Subgroups — C₃₁×#25056 #21481815

Order 7936.

766 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2538	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2093	C₂	1	2	yes	maximal
4	C₃₁×#2083	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal
8	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#396	—	2	2	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	—	2	2	no	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#384	C₂²	1	2	yes	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#380	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	—	2	1	no	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₂²⋊Q₈	—	4	4	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×Q₈×C₃₁	—	4	4	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#107	—	2	4	no	—
75	C₃₁×#107	—	4	2	no	—
76	C₃₁×#107	—	4	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#102	—	4	2	no	—
80	C₃₁×#102	Q₈	1	4	yes	complemented
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₂⁴×C₃₁	—	2	1	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	D₈	1	4	yes	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	4	2	no	—
101	C₂²×C₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	—	4	2	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₃₁	#41	1	2	yes	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	4	4	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	4	no	—
132	C₄⋊C₄×C₃₁	—	8	1	no	—
133	C₄⋊C₄×C₃₁	—	8	1	no	—
134	C₄⋊C₄×C₃₁	—	8	2	no	—
135	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	8	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	4	4	no	—
142	C₄²×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	#41	1	2	yes	—
144	C₄²×C₃₁	#41	1	2	yes	—
145	#25056	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	8	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	4	1	no	—
158	D₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	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	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	8	4	no	—
182	C₂×C₄×C₃₁	#123	1	2	yes	—
183	C₂×C₄×C₃₁	#123	1	2	yes	—
184	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
185	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
186	C₂×C₄×C₃₁	#137	1	2	yes	—
187	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	4	2	no	—
196	#2538	C₂×C₃₁	1	1	yes	characteristic
197	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2093	C₂×C₃₁	1	2	yes	—
199	#2083	C₂×C₃₁	1	2	yes	complemented
200	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1925	C₂×C₃₁	1	2	yes	—
203	#1918	C₂×C₃₁	1	2	yes	complemented
204	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1802	C₂×C₃₁	1	1	yes	characteristic
206	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
215	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
216	C₂²×C₃₁	#550	1	1	yes	characteristic
217	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
218	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
219	C₂²×C₃₁	#576	1	1	yes	characteristic
220	C₂²×C₃₁	#578	1	1	yes	characteristic
221	C₄×C₃₁	—	2	4	no	—
222	C₄×C₃₁	—	2	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	8	1	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	—	8	2	no	—
229	C₄×C₃₁	#480	1	4	yes	—
230	C₄×C₃₁	#533	1	1	yes	characteristic
231	C₄×C₃₁	#542	1	1	yes	characteristic
232	C₄×C₃₁	#552	1	1	yes	characteristic
233	C₄×C₃₁	#554	1	1	yes	characteristic
234	C₂×C₂²⋊Q₈	—	2	1	no	—
235	C₂×C₄×Q₈	—	2	1	no	—
236	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#107	C₂²×C₃₁	1	2	yes	—
239	C₂×#107	C₂²×C₃₁	1	2	yes	—
240	C₂×#107	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#102	C₂²×C₃₁	1	2	yes	—
244	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	#396	—	2	2	no	—
247	#390	C₂²×C₃₁	1	2	yes	—
248	#389	—	2	1	no	—
249	#389	—	2	2	no	—
250	#389	C₂²×C₃₁	1	2	yes	—
251	#384	C₂²×C₃₁	1	1	yes	characteristic
252	#384	C₂²×C₃₁	1	2	yes	—
253	#384	C₂²×C₃₁	1	1	yes	characteristic
254	#380	—	2	1	no	—
255	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
256	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
257	#340	C₂²×C₃₁	1	2	yes	—
258	#340	C₂²×C₃₁	1	1	yes	characteristic
259	#340	C₂²×C₃₁	1	1	yes	characteristic
260	#340	C₂²×C₃₁	1	2	yes	—
