Subgroups — C₃₁×#17833 #21474592

Order 7936.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	2	yes	maximal
3	C₃₁×#1574	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1573	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1572	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1569	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1565	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#446	C₄	1	1	yes	characteristic
12	C₃₁×#446	C₄	1	1	yes	characteristic
13	C₃₁×#436	C₄	1	1	yes	characteristic
14	C₃₁×#436	C₄	1	1	yes	characteristic
15	C₃₁×#435	C₄	1	2	yes	—
16	C₃₁×#435	C₄	1	2	yes	—
17	C₃₁×#433	—	2	2	no	—
18	C₃₁×#433	—	2	2	no	—
19	C₃₁×#432	C₂²	1	2	yes	—
20	C₃₁×#432	C₂²	1	2	yes	—
21	C₃₁×#431	C₂²	1	2	yes	—
22	C₃₁×#431	C₂²	1	2	yes	—
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₃₁×#423	C₂²	1	2	yes	—
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₃₁×#408	—	2	2	no	—
28	C₃₁×#405	—	2	2	no	—
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#335	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	1	yes	characteristic
33	C₃₁×#335	C₂²	1	2	yes	—
34	C₃₁×#335	C₂²	1	1	yes	characteristic
35	C₃₁×#335	C₂²	1	1	yes	characteristic
36	C₃₁×#335	C₂²	1	2	yes	—
37	C₃₁×#334	C₂²	1	2	yes	—
38	C₃₁×#334	C₂²	1	2	yes	—
39	C₃₁×#334	C₂²	1	2	yes	—
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
50	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₃₁	Q₈	1	1	yes	characteristic
53	C₄×D₄×C₃₁	Q₈	1	2	yes	—
54	C₃₁×#117	D₄	1	2	yes	—
55	C₃₁×#117	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×C₄⋊C₈	—	2	2	no	—
60	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
65	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
68	C₃₁×C₂²⋊C₈	—	2	2	no	—
69	C₃₁×C₂²⋊C₈	—	2	2	no	—
70	C₃₁×C₂²⋊C₈	—	2	2	no	—
71	C₃₁×C₂²⋊C₈	—	2	2	no	—
72	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
74	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₃₁×#97	—	2	2	no	—
76	C₃₁×#97	—	2	2	no	—
77	C₃₁×#97	C₂×C₄	1	2	yes	—
78	C₃₁×#97	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₃₁	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂³	1	2	yes	—
86	#41×C₃₁	—	4	1	no	—
87	#41×C₃₁	—	4	1	no	—
88	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
89	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
90	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
97	M₁₆×C₃₁	—	2	4	no	—
98	M₁₆×C₃₁	—	4	2	no	—
99	M₁₆×C₃₁	—	4	4	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	4	1	no	—
106	C₂×C₈×C₃₁	—	4	4	no	—
107	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
120	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
122	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂×D₄	1	2	yes	—
124	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
125	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
126	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
127	#17833	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
130	Q₈×C₃₁	—	4	1	no	—
131	D₄×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
152	C₈×C₃₁	—	2	1	no	—
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	—	2	1	no	—
156	C₈×C₃₁	—	2	2	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₈×C₃₁	—	4	2	no	—
160	C₈×C₃₁	—	4	2	no	—
161	C₈×C₃₁	#131	1	2	yes	—
162	#2965	C₂×C₃₁	1	1	yes	characteristic
163	#2925	C₂×C₃₁	1	2	yes	—
164	#1574	C₂×C₃₁	1	2	yes	complemented
165	#1573	C₂×C₃₁	1	2	yes	complemented
166	#1572	C₂×C₃₁	1	2	yes	complemented
167	#1569	C₂×C₃₁	1	2	yes	complemented
168	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1565	C₂×C₃₁	1	2	yes	complemented
170	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	1	no	—
175	C₂²×C₃₁	—	2	1	no	—
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	1	no	—
182	C₄×C₃₁	—	2	1	no	—
183	C₄×C₃₁	—	2	1	no	—
184	C₄×C₃₁	—	2	2	no	—
185	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
186	C₄×C₃₁	#568	1	1	yes	characteristic
187	C₄×C₃₁	#575	1	1	yes	characteristic
188	C₄×C₃₁	#575	1	1	yes	characteristic
189	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
190	#446	C₄×C₃₁	1	1	yes	characteristic
191	#446	C₄×C₃₁	1	1	yes	characteristic
192	#436	C₄×C₃₁	1	1	yes	characteristic
193	#436	C₄×C₃₁	1	1	yes	characteristic
194	#435	C₄×C₃₁	1	2	yes	—
195	#435	C₄×C₃₁	1	2	yes	—
196	#433	—	2	2	no	—
197	#433	—	2	2	no	—
198	#432	C₂²×C₃₁	1	2	yes	—
199	#432	C₂²×C₃₁	1	2	yes	—
200	#431	C₂²×C₃₁	1	2	yes	—
201	#431	C₂²×C₃₁	1	2	yes	—
202	#431	C₂²×C₃₁	1	1	yes	characteristic
203	#431	C₂²×C₃₁	1	2	yes	—
204	#423	C₂²×C₃₁	1	2	yes	—
205	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
206	#408	—	2	2	no	—
207	#405	—	2	2	no	—
