Subgroups — C₃₁×#33147 #21489906

Order 7936.

892 conjugacy classes of subgroups.

The subgroup lattice has 408 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	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1988	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1948	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1919	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1853	C₂	1	2	yes	maximal
7	C₃₁×#1824	C₂	1	2	yes	maximal
8	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₃₁	—	2	1	no	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#106	C₂²	1	4	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₃₁×#391	—	2	1	no	—
19	C₃₁×#389	—	2	1	no	—
20	C₃₁×#389	C₂²	1	2	yes	—
21	C₃₁×#384	C₂²	1	1	yes	characteristic
22	C₃₁×#380	—	2	1	no	—
23	C₃₁×#380	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
25	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#95	—	2	1	no	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂³×C₄×C₃₁	—	2	1	no	—
35	C₂³×C₄×C₃₁	—	2	1	no	—
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
40	C₃₁×C₄⋊Q₈	—	2	4	no	—
41	C₃₁×C₂²⋊Q₈	—	4	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₃₁	Q₈	1	4	yes	complemented
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄²×C₃₁	—	2	4	no	—
57	C₂×C₄²×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	4	4	no	—
60	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#106	—	2	8	no	—
65	C₃₁×C₄⋊C₈	—	2	4	no	—
66	C₃₁×C₂²⋊C₈	—	2	4	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	—	2	2	no	—
71	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×Q₈×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	4	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	—	4	2	no	—
92	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₃₁	#41	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	2	4	no	—
105	C₄⋊C₄×C₃₁	—	2	4	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	—	4	4	no	—
114	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
115	C₄⋊C₄×C₃₁	#41	1	4	yes	—
116	C₂²⋊C₄×C₃₁	—	4	4	no	—
117	C₄²×C₃₁	—	2	2	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	4	4	no	—
121	C₄²×C₃₁	—	4	4	no	—
122	C₄²×C₃₁	#41	1	4	yes	—
123	#33147	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	—	2	1	no	—
125	C₂³×C₃₁	—	2	1	no	—
126	C₂³×C₃₁	—	2	1	no	—
127	C₂³×C₃₁	—	2	1	no	—
128	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
129	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
130	C₂³×C₃₁	#137	1	2	yes	—
131	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	4	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	8	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	4	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	4	4	no	—
148	C₂×C₄×C₃₁	—	4	8	no	—
149	C₂×C₄×C₃₁	—	4	8	no	—
150	C₂×C₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
161	C₂×C₄×C₃₁	#123	1	2	yes	—
162	C₂×C₄×C₃₁	#123	1	2	yes	—
163	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
165	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
170	C₈×C₃₁	—	4	4	no	—
171	#2857	C₂×C₃₁	1	1	yes	characteristic
172	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
173	#1988	C₂×C₃₁	1	2	yes	complemented
174	#1948	C₂×C₃₁	1	2	yes	complemented
175	#1919	C₂×C₃₁	1	4	yes	complemented
176	#1853	C₂×C₃₁	1	2	yes	—
177	#1824	C₂×C₃₁	1	2	yes	—
178	#1797	C₂×C₃₁	1	1	yes	characteristic
179	C₂²×C₃₁	—	2	2	no	—
180	C₂²×C₃₁	—	2	2	no	—
181	C₂²×C₃₁	—	2	2	no	—
182	C₂²×C₃₁	—	2	2	no	—
183	C₂²×C₃₁	—	2	2	no	—
184	C₂²×C₃₁	—	2	2	no	—
185	C₂²×C₃₁	#475	1	4	yes	—
186	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
188	C₂²×C₃₁	#525	1	1	yes	characteristic
189	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
190	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
192	C₂²×C₃₁	#575	1	1	yes	characteristic
193	C₄×C₃₁	—	2	2	no	—
194	C₄×C₃₁	—	2	2	no	—
195	C₄×C₃₁	—	2	2	no	—
196	C₄×C₃₁	—	4	4	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	4	no	—
199	C₄×C₃₁	—	8	2	no	—
200	C₄×C₃₁	—	8	2	no	—
201	C₄×C₃₁	#481	1	4	yes	—
202	C₄×C₃₁	#529	1	2	yes	—
203	C₄×C₃₁	#533	1	2	yes	—
204	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₂²⋊Q₈	—	2	1	no	—
206	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
207	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
208	C₂²×C₄²	—	2	1	no	—
