3	no	—
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	4	12	no	—
84	#41×C₃₁	—	4	6	no	—
85	#41×C₃₁	—	4	6	no	—
86	#41×C₃₁	D₈	1	4	yes	—
87	#41×C₃₁	SD₁₆	1	4	yes	—
88	C₂×Q₈×C₃₁	—	2	3	no	—
89	C₂×Q₈×C₃₁	—	4	4	no	—
90	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×D₄×C₃₁	—	2	6	no	—
92	C₂×D₄×C₃₁	—	2	6	no	—
93	C₂×D₄×C₃₁	—	2	3	no	—
94	C₂×D₄×C₃₁	—	2	3	no	—
95	C₂×D₄×C₃₁	—	2	3	no	—
96	C₂×D₄×C₃₁	—	4	3	no	—
97	C₂×D₄×C₃₁	—	4	12	no	—
98	C₂×D₄×C₃₁	C₂²⋊C₄	1	3	yes	—
99	C₂×D₄×C₃₁	C₂²×C₄	1	3	yes	—
100	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	—	2	3	no	—
102	C₂²×C₄×C₃₁	—	2	3	no	—
103	C₂²×C₄×C₃₁	—	2	3	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	3	no	—
106	C₂²×C₄×C₃₁	—	4	6	no	—
107	C₂²×C₄×C₃₁	—	4	6	no	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
111	M₁₆×C₃₁	—	4	6	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	3	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	3	no	—
116	C₂×C₈×C₃₁	—	4	6	no	—
117	C₂×C₈×C₃₁	—	4	6	no	—
118	C₄⋊C₄×C₃₁	—	2	3	no	—
119	C₄⋊C₄×C₃₁	—	2	3	no	—
120	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	3	yes	—
121	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	3	yes	—
122	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₃₁	C₂²×C₄	1	3	yes	—
124	C₂²⋊C₄×C₃₁	—	2	3	no	—
125	C₂²⋊C₄×C₃₁	—	2	3	no	—
126	C₂²⋊C₄×C₃₁	—	2	6	no	—
127	C₂²⋊C₄×C₃₁	—	2	1	no	—
128	C₂²⋊C₄×C₃₁	—	2	3	no	—
129	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
130	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
131	C₄²×C₃₁	C₂×D₄	1	3	yes	—
132	C₄²×C₃₁	C₂×D₄	1	3	yes	—
133	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
134	#17507	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	2	1	no	—
136	C₂³×C₃₁	—	2	3	no	—
137	C₂³×C₃₁	—	4	6	no	—
138	C₂³×C₃₁	—	4	6	no	—
139	C₂³×C₃₁	—	8	3	no	—
140	C₂³×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
141	Q₈×C₃₁	—	4	6	no	—
142	Q₈×C₃₁	#102	1	4	yes	—
143	D₄×C₃₁	—	2	3	no	—
144	D₄×C₃₁	—	2	1	no	—
145	D₄×C₃₁	—	2	3	no	—
146	D₄×C₃₁	—	4	6	no	—
147	D₄×C₃₁	—	4	12	no	—
148	D₄×C₃₁	—	4	6	no	—
149	D₄×C₃₁	—	8	3	no	—
150	D₄×C₃₁	#102	1	12	yes	—
151	D₄×C₃₁	#131	1	2	yes	complemented
152	C₂×C₄×C₃₁	—	2	3	no	—
153	C₂×C₄×C₃₁	—	2	3	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	3	no	—
156	C₂×C₄×C₃₁	—	4	3	no	—
157	C₂×C₄×C₃₁	—	4	6	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	3	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	3	no	—
162	C₂×C₄×C₃₁	—	4	6	no	—
163	C₂×C₄×C₃₁	—	4	6	no	—
164	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
165	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
166	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
167	C₂×C₄×C₃₁	C₂×D₈	1	6	yes	—
168	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
169	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	4	3	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	—	4	3	no	—
176	D₄×#41	C₂×C₃₁	1	1	yes	characteristic
177	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1492	C₂×C₃₁	1	3	yes	complemented
180	#1486	C₂×C₃₁	1	3	yes	complemented
181	#1480	C₂×C₃₁	1	3	yes	complemented
182	C₂×D₄⋊C₈	C₂×C₃₁	1	3	yes	complemented
183	C₂²×C₃₁	—	2	3	no	—
184	C₂²×C₃₁	—	2	3	no	—
185	C₂²×C₃₁	—	2	2	no	—
186	C₂²×C₃₁	—	4	1	no	—
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	3	no	—
189	C₂²×C₃₁	—	4	3	no	—
190	C₂²×C₃₁	—	4	3	no	—
191	C₂²×C₃₁	—	8	12	no	—
192	C₂²×C₃₁	C₂×#102	1	6	yes	—
193	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
194	C₄×C₃₁	—	2	3	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	3	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	3	no	—
199	C₄×C₃₁	C₂×#102	1	6	yes	—
200	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
