Subgroups — C₃₁×#23045 #21479804

Order 7936.

844 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2947	C₂	1	2	yes	maximal
4	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1930	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1882	C₂	1	2	yes	maximal
9	C₃₁×#1857	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₃₁×#520	C₂²	1	2	yes	—
16	C₃₁×#519	C₂²	1	2	yes	—
17	C₃₁×#473	C₄	1	8	yes	complemented
18	C₃₁×#441	—	2	2	no	—
19	C₃₁×#441	—	2	2	no	—
20	C₃₁×#440	—	2	2	no	—
21	C₄×Q₁₆×C₃₁	—	2	1	no	—
22	C₄×Q₁₆×C₃₁	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	—	2	2	no	—
24	C₄×SD₁₆×C₃₁	—	2	2	no	—
25	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
26	C₃₁×#428	—	2	2	no	—
27	C₃₁×#428	C₂²	1	2	yes	—
28	C₃₁×#428	C₂²	1	2	yes	—
29	C₃₁×#427	—	2	1	no	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	—	2	1	no	—
32	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
34	C₃₁×#419	C₂²	1	2	yes	—
35	C₃₁×#416	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#103	C₂²	1	2	yes	—
37	C₂×C₃₁×#101	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	2	yes	—
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×#341	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₃₁×#339	C₂²	1	2	yes	—
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₃₁×#137	—	2	4	no	—
49	C₃₁×#137	C₂×C₄	1	8	yes	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	1	no	—
52	C₂×Q₁₆×C₃₁	—	2	1	no	—
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₃₁×#131	—	2	2	no	—
57	C₃₁×#131	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	4	2	no	—
72	C₃₁×#117	—	2	2	no	—
73	C₃₁×#117	—	4	2	no	—
74	C₃₁×#117	C₂³	1	1	yes	characteristic
75	C₃₁×#117	C₂³	1	2	yes	—
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	C₂×C₄	1	4	yes	complemented
80	C₃₁×#107	—	2	1	no	—
81	C₃₁×#107	—	2	1	no	—
82	C₃₁×#107	—	2	2	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	—	2	2	no	—
88	C₃₁×#106	D₄	1	2	yes	—
89	C₃₁×#104	—	2	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	D₄	1	1	yes	characteristic
95	C₃₁×#103	D₄	1	1	yes	characteristic
96	C₃₁×#102	D₄	1	1	yes	characteristic
97	C₃₁×#102	D₄	1	1	yes	characteristic
98	C₃₁×#101	C₂×C₄	1	8	yes	complemented
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	2	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	1	no	—
106	#41×C₃₁	—	2	1	no	—
107	#41×C₃₁	C₂²×C₄	1	2	yes	—
108	#41×C₃₁	#41	1	2	yes	—
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	C₂²×C₄	1	4	yes	—
115	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	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	8	no	—
125	Q₁₆×C₃₁	—	2	4	no	—
126	Q₁₆×C₃₁	—	2	4	no	—
127	SD₁₆×C₃₁	—	2	8	no	—
128	M₁₆×C₃₁	—	2	1	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	2	4	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
134	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	4	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
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	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²⋊C₄×C₃₁	—	2	1	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	—	4	2	no	—
165	C₄²×C₃₁	—	4	2	no	—
166	#23045	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
169	Q₈×C₃₁	—	2	1	no	—
170	Q₈×C₃₁	—	2	4	no	—
171	Q₈×C₃₁	—	2	4	no	—
172	Q₈×C₃₁	—	4	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	C₄×D₄	1	2	yes	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	2	2	no	—
177	D₄×C₃₁	—	2	1	no	—
