Subgroups — C₃₁×#3563 #16843496

Order 3968.

912 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#564	C₂	1	2	yes	maximal, complemented
2	C₃₁×#563	C₂	1	2	yes	maximal, complemented
3	C₃₁×#557	C₂	1	2	yes	maximal, complemented
4	C₃₁×#554	C₂	1	2	yes	maximal, complemented
5	C₃₁×#552	C₂	1	4	yes	maximal, complemented
6	C₃₁×#552	C₂	1	2	yes	maximal, complemented
7	C₃₁×#552	C₂	1	2	yes	maximal, complemented
8	C₃₁×#544	C₂	1	2	yes	maximal, complemented
9	C₃₁×#544	C₂	1	2	yes	maximal, complemented
10	C₃₁×#541	C₂	1	2	yes	maximal, complemented
11	C₃₁×#541	C₂	1	4	yes	maximal, complemented
12	C₃₁×#539	C₂	1	2	yes	maximal
13	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#529	C₂	1	1	yes	maximal, characteristic
15	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
19	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
21	C₃₁×#126	C₂²	1	2	yes	—
22	C₃₁×#126	C₂²	1	4	yes	—
23	C₃₁×#126	C₂²	1	4	yes	—
24	C₃₁×#126	C₂²	1	4	yes	—
25	C₃₁×#126	C₂²	1	4	yes	—
26	C₃₁×#125	C₂²	1	2	yes	—
27	C₃₁×#125	C₂²	1	2	yes	—
28	C₃₁×#125	C₂²	1	2	yes	—
29	C₃₁×#125	C₂²	1	2	yes	—
30	C₃₁×#125	C₂²	1	1	yes	characteristic
31	C₃₁×#125	C₂²	1	2	yes	—
32	C₃₁×#125	C₂²	1	4	yes	—
33	C₃₁×#125	C₂²	1	4	yes	—
34	C₃₁×#125	C₂²	1	4	yes	—
35	C₃₁×#125	C₂²	1	4	yes	—
36	C₃₁×#125	C₂²	1	4	yes	—
37	C₃₁×#125	C₂²	1	4	yes	—
38	C₃₁×#125	C₂²	1	2	yes	—
39	C₃₁×#125	C₂²	1	4	yes	—
40	C₃₁×#125	C₂²	1	4	yes	—
41	C₃₁×#124	C₂²	1	1	yes	characteristic
42	C₃₁×#123	C₂²	1	2	yes	—
43	C₃₁×#123	C₂²	1	4	yes	—
44	C₃₁×#123	C₂²	1	4	yes	—
45	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
46	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
47	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
48	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
49	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
50	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
51	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
52	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
54	C₃₁×#121	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
57	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
58	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
59	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
60	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
61	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
62	C₄×D₄×C₃₁	C₂²	1	2	yes	—
63	C₄×D₄×C₃₁	C₂²	1	4	yes	—
64	C₃₁×#117	C₂²	1	2	yes	—
65	C₃₁×#117	C₂²	1	2	yes	—
66	C₃₁×#117	C₂²	1	4	yes	—
67	C₃₁×#117	C₂²	1	2	yes	—
68	C₃₁×#117	C₂²	1	4	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
76	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
77	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
78	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₃₁	C₂³	1	2	yes	—
80	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
88	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₄²×C₃₁	C₂³	1	1	yes	characteristic
129	C₄²×C₃₁	C₂³	1	4	yes	—
130	C₄²×C₃₁	C₂³	1	4	yes	—
131	C₄²×C₃₁	C₂³	1	2	yes	—
132	C₄²×C₃₁	C₂³	1	4	yes	—
133	C₄²×C₃₁	C₂³	1	4	yes	—
134	C₂³×C₃₁	C₂⁴	1	2	yes	—
135	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
136	Q₈×C₃₁	—	2	4	no	—
137	Q₈×C₃₁	—	4	4	no	—
138	Q₈×C₃₁	—	4	1	no	—
139	Q₈×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	2	4	no	—
141	D₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	4	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	4	no	—
148	C₂×C₄×C₃₁	—	4	1	no	—
149	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	#3563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	2	no	—
166	C₂²×C₃₁	—	2	2	no	—
167	C₂²×C₃₁	—	2	2	no	—
168	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
169	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	2	4	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	2	4	no	—
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	2	4	no	—
180	C₄×C₃₁	—	2	1	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
183	#564	C₂×C₃₁	1	2	yes	complemented
184	#563	C₂×C₃₁	1	2	yes	complemented
185	#557	C₂×C₃₁	1	2	yes	complemented
186	#554	C₂×C₃₁	1	2	yes	complemented
187	#552	C₂×C₃₁	1	4	yes	complemented
188	#552	C₂×C₃₁	1	2	yes	complemented
189	#552	C₂×C₃₁	1	2	yes	complemented
190	#544	C₂×C₃₁	1	2	yes	complemented
191	#544	C₂×C₃₁	1	2	yes	complemented
192	#541	C₂×C₃₁	1	2	yes	complemented
193	#541	C₂×C₃₁	1	4	yes	complemented
194	#539	C₂×C₃₁	1	2	yes	—
195	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#529	C₂×C₃₁	1	1	yes	characteristic
197	C₂×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×C₃₁	—	2	2	no	—
199	C₂×C₃₁	—	4	2	no	—
200	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
201	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×C₃₁	#585	1	1	yes	characteristic
203	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
204	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
205	C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
206	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
207	C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
208	#126	C₂²×C₃₁	1	2	yes	—
209	#126	C₂²×C₃₁	1	4	yes	—
210	#126	C₂²×C₃₁	1	4	yes	—
211	#126	C₂²×C₃₁	1	4	yes	—
212	#126	C₂²×C₃₁	1	4	yes	—
213	#125	C₂²×C₃₁	1	2	yes	—
214	#125	C₂²×C₃₁	1	2	yes	—
215	#125	C₂²×C₃₁	1	2	yes	—
216	#125	C₂²×C₃₁	1	2	yes	—
217	#125	C₂²×C₃₁	1	1	yes	characteristic
218	#125	C₂²×C₃₁	1	2	yes	—
219	#125	C₂²×C₃₁	1	4	yes	—
220	#125	C₂²×C₃₁	1	4	yes	—
221	#125	C₂²×C₃₁	1	4	yes	—
222	#125	C₂²×C₃₁	1	4	yes	—
223	#125	C₂²×C₃₁	1	4	yes	—
224	#125	C₂²×C₃₁	1	4	yes	—
225	#125	C₂²×C₃₁	1	2	yes	—
226	#125	C₂²×C₃₁	1	4	yes	—
227	#125	C₂²×C₃₁	1	4	yes	—
228	#124	C₂²×C₃₁	1	1	yes	characteristic
229	#123	C₂²×C₃₁	1	2	yes	—
