Subgroups — C₃₁×#23043 #21479802

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3067	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2956	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1932	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1881	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1858	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1778	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₃₁×#568	C₂²	1	2	yes	—
15	C₃₁×#568	C₂²	1	2	yes	—
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₃₁×#473	C₄	1	2	yes	complemented
18	C₃₁×#472	C₄	1	2	yes	complemented
19	C₃₁×#471	C₄	1	4	yes	complemented
20	C₃₁×#444	—	2	2	no	—
21	C₃₁×#444	—	2	4	no	—
22	C₃₁×#443	—	2	2	no	—
23	C₃₁×#443	—	2	4	no	—
24	C₃₁×#430	—	2	2	no	—
25	C₃₁×#430	C₂²	1	2	yes	—
26	C₃₁×#430	C₂²	1	2	yes	—
27	C₂×C₃₁×#108	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
31	C₃₁×#421	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
34	C₃₁×#416	C₂²	1	1	yes	characteristic
35	C₃₁×#413	C₂²	1	2	yes	—
36	C₃₁×#408	C₂²	1	2	yes	—
37	C₃₁×#405	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
39	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
40	C₃₁×#344	C₂²	1	2	yes	—
41	C₃₁×#341	C₂²	1	1	yes	characteristic
42	C₃₁×#341	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₂×#41×C₃₁	C₂³	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#137	—	2	2	no	—
47	C₃₁×#137	C₂×C₄	1	4	yes	—
48	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
49	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
50	C₃₁×#135	—	2	4	no	—
51	C₃₁×#135	—	2	1	no	—
52	C₃₁×#135	—	2	1	no	—
53	C₃₁×#135	—	2	1	no	—
54	C₃₁×#135	—	2	1	no	—
55	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₃₁×#131	—	2	1	no	—
59	C₃₁×#131	—	2	1	no	—
60	C₃₁×#131	—	2	4	no	—
61	C₃₁×#131	—	2	4	no	—
62	C₃₁×#131	—	2	4	no	—
63	C₃₁×#131	—	4	2	no	—
64	C₃₁×#131	C₂×C₄	1	2	yes	—
65	C₃₁×#131	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₃₁	—	2	1	no	—
67	C₂×M₁₆×C₃₁	—	2	4	no	—
68	C₂×M₁₆×C₃₁	—	4	2	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#117	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#108	—	2	4	no	—
80	C₃₁×#108	—	2	4	no	—
81	C₃₁×#108	—	2	1	no	—
82	C₃₁×#108	—	2	1	no	—
83	C₃₁×#108	C₂×C₄	1	2	yes	—
84	C₃₁×#108	C₂×C₄	1	2	yes	—
85	C₃₁×#108	D₄	1	2	yes	—
86	C₃₁×#108	D₄	1	2	yes	—
87	C₃₁×#104	—	2	4	no	—
88	C₃₁×#104	—	2	2	no	—
89	C₃₁×#104	—	2	2	no	—
90	C₃₁×#104	D₄	1	4	yes	—
91	C₃₁×#103	—	2	1	no	—
92	C₃₁×#102	—	2	1	no	—
93	C₃₁×#101	C₂×C₄	1	4	yes	—
94	C₃₁×#100	C₂×C₄	1	4	yes	—
95	C₃₁×C₂²⋊C₈	—	4	2	no	—
96	C₃₁×#97	—	2	1	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	#41×C₃₁	—	2	1	no	—
103	#41×C₃₁	—	2	1	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	C₂²×C₄	1	2	yes	—
109	#41×C₃₁	C₂×D₄	1	2	yes	—
110	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	Q₁₆×C₃₁	—	2	4	no	—
121	Q₁₆×C₃₁	—	2	1	no	—
122	Q₁₆×C₃₁	—	2	1	no	—
123	SD₁₆×C₃₁	—	2	4	no	—
124	SD₁₆×C₃₁	—	2	4	no	—
125	SD₁₆×C₃₁	—	2	2	no	—
126	SD₁₆×C₃₁	—	2	2	no	—
127	D₈×C₃₁	—	2	4	no	—
128	D₈×C₃₁	—	2	1	no	—
129	D₈×C₃₁	—	2	1	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	4	4	no	—
139	M₁₆×C₃₁	—	4	4	no	—
140	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
141	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
142	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
143	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	—	4	4	no	—
156	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
161	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₂²⋊C₄×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	—	2	1	no	—
166	C₄²×C₃₁	C₂×D₄	1	2	yes	—
167	#23043	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
171	Q₈×C₃₁	—	2	1	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	2	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	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	1	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	8	1	no	—
