Subgroups — C₃₁×#26271 #21483030

Order 7936.

752 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2951	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2052	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2038	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1882	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#536	C₂²	1	2	yes	—
16	C₃₁×#520	C₂²	1	2	yes	—
17	C₃₁×#497	—	2	2	no	—
18	C₃₁×#495	—	2	1	no	—
19	C₃₁×#494	—	2	1	no	—
20	C₃₁×#485	—	2	2	no	—
21	C₃₁×#485	—	2	2	no	—
22	C₃₁×#484	—	2	2	no	—
23	C₃₁×#483	—	2	2	no	—
24	C₃₁×#440	—	2	2	no	—
25	C₄×Q₁₆×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₃₁×#419	C₂²	1	2	yes	—
34	C₃₁×#417	C₂²	1	2	yes	—
35	C₂×C₃₁×#103	C₂²	1	2	yes	—
36	C₂×C₃₁×#101	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₃₁×#405	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#391	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#137	—	2	4	no	—
48	C₃₁×#137	—	4	2	no	—
49	C₃₁×#135	—	2	2	no	—
50	C₃₁×#135	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	—	2	1	no	—
53	C₂×Q₁₆×C₃₁	—	2	1	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
59	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×#124	—	2	2	no	—
67	C₃₁×#123	—	4	2	no	—
68	C₃₁×C₂²⋊Q₈	—	2	2	no	—
69	C₃₁×C₂²⋊Q₈	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	4	2	no	—
73	C₃₁×#117	—	2	2	no	—
74	C₃₁×#117	C₂³	1	1	yes	characteristic
75	C₃₁×#117	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#108	—	2	2	no	—
80	C₃₁×#108	—	2	2	no	—
81	C₃₁×#107	—	2	1	no	—
82	C₃₁×#107	—	2	1	no	—
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#107	D₄	1	1	yes	characteristic
85	C₃₁×#107	D₄	1	1	yes	characteristic
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×#106	D₄	1	1	yes	characteristic
88	C₃₁×#106	D₄	1	1	yes	characteristic
89	C₃₁×#104	—	4	2	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	2	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#103	—	2	2	no	—
94	C₃₁×#103	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#101	—	4	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×#97	—	2	1	no	—
100	C₃₁×#97	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	—	2	2	no	—
104	C₃₁×#95	—	4	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₃₁	—	2	2	no	—
125	Q₁₆×C₃₁	—	4	4	no	—
126	Q₁₆×C₃₁	—	4	2	no	—
127	Q₁₆×C₃₁	—	4	2	no	—
128	SD₁₆×C₃₁	—	2	4	no	—
129	SD₁₆×C₃₁	—	4	4	no	—
130	D₈×C₃₁	—	2	2	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	2	1	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	#41	1	2	yes	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	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	C₂×C₈×C₃₁	#41	1	2	yes	—
147	C₂×C₈×C₃₁	#41	1	2	yes	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²⋊C₄×C₃₁	—	2	1	no	—
158	C₂²⋊C₄×C₃₁	—	2	1	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₂²⋊C₄×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	#26271	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	—	4	2	no	—
175	Q₈×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	4	1	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	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	8	2	no	—
195	C₂×C₄×C₃₁	—	16	2	no	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
202	C₈×C₃₁	—	2	1	no	—
203	C₈×C₃₁	—	2	1	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	—	4	2	no	—
207	C₈×C₃₁	#123	1	2	yes	—
208	C₈×C₃₁	#123	1	2	yes	—
209	#3156	C₂×C₃₁	1	1	yes	characteristic
210	#3097	C₂×C₃₁	1	2	yes	complemented
211	#2951	C₂×C₃₁	1	1	yes	characteristic
212	#2052	C₂×C₃₁	1	2	yes	complemented
213	#2038	C₂×C₃₁	1	2	yes	complemented
214	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1974	C₂×C₃₁	1	1	yes	characteristic
216	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1882	C₂×C₃₁	1	2	yes	complemented
219	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	2	1	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	#540	1	1	yes	characteristic
226	C₂²×C₃₁	D₄²	1	1	yes	characteristic
227	C₂²×C₃₁	#559	1	1	yes	characteristic
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	4	1	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
237	C₄×C₃₁	#532	1	1	yes	characteristic
238	C₄×C₃₁	#546	1	1	yes	characteristic
239	C₄×C₃₁	#550	1	1	yes	characteristic
240	C₂×#137	C₂²×C₃₁	1	2	yes	—
241	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
242	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
243	#536	C₂²×C₃₁	1	2	yes	—
244	#520	C₂²×C₃₁	1	2	yes	—
245	#497	—	2	2	no	—
246	#495	—	2	1	no	—
247	#494	—	2	1	no	—
248	#485	—	2	2	no	—
249	#485	—	2	2	no	—
250	#484	—	2	2	no	—
251	#483	—	2	2	no	—
