Subgroups — C₃₁×#22193 #21478952

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₃₁×#466	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#390	—	2	2	no	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#385	—	2	2	no	—
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×D₈×C₃₁	—	4	1	no	—
41	C₂×D₈×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	4	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#121	—	2	4	no	—
53	C₃₁×#121	C₂×C₄	1	8	yes	complemented
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	4	1	no	—
57	C₃₁×#117	—	4	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₃₁×#107	C₂×C₄	1	4	yes	complemented
73	C₃₁×#106	—	4	1	no	—
74	C₃₁×#106	—	4	1	no	—
75	C₃₁×#102	—	4	2	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	C₂×C₄	1	8	yes	complemented
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₂⁴×C₃₁	—	2	2	no	—
82	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	8	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	8	2	no	—
90	C₂×D₄×C₃₁	—	8	2	no	—
91	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	4	4	no	—
100	C₂²×C₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	8	2	no	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	#41	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	D₈×C₃₁	—	4	4	no	—
109	M₁₆×C₃₁	—	4	4	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	4	4	no	—
118	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₄⋊C₄×C₃₁	—	8	2	no	—
127	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
128	C₂²⋊C₄×C₃₁	—	2	8	no	—
129	C₂²⋊C₄×C₃₁	—	4	1	no	—
130	C₂²⋊C₄×C₃₁	—	4	1	no	—
131	C₂²⋊C₄×C₃₁	—	4	4	no	—
132	C₂²⋊C₄×C₃₁	—	8	2	no	—
133	C₂²⋊C₄×C₃₁	—	8	2	no	—
134	C₂²⋊C₄×C₃₁	—	8	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	8	2	no	—
138	#22193	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	1	no	—
140	C₂³×C₃₁	—	2	1	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	—	2	4	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	C₄×D₄	1	2	yes	—
151	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	D₄×C₃₁	—	2	4	no	—
153	D₄×C₃₁	—	2	4	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	8	8	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	8	8	no	—
179	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
190	#2324	C₂×C₃₁	1	2	yes	complemented
191	#1926	C₂×C₃₁	1	2	yes	complemented
192	#1918	C₂×C₃₁	1	2	yes	complemented
193	#1895	C₂×C₃₁	1	2	yes	complemented
194	#1879	C₂×C₃₁	1	2	yes	complemented
195	#1864	C₂×C₃₁	1	2	yes	complemented
196	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	—	4	4	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₃₁	#534	1	1	yes	characteristic
212	C₂²×C₃₁	#537	1	1	yes	characteristic
213	C₂²×C₃₁	#576	1	2	yes	—
214	C₂²×C₃₁	#577	1	2	yes	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	4	no	—
219	C₄×C₃₁	—	8	2	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	#518	1	1	yes	characteristic
222	C₄×C₃₁	#520	1	1	yes	characteristic
223	C₄×C₃₁	#538	1	1	yes	characteristic
224	C₄×C₃₁	#539	1	1	yes	characteristic
225	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
227	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#121	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
230	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
231	#466	C₄×C₃₁	1	8	yes	complemented
232	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#102	C₂²×C₃₁	1	2	yes	—
236	C₂×#102	C₂²×C₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₃₁	1	2	yes	—
238	#390	—	2	2	no	—
239	#390	C₂²×C₃₁	1	2	yes	—
240	#389	C₂²×C₃₁	1	2	yes	—
241	#388	—	2	2	no	—
242	#386	—	2	2	no	—
243	#385	—	2	2	no	—
244	#380	C₂²×C₃₁	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
246	#340	C₂²×C₃₁	1	2	yes	—
247	#340	C₂²×C₃₁	1	2	yes	—
248	#340	C₂²×C₃₁	1	2	yes	—
249	#336	C₂²×C₃₁	1	2	yes	—
250	#336	C₂²×C₃₁	1	2	yes	—
251	C₂×C₃₁	—	4	2	no	—
252	C₂×C₃₁	—	8	4	no	—
253	C₂×C₃₁	#2325	1	1	yes	characteristic
254	C₂×C₃₁	#2948	1	1	yes	characteristic
255	C₂×C₃₁	#2952	1	1	yes	characteristic
256	C₂×C₃₁	#3198	1	2	yes	—
257	C₂×C₃₁	#3257	1	2	yes	—
258	C₂²×D₄	—	2	2	no	—
