Subgroups — C₃₁×#21370 #21478129

Order 7936.

780 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×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1950	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1855	C₂	1	4	yes	maximal
6	C₄×C₃₁×#108	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#432	—	2	1	no	—
10	C₃₁×#432	—	2	1	no	—
11	C₃₁×#431	—	2	1	no	—
12	C₃₁×#431	—	2	1	no	—
13	C₃₁×#431	C₄	1	2	yes	—
14	C₃₁×#431	C₄	1	2	yes	—
15	C₃₁×#431	C₄	1	2	yes	—
16	C₃₁×#431	C₄	1	2	yes	—
17	C₂×C₃₁×#108	C₂²	1	8	yes	—
18	C₃₁×#423	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₃₁×#407	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₄×M₁₆×C₃₁	—	2	1	no	—
23	C₄×M₁₆×C₃₁	—	2	1	no	—
24	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
27	C₂×C₃₁×#97	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	—	2	1	no	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#341	C₂²	1	8	yes	—
31	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	—	2	8	no	—
36	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
37	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#108	—	2	16	no	—
53	C₃₁×C₄⋊C₈	—	2	2	no	—
54	C₃₁×C₄⋊C₈	—	2	2	no	—
55	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₃₁×C₂²⋊C₈	—	2	2	no	—
58	C₃₁×C₂²⋊C₈	—	2	2	no	—
59	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
61	C₃₁×#97	—	2	2	no	—
62	C₃₁×#97	—	2	2	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	C₄×C₈×C₃₁	—	2	2	no	—
66	C₄×C₈×C₃₁	—	2	2	no	—
67	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
69	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₃₁	—	2	8	no	—
92	M₁₆×C₃₁	—	2	4	no	—
93	M₁₆×C₃₁	—	2	8	no	—
94	M₁₆×C₃₁	#41	1	8	yes	—
95	C₂×C₈×C₃₁	—	2	8	no	—
96	C₂×C₈×C₃₁	—	2	4	no	—
97	C₂×C₈×C₃₁	—	2	8	no	—
98	C₂×C₈×C₃₁	—	2	4	no	—
99	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
100	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	#21370	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	2	2	no	—
118	C₂³×C₃₁	—	2	2	no	—
119	C₂³×C₃₁	—	4	1	no	—
120	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	#108	1	4	yes	complemented
140	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
153	C₈×C₃₁	—	2	8	no	—
154	C₈×C₃₁	—	2	8	no	—
155	C₈×C₃₁	—	4	4	no	—
156	#2920	C₂×C₃₁	1	1	yes	characteristic
157	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
158	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
159	#1950	C₂×C₃₁	1	4	yes	complemented
160	#1855	C₂×C₃₁	1	4	yes	—
161	C₄×#108	C₂×C₃₁	1	4	yes	complemented
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	2	1	no	—
166	C₂²×C₃₁	—	2	1	no	—
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	4	4	no	—
170	C₂²×C₃₁	C₂×#108	1	2	yes	—
171	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₃₁	#533	1	2	yes	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	2	1	no	—
178	C₄×C₃₁	—	2	1	no	—
179	C₄×C₃₁	—	2	1	no	—
180	C₄×C₃₁	C₂×#108	1	2	yes	—
181	C₄×C₃₁	C₂×#108	1	2	yes	—
182	C₄×C₃₁	#430	1	2	yes	—
183	C₄×C₃₁	C₄×#41	1	2	yes	—
184	C₄×C₃₁	#525	1	2	yes	—
185	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
186	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
187	#432	—	2	1	no	—
188	#432	—	2	1	no	—
189	#431	—	2	1	no	—
190	#431	—	2	1	no	—
191	#431	C₄×C₃₁	1	2	yes	—
192	#431	C₄×C₃₁	1	2	yes	—
193	#431	C₄×C₃₁	1	2	yes	—
194	#431	C₄×C₃₁	1	2	yes	—
195	C₂×#108	C₂²×C₃₁	1	8	yes	—
196	#423	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
198	#407	C₂²×C₃₁	1	2	yes	—
199	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
200	C₄×M₁₆	—	2	1	no	—
201	C₄×M₁₆	—	2	1	no	—
202	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
203	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
204	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
205	C₂×#97	—	2	1	no	—
206	C₂×C₄×C₈	—	2	1	no	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
208	#341	C₂²×C₃₁	1	8	yes	—
209	C₂×C₃₁	—	2	2	no	—
210	C₂×C₃₁	—	2	2	no	—
211	C₂×C₃₁	—	4	1	no	—
212	C₂×C₃₁	#2296	1	1	yes	characteristic
213	C₂×C₃₁	C₂²×#108	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
215	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
216	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
217	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	8	no	—
220	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
221	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
222	C₂²×C₈	—	2	2	no	—
223	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
224	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
233	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
236	#108	—	2	16	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
240	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
241	C₂²⋊C₈	—	2	2	no	—
242	C₂²⋊C₈	—	2	2	no	—
243	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
244	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
245	#97	—	2	2	no	—
246	#97	—	2	2	no	—
247	#97	—	2	2	no	—
248	C₄×C₈	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
252	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
253	C₃₁	#21370	1	1	yes	characteristic, direct product factor, Sylow
254	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
276	M₁₆	—	2	8	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	8	no	—
279	M₁₆	#41×C₃₁	1	8	yes	—
280	C₂×C₈	—	2	8	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	8	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
285	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
286	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
287	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
288	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
289	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
290	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
291	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
293	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
294	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
296	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
305	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
306	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	C₃₁×#108	1	4	yes	complemented
324	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
337	C₈	—	2	8	no	—
338	C₈	—	2	8	no	—
339	C₈	—	4	4	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×C₃₁×#108	1	2	yes	—
349	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
350	C₂²	C₃₁×#533	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	C₂×C₃₁×#108	1	2	yes	—
359	C₄	C₂×C₃₁×#108	1	2	yes	—
360	C₄	C₃₁×#430	1	2	yes	—
361	C₄	C₄×#41×C₃₁	1	2	yes	—
362	C₄	C₃₁×#525	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	C₃₁×#2296	1	1	yes	characteristic
367	C₂	C₂²×C₃₁×#108	1	1	yes	characteristic
368	C₂	C₂×C₃₁×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1950	C₂	semidirect
6. C₄×C₃₁×#108	C₂	semidirect
116. #21370	C₃₁	direct (N × H)
139. C₂×C₄×C₃₁	#108	semidirect
159. #1950	C₂×C₃₁	semidirect
161. C₄×#108	C₂×C₃₁	semidirect
253. C₃₁	#21370	direct (N × H)
323. C₂×C₄	C₃₁×#108	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2920	2
C₂×C₃₁×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1950	2
C₃₁×#1855	2
C₄×C₃₁×#108	2
#21370	31

Subgroups — C31×#21370 #21478129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21370 #21478129