201	C₄×C₃₁	#568	1	1	yes	characteristic
202	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
204	C₂²×#41	—	2	1	no	—
205	C₂×#131	—	2	1	no	—
206	#550	C₄×C₃₁	1	1	yes	characteristic
207	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
208	#548	—	2	1	no	—
209	#547	C₄×C₃₁	1	3	yes	—
210	#546	—	2	3	no	—
211	D₄²	C₄×C₃₁	1	3	yes	—
212	#532	C₂²×C₃₁	1	3	yes	—
213	#525	—	2	3	no	—
214	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
216	C₈×Q₈	—	2	1	no	—
217	C₈×D₄	—	2	3	no	—
218	#431	C₂²×C₃₁	1	3	yes	complemented
219	#424	—	2	3	no	—
220	#423	C₂²×C₃₁	1	3	yes	complemented
221	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
222	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	complemented
223	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
224	D₄⋊C₈	C₂²×C₃₁	1	3	yes	—
225	D₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
226	D₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
227	D₄⋊C₈	C₂²×C₃₁	1	6	yes	complemented
228	C₂×C₃₁	—	2	6	no	—
229	C₂×C₃₁	—	4	2	no	—
230	C₂×C₃₁	—	8	3	no	—
231	C₂×C₃₁	#2880	1	1	yes	characteristic
232	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
233	C₂×C₃₁	#2901	1	1	yes	characteristic
234	C₂×#41	—	2	1	no	—
235	C₂×#41	—	2	3	no	—
236	C₂×#41	—	2	3	no	—
237	C₂×#41	—	4	4	no	—
238	C₂×#41	—	4	4	no	—
239	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×D₄	—	2	3	no	—
242	C₂²×D₄	C₂×C₄×C₃₁	1	3	yes	—
243	C₂³×C₄	—	2	3	no	—
244	#131	—	4	4	no	—
245	C₂×M₁₆	—	2	3	no	—
246	C₂²×C₈	—	2	3	no	—
247	C₂²×C₈	—	2	3	no	—
248	C₄⋊Q₈	C₂×C₄×C₃₁	1	3	yes	—
249	#127	C₂×C₄×C₃₁	1	3	yes	—
250	#124	—	2	3	no	—
251	#123	—	2	6	no	—
252	C₂²⋊Q₈	—	2	3	no	—
253	C₂²⋊Q₈	—	2	3	no	—
254	#121	—	2	3	no	—
255	#121	—	2	3	no	—
256	#121	—	2	3	no	—
257	#121	C₂×C₄×C₃₁	1	6	yes	—
258	#120	—	2	6	no	—
259	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
261	C₄×D₄	—	2	3	no	—
262	C₄×D₄	—	2	3	no	—
263	C₄×D₄	—	2	3	no	—
264	C₄×D₄	C₂×C₄×C₃₁	1	3	yes	—
265	C₄×D₄	C₂×C₄×C₃₁	1	3	yes	—
266	C₄×D₄	C₂×C₄×C₃₁	1	3	yes	—
267	C₄×D₄	D₄×C₃₁	1	3	yes	—
268	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×D₄	C₂³×C₃₁	1	3	yes	—
270	#117	D₄×C₃₁	1	3	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
272	C₂×C₂²⋊C₄	—	2	3	no	—
273	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
274	C₄⋊C₈	—	2	3	no	—
275	C₄⋊C₈	—	2	3	no	—
276	C₄⋊C₈	—	2	3	no	—
277	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
279	C₂²⋊C₈	—	2	3	no	—
280	C₄×C₈	—	2	3	no	—
281	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄×C₈	C₂³×C₃₁	1	3	yes	—
283	C₃₁	#17507	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁴	—	2	3	no	—
285	#41	—	4	4	no	—
286	#41	—	4	12	no	—
287	#41	—	4	6	no	—
288	#41	—	4	6	no	—
289	#41	D₈×C₃₁	1	4	yes	—
290	#41	SD₁₆×C₃₁	1	4	yes	—
291	C₂×Q₈	—	2	3	no	—
292	C₂×Q₈	—	4	4	no	—
293	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
294	C₂×D₄	—	2	6	no	—
295	C₂×D₄	—	2	6	no	—
296	C₂×D₄	—	2	3	no	—
297	C₂×D₄	—	2	3	no	—
298	C₂×D₄	—	2	3	no	—
299	C₂×D₄	—	4	3	no	—
300	C₂×D₄	—	4	12	no	—
301	C₂×D₄	C₂²⋊C₄×C₃₁	1	3	yes	—
302	C₂×D₄	C₂²×C₄×C₃₁	1	3	yes	—
303	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	—	2	3	no	—
305	C₂²×C₄	—	2	3	no	—
306	C₂²×C₄	—	2	3	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	3	no	—
309	C₂²×C₄	—	4	6	no	—
310	C₂²×C₄	—	4	6	no	—
311	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
314	M₁₆	—	4	6	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	3	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	3	no	—
319	C₂×C₈	—	4	6	no	—
320	C₂×C₈	—	4	6	no	—
321	C₄⋊C₄	—	2	3	no	—
322	C₄⋊C₄	—	2	3	no	—
323	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	3	yes	—
