Subgroups — C₃₁×#23094 #21479853

Order 7936.

902 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1968	C₂	1	2	yes	maximal
5	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1927	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1880	C₂	1	2	yes	maximal
9	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#103	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	—	2	1	no	—
17	C₃₁×#461	C₄	1	8	yes	complemented
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#391	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#388	—	2	2	no	—
31	C₃₁×#387	—	2	2	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#380	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂×SD₁₆×C₃₁	—	4	2	no	—
46	C₂×SD₁₆×C₃₁	—	4	2	no	—
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
53	C₄×Q₈×C₃₁	—	2	4	no	—
54	C₄×Q₈×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
57	C₃₁×#117	—	4	4	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#106	—	4	2	no	—
78	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	4	2	no	—
81	C₃₁×#103	C₂×C₄	1	8	yes	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₄×C₈×C₃₁	—	4	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	D₄	1	2	yes	—
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₃₁	—	4	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₃₁	#41	1	2	yes	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	4	4	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	8	1	no	—
139	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
140	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	#41	1	2	yes	—
142	C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	8	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	4	4	no	—
148	C₄²×C₃₁	—	4	4	no	—
149	C₄²×C₃₁	—	4	2	no	—
150	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
151	C₄²×C₃₁	#41	1	2	yes	—
152	#23094	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₃₁	—	2	4	no	—
162	Q₈×C₃₁	—	2	4	no	—
163	Q₈×C₃₁	—	4	2	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	4	1	no	—
168	D₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	8	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
195	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
196	C₂×C₄×C₃₁	#135	1	2	yes	—
197	C₂×C₄×C₃₁	C₂³×C₄	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	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
203	C₈×C₃₁	—	2	4	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
207	#2365	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2354	C₂×C₃₁	1	1	yes	characteristic
209	#1968	C₂×C₃₁	1	2	yes	—
210	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1927	C₂×C₃₁	1	2	yes	complemented
213	#1880	C₂×C₃₁	1	2	yes	—
214	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₄×#103	C₂×C₃₁	1	2	yes	complemented
216	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	8	1	no	—
221	C₂²×C₃₁	—	8	1	no	—
222	C₂²×C₃₁	—	8	1	no	—
223	C₂²×C₃₁	—	8	1	no	—
224	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	#546	1	1	yes	characteristic
226	C₂²×C₃₁	#550	1	1	yes	characteristic
227	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
228	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
229	C₂²×C₃₁	#577	1	1	yes	characteristic
230	C₂²×C₃₁	#578	1	1	yes	characteristic
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	4	4	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	4	2	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	8	2	no	—
238	C₄×C₃₁	—	8	1	no	—
239	C₄×C₃₁	C₄×SD₁₆	1	4	yes	—
240	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
241	C₄×C₃₁	#518	1	1	yes	characteristic
242	C₄×C₃₁	#553	1	1	yes	characteristic
243	C₄×C₃₁	#555	1	1	yes	characteristic
244	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#117	—	2	1	no	—
249	#461	C₄×C₃₁	1	8	yes	complemented
250	C₂×#106	C₂²×C₃₁	1	2	yes	—
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#103	C₂²×C₃₁	1	2	yes	—
253	C₂×#103	C₂²×C₃₁	1	2	yes	—
254	C₂×#103	C₂²×C₃₁	1	2	yes	—
255	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
259	#391	C₂²×C₃₁	1	2	yes	—
260	#390	C₂²×C₃₁	1	2	yes	—
261	#389	—	2	1	no	—
262	#388	—	2	2	no	—
263	#387	—	2	2	no	—
264	#384	C₂²×C₃₁	1	1	yes	characteristic
265	#380	—	2	1	no	—
266	C₄×C₄⋊C₄	—	2	1	no	—
267	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
268	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	#340	C₂²×C₃₁	1	2	yes	—
271	#340	C₂²×C₃₁	1	1	yes	characteristic
272	#340	C₂²×C₃₁	1	1	yes	characteristic
