Subgroups — C₃₁×#19270 #21476029

Order 7936.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1672	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1671	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#498	—	2	1	no	—
15	C₃₁×#498	—	2	1	no	—
16	C₃₁×#498	C₂²	1	4	yes	—
17	C₃₁×#477	C₂²	1	2	yes	—
18	C₃₁×#476	C₂²	1	2	yes	—
19	C₃₁×#475	—	2	1	no	—
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#462	—	2	1	no	—
22	C₃₁×#462	C₂²	1	2	yes	—
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	C₂²	1	2	yes	—
25	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₈×D₄×C₃₁	—	2	2	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#428	—	2	1	no	—
31	C₂×C₃₁×#106	—	2	2	no	—
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₃₁×#423	C₂²	1	1	yes	characteristic
34	C₃₁×#419	—	2	1	no	—
35	C₃₁×#417	—	2	1	no	—
36	C₂×C₄×C₈×C₃₁	—	2	1	no	—
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	4	yes	—
39	C₃₁×#333	C₂²	1	4	yes	—
40	C₃₁×#332	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₃₁×#137	—	4	4	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	D₄	1	2	yes	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	—	2	1	no	—
53	C₃₁×C₄⋊Q₈	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×D₄×C₃₁	—	2	1	no	—
62	C₄×D₄×C₃₁	—	2	1	no	—
63	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
64	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#117	—	2	1	no	—
66	C₃₁×#117	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#106	—	2	4	no	—
75	C₃₁×#106	—	2	4	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	—	2	1	no	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
92	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	2	yes	—
94	#41×C₃₁	—	2	2	no	—
95	#41×C₃₁	SD₁₆	1	4	yes	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	4	1	no	—
99	C₂×Q₈×C₃₁	—	4	1	no	—
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂×Q₈×C₃₁	—	4	2	no	—
102	C₂×Q₈×C₃₁	—	8	2	no	—
103	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	4	1	no	—
110	C₂²×C₄×C₃₁	—	4	1	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
114	Q₁₆×C₃₁	—	4	4	no	—
115	SD₁₆×C₃₁	—	4	4	no	—
116	M₁₆×C₃₁	—	4	2	no	—
117	M₁₆×C₃₁	—	4	2	no	—
118	C₂×C₈×C₃₁	—	2	4	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	4	4	no	—
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₄⋊C₄×C₃₁	—	4	1	no	—
133	C₄⋊C₄×C₃₁	—	4	1	no	—
134	C₄⋊C₄×C₃₁	—	4	1	no	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	2	1	no	—
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₄²×C₃₁	—	2	1	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₃₁	#41	1	1	yes	characteristic
146	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
147	#19270	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	#121	1	1	yes	characteristic
150	Q₈×C₃₁	—	2	1	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	2	no	—
153	Q₈×C₃₁	—	8	2	no	—
154	Q₈×C₃₁	—	8	2	no	—
155	Q₈×C₃₁	—	8	2	no	—
156	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	1	no	—
159	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	8	2	no	—
171	C₂×C₄×C₃₁	—	8	2	no	—
172	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₃₁	#135	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	#137	1	4	yes	complemented
185	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#3079	C₂×C₃₁	1	1	yes	characteristic
187	#3039	C₂×C₃₁	1	1	yes	characteristic
188	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
189	#1672	C₂×C₃₁	1	4	yes	complemented
190	#1671	C₂×C₃₁	1	4	yes	complemented
191	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	4	1	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	#465	1	2	yes	—
200	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	#465	1	2	yes	—
209	C₄×C₃₁	#471	1	2	yes	—
210	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
211	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
212	C₄×C₃₁	C₂×#137	1	2	yes	—
213	C₂×#137	—	2	1	no	—
214	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
217	#498	—	2	1	no	—
218	#498	—	2	1	no	—
219	#498	C₂²×C₃₁	1	4	yes	—
220	#477	C₂²×C₃₁	1	2	yes	—
221	#476	C₂²×C₃₁	1	2	yes	—
222	#475	—	2	1	no	—
223	#474	—	2	1	no	—
224	#462	—	2	1	no	—
225	#462	C₂²×C₃₁	1	2	yes	—
226	#461	—	2	1	no	—
227	#461	C₂²×C₃₁	1	2	yes	—
228	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
229	C₈×D₄	—	2	2	no	—
230	C₈×D₄	C₂²×C₃₁	1	2	yes	—
231	#431	—	2	1	no	—
232	#431	C₂²×C₃₁	1	1	yes	characteristic
233	#428	—	2	1	no	—
234	C₂×#106	—	2	2	no	—
235	#423	C₂²×C₃₁	1	1	yes	characteristic
236	#423	C₂²×C₃₁	1	1	yes	characteristic
237	#419	—	2	1	no	—
238	#417	—	2	1	no	—
239	C₂×C₄×C₈	—	2	1	no	—