324	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	3	yes	—
325	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₄⋊C₄	C₂²×C₄×C₃₁	1	3	yes	—
327	C₂²⋊C₄	—	2	3	no	—
328	C₂²⋊C₄	—	2	3	no	—
329	C₂²⋊C₄	—	2	6	no	—
330	C₂²⋊C₄	—	2	1	no	—
331	C₂²⋊C₄	—	2	3	no	—
332	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
333	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
334	C₄²	C₂×D₄×C₃₁	1	3	yes	—
335	C₄²	C₂×D₄×C₃₁	1	3	yes	—
336	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
337	C₂³	—	2	1	no	—
338	C₂³	—	2	3	no	—
339	C₂³	—	4	6	no	—
340	C₂³	—	4	6	no	—
341	C₂³	—	8	3	no	—
342	C₂³	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
343	Q₈	—	4	6	no	—
344	Q₈	C₃₁×#102	1	4	yes	—
345	D₄	—	2	3	no	—
346	D₄	—	2	1	no	—
347	D₄	—	2	3	no	—
348	D₄	—	4	6	no	—
349	D₄	—	4	12	no	—
350	D₄	—	4	6	no	—
351	D₄	—	8	3	no	—
352	D₄	C₃₁×#102	1	12	yes	—
353	D₄	C₃₁×#131	1	2	yes	complemented
354	C₂×C₄	—	2	3	no	—
355	C₂×C₄	—	2	3	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	3	no	—
358	C₂×C₄	—	4	3	no	—
359	C₂×C₄	—	4	6	no	—
360	C₂×C₄	—	4	6	no	—
361	C₂×C₄	—	4	6	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	3	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	3	no	—
366	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
367	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
369	C₂×C₄	C₂×D₈×C₃₁	1	6	yes	—
370	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
371	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	4	3	no	—
376	C₈	—	4	1	no	—
377	C₈	—	4	3	no	—
378	C₂²	—	2	3	no	—
379	C₂²	—	2	3	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	3	no	—
384	C₂²	—	4	3	no	—
385	C₂²	—	4	3	no	—
386	C₂²	—	8	12	no	—
387	C₂²	C₂×C₃₁×#102	1	6	yes	—
388	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
389	C₄	—	2	3	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	3	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	3	no	—
394	C₄	C₂×C₃₁×#102	1	6	yes	—
395	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
396	C₄	C₃₁×#568	1	1	yes	characteristic
397	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
398	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
399	C₂	—	2	6	no	—
400	C₂	—	4	2	no	—
401	C₂	—	8	3	no	—
402	C₂	C₃₁×#2880	1	1	yes	characteristic
403	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic
404	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	D₄×#41×C₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×#41×C₃₁	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1492	C₂	semidirect
5. C₃₁×#1486	C₂	semidirect
6. C₃₁×#1480	C₂	semidirect
7. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
22. C₃₁×#431	C₂²	semidirect
24. C₃₁×#423	C₂²	semidirect
25. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
26. C₂×C₄×C₈×C₃₁	C₂²	semidirect
29. C₃₁×D₄⋊C₈	C₂²	semidirect
30. C₃₁×D₄⋊C₈	C₂²	semidirect
31. C₃₁×D₄⋊C₈	C₂²	semidirect
134. #17507	C₃₁	direct (N × H)
151. D₄×C₃₁	#131	semidirect
177. C₈×#41	C₂×C₃₁	semidirect
178. #2906	C₂×C₃₁	semidirect
179. #1492	C₂×C₃₁	semidirect
180. #1486	C₂×C₃₁	semidirect
181. #1480	C₂×C₃₁	semidirect
182. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
218. #431	C₂²×C₃₁	semidirect
220. #423	C₂²×C₃₁	semidirect
221. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
222. C₂×C₄×C₈	C₂²×C₃₁	semidirect
225. D₄⋊C₈	C₂²×C₃₁	semidirect
226. D₄⋊C₈	C₂²×C₃₁	semidirect
227. D₄⋊C₈	C₂²×C₃₁	semidirect
283. C₃₁	#17507	direct (N × H)
353. D₄	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
D₄×#41×C₃₁	2
C₈×#41×C₃₁	2
C₃₁×#2906	2
C₃₁×#1492	2
C₃₁×#1486	2
C₃₁×#1480	2
C₂×C₃₁×D₄⋊C₈	2
#17507	31

Subgroups — C31×#17507 #21474266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17507 #21474266