Subgroups — C₃₁×#22718 #21479477

Order 7936.

796 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₃₁	C₂	1	2	yes	maximal
2	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1950	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1855	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1855	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1838	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1811	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1738	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#436	—	2	2	no	—
18	C₃₁×#435	—	2	2	no	—
19	C₃₁×#435	—	2	2	no	—
20	C₃₁×#435	C₄	1	4	yes	—
21	C₃₁×#435	C₄	1	4	yes	—
22	C₈×D₄×C₃₁	—	2	2	no	—
23	C₂×C₃₁×#108	C₂²	1	2	yes	—
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₂×C₃₁×#108	C₂²	1	2	yes	—
26	C₃₁×#423	C₂²	1	2	yes	—
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃₁×#407	C₂²	1	2	yes	—
29	C₃₁×#407	C₂²	1	2	yes	—
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#341	C₂²	1	1	yes	characteristic
35	C₃₁×#341	C₂²	1	1	yes	characteristic
36	C₃₁×#341	C₂²	1	1	yes	characteristic
37	C₃₁×#341	C₂²	1	2	yes	—
38	C₃₁×#341	C₂²	1	2	yes	—
39	C₃₁×#341	C₂²	1	2	yes	—
40	C₃₁×#341	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	4	2	no	—
49	C₂×M₁₆×C₃₁	—	4	2	no	—
50	C₂×M₁₆×C₃₁	—	4	2	no	—
51	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	4	2	no	—
59	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×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	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₃₁	D₄	1	2	yes	—
68	C₄×D₄×C₃₁	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#108	—	2	4	no	—
74	C₃₁×#108	—	4	2	no	—
75	C₃₁×#108	—	4	2	no	—
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×C₂²⋊C₈	—	2	2	no	—
82	C₃₁×C₂²⋊C₈	—	2	2	no	—
83	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
84	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
85	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
86	C₃₁×#97	—	2	2	no	—
87	C₃₁×#97	C₂×C₄	1	4	yes	—
88	C₄×C₈×C₃₁	—	2	2	no	—
89	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
90	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
93	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	M₁₆×C₃₁	—	2	2	no	—
110	M₁₆×C₃₁	—	2	2	no	—
111	M₁₆×C₃₁	—	2	2	no	—
112	M₁₆×C₃₁	—	2	2	no	—
113	M₁₆×C₃₁	—	2	2	no	—
114	M₁₆×C₃₁	—	2	2	no	—
115	M₁₆×C₃₁	—	2	2	no	—
116	M₁₆×C₃₁	—	4	4	no	—
117	M₁₆×C₃₁	—	4	4	no	—
118	M₁₆×C₃₁	—	4	4	no	—
119	M₁₆×C₃₁	#41	1	4	yes	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	—	4	2	no	—
130	C₂×C₈×C₃₁	—	4	4	no	—
131	C₂×C₈×C₃₁	—	4	4	no	—
132	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
137	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
138	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
139	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
140	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
141	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
142	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
143	C₄²×C₃₁	C₂×D₄	1	2	yes	—
144	#22718	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
152	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	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	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	2	2	no	—
182	C₈×C₃₁	—	2	2	no	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	2	4	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₈×C₃₁	—	4	2	no	—
188	D₄×M₁₆	C₂×C₃₁	1	2	yes	—
189	C₂×#435	C₂×C₃₁	1	1	yes	characteristic
190	#1950	C₂×C₃₁	1	2	yes	complemented
191	#1855	C₂×C₃₁	1	2	yes	complemented
192	#1855	C₂×C₃₁	1	2	yes	complemented
193	#1838	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1811	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1778	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1738	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×#341	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	4	4	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	#430	1	2	yes	—
212	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₃₁	#554	1	2	yes	—
214	C₄×C₃₁	—	2	1	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
220	C₄×C₃₁	#518	1	1	yes	characteristic
221	C₄×C₃₁	#546	1	2	yes	—
222	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
223	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
224	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
225	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	#436	—	2	2	no	—
228	#435	—	2	2	no	—
229	#435	—	2	2	no	—
230	#435	C₄×C₃₁	1	4	yes	—
231	#435	C₄×C₃₁	1	4	yes	—
232	C₈×D₄	—	2	2	no	—
233	C₂×#108	C₂²×C₃₁	1	2	yes	—
234	C₂×#108	C₂²×C₃₁	1	2	yes	—
235	C₂×#108	C₂²×C₃₁	1	2	yes	—
236	#423	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	#407	C₂²×C₃₁	1	2	yes	—
239	#407	C₂²×C₃₁	1	2	yes	—
240	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
243	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