192	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	2	4	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	2	4	no	—
202	C₈×C₃₁	—	4	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
205	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
206	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
207	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
208	#3067	C₂×C₃₁	1	1	yes	characteristic
209	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2956	C₂×C₃₁	1	2	yes	complemented
211	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1975	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1953	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1932	C₂×C₃₁	1	2	yes	complemented
215	#1881	C₂×C₃₁	1	2	yes	complemented
216	#1858	C₂×C₃₁	1	2	yes	complemented
217	#1778	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1765	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	8	1	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	#520	1	1	yes	characteristic
225	C₂²×C₃₁	#547	1	1	yes	characteristic
226	C₂²×C₃₁	#548	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	8	1	no	—
231	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
232	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
233	C₄×C₃₁	D₄²	1	1	yes	characteristic
234	C₄×C₃₁	#553	1	1	yes	characteristic
235	#575	C₂²×C₃₁	1	2	yes	—
236	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
237	#568	C₂²×C₃₁	1	2	yes	—
238	#568	C₂²×C₃₁	1	2	yes	—
239	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
240	#473	C₄×C₃₁	1	2	yes	complemented
241	#472	C₄×C₃₁	1	2	yes	complemented
242	#471	C₄×C₃₁	1	4	yes	complemented
243	#444	—	2	2	no	—
244	#444	—	2	4	no	—
245	#443	—	2	2	no	—
246	#443	—	2	4	no	—
247	#430	—	2	2	no	—
248	#430	C₂²×C₃₁	1	2	yes	—
249	#430	C₂²×C₃₁	1	2	yes	—
250	C₂×#108	—	2	2	no	—
251	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
254	#421	C₂²×C₃₁	1	2	yes	—
255	C₂×#104	C₂²×C₃₁	1	2	yes	—
256	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
257	#416	C₂²×C₃₁	1	1	yes	characteristic
258	#413	C₂²×C₃₁	1	2	yes	—
259	#408	C₂²×C₃₁	1	2	yes	—
260	#405	C₂²×C₃₁	1	1	yes	characteristic
261	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
262	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
263	#344	C₂²×C₃₁	1	2	yes	—
264	#341	C₂²×C₃₁	1	1	yes	characteristic
265	#341	C₂²×C₃₁	1	2	yes	—
266	#339	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₃₁	—	2	1	no	—
268	C₂×C₃₁	—	2	1	no	—
269	C₂×C₃₁	—	2	1	no	—
270	C₂×C₃₁	—	4	2	no	—
271	C₂×C₃₁	—	8	2	no	—
272	C₂×C₃₁	#2361	1	1	yes	characteristic
273	C₂×#41	C₂³×C₃₁	1	2	yes	—
274	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
275	#137	—	2	2	no	—
276	#137	C₂×C₄×C₃₁	1	4	yes	—
277	AGL(1,ℤ/8)	—	2	2	no	—
278	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	4	yes	—
279	#135	—	2	4	no	—
280	#135	—	2	1	no	—
281	#135	—	2	1	no	—
282	#135	—	2	1	no	—
283	#135	—	2	1	no	—
284	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
286	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
287	#131	—	2	1	no	—
288	#131	—	2	1	no	—
289	#131	—	2	4	no	—
290	#131	—	2	4	no	—
291	#131	—	2	4	no	—
292	#131	—	4	2	no	—
293	#131	C₂×C₄×C₃₁	1	2	yes	—
294	#131	C₂×C₄×C₃₁	1	2	yes	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂×M₁₆	—	2	4	no	—
297	C₂×M₁₆	—	4	2	no	—
298	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
299	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
301	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
302	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	—	2	1	no	—
304	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	#117	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
308	#108	—	2	4	no	—
309	#108	—	2	4	no	—
310	#108	—	2	1	no	—
311	#108	—	2	1	no	—
312	#108	C₂×C₄×C₃₁	1	2	yes	—
313	#108	C₂×C₄×C₃₁	1	2	yes	—
314	#108	D₄×C₃₁	1	2	yes	—
315	#108	D₄×C₃₁	1	2	yes	—
316	#104	—	2	4	no	—
317	#104	—	2	2	no	—
318	#104	—	2	2	no	—
319	#104	D₄×C₃₁	1	4	yes	—
320	#103	—	2	1	no	—
321	#102	—	2	1	no	—
