Subgroups — C₃₁×#33154 #21489913

Order 7936.

804 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1920	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1854	C₂	1	2	yes	maximal
10	C₃₁×#1825	C₂	1	2	yes	maximal
11	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃₁×#391	—	2	1	no	—
23	C₃₁×#389	—	2	1	no	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₃₁×#380	—	2	1	no	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
33	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
34	C₃₁×#376	—	2	1	no	—
35	C₄³×C₃₁	—	2	1	no	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	4	no	—
52	C₃₁×#117	—	2	2	no	—
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	Q₈	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
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	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×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	2	4	no	—
82	C₃₁×#106	—	2	4	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	—	2	4	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×Q₈×C₃₁	—	4	2	no	—
94	C₂×Q₈×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	1	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	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	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
140	C₄⋊C₄×C₃₁	#41	1	2	yes	—
141	C₄⋊C₄×C₃₁	#41	1	2	yes	—
142	C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂²⋊C₄×C₃₁	—	2	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₄²×C₃₁	—	2	4	no	—
146	C₄²×C₃₁	—	2	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	4	2	no	—
152	C₄²×C₃₁	—	4	2	no	—
153	C₄²×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	4	2	no	—
155	C₄²×C₃₁	#41	1	2	yes	—
156	C₄²×C₃₁	#41	1	2	yes	—
157	#33154	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
165	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	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	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	4	4	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
201	C₂×C₄×C₃₁	#123	1	2	yes	—
202	C₂×C₄×C₃₁	#123	1	2	yes	—
203	C₂×C₄×C₃₁	#135	1	2	yes	—
204	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₃₁	#137	1	2	yes	—
206	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₈×C₃₁	—	4	2	no	—
213	C₈×C₃₁	—	4	2	no	—
214	#2857	C₂×C₃₁	1	1	yes	characteristic
215	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
216	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1921	C₂×C₃₁	1	2	yes	complemented
221	#1920	C₂×C₃₁	1	2	yes	complemented
222	#1854	C₂×C₃₁	1	2	yes	—
223	#1825	C₂×C₃₁	1	2	yes	—
224	#1797	C₂×C₃₁	1	1	yes	characteristic
225	C₂²×C₃₁	—	2	2	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	1	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
234	C₂²×C₃₁	#525	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
236	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
238	C₂²×C₃₁	#575	1	1	yes	characteristic
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	1	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	8	2	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	#482	1	4	yes	—
250	C₄×C₃₁	#529	1	1	yes	characteristic
251	C₄×C₃₁	#529	1	1	yes	characteristic
252	C₄×C₃₁	#533	1	1	yes	characteristic
253	C₄×C₃₁	#533	1	1	yes	characteristic
254	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₂²⋊Q₈	—	2	1	no	—
256	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#107	C₂²×C₃₁	1	2	yes	—
260	C₂×#106	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
264	#391	—	2	1	no	—
265	#389	—	2	1	no	—
266	#389	C₂²×C₃₁	1	2	yes	—
267	#384	C₂²×C₃₁	1	1	yes	characteristic
268	#380	—	2	1	no	—
269	#380	C₂²×C₃₁	1	2	yes	—
270	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
271	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₄×C₂²⋊C₄	—	2	1	no	—
275	C₄×C₂²⋊C₄	—	2	1	no	—