178	D₄×C₃₁	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	8	2	no	—
194	C₂×C₄×C₃₁	—	16	2	no	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
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	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	1	no	—
204	C₈×C₃₁	—	2	1	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	C₄×D₄	1	2	yes	—
207	C₈×C₃₁	C₄×D₄	1	2	yes	—
208	C₂×#473	C₂×C₃₁	1	1	yes	characteristic
209	#2950	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2947	C₂×C₃₁	1	2	yes	—
211	#2913	C₂×C₃₁	1	1	yes	characteristic
212	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1930	C₂×C₃₁	1	2	yes	complemented
215	#1882	C₂×C₃₁	1	2	yes	—
216	#1857	C₂×C₃₁	1	2	yes	complemented
217	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1764	C₂×C₃₁	1	1	yes	characteristic
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	#520	1	1	yes	characteristic
225	C₂²×C₃₁	D₄²	1	1	yes	characteristic
226	C₂²×C₃₁	#553	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	4	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
237	C₄×C₃₁	#547	1	1	yes	characteristic
238	C₄×C₃₁	#548	1	1	yes	characteristic
239	C₂×#137	C₂²×C₃₁	1	2	yes	—
240	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
242	#520	C₂²×C₃₁	1	2	yes	—
243	#519	C₂²×C₃₁	1	2	yes	—
244	#473	C₄×C₃₁	1	8	yes	complemented
245	#441	—	2	2	no	—
246	#441	—	2	2	no	—
247	#440	—	2	2	no	—
248	C₄×Q₁₆	—	2	1	no	—
249	C₄×Q₁₆	—	2	1	no	—
250	C₄×Q₁₆	—	2	2	no	—
251	C₄×SD₁₆	—	2	2	no	—
252	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
253	#428	—	2	2	no	—
254	#428	C₂²×C₃₁	1	2	yes	—
255	#428	C₂²×C₃₁	1	2	yes	—
256	#427	—	2	1	no	—
257	#427	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#107	—	2	1	no	—
259	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
261	#419	C₂²×C₃₁	1	2	yes	—
262	#416	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#103	C₂²×C₃₁	1	2	yes	—
264	C₂×#101	C₂²×C₃₁	1	2	yes	—
265	#405	C₂²×C₃₁	1	2	yes	—
266	#405	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
268	#391	C₂²×C₃₁	1	2	yes	—
269	#343	C₂²×C₃₁	1	2	yes	—
270	#341	C₂²×C₃₁	1	1	yes	characteristic
271	#339	C₂²×C₃₁	1	1	yes	characteristic
272	#339	C₂²×C₃₁	1	2	yes	—
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	4	2	no	—
276	C₂×C₃₁	#2361	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	1	yes	characteristic
280	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
281	#137	—	2	4	no	—
282	#137	C₂×C₄×C₃₁	1	8	yes	—
283	C₂×Q₁₆	—	2	2	no	—
284	C₂×Q₁₆	—	2	1	no	—
285	C₂×Q₁₆	—	2	1	no	—
286	C₂×Q₁₆	—	2	2	no	—
287	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
288	C₂×SD₁₆	—	2	2	no	—
289	#131	—	2	2	no	—
290	#131	C₂×C₄×C₃₁	1	4	yes	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
295	C₂²×C₈	—	2	1	no	—
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄×Q₈	—	2	2	no	—
299	C₄×Q₈	—	2	2	no	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×Q₈	—	2	2	no	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	—	4	2	no	—
305	#117	—	2	2	no	—
306	#117	—	4	2	no	—
307	#117	C₂³×C₃₁	1	1	yes	characteristic
308	#117	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	#108	C₂×C₄×C₃₁	1	4	yes	complemented
313	#107	—	2	1	no	—
314	#107	—	2	1	no	—
315	#107	—	2	2	no	—
316	#107	—	2	2	no	—
317	#107	D₄×C₃₁	1	1	yes	characteristic
318	#107	D₄×C₃₁	1	1	yes	characteristic
319	#106	—	2	2	no	—
320	#106	—	2	2	no	—
321	#106	D₄×C₃₁	1	2	yes	—
322	#104	—	2	2	no	—
323	#103	—	2	2	no	—
324	#103	—	2	2	no	—
325	#103	—	2	2	no	—
326	#103	—	2	2	no	—
