Subgroups — C₃₁×#20655 #21477414

Order 7936.

1426 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#421	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#104	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1806	C₂	1	8	yes	maximal, complemented
7	C₃₁×#1733	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#339	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#520	—	2	2	no	—
13	C₃₁×#520	C₄	1	4	yes	complemented
14	C₃₁×#519	C₄	1	2	yes	complemented
15	C₃₁×#518	—	2	2	no	—
16	C₃₁×#518	C₄	1	2	yes	complemented
17	C₄×#41×C₃₁	—	2	2	no	—
18	C₄×#41×C₃₁	C₂²	1	4	yes	—
19	C₂×C₃₁×#117	—	2	2	no	—
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#424	—	2	2	no	—
25	C₃₁×#424	—	2	2	no	—
26	C₃₁×#423	C₂²	1	4	yes	complemented
27	C₃₁×#421	—	2	4	no	—
28	C₂×C₃₁×#104	—	2	4	no	—
29	C₂×C₃₁×#104	C₂²	1	4	yes	—
30	C₂×C₃₁×#104	C₂²	1	4	yes	—
31	C₃₁×#389	—	2	2	no	—
32	C₃₁×#385	—	2	2	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
34	C₃₁×#339	C₂²	1	4	yes	complemented
35	C₃₁×#339	C₂²	1	4	yes	complemented
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×#41×C₃₁	D₄	1	4	yes	—
42	C₂×#41×C₃₁	Q₈	1	4	yes	—
43	C₂×#41×C₃₁	C₂³	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	1	no	—
51	C₂×M₁₆×C₃₁	—	2	1	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₄×Q₈×C₃₁	—	2	4	no	—
61	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
62	C₄×D₄×C₃₁	—	2	8	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	8	no	—
65	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₃₁×#117	—	2	1	no	—
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	—	2	4	no	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	—	2	4	no	—
73	C₃₁×#117	C₂×C₄	1	2	yes	—
74	C₃₁×#117	C₂×C₄	1	2	yes	—
75	C₃₁×#117	C₂×C₄	1	2	yes	—
76	C₃₁×#117	D₄	1	2	yes	—
77	C₃₁×#117	D₄	1	2	yes	—
78	C₃₁×#117	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	—	2	4	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	—	2	1	no	—
93	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
96	C₃₁×C₄⋊C₈	—	2	8	no	—
97	C₃₁×#104	—	2	8	no	—
98	C₃₁×#104	—	2	8	no	—
99	C₃₁×#95	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	1	no	—
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	#41×C₃₁	—	2	4	no	—
103	#41×C₃₁	—	2	4	no	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	4	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	C₄⋊C₄	1	8	yes	complemented
110	#41×C₃₁	C₂×D₄	1	4	yes	—
111	#41×C₃₁	C₂×Q₈	1	4	yes	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
115	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	4	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	complemented
123	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	8	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	4	4	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
143	C₂²×C₄×C₃₁	#41	1	2	yes	—
144	C₂²×C₄×C₃₁	#41	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	M₁₆×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	4	2	no	—
151	M₁₆×C₃₁	—	4	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	2	no	—
155	C₂×C₈×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	4	8	no	—
159	C₄⋊C₄×C₃₁	—	4	2	no	—
160	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
161	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
162	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	2	8	no	—
166	C₂²⋊C₄×C₃₁	—	4	2	no	—
167	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
168	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
169	C₄²×C₃₁	—	2	8	no	—
170	C₄²×C₃₁	—	2	2	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	—	2	2	no	—
173	C₄²×C₃₁	—	2	2	no	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	—	4	2	no	—
177	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
178	C₄²×C₃₁	C₂×D₄	1	4	yes	—
179	#20655	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	8	2	no	—
184	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂³×C₃₁	#121	1	2	yes	—
187	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
188	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₃₁	—	2	2	no	—
190	Q₈×C₃₁	—	2	4	no	—