208	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
209	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
210	#335	C₂²×C₃₁	1	1	yes	characteristic
211	#335	C₂²×C₃₁	1	1	yes	characteristic
212	#335	C₂²×C₃₁	1	2	yes	—
213	#335	C₂²×C₃₁	1	1	yes	characteristic
214	#335	C₂²×C₃₁	1	1	yes	characteristic
215	#335	C₂²×C₃₁	1	2	yes	—
216	#334	C₂²×C₃₁	1	2	yes	—
217	#334	C₂²×C₃₁	1	2	yes	—
218	#334	C₂²×C₃₁	1	2	yes	—
219	#334	C₂²×C₃₁	1	2	yes	—
220	C₂×C₃₁	—	4	1	no	—
221	C₂×C₃₁	—	4	2	no	—
222	C₂×C₃₁	#2906	1	1	yes	characteristic
223	C₂×C₃₁	#2911	1	1	yes	characteristic
224	C₂×C₃₁	#2916	1	1	yes	characteristic
225	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
228	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
233	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
234	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
235	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
236	C₄×D₄	Q₈×C₃₁	1	1	yes	characteristic
237	C₄×D₄	Q₈×C₃₁	1	2	yes	—
238	#117	D₄×C₃₁	1	2	yes	—
239	#117	D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
245	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
246	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
249	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
250	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
251	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
257	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
258	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
259	#97	—	2	2	no	—
260	#97	—	2	2	no	—
261	#97	C₂×C₄×C₃₁	1	2	yes	—
262	#97	C₂×C₄×C₃₁	1	2	yes	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×C₈	C₂³×C₃₁	1	2	yes	—
270	C₃₁	#17833	1	1	yes	characteristic, direct product factor, Sylow
271	#41	—	4	1	no	—
272	#41	—	4	1	no	—
273	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
274	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
275	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
280	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
282	M₁₆	—	2	4	no	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	4	no	—
292	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
293	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
296	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
297	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
298	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
299	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
300	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
301	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
302	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
303	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
304	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
305	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
306	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
307	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₃₁	1	2	yes	—
309	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
310	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
311	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
312	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
313	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
314	Q₈	—	4	1	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
329	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
331	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
333	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	C₃₁×#131	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	2	no	—
360	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
361	C₄	C₃₁×#568	1	1	yes	characteristic
362	C₄	C₃₁×#575	1	1	yes	characteristic
363	C₄	C₃₁×#575	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	4	2	no	—
366	C₂	C₃₁×#2906	1	1	yes	characteristic
367	C₂	C₃₁×#2911	1	1	yes	characteristic
368	C₂	C₃₁×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1574	C₂	semidirect
4. C₃₁×#1573	C₂	semidirect
5. C₃₁×#1572	C₂	semidirect
6. C₃₁×#1569	C₂	semidirect
7. C₃₁×#1568	C₂	semidirect
8. C₃₁×#1565	C₂	semidirect
9. C₃₁×#1564	C₂	semidirect
127. #17833	C₃₁	direct (N × H)
164. #1574	C₂×C₃₁	semidirect
165. #1573	C₂×C₃₁	semidirect
166. #1572	C₂×C₃₁	semidirect
167. #1569	C₂×C₃₁	semidirect
168. #1568	C₂×C₃₁	semidirect
169. #1565	C₂×C₃₁	semidirect
170. #1564	C₂×C₃₁	semidirect
270. C₃₁	#17833	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×#2965	2
C₃₁×#2925	2
C₃₁×#1574	2
C₃₁×#1573	2
C₃₁×#1572	2
C₃₁×#1569	2
C₃₁×#1568	2
C₃₁×#1565	2
C₃₁×#1564	2
#17833	31

Subgroups — C31×#17833 #21474592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17833 #21474592