Subgroups — C₃₁×#26318 #21483077

Order 7936.

802 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2094	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2071	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	—	2	1	no	—
14	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	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	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#394	—	2	2	no	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#389	—	2	2	no	—
31	C₃₁×#384	C₂²	1	2	yes	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	2	no	—
50	C₃₁×#124	—	4	4	no	—
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	4	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
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	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	1	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×#106	—	2	2	no	—
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×#106	—	4	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	—	2	4	no	—
80	C₃₁×#102	—	4	2	no	—
81	C₃₁×#102	—	4	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₂⁴×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	8	1	no	—
101	C₂×D₄×C₃₁	—	8	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	#41	1	2	yes	—
121	C₄⋊C₄×C₃₁	—	2	4	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	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	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	8	1	no	—
135	C₄⋊C₄×C₃₁	—	8	2	no	—
136	C₄⋊C₄×C₃₁	#41	1	2	yes	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₂²⋊C₄×C₃₁	—	8	2	no	—
140	C₂²⋊C₄×C₃₁	—	8	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	4	4	no	—
144	C₄²×C₃₁	—	4	2	no	—
145	C₄²×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	#41	1	2	yes	—
147	C₄²×C₃₁	#41	1	2	yes	—
148	#26318	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	8	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	8	4	no	—
163	D₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	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	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	#123	1	2	yes	—
187	C₂×C₄×C₃₁	#123	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
189	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
190	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	2	no	—
200	#2548	C₂×C₃₁	1	1	yes	characteristic
201	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2094	C₂×C₃₁	1	2	yes	complemented
203	#2071	C₂×C₃₁	1	2	yes	complemented
204	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1982	C₂×C₃₁	1	1	yes	characteristic
206	#1957	C₂×C₃₁	1	1	yes	characteristic
207	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1926	C₂×C₃₁	1	2	yes	complemented
209	#1926	C₂×C₃₁	1	2	yes	complemented
210	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
219	C₂²×C₃₁	#548	1	1	yes	characteristic
220	C₂²×C₃₁	#550	1	1	yes	characteristic
221	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
222	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₃₁	#576	1	1	yes	characteristic
224	C₂²×C₃₁	#577	1	1	yes	characteristic
225	C₄×C₃₁	—	2	4	no	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	—	8	1	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	#481	1	4	yes	—
234	C₄×C₃₁	#529	1	1	yes	characteristic
235	C₄×C₃₁	#542	1	1	yes	characteristic
236	C₄×C₃₁	#556	1	1	yes	characteristic
237	C₄×C₃₁	#559	1	1	yes	characteristic
238	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#124	—	2	1	no	—
240	C₂×C₄×Q₈	—	2	1	no	—
241	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#106	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#102	C₂²×C₃₁	1	2	yes	—
247	C₂×#102	C₂²×C₃₁	1	2	yes	—
248	C₂×#102	C₂²×C₃₁	1	2	yes	—
249	C₂×#102	C₂²×C₃₁	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
251	#394	—	2	2	no	—
252	#390	—	2	1	no	—
253	#390	C₂²×C₃₁	1	2	yes	—
254	#390	C₂²×C₃₁	1	2	yes	—
255	#389	—	2	1	no	—
256	#389	—	2	2	no	—
257	#384	C₂²×C₃₁	1	2	yes	—
258	#384	C₂²×C₃₁	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
260	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	#340	C₂²×C₃₁	1	1	yes	characteristic