261	#336	C₂²×C₃₁	1	2	yes	—
262	#336	C₂²×C₃₁	1	2	yes	—
263	C₂×C₃₁	—	8	2	no	—
264	C₂×C₃₁	#2423	1	1	yes	characteristic
265	C₂×C₃₁	C₂×#480	1	1	yes	characteristic
266	C₂×C₃₁	#3098	1	1	yes	characteristic
267	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
268	C₂×C₃₁	#3385	1	1	yes	characteristic
269	C₂×C₃₁	#3391	1	1	yes	characteristic
270	C₂×C₃₁	#3398	1	1	yes	characteristic
271	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
272	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
273	C₂³×C₄	—	2	1	no	—
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²⋊Q₈	—	4	4	no	—
278	C₄×Q₈	—	2	2	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	—	4	4	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	#107	—	2	4	no	—
305	#107	—	4	2	no	—
306	#107	—	4	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	4	2	no	—
310	#102	Q₈×C₃₁	1	4	yes	complemented
311	C₄×C₈	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	2	no	—
315	C₃₁	#25056	1	1	yes	characteristic, direct product factor, Sylow
316	C₂⁴	—	2	1	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	D₈×C₃₁	1	4	yes	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
347	C₂×C₈	#41×C₃₁	1	2	yes	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	4	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	4	no	—
363	C₄⋊C₄	—	8	1	no	—
364	C₄⋊C₄	—	8	1	no	—
365	C₄⋊C₄	—	8	2	no	—
366	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	4	no	—
369	C₂²⋊C₄	—	8	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	4	no	—
373	C₄²	—	4	2	no	—
374	C₄²	#41×C₃₁	1	2	yes	—
375	C₄²	#41×C₃₁	1	2	yes	—
376	C₂³	—	2	2	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	1	no	—
380	C₂³	—	4	1	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
384	Q₈	—	2	8	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	1	no	—
388	D₄	—	4	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	8	4	no	—
412	C₂×C₄	C₃₁×#123	1	2	yes	—
413	C₂×C₄	C₃₁×#123	1	2	yes	—
414	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
415	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
416	C₂×C₄	C₃₁×#137	1	2	yes	—
417	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
422	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
423	C₈	—	2	2	no	—
424	C₈	—	2	2	no	—
425	C₈	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	1	no	—
433	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
434	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
435	C₂²	C₃₁×#550	1	1	yes	characteristic
436	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
437	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
438	C₂²	C₃₁×#576	1	1	yes	characteristic
439	C₂²	C₃₁×#578	1	1	yes	characteristic
440	C₄	—	2	4	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	8	1	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	2	no	—
448	C₄	C₃₁×#480	1	4	yes	—
449	C₄	C₃₁×#533	1	1	yes	characteristic
450	C₄	C₃₁×#542	1	1	yes	characteristic
451	C₄	C₃₁×#552	1	1	yes	characteristic
452	C₄	C₃₁×#554	1	1	yes	characteristic
453	C₂	—	8	2	no	—
454	C₂	C₃₁×#2423	1	1	yes	characteristic
455	C₂	C₂×C₃₁×#480	1	1	yes	characteristic
456	C₂	C₃₁×#3098	1	1	yes	characteristic
457	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
458	C₂	C₃₁×#3385	1	1	yes	characteristic
459	C₂	C₃₁×#3391	1	1	yes	characteristic
460	C₂	C₃₁×#3398	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#2083	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1947	C₂	semidirect
8. C₃₁×#1918	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
80. C₃₁×#102	Q₈	semidirect
145. #25056	C₃₁	direct (N × H)
197. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
199. #2083	C₂×C₃₁	semidirect
200. #1984	C₂×C₃₁	semidirect
201. #1947	C₂×C₃₁	semidirect
203. #1918	C₂×C₃₁	semidirect
204. #1845	C₂×C₃₁	semidirect
310. #102	Q₈×C₃₁	semidirect
315. C₃₁	#25056	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×#2538	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2093	2
C₃₁×#2083	2
C₃₁×#1984	2
C₃₁×#1947	2
C₃₁×#1925	2
C₃₁×#1918	2
C₃₁×#1845	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
#25056	31

Subgroups — C31×#25056 #21481815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25056 #21481815