230	#123	C₂²×C₃₁	1	4	yes	—
231	#123	C₂²×C₃₁	1	4	yes	—
232	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
234	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
235	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
236	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
237	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
238	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
239	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
240	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
241	#121	C₂²×C₃₁	1	1	yes	characteristic
242	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
244	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
245	C₄×Q₈	C₂²×C₃₁	1	4	yes	—
246	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
248	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₄×D₄	C₂²×C₃₁	1	2	yes	—
250	C₄×D₄	C₂²×C₃₁	1	4	yes	—
251	#117	C₂²×C₃₁	1	2	yes	—
252	#117	C₂²×C₃₁	1	2	yes	—
253	#117	C₂²×C₃₁	1	4	yes	—
254	#117	C₂²×C₃₁	1	2	yes	—
255	#117	C₂²×C₃₁	1	4	yes	—
256	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
262	C₃₁	#3563	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
264	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
265	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
266	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×D₄	C₂³×C₃₁	1	2	yes	—
268	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
272	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
273	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
276	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
277	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
278	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
280	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
282	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
283	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
284	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
285	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
287	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
288	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
289	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
291	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
292	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
293	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
295	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
299	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
300	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
301	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
302	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
303	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
304	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
305	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
306	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
307	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
308	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
309	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
311	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
312	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	C₄²	C₂³×C₃₁	1	1	yes	characteristic
317	C₄²	C₂³×C₃₁	1	4	yes	—
318	C₄²	C₂³×C₃₁	1	4	yes	—
319	C₄²	C₂³×C₃₁	1	2	yes	—
320	C₄²	C₂³×C₃₁	1	4	yes	—
321	C₄²	C₂³×C₃₁	1	4	yes	—
322	C₂³	C₂⁴×C₃₁	1	2	yes	—
323	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	4	4	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	2	no	—
328	D₄	—	2	4	no	—
329	D₄	—	4	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
339	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
340	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
341	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
344	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
345	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
347	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
349	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
356	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	4	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	4	no	—
364	C₄	—	2	4	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	4	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	2	no	—
369	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	4	2	no	—
372	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
373	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
374	C₂	C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
prime radical	C₃₁×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#564	C₂	semidirect
2. C₃₁×#563	C₂	semidirect
3. C₃₁×#557	C₂	semidirect
4. C₃₁×#554	C₂	semidirect
5. C₃₁×#552	C₂	semidirect
6. C₃₁×#552	C₂	semidirect
7. C₃₁×#552	C₂	semidirect
8. C₃₁×#544	C₂	semidirect
9. C₃₁×#544	C₂	semidirect
10. C₃₁×#541	C₂	semidirect
11. C₃₁×#541	C₂	semidirect
13. C₃₁×#533	C₂	semidirect
15. C₂×C₃₁×#125	C₂	semidirect
162. #3563	C₃₁	direct (N × H)
183. #564	C₂×C₃₁	semidirect
184. #563	C₂×C₃₁	semidirect
185. #557	C₂×C₃₁	semidirect
186. #554	C₂×C₃₁	semidirect
187. #552	C₂×C₃₁	semidirect
188. #552	C₂×C₃₁	semidirect
189. #552	C₂×C₃₁	semidirect
190. #544	C₂×C₃₁	semidirect
191. #544	C₂×C₃₁	semidirect
192. #541	C₂×C₃₁	semidirect
193. #541	C₂×C₃₁	semidirect
195. #533	C₂×C₃₁	semidirect
197. C₂×#125	C₂×C₃₁	semidirect
262. C₃₁	#3563	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×#564	2
C₃₁×#563	2
C₃₁×#557	2
C₃₁×#554	2
C₃₁×#552	2
C₃₁×#544	2
C₃₁×#541	2
C₃₁×#539	2
C₃₁×#533	2
C₃₁×#529	2
C₂×C₃₁×#125	2
#3563	31

Subgroups — C31×#3563 #16843496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3563 #16843496