Subgroups — C₃₁×#19128 #21475887

Order 7936.

664 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3106	C₂	1	2	yes	maximal, complemented
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1543	C₂	1	2	yes	maximal, complemented
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	1	yes	characteristic
16	C₃₁×#486	C₂²	1	2	yes	—
17	C₃₁×#482	—	2	2	no	—
18	C₃₁×#477	C₂²	1	2	yes	—
19	C₃₁×#476	C₂²	1	2	yes	—
20	C₃₁×#466	—	2	1	no	—
21	C₃₁×#466	—	2	1	no	—
22	C₃₁×#460	C₂²	1	2	yes	—
23	C₃₁×#459	C₂²	1	2	yes	—
24	C₃₁×#449	—	2	2	no	—
25	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₃₁	—	2	2	no	—
28	C₄×D₈×C₃₁	—	2	2	no	—
29	C₄×D₈×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#431	—	2	1	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₃₁	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#335	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂²×C₈×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#127	C₂³	1	2	yes	—
59	C₃₁×#124	—	2	2	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	—	2	1	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	1	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₄×C₈×C₃₁	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	#41×C₃₁	—	2	2	no	—
103	#41×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	8	4	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
120	SD₁₆×C₃₁	—	4	4	no	—
121	D₈×C₃₁	—	4	2	no	—
122	D₈×C₃₁	—	4	4	no	—
123	D₈×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	4	1	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	#41	1	1	yes	characteristic
153	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
154	#19128	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	#121	1	1	yes	characteristic
158	Q₈×C₃₁	—	2	1	no	—
159	Q₈×C₃₁	—	8	2	no	—
160	Q₈×C₃₁	#135	1	2	yes	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	2	1	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	8	4	no	—
169	D₄×C₃₁	#135	1	2	yes	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	8	4	no	—
184	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
186	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	2	1	no	—
191	C₈×C₃₁	—	2	1	no	—
192	C₈×C₃₁	—	2	1	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₈×C₃₁	—	4	2	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	#135	1	2	yes	—
198	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#3106	C₂×C₃₁	1	2	yes	complemented
200	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1678	C₂×C₃₁	1	2	yes	complemented
202	#1677	C₂×C₃₁	1	2	yes	complemented
203	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1543	C₂×C₃₁	1	2	yes	complemented
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	2	1	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	4	1	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	#472	1	2	yes	—
227	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
228	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
229	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
230	C₄×C₃₁	#575	1	1	yes	characteristic
231	#532	C₂²×C₃₁	1	2	yes	—
232	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
233	#500	C₂²×C₃₁	1	1	yes	characteristic
234	#493	C₂²×C₃₁	1	1	yes	characteristic
235	#486	C₂²×C₃₁	1	2	yes	—
236	#482	—	2	2	no	—
237	#477	C₂²×C₃₁	1	2	yes	—
238	#476	C₂²×C₃₁	1	2	yes	—
239	#466	—	2	1	no	—
240	#466	—	2	1	no	—
241	#460	C₂²×C₃₁	1	2	yes	—
242	#459	C₂²×C₃₁	1	2	yes	—
243	#449	—	2	2	no	—
244	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₄×SD₁₆	—	2	2	no	—
247	C₄×D₈	—	2	2	no	—
248	C₄×D₈	—	2	1	no	—
249	C₄×D₈	—	2	1	no	—
250	C₈×D₄	—	2	1	no	—
251	C₈×D₄	—	2	1	no	—
252	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
253	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
254	#431	—	2	1	no	—
255	#431	C₂²×C₃₁	1	1	yes	characteristic
256	#431	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×C₈	—	2	1	no	—
258	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	#335	C₂²×C₃₁	1	1	yes	characteristic
260	#335	C₂²×C₃₁	1	1	yes	characteristic
261	#334	C₂²×C₃₁	1	2	yes	—
262	#333	C₂²×C₃₁	1	2	yes	—
263	#333	C₂²×C₃₁	1	2	yes	—
264	#331	C₂²×C₃₁	1	2	yes	—
265	#331	C₂²×C₃₁	1	2	yes	—
266	C₂×C₃₁	—	4	1	no	—