191	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	8	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	4	no	—
197	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	8	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	4	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	8	8	no	—
212	C₂×C₄×C₃₁	—	8	2	no	—
213	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
214	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
218	C₂×C₄×C₃₁	#117	1	2	yes	—
219	C₂×C₄×C₃₁	#117	1	2	yes	—
220	C₂×C₄×C₃₁	#121	1	2	yes	—
221	C₂×C₄×C₃₁	#121	1	2	yes	—
222	C₂×C₄×C₃₁	#121	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
225	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
226	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
232	C₈×C₃₁	—	4	2	no	—
233	C₈×C₃₁	—	4	2	no	—
234	#3428	C₂×C₃₁	1	1	yes	characteristic
235	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#421	C₂×C₃₁	1	1	yes	characteristic
237	C₂²×#104	C₂×C₃₁	1	1	yes	characteristic
238	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1806	C₂×C₃₁	1	8	yes	complemented
240	#1733	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#339	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	4	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	8	4	no	—
250	C₂²×C₃₁	#380	1	2	yes	—
251	C₂²×C₃₁	#380	1	2	yes	—
252	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
255	C₄×C₃₁	—	2	8	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	4	4	no	—
258	C₄×C₃₁	—	4	4	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	#380	1	2	yes	—
261	C₄×C₃₁	#380	1	2	yes	—
262	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
263	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
264	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
265	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	#520	—	2	2	no	—
270	#520	C₄×C₃₁	1	4	yes	complemented
271	#519	C₄×C₃₁	1	2	yes	complemented
272	#518	—	2	2	no	—
273	#518	C₄×C₃₁	1	2	yes	complemented
274	C₄×#41	—	2	2	no	—
275	C₄×#41	C₂²×C₃₁	1	4	yes	—
276	C₂×#117	—	2	2	no	—
277	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
281	#424	—	2	2	no	—
282	#424	—	2	2	no	—
283	#423	C₂²×C₃₁	1	4	yes	complemented
284	#421	—	2	4	no	—
285	C₂×#104	—	2	4	no	—
286	C₂×#104	C₂²×C₃₁	1	4	yes	—
287	C₂×#104	C₂²×C₃₁	1	4	yes	—
288	#389	—	2	2	no	—
289	#385	—	2	2	no	—
290	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
291	#339	C₂²×C₃₁	1	4	yes	complemented
292	#339	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₃₁	—	2	1	no	—
294	C₂×C₃₁	—	2	2	no	—
295	C₂×C₃₁	—	2	1	no	—
296	C₂×C₃₁	—	2	2	no	—
297	C₂×C₃₁	—	4	4	no	—
298	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
299	C₂×C₃₁	#2901	1	2	yes	—
300	C₂×#41	—	2	1	no	—
301	C₂×#41	—	2	1	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	2	no	—
304	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×#41	D₄×C₃₁	1	4	yes	—
306	C₂×#41	Q₈×C₃₁	1	4	yes	—
307	C₂×#41	C₂³×C₃₁	1	2	yes	—
308	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×M₁₆	—	2	1	no	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	—	2	2	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	2	no	—
320	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
321	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
322	C₂²×C₈	—	2	1	no	—
323	C₂²×C₈	—	2	1	no	—
324	C₄×Q₈	—	2	4	no	—
325	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
326	C₄×D₄	—	2	8	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
330	#117	—	2	1	no	—
331	#117	—	2	1	no	—
332	#117	—	2	4	no	—
333	#117	—	2	4	no	—
334	#117	—	2	2	no	—
335	#117	—	2	2	no	—
336	#117	—	2	4	no	—
337	#117	C₂×C₄×C₃₁	1	2	yes	—
338	#117	C₂×C₄×C₃₁	1	2	yes	—
339	#117	C₂×C₄×C₃₁	1	2	yes	—
340	#117	D₄×C₃₁	1	2	yes	—
341	#117	D₄×C₃₁	1	2	yes	—
342	#117	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	—	2	4	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	4	no	—
346	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
347	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	4	no	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	—	2	4	no	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	—	2	2	no	—
356	C₂×C₄²	—	2	1	no	—