263	#340	C₂²×C₃₁	1	2	yes	—
264	#340	C₂²×C₃₁	1	1	yes	characteristic
265	#336	C₂²×C₃₁	1	2	yes	—
266	#336	C₂²×C₃₁	1	2	yes	—
267	C₂×C₃₁	—	16	2	no	—
268	C₂×C₃₁	#2448	1	1	yes	characteristic
269	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
270	C₂×C₃₁	#3096	1	1	yes	characteristic
271	C₂×C₃₁	#3362	1	1	yes	characteristic
272	C₂×C₃₁	#3373	1	1	yes	characteristic
273	C₂×C₃₁	#3392	1	1	yes	characteristic
274	C₂×C₃₁	#3399	1	1	yes	characteristic
275	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
276	C₂²×D₄	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	#127	—	2	2	no	—
283	#127	—	2	2	no	—
284	#124	—	4	4	no	—
285	C₄×Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	2	no	—
287	C₄×Q₈	—	4	4	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	1	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
307	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
308	#106	—	2	2	no	—
309	#106	—	2	2	no	—
310	#106	—	4	2	no	—
311	C₄⋊C₈	—	2	2	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#102	—	4	2	no	—
315	#102	—	4	2	no	—
316	C₄×C₈	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	1	no	—
320	C₃₁	#26318	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	4	2	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	8	1	no	—
336	C₂×D₄	—	8	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	#41×C₃₁	1	2	yes	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	8	1	no	—
370	C₄⋊C₄	—	8	2	no	—
371	C₄⋊C₄	#41×C₃₁	1	2	yes	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	8	2	no	—
375	C₂²⋊C₄	—	8	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	4	4	no	—
379	C₄²	—	4	2	no	—
380	C₄²	—	4	2	no	—
381	C₄²	#41×C₃₁	1	2	yes	—
382	C₄²	#41×C₃₁	1	2	yes	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	Q₈	—	2	8	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	8	4	no	—
397	D₄	—	8	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	C₃₁×#123	1	2	yes	—
421	C₂×C₄	C₃₁×#123	1	2	yes	—
422	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
423	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
424	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
425	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
430	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
442	C₂²	C₃₁×#548	1	1	yes	characteristic
443	C₂²	C₃₁×#550	1	1	yes	characteristic
444	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
445	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
446	C₂²	C₃₁×#576	1	1	yes	characteristic
447	C₂²	C₃₁×#577	1	1	yes	characteristic
448	C₄	—	2	4	no	—
449	C₄	—	2	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	2	no	—
455	C₄	—	8	1	no	—
456	C₄	C₃₁×#481	1	4	yes	—
457	C₄	C₃₁×#529	1	1	yes	characteristic
458	C₄	C₃₁×#542	1	1	yes	characteristic
459	C₄	C₃₁×#556	1	1	yes	characteristic
460	C₄	C₃₁×#559	1	1	yes	characteristic
461	C₂	—	16	2	no	—
462	C₂	C₃₁×#2448	1	1	yes	characteristic
463	C₂	C₂×C₃₁×#481	1	1	yes	characteristic
464	C₂	C₃₁×#3096	1	1	yes	characteristic
465	C₂	C₃₁×#3362	1	1	yes	characteristic
466	C₂	C₃₁×#3373	1	1	yes	characteristic
467	C₂	C₃₁×#3392	1	1	yes	characteristic
468	C₂	C₃₁×#3399	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
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
3. C₃₁×#2094	C₂	semidirect
4. C₃₁×#2071	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
8. C₃₁×#1948	C₂	semidirect
9. C₃₁×#1926	C₂	semidirect
10. C₃₁×#1926	C₂	semidirect
11. C₄×C₃₁×#106	C₂	semidirect
148. #26318	C₃₁	direct (N × H)
201. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
202. #2094	C₂×C₃₁	semidirect
203. #2071	C₂×C₃₁	semidirect
204. #1984	C₂×C₃₁	semidirect
207. #1948	C₂×C₃₁	semidirect
208. #1926	C₂×C₃₁	semidirect
209. #1926	C₂×C₃₁	semidirect
210. C₄×#106	C₂×C₃₁	semidirect
320. C₃₁	#26318	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×#2548	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2094	2
C₃₁×#2071	2
C₃₁×#1984	2
C₃₁×#1982	2
C₃₁×#1957	2
C₃₁×#1948	2
C₃₁×#1926	2
C₄×C₃₁×#106	2
#26318	31

Subgroups — C31×#26318 #21483077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26318 #21483077