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	—	16	2	no	—
269	C₂×C₃₁	#3035	1	1	yes	characteristic
270	C₂×C₃₁	#3050	1	1	yes	characteristic
271	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
272	C₂×#41	—	2	2	no	—
273	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
274	C₂×SD₁₆	—	2	2	no	—
275	C₂×D₈	—	2	2	no	—
276	C₂×D₈	—	2	1	no	—
277	C₂×D₈	—	2	1	no	—
278	C₂²×C₈	—	2	1	no	—
279	C₂²×C₈	—	2	1	no	—
280	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
282	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
283	#127	C₂³×C₃₁	1	2	yes	—
284	#124	—	2	2	no	—
285	#121	—	2	2	no	—
286	#121	—	2	2	no	—
287	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
288	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄×D₄	—	2	1	no	—
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	—	2	2	no	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
294	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
295	#117	—	2	1	no	—
296	#117	D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	#107	—	2	1	no	—
300	#107	—	2	1	no	—
301	#107	—	2	2	no	—
302	#106	—	2	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
308	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
309	#103	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	—	2	1	no	—
318	C₂²⋊C₈	—	2	1	no	—
319	C₄×C₈	—	2	1	no	—
320	C₄×C₈	—	2	1	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
324	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₃₁	#19128	1	1	yes	characteristic, direct product factor, Sylow
328	#41	—	2	2	no	—
329	#41	—	2	2	no	—
330	#41	—	8	4	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
346	SD₁₆	—	4	4	no	—
347	D₈	—	4	2	no	—
348	D₈	—	4	4	no	—
349	D₈	—	4	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	4	1	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₄²	#41×C₃₁	1	1	yes	characteristic
379	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₂³	—	2	1	no	—
381	C₂³	—	4	2	no	—
382	C₂³	C₃₁×#121	1	1	yes	characteristic
383	Q₈	—	2	1	no	—
384	Q₈	—	8	2	no	—
385	Q₈	C₃₁×#135	1	2	yes	—
386	D₄	—	2	4	no	—
387	D₄	—	2	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	4	2	no	—
392	D₄	—	8	2	no	—
393	D₄	—	8	4	no	—
394	D₄	C₃₁×#135	1	2	yes	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	8	4	no	—
409	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
410	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
411	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
415	C₈	—	2	1	no	—
416	C₈	—	2	1	no	—
417	C₈	—	2	1	no	—
418	C₈	—	4	1	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	1	no	—
422	C₈	C₃₁×#135	1	2	yes	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	8	2	no	—
432	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	8	2	no	—
440	C₄	C₃₁×#472	1	2	yes	—
441	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
442	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
443	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
444	C₄	C₃₁×#575	1	1	yes	characteristic
445	C₂	—	4	1	no	—
446	C₂	—	4	2	no	—
447	C₂	—	16	2	no	—
448	C₂	C₃₁×#3035	1	1	yes	characteristic
449	C₂	C₃₁×#3050	1	1	yes	characteristic
450	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
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×#3169	C₂	semidirect
2. C₃₁×#3106	C₂	semidirect
3. C₈×#41×C₃₁	C₂	semidirect
4. C₃₁×#1678	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1675	C₂	semidirect
8. C₃₁×#1668	C₂	semidirect
9. C₃₁×#1667	C₂	semidirect
10. C₃₁×#1564	C₂	semidirect
11. C₃₁×#1543	C₂	semidirect
154. #19128	C₃₁	direct (N × H)
198. #3169	C₂×C₃₁	semidirect
199. #3106	C₂×C₃₁	semidirect
200. C₈×#41	C₂×C₃₁	semidirect
201. #1678	C₂×C₃₁	semidirect
202. #1677	C₂×C₃₁	semidirect
203. #1676	C₂×C₃₁	semidirect
204. #1675	C₂×C₃₁	semidirect
205. #1668	C₂×C₃₁	semidirect
206. #1667	C₂×C₃₁	semidirect
207. #1564	C₂×C₃₁	semidirect
208. #1543	C₂×C₃₁	semidirect
327. C₃₁	#19128	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×#3169	2
C₃₁×#3106	2
C₈×#41×C₃₁	2
C₃₁×#1678	2
C₃₁×#1677	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1668	2
C₃₁×#1667	2
C₃₁×#1564	2
C₃₁×#1543	2
#19128	31

Subgroups — C31×#19128 #21475887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19128 #21475887