244	#341	C₂²×C₃₁	1	1	yes	characteristic
245	#341	C₂²×C₃₁	1	1	yes	characteristic
246	#341	C₂²×C₃₁	1	1	yes	characteristic
247	#341	C₂²×C₃₁	1	2	yes	—
248	#341	C₂²×C₃₁	1	2	yes	—
249	#341	C₂²×C₃₁	1	2	yes	—
250	#341	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₃₁	—	2	2	no	—
252	C₂×C₃₁	—	2	2	no	—
253	C₂×C₃₁	—	4	1	no	—
254	C₂×C₃₁	—	4	2	no	—
255	C₂×C₃₁	#2349	1	1	yes	characteristic
256	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
257	C₂×C₃₁	#2916	1	1	yes	characteristic
258	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
259	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
260	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	4	no	—
264	C₂×M₁₆	—	2	4	no	—
265	C₂×M₁₆	—	4	2	no	—
266	C₂×M₁₆	—	4	2	no	—
267	C₂×M₁₆	—	4	2	no	—
268	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
269	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
270	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
271	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
272	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	—	2	2	no	—
275	C₂²×C₈	—	4	2	no	—
276	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
278	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
283	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
284	C₄×D₄	D₄×C₃₁	1	2	yes	—
285	C₄×D₄	D₄×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	#108	—	2	4	no	—
291	#108	—	4	2	no	—
292	#108	—	4	2	no	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	2	2	no	—
300	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
301	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
302	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
303	#97	—	2	2	no	—
304	#97	C₂×C₄×C₃₁	1	4	yes	—
305	C₄×C₈	—	2	2	no	—
306	C₃₁	#22718	1	1	yes	characteristic, direct product factor, Sylow
307	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
308	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
311	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	2	no	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	2	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	4	4	no	—
335	M₁₆	—	4	4	no	—
336	M₁₆	—	4	4	no	—
337	M₁₆	#41×C₃₁	1	4	yes	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	4	2	no	—
347	C₂×C₈	—	4	2	no	—
348	C₂×C₈	—	4	4	no	—
349	C₂×C₈	—	4	4	no	—
350	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
352	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
353	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
354	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
355	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
356	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
357	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
358	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
359	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
360	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
361	C₄²	C₂×D₄×C₃₁	1	2	yes	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
369	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
371	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
372	D₄	—	2	4	no	—
373	D₄	—	4	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	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	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
393	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
396	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	2	4	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	1	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	8	2	no	—
417	C₂²	C₃₁×#430	1	2	yes	—
418	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
419	C₂²	C₃₁×#554	1	2	yes	—
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
426	C₄	C₃₁×#518	1	1	yes	characteristic
427	C₄	C₃₁×#546	1	2	yes	—
428	C₂	—	2	2	no	—
429	C₂	—	2	2	no	—
430	C₂	—	4	1	no	—
431	C₂	—	4	2	no	—
432	C₂	C₃₁×#2349	1	1	yes	characteristic
433	C₂	C₂×C₃₁×#430	1	1	yes	characteristic
434	C₂	C₃₁×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₃₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1950	C₂	semidirect
4. C₃₁×#1855	C₂	semidirect
5. C₃₁×#1855	C₂	semidirect
6. C₃₁×#1838	C₂	semidirect
7. C₃₁×#1811	C₂	semidirect
8. C₃₁×#1810	C₂	semidirect
9. C₃₁×#1778	C₂	semidirect
10. C₃₁×#1738	C₂	semidirect
11. C₂×C₃₁×#341	C₂	semidirect
144. #22718	C₃₁	direct (N × H)
190. #1950	C₂×C₃₁	semidirect
191. #1855	C₂×C₃₁	semidirect
192. #1855	C₂×C₃₁	semidirect
193. #1838	C₂×C₃₁	semidirect
194. #1811	C₂×C₃₁	semidirect
195. #1810	C₂×C₃₁	semidirect
196. #1778	C₂×C₃₁	semidirect
197. #1738	C₂×C₃₁	semidirect
198. C₂×#341	C₂×C₃₁	semidirect
306. C₃₁	#22718	direct (N × H)

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₃₁	2
C₂×C₃₁×#435	2
C₃₁×#1950	2
C₃₁×#1855	2
C₃₁×#1838	2
C₃₁×#1811	2
C₃₁×#1810	2
C₃₁×#1778	2
C₃₁×#1738	2
C₂×C₃₁×#341	2
#22718	31

Subgroups — C31×#22718 #21479477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22718 #21479477