252	#440	—	2	2	no	—
253	C₄×Q₁₆	—	2	1	no	—
254	C₄×D₈	—	2	1	no	—
255	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
257	#428	C₂²×C₃₁	1	2	yes	—
258	#427	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#104	C₂²×C₃₁	1	2	yes	—
261	#419	C₂²×C₃₁	1	2	yes	—
262	#417	C₂²×C₃₁	1	2	yes	—
263	C₂×#103	C₂²×C₃₁	1	2	yes	—
264	C₂×#101	C₂²×C₃₁	1	2	yes	—
265	#408	C₂²×C₃₁	1	2	yes	—
266	#405	C₂²×C₃₁	1	1	yes	characteristic
267	#405	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
269	#391	C₂²×C₃₁	1	2	yes	—
270	#343	C₂²×C₃₁	1	2	yes	—
271	#339	C₂²×C₃₁	1	1	yes	characteristic
272	#339	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	8	2	no	—
276	C₂×C₃₁	#2445	1	1	yes	characteristic
277	C₂×C₃₁	#3290	1	1	yes	characteristic
278	C₂×C₃₁	#3293	1	1	yes	characteristic
279	C₂×#41	C₂³×C₃₁	1	2	yes	—
280	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
281	#137	—	2	4	no	—
282	#137	—	4	2	no	—
283	#135	—	2	2	no	—
284	#135	—	2	2	no	—
285	C₂×Q₁₆	—	2	2	no	—
286	C₂×Q₁₆	—	2	1	no	—
287	C₂×Q₁₆	—	2	1	no	—
288	C₂×Q₁₆	—	2	2	no	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
293	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
296	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄⋊Q₈	—	2	2	no	—
300	#124	—	2	2	no	—
301	#123	—	4	2	no	—
302	C₂²⋊Q₈	—	2	2	no	—
303	C₂²⋊Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	4	2	no	—
307	#117	—	2	2	no	—
308	#117	C₂³×C₃₁	1	1	yes	characteristic
309	#117	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	#108	—	2	2	no	—
314	#108	—	2	2	no	—
315	#107	—	2	1	no	—
316	#107	—	2	1	no	—
317	#107	—	2	2	no	—
318	#107	D₄×C₃₁	1	1	yes	characteristic
319	#107	D₄×C₃₁	1	1	yes	characteristic
320	#106	—	2	2	no	—
321	#106	D₄×C₃₁	1	1	yes	characteristic
322	#106	D₄×C₃₁	1	1	yes	characteristic
323	#104	—	4	2	no	—
324	#103	—	2	2	no	—
325	#103	—	2	2	no	—
326	#103	—	2	2	no	—
327	#103	—	2	2	no	—
328	#103	—	2	2	no	—
329	#102	—	2	2	no	—
330	#101	—	4	2	no	—
331	C₂²⋊C₈	—	2	2	no	—
332	C₂²⋊C₈	—	2	2	no	—
333	#97	—	2	1	no	—
334	#97	—	2	1	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	—	2	1	no	—
337	C₄×C₈	—	2	2	no	—
338	#95	—	4	2	no	—
339	C₃₁	#26271	1	1	yes	characteristic, direct product factor, Sylow
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	2	2	no	—
343	#41	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
359	Q₁₆	—	2	2	no	—
360	Q₁₆	—	4	4	no	—
361	Q₁₆	—	4	2	no	—
362	Q₁₆	—	4	2	no	—
363	SD₁₆	—	2	4	no	—
364	SD₁₆	—	4	4	no	—
365	D₈	—	2	2	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	2	2	no	—
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	#41×C₃₁	1	2	yes	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₈	#41×C₃₁	1	2	yes	—
382	C₂×C₈	#41×C₃₁	1	2	yes	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₂²⋊C₄	—	4	2	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	4	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
404	Q₈	—	2	2	no	—
405	Q₈	—	2	2	no	—
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	4	2	no	—
409	Q₈	—	4	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	16	2	no	—
430	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
436	C₈	—	2	1	no	—
437	C₈	—	2	1	no	—
438	C₈	—	2	2	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	C₃₁×#123	1	2	yes	—
442	C₈	C₃₁×#123	1	2	yes	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	C₃₁×#540	1	1	yes	characteristic
449	C₂²	D₄²×C₃₁	1	1	yes	characteristic
450	C₂²	C₃₁×#559	1	1	yes	characteristic
451	C₄	—	4	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
460	C₄	C₃₁×#532	1	1	yes	characteristic
461	C₄	C₃₁×#546	1	1	yes	characteristic
462	C₄	C₃₁×#550	1	1	yes	characteristic
463	C₂	—	2	1	no	—
464	C₂	—	2	1	no	—
465	C₂	—	8	2	no	—
466	C₂	C₃₁×#2445	1	1	yes	characteristic
467	C₂	C₃₁×#3290	1	1	yes	characteristic
468	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3097	C₂	semidirect
4. C₃₁×#2052	C₂	semidirect
5. C₃₁×#2038	C₂	semidirect
6. C₃₁×#1979	C₂	semidirect
8. C₃₁×#1969	C₂	semidirect
9. C₃₁×#1952	C₂	semidirect
10. C₃₁×#1882	C₂	semidirect
11. C₃₁×#1772	C₂	semidirect
167. #26271	C₃₁	direct (N × H)
210. #3097	C₂×C₃₁	semidirect
212. #2052	C₂×C₃₁	semidirect
213. #2038	C₂×C₃₁	semidirect
214. #1979	C₂×C₃₁	semidirect
216. #1969	C₂×C₃₁	semidirect
217. #1952	C₂×C₃₁	semidirect
218. #1882	C₂×C₃₁	semidirect
219. #1772	C₂×C₃₁	semidirect
339. C₃₁	#26271	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×#3156	2
C₃₁×#3097	2
C₃₁×#2951	2
C₃₁×#2052	2
C₃₁×#2038	2
C₃₁×#1979	2
C₃₁×#1974	2
C₃₁×#1969	2
C₃₁×#1952	2
C₃₁×#1882	2
C₃₁×#1772	2
#26271	31

Subgroups — C31×#26271 #21483030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26271 #21483030