273	#336	C₂²×C₃₁	1	2	yes	—
274	C₂×C₃₁	—	8	2	no	—
275	C₂×C₃₁	#2363	1	1	yes	characteristic
276	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
277	C₂×C₃₁	#2947	1	1	yes	characteristic
278	C₂×C₃₁	#3298	1	1	yes	characteristic
279	C₂×C₃₁	#3303	1	1	yes	characteristic
280	C₂×C₃₁	#3392	1	1	yes	characteristic
281	C₂×C₃₁	#3393	1	1	yes	characteristic
282	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
283	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	C₂×SD₁₆	—	4	2	no	—
286	C₂×SD₁₆	—	4	2	no	—
287	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	—
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
293	C₄×Q₈	—	2	4	no	—
294	C₄×Q₈	—	4	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
297	#117	—	4	4	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
316	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	#106	—	4	2	no	—
318	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
319	#103	—	2	4	no	—
320	#103	—	4	2	no	—
321	#103	C₂×C₄×C₃₁	1	8	yes	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	C₄×C₈	—	2	2	no	—
325	C₄×C₈	—	4	2	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	#95	—	2	1	no	—
329	#95	—	2	2	no	—
330	#95	D₄×C₃₁	1	2	yes	—
331	C₃₁	#23094	1	1	yes	characteristic, direct product factor, Sylow
332	C₂⁴	—	2	1	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
358	SD₁₆	—	4	8	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
367	C₂×C₈	#41×C₃₁	1	2	yes	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	8	1	no	—
380	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
381	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
382	C₄⋊C₄	#41×C₃₁	1	2	yes	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	8	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	4	4	no	—
389	C₄²	—	4	4	no	—
390	C₄²	—	4	2	no	—
391	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
392	C₄²	#41×C₃₁	1	2	yes	—
393	C₂³	—	2	2	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	Q₈	—	2	4	no	—
402	Q₈	—	2	4	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	4	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	4	1	no	—
408	D₄	—	4	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	8	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
434	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
435	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
436	C₂×C₄	C₃₁×#135	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
442	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
443	C₈	—	2	4	no	—
444	C₈	—	2	2	no	—
445	C₈	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
454	C₂²	C₃₁×#546	1	1	yes	characteristic
455	C₂²	C₃₁×#550	1	1	yes	characteristic
456	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
457	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
458	C₂²	C₃₁×#577	1	1	yes	characteristic
459	C₂²	C₃₁×#578	1	1	yes	characteristic
460	C₄	—	2	2	no	—
461	C₄	—	2	4	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	8	2	no	—
467	C₄	—	8	1	no	—
468	C₄	C₄×SD₁₆×C₃₁	1	4	yes	—
469	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
470	C₄	C₃₁×#518	1	1	yes	characteristic
471	C₄	C₃₁×#553	1	1	yes	characteristic
472	C₄	C₃₁×#555	1	1	yes	characteristic
473	C₂	—	8	2	no	—
474	C₂	C₃₁×#2363	1	1	yes	characteristic
475	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
476	C₂	C₃₁×#2947	1	1	yes	characteristic
477	C₂	C₃₁×#3298	1	1	yes	characteristic
478	C₂	C₃₁×#3303	1	1	yes	characteristic
479	C₂	C₃₁×#3392	1	1	yes	characteristic
480	C₂	C₃₁×#3393	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2365	C₂	semidirect
5. C₃₁×#1956	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1927	C₂	semidirect
9. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₄×C₃₁×#103	C₂	semidirect
17. C₃₁×#461	C₄	semidirect
139. C₄⋊C₄×C₃₁	SD₁₆	semidirect
152. #23094	C₃₁	direct (N × H)
207. #2365	C₂×C₃₁	semidirect
210. #1956	C₂×C₃₁	semidirect
211. #1948	C₂×C₃₁	semidirect
212. #1927	C₂×C₃₁	semidirect
214. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
215. C₄×#103	C₂×C₃₁	semidirect
249. #461	C₄×C₃₁	semidirect
331. C₃₁	#23094	direct (N × H)
380. C₄⋊C₄	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#461	2
C₃₁×#2365	2
C₃₁×#2354	2
C₃₁×#1968	2
C₃₁×#1956	2
C₃₁×#1948	2
C₃₁×#1927	2
C₃₁×#1880	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×#103	2
C₄×C₃₁×#102	2
#23094	31

Subgroups — C31×#23094 #21479853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23094 #21479853