322	#101	C₂×C₄×C₃₁	1	4	yes	—
323	#100	C₂×C₄×C₃₁	1	4	yes	—
324	C₂²⋊C₈	—	4	2	no	—
325	#97	—	2	1	no	—
326	#97	—	2	1	no	—
327	#97	—	2	1	no	—
328	#97	—	2	2	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	—	2	2	no	—
331	C₃₁	#23043	1	1	yes	characteristic, direct product factor, Sylow
332	#41	—	2	1	no	—
333	#41	—	2	1	no	—
334	#41	—	2	2	no	—
335	#41	—	2	2	no	—
336	#41	—	2	2	no	—
337	#41	—	2	2	no	—
338	#41	C₂²×C₄×C₃₁	1	2	yes	—
339	#41	C₂×D₄×C₃₁	1	2	yes	—
340	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
341	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
350	Q₁₆	—	2	4	no	—
351	Q₁₆	—	2	1	no	—
352	Q₁₆	—	2	1	no	—
353	SD₁₆	—	2	4	no	—
354	SD₁₆	—	2	4	no	—
355	SD₁₆	—	2	2	no	—
356	SD₁₆	—	2	2	no	—
357	D₈	—	2	4	no	—
358	D₈	—	2	1	no	—
359	D₈	—	2	1	no	—
360	M₁₆	—	2	1	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	2	2	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	4	4	no	—
369	M₁₆	—	4	4	no	—
370	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
371	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
372	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
373	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	4	4	no	—
385	C₂×C₈	—	4	4	no	—
386	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
391	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
392	C₄⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	C₂×D₄×C₃₁	1	2	yes	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	2	no	—
399	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	2	no	—
402	Q₈	—	2	2	no	—
403	Q₈	C₄×D₄×C₃₁	1	2	yes	—
404	D₄	—	2	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	1	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	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	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	8	1	no	—
421	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	2	4	no	—
429	C₈	—	2	2	no	—
430	C₈	—	2	4	no	—
431	C₈	—	4	2	no	—
432	C₈	—	4	1	no	—
433	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
434	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
435	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
436	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	8	1	no	—
441	C₂²	—	8	2	no	—
442	C₂²	C₃₁×#520	1	1	yes	characteristic
443	C₂²	C₃₁×#547	1	1	yes	characteristic
444	C₂²	C₃₁×#548	1	1	yes	characteristic
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	8	1	no	—
449	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
450	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
451	C₄	D₄²×C₃₁	1	1	yes	characteristic
452	C₄	C₃₁×#553	1	1	yes	characteristic
453	C₂	—	2	1	no	—
454	C₂	—	2	1	no	—
455	C₂	—	2	1	no	—
456	C₂	—	4	2	no	—
457	C₂	—	8	2	no	—
458	C₂	C₃₁×#2361	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2361
prime radical	C₃₁×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2957	C₂	semidirect
3. C₃₁×#2956	C₂	semidirect
4. C₃₁×#2916	C₂	semidirect
5. C₃₁×#1975	C₂	semidirect
6. C₃₁×#1953	C₂	semidirect
7. C₃₁×#1932	C₂	semidirect
8. C₃₁×#1881	C₂	semidirect
9. C₃₁×#1858	C₂	semidirect
10. C₃₁×#1778	C₂	semidirect
11. C₃₁×#1765	C₂	semidirect
17. C₃₁×#473	C₄	semidirect
18. C₃₁×#472	C₄	semidirect
19. C₃₁×#471	C₄	semidirect
167. #23043	C₃₁	direct (N × H)
209. #2957	C₂×C₃₁	semidirect
210. #2956	C₂×C₃₁	semidirect
211. #2916	C₂×C₃₁	semidirect
212. #1975	C₂×C₃₁	semidirect
213. #1953	C₂×C₃₁	semidirect
214. #1932	C₂×C₃₁	semidirect
215. #1881	C₂×C₃₁	semidirect
216. #1858	C₂×C₃₁	semidirect
217. #1778	C₂×C₃₁	semidirect
218. #1765	C₂×C₃₁	semidirect
240. #473	C₄×C₃₁	semidirect
241. #472	C₄×C₃₁	semidirect
242. #471	C₄×C₃₁	semidirect
331. C₃₁	#23043	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×#3067	2
C₃₁×#2957	2
C₃₁×#2956	2
C₃₁×#2916	2
C₃₁×#1975	2
C₃₁×#1953	2
C₃₁×#1932	2
C₃₁×#1881	2
C₃₁×#1858	2
C₃₁×#1778	2
C₃₁×#1765	2
#23043	31

Subgroups — C31×#23043 #21479802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23043 #21479802