327	#103	D₄×C₃₁	1	1	yes	characteristic
328	#103	D₄×C₃₁	1	1	yes	characteristic
329	#102	D₄×C₃₁	1	1	yes	characteristic
330	#102	D₄×C₃₁	1	1	yes	characteristic
331	#101	C₂×C₄×C₃₁	1	8	yes	complemented
332	#97	—	2	2	no	—
333	#97	—	2	1	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	—	2	2	no	—
337	#95	—	4	2	no	—
338	C₃₁	#23045	1	1	yes	characteristic, direct product factor, Sylow
339	#41	—	2	1	no	—
340	#41	—	2	1	no	—
341	#41	C₂²×C₄×C₃₁	1	2	yes	—
342	#41	#41×C₃₁	1	2	yes	—
343	C₂×Q₈	—	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	C₂²×C₄×C₃₁	1	4	yes	—
349	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
358	Q₁₆	—	2	8	no	—
359	Q₁₆	—	2	4	no	—
360	Q₁₆	—	2	4	no	—
361	SD₁₆	—	2	8	no	—
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	2	4	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
368	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	4	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	#41×C₃₁	1	2	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	4	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	4	2	no	—
398	C₄²	—	4	2	no	—
399	C₄²	—	4	2	no	—
400	C₂³	—	2	1	no	—
401	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
402	Q₈	—	2	1	no	—
403	Q₈	—	2	4	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	2	no	—
407	Q₈	C₄×D₄×C₃₁	1	2	yes	—
408	D₄	—	2	2	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	1	no	—
411	D₄	C₄×D₄×C₃₁	1	2	yes	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	8	2	no	—
427	C₂×C₄	—	16	2	no	—
428	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	2	4	no	—
436	C₈	—	2	1	no	—
437	C₈	—	2	1	no	—
438	C₈	—	4	1	no	—
439	C₈	C₄×D₄×C₃₁	1	2	yes	—
440	C₈	C₄×D₄×C₃₁	1	2	yes	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	4	1	no	—
446	C₂²	C₃₁×#520	1	1	yes	characteristic
447	C₂²	D₄²×C₃₁	1	1	yes	characteristic
448	C₂²	C₃₁×#553	1	1	yes	characteristic
449	C₄	—	2	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	4	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	8	2	no	—
457	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
458	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
459	C₄	C₃₁×#547	1	1	yes	characteristic
460	C₄	C₃₁×#548	1	1	yes	characteristic
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	4	2	no	—
464	C₂	C₃₁×#2361	1	1	yes	characteristic
465	C₂	C₃₁×#3290	1	1	yes	characteristic
466	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂×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×#2950	C₂	semidirect
5. C₃₁×#1969	C₂	semidirect
6. C₃₁×#1952	C₂	semidirect
7. C₃₁×#1930	C₂	semidirect
9. C₃₁×#1857	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
17. C₃₁×#473	C₄	semidirect
54. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
79. C₃₁×#108	C₂×C₄	semidirect
98. C₃₁×#101	C₂×C₄	semidirect
166. #23045	C₃₁	direct (N × H)
209. #2950	C₂×C₃₁	semidirect
212. #1969	C₂×C₃₁	semidirect
213. #1952	C₂×C₃₁	semidirect
214. #1930	C₂×C₃₁	semidirect
216. #1857	C₂×C₃₁	semidirect
217. #1772	C₂×C₃₁	semidirect
244. #473	C₄×C₃₁	semidirect
287. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
312. #108	C₂×C₄×C₃₁	semidirect
331. #101	C₂×C₄×C₃₁	semidirect
338. C₃₁	#23045	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#473	2
C₃₁×#2950	2
C₃₁×#2947	2
C₃₁×#2913	2
C₃₁×#1969	2
C₃₁×#1952	2
C₃₁×#1930	2
C₃₁×#1882	2
C₃₁×#1857	2
C₃₁×#1772	2
C₃₁×#1764	2
#23045	31

Subgroups — C31×#23045 #21479804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23045 #21479804