240	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	#334	C₂²×C₃₁	1	4	yes	—
242	#333	C₂²×C₃₁	1	4	yes	—
243	#332	C₂²×C₃₁	1	4	yes	—
244	C₂×C₃₁	—	2	2	no	—
245	C₂×C₃₁	—	4	2	no	—
246	C₂×C₃₁	#3039	1	1	yes	characteristic
247	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
248	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
249	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×Q₈	—	2	1	no	—
252	#137	—	4	4	no	—
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	1	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	D₄×C₃₁	1	2	yes	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
266	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
267	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
268	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×D₄	—	2	1	no	—
270	C₄×D₄	—	2	1	no	—
271	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
272	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
273	#117	—	2	1	no	—
274	#117	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
281	#107	—	2	2	no	—
282	#106	—	2	4	no	—
283	#106	—	2	4	no	—
284	#106	—	2	2	no	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
289	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
290	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#103	—	2	2	no	—
294	#102	—	2	2	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₄×C₈	—	2	2	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
300	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₃₁	1	2	yes	—
302	C₃₁	#19270	1	1	yes	characteristic, direct product factor, Sylow
303	#41	—	2	2	no	—
304	#41	SD₁₆×C₃₁	1	4	yes	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	4	1	no	—
308	C₂×Q₈	—	4	1	no	—
309	C₂×Q₈	—	4	2	no	—
310	C₂×Q₈	—	4	2	no	—
311	C₂×Q₈	—	8	2	no	—
312	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	—	4	1	no	—
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
323	Q₁₆	—	4	4	no	—
324	SD₁₆	—	4	4	no	—
325	M₁₆	—	4	2	no	—
326	M₁₆	—	4	2	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	4	4	no	—
332	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
333	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	2	1	no	—
352	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₄²	#41×C₃₁	1	1	yes	characteristic
355	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
356	C₂³	—	2	1	no	—
357	C₂³	C₃₁×#121	1	1	yes	characteristic
358	Q₈	—	2	1	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	8	2	no	—
362	Q₈	—	8	2	no	—
363	Q₈	—	8	2	no	—
364	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
365	D₄	—	2	4	no	—
366	D₄	—	2	1	no	—
367	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	8	2	no	—
379	C₂×C₄	—	8	2	no	—
380	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
381	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
382	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
383	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
384	C₂×C₄	C₃₁×#135	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
388	C₈	—	2	2	no	—
389	C₈	—	4	2	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	2	no	—
392	C₈	C₃₁×#137	1	4	yes	complemented
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	1	no	—
398	C₂²	C₃₁×#465	1	2	yes	—
399	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	2	no	—
402	C₄	—	2	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	2	no	—
407	C₄	C₃₁×#465	1	2	yes	—
408	C₄	C₃₁×#471	1	2	yes	—
409	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
410	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
411	C₄	C₂×C₃₁×#137	1	2	yes	—
412	C₂	—	2	2	no	—
413	C₂	—	4	2	no	—
414	C₂	C₃₁×#3039	1	1	yes	characteristic
415	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
416	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
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. C₂×C₃₁×#498	C₂	semidirect
5. C₃₁×#1672	C₂	semidirect
6. C₃₁×#1671	C₂	semidirect
7. C₃₁×#1566	C₂	semidirect
8. C₃₁×#1548	C₂	semidirect
9. C₃₁×#1546	C₂	semidirect
147. #19270	C₃₁	direct (N × H)
184. C₈×C₃₁	#137	semidirect
185. C₂×#498	C₂×C₃₁	semidirect
189. #1672	C₂×C₃₁	semidirect
190. #1671	C₂×C₃₁	semidirect
191. #1566	C₂×C₃₁	semidirect
192. #1548	C₂×C₃₁	semidirect
193. #1546	C₂×C₃₁	semidirect
302. C₃₁	#19270	direct (N × H)
392. C₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#498	2
C₃₁×#3079	2
C₃₁×#3039	2
C₈×#41×C₃₁	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1566	2
C₃₁×#1548	2
C₃₁×#1546	2
#19270	31

Subgroups — C31×#19270 #21476029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19270 #21476029