276	#376	—	2	1	no	—
277	C₄³	—	2	1	no	—
278	#340	C₂²×C₃₁	1	2	yes	—
279	#340	C₂²×C₃₁	1	2	yes	—
280	#340	C₂²×C₃₁	1	2	yes	—
281	#340	C₂²×C₃₁	1	2	yes	—
282	#340	C₂²×C₃₁	1	2	yes	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	C₂×C₃₁	—	4	2	no	—
286	C₂×C₃₁	#2855	1	1	yes	characteristic
287	C₂×C₃₁	#3076	1	1	yes	characteristic
288	C₂×C₃₁	#3079	1	1	yes	characteristic
289	C₂×C₃₁	C₂×#482	1	1	yes	characteristic
290	C₂×C₃₁	#3095	1	1	yes	characteristic
291	C₂×C₃₁	#3107	1	1	yes	characteristic
292	C₂×C₃₁	#3115	1	1	yes	characteristic
293	C₂²×Q₈	—	2	1	no	—
294	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
295	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	C₄⋊Q₈	—	2	2	no	—
300	C₄⋊Q₈	—	2	2	no	—
301	C₂²⋊Q₈	—	4	4	no	—
302	#117	—	2	2	no	—
303	#117	—	2	2	no	—
304	#117	—	2	2	no	—
305	#117	Q₈×C₃₁	1	4	yes	complemented
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	#107	—	2	4	no	—
332	#106	—	2	4	no	—
333	C₄⋊C₈	—	2	2	no	—
334	C₄⋊C₈	—	2	2	no	—
335	C₂²⋊C₈	—	2	4	no	—
336	#95	—	2	1	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	#95	—	2	2	no	—
341	#95	—	2	2	no	—
342	C₃₁	#33154	1	1	yes	characteristic, direct product factor, Sylow
343	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×Q₈	—	4	2	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	4	1	no	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
369	C₂²×C₄	#41×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
390	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
391	C₄⋊C₄	#41×C₃₁	1	2	yes	—
392	C₄⋊C₄	#41×C₃₁	1	2	yes	—
393	C₂²⋊C₄	—	2	4	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	2	4	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	4	2	no	—
403	C₄²	—	4	2	no	—
404	C₄²	—	4	2	no	—
405	C₄²	—	4	2	no	—
406	C₄²	#41×C₃₁	1	2	yes	—
407	C₄²	#41×C₃₁	1	2	yes	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
415	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	Q₈	—	8	4	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
451	C₂×C₄	C₃₁×#123	1	2	yes	—
452	C₂×C₄	C₃₁×#123	1	2	yes	—
453	C₂×C₄	C₃₁×#135	1	2	yes	—
454	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
455	C₂×C₄	C₃₁×#137	1	2	yes	—
456	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
472	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
473	C₂²	C₃₁×#525	1	1	yes	characteristic
474	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
475	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
476	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
477	C₂²	C₃₁×#575	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	2	no	—
487	C₄	—	8	2	no	—
488	C₄	C₃₁×#482	1	4	yes	—
489	C₄	C₃₁×#529	1	1	yes	characteristic
490	C₄	C₃₁×#529	1	1	yes	characteristic
491	C₄	C₃₁×#533	1	1	yes	characteristic
492	C₄	C₃₁×#533	1	1	yes	characteristic
493	C₂	—	4	2	no	—
494	C₂	C₃₁×#2855	1	1	yes	characteristic
495	C₂	C₃₁×#3076	1	1	yes	characteristic
496	C₂	C₃₁×#3079	1	1	yes	characteristic
497	C₂	C₂×C₃₁×#482	1	1	yes	characteristic
498	C₂	C₃₁×#3095	1	1	yes	characteristic
499	C₂	C₃₁×#3107	1	1	yes	characteristic
500	C₂	C₃₁×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1988	C₂	semidirect
4. C₃₁×#1988	C₂	semidirect
5. C₃₁×#1948	C₂	semidirect
6. C₃₁×#1947	C₂	semidirect
7. C₃₁×#1921	C₂	semidirect
8. C₃₁×#1920	C₂	semidirect
55. C₃₁×#117	Q₈	semidirect
157. #33154	C₃₁	direct (N × H)
216. #1988	C₂×C₃₁	semidirect
217. #1988	C₂×C₃₁	semidirect
218. #1948	C₂×C₃₁	semidirect
219. #1947	C₂×C₃₁	semidirect
220. #1921	C₂×C₃₁	semidirect
221. #1920	C₂×C₃₁	semidirect
305. #117	Q₈×C₃₁	semidirect
342. C₃₁	#33154	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×#2857	2
C₄×C₃₁×#117	2
C₃₁×#1988	2
C₃₁×#1948	2
C₃₁×#1947	2
C₃₁×#1921	2
C₃₁×#1920	2
C₃₁×#1854	2
C₃₁×#1825	2
C₃₁×#1797	2
#33154	31

Subgroups — C31×#33154 #21489913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33154 #21489913