259	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
260	C₂³×C₄	—	2	2	no	—
261	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×D₈	—	4	1	no	—
263	C₂×D₈	—	4	1	no	—
264	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	—	4	1	no	—
267	C₂×M₁₆	—	4	1	no	—
268	C₂²×C₈	—	4	1	no	—
269	C₂²×C₈	—	4	1	no	—
270	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
271	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	#121	—	2	4	no	—
275	#121	C₂×C₄×C₃₁	1	8	yes	complemented
276	C₄×D₄	—	4	2	no	—
277	C₄×D₄	—	4	2	no	—
278	#117	—	4	1	no	—
279	#117	—	4	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
294	#107	C₂×C₄×C₃₁	1	4	yes	complemented
295	#106	—	4	1	no	—
296	#106	—	4	1	no	—
297	#102	—	4	2	no	—
298	#102	—	4	2	no	—
299	#102	C₂×C₄×C₃₁	1	8	yes	complemented
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	C₃₁	#22193	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	—	2	2	no	—
305	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂×D₄	—	2	8	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	8	2	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	4	4	no	—
323	C₂²×C₄	—	8	2	no	—
324	C₂²×C₄	—	8	2	no	—
325	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	#41×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
331	D₈	—	4	4	no	—
332	M₁₆	—	4	4	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
342	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	8	2	no	—
350	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₂²⋊C₄	—	4	1	no	—
353	C₂²⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₂²⋊C₄	—	8	2	no	—
356	C₂²⋊C₄	—	8	2	no	—
357	C₂²⋊C₄	—	8	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	8	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	1	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	C₄×D₄×C₃₁	1	2	yes	—
373	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	D₄	—	2	4	no	—
375	D₄	—	2	4	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	8	8	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	8	no	—
401	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
402	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
405	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
406	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
407	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
408	C₈	—	2	2	no	—
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
424	C₂²	C₃₁×#534	1	1	yes	characteristic
425	C₂²	C₃₁×#537	1	1	yes	characteristic
426	C₂²	C₃₁×#576	1	2	yes	—
427	C₂²	C₃₁×#577	1	2	yes	—
428	C₄	—	2	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	4	no	—
432	C₄	—	8	2	no	—
433	C₄	—	8	2	no	—
434	C₄	C₃₁×#518	1	1	yes	characteristic
435	C₄	C₃₁×#520	1	1	yes	characteristic
436	C₄	C₃₁×#538	1	1	yes	characteristic
437	C₄	C₃₁×#539	1	1	yes	characteristic
438	C₂	—	4	2	no	—
439	C₂	—	8	4	no	—
440	C₂	C₃₁×#2325	1	1	yes	characteristic
441	C₂	C₃₁×#2948	1	1	yes	characteristic
442	C₂	C₃₁×#2952	1	1	yes	characteristic
443	C₂	C₃₁×#3198	1	2	yes	—
444	C₂	C₃₁×#3257	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2324	C₂	semidirect
3. C₃₁×#1926	C₂	semidirect
4. C₃₁×#1918	C₂	semidirect
5. C₃₁×#1895	C₂	semidirect
6. C₃₁×#1879	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
16. C₃₁×#466	C₄	semidirect
48. C₂²×C₈×C₃₁	C₂×C₄	semidirect
53. C₃₁×#121	C₂×C₄	semidirect
72. C₃₁×#107	C₂×C₄	semidirect
77. C₃₁×#102	C₂×C₄	semidirect
138. #22193	C₃₁	direct (N × H)
190. #2324	C₂×C₃₁	semidirect
191. #1926	C₂×C₃₁	semidirect
192. #1918	C₂×C₃₁	semidirect
193. #1895	C₂×C₃₁	semidirect
194. #1879	C₂×C₃₁	semidirect
195. #1864	C₂×C₃₁	semidirect
196. #1807	C₂×C₃₁	semidirect
197. #1729	C₂×C₃₁	semidirect
231. #466	C₄×C₃₁	semidirect
270. C₂²×C₈	C₂×C₄×C₃₁	semidirect
275. #121	C₂×C₄×C₃₁	semidirect
294. #107	C₂×C₄×C₃₁	semidirect
299. #102	C₂×C₄×C₃₁	semidirect
303. C₃₁	#22193	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×#466	2
C₃₁×#2324	2
C₃₁×#1926	2
C₃₁×#1918	2
C₃₁×#1895	2
C₃₁×#1879	2
C₃₁×#1864	2
C₃₁×#1807	2
C₃₁×#1729	2
#22193	31

Subgroups — C31×#22193 #21478952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22193 #21478952