209	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×#106	C₂²×C₃₁	1	4	yes	—
211	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
212	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
213	#391	—	2	1	no	—
214	#389	—	2	1	no	—
215	#389	C₂²×C₃₁	1	2	yes	—
216	#384	C₂²×C₃₁	1	1	yes	characteristic
217	#380	—	2	1	no	—
218	#380	C₂²×C₃₁	1	2	yes	—
219	C₄×C₄⋊C₄	—	2	1	no	—
220	C₄×C₄⋊C₄	—	2	1	no	—
221	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
222	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
223	C₂×#95	—	2	1	no	—
224	#340	C₂²×C₃₁	1	4	yes	—
225	#340	C₂²×C₃₁	1	4	yes	—
226	#340	C₂²×C₃₁	1	4	yes	—
227	#336	C₂²×C₃₁	1	2	yes	—
228	C₂×C₃₁	—	2	4	no	—
229	C₂×C₃₁	#2855	1	1	yes	characteristic
230	C₂×C₃₁	C₂×#475	1	1	yes	characteristic
231	C₂×C₃₁	#3077	1	1	yes	characteristic
232	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
233	C₂×C₃₁	#3095	1	1	yes	characteristic
234	C₂×C₃₁	#3103	1	1	yes	characteristic
235	C₂×C₃₁	#3110	1	1	yes	characteristic
236	C₂²×Q₈	—	2	1	no	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
240	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
243	C₄⋊Q₈	—	2	4	no	—
244	C₂²⋊Q₈	—	4	4	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	4	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	Q₈×C₃₁	1	4	yes	complemented
254	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	4	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	4	4	no	—
263	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	#106	—	2	8	no	—
268	C₄⋊C₈	—	2	4	no	—
269	C₂²⋊C₈	—	2	4	no	—
270	#95	—	2	2	no	—
271	#95	—	2	2	no	—
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	C₃₁	#33147	1	1	yes	characteristic, direct product factor, Sylow
275	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
297	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
303	C₂²×C₄	#41×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	2	4	no	—
307	C₂×C₈	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
319	C₄⋊C₄	#41×C₃₁	1	4	yes	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	4	4	no	—
325	C₄²	—	4	4	no	—
326	C₄²	#41×C₃₁	1	4	yes	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	—	2	1	no	—
331	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
332	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
333	C₂³	C₃₁×#137	1	2	yes	—
334	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
335	Q₈	—	8	4	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
364	C₂×C₄	C₃₁×#123	1	2	yes	—
365	C₂×C₄	C₃₁×#123	1	2	yes	—
366	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
367	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
368	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
369	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
370	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
372	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
373	C₈	—	4	4	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	C₃₁×#475	1	4	yes	—
381	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
382	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
383	C₂²	C₃₁×#525	1	1	yes	characteristic
384	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
385	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
386	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
387	C₂²	C₃₁×#575	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	C₃₁×#481	1	4	yes	—
397	C₄	C₃₁×#529	1	2	yes	—
398	C₄	C₃₁×#533	1	2	yes	—
399	C₂	—	2	4	no	—
400	C₂	C₃₁×#2855	1	1	yes	characteristic
401	C₂	C₂×C₃₁×#475	1	1	yes	characteristic
402	C₂	C₃₁×#3077	1	1	yes	characteristic
403	C₂	C₂×C₃₁×#481	1	1	yes	characteristic
404	C₂	C₃₁×#3095	1	1	yes	characteristic
405	C₂	C₃₁×#3103	1	1	yes	characteristic
406	C₂	C₃₁×#3110	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×#1948	C₂	semidirect
5. C₃₁×#1919	C₂	semidirect
50. C₂×C₄⋊C₄×C₃₁	Q₈	semidirect
123. #33147	C₃₁	direct (N × H)
173. #1988	C₂×C₃₁	semidirect
174. #1948	C₂×C₃₁	semidirect
175. #1919	C₂×C₃₁	semidirect
253. C₂×C₄⋊C₄	Q₈×C₃₁	semidirect
274. C₃₁	#33147	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
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1988	2
C₃₁×#1948	2
C₃₁×#1919	2
C₃₁×#1853	2
C₃₁×#1824	2
C₃₁×#1797	2
#33147	31

Subgroups — C31×#33147 #21489906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33147 #21489906