357	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
360	C₄⋊C₈	—	2	8	no	—
361	#104	—	2	8	no	—
362	#104	—	2	8	no	—
363	#95	—	2	4	no	—
364	C₃₁	#20655	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
367	#41	—	2	4	no	—
368	#41	—	2	4	no	—
369	#41	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	2	2	no	—
373	#41	—	2	2	no	—
374	#41	C₄⋊C₄×C₃₁	1	8	yes	complemented
375	#41	C₂×D₄×C₃₁	1	4	yes	—
376	#41	C₂×Q₈×C₃₁	1	4	yes	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	complemented
380	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	complemented
388	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
407	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
408	C₂²×C₄	#41×C₃₁	1	2	yes	—
409	C₂²×C₄	#41×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	—	4	2	no	—
416	M₁₆	—	4	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	—	4	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	4	8	no	—
424	C₄⋊C₄	—	4	2	no	—
425	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
426	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
427	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
433	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
434	C₄²	—	2	8	no	—
435	C₄²	—	2	2	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	4	2	no	—
442	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
443	C₄²	C₂×D₄×C₃₁	1	4	yes	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	8	2	no	—
448	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂³	C₃₁×#121	1	2	yes	—
451	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
452	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	4	no	—
455	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	8	no	—
459	D₄	—	2	2	no	—
460	D₄	—	2	4	no	—
461	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	8	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	8	8	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
482	C₂×C₄	C₃₁×#117	1	2	yes	—
483	C₂×C₄	C₃₁×#117	1	2	yes	—
484	C₂×C₄	C₃₁×#121	1	2	yes	—
485	C₂×C₄	C₃₁×#121	1	2	yes	—
486	C₂×C₄	C₃₁×#121	1	2	yes	—
487	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
488	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
489	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
496	C₈	—	4	2	no	—
497	C₈	—	4	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	8	4	no	—
506	C₂²	C₃₁×#380	1	2	yes	—
507	C₂²	C₃₁×#380	1	2	yes	—
508	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
509	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
510	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
511	C₄	—	2	8	no	—
512	C₄	—	2	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	8	2	no	—
516	C₄	C₃₁×#380	1	2	yes	—
517	C₄	C₃₁×#380	1	2	yes	—
518	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
519	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
520	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
521	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
522	C₂	—	2	1	no	—
523	C₂	—	2	2	no	—
524	C₂	—	2	1	no	—
525	C₂	—	2	2	no	—
526	C₂	—	4	4	no	—
527	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
528	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2905	C₂	semidirect
5. C₃₁×#2296	C₂	semidirect
6. C₃₁×#1806	C₂	semidirect
7. C₃₁×#1733	C₂	semidirect
8. C₂×C₃₁×#339	C₂	semidirect
13. C₃₁×#520	C₄	semidirect
14. C₃₁×#519	C₄	semidirect
16. C₃₁×#518	C₄	semidirect
26. C₃₁×#423	C₂²	semidirect
33. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
34. C₃₁×#339	C₂²	semidirect
35. C₃₁×#339	C₂²	semidirect
61. C₄×Q₈×C₃₁	C₂×C₄	semidirect
65. C₄×D₄×C₃₁	C₂×C₄	semidirect
109. #41×C₃₁	C₄⋊C₄	semidirect
114. C₂×Q₈×C₃₁	C₄⋊C₄	semidirect
122. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
179. #20655	C₃₁	direct (N × H)
235. #2905	C₂×C₃₁	semidirect
238. #2296	C₂×C₃₁	semidirect
239. #1806	C₂×C₃₁	semidirect
240. #1733	C₂×C₃₁	semidirect
241. C₂×#339	C₂×C₃₁	semidirect
270. #520	C₄×C₃₁	semidirect
271. #519	C₄×C₃₁	semidirect
273. #518	C₄×C₃₁	semidirect
283. #423	C₂²×C₃₁	semidirect
290. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
291. #339	C₂²×C₃₁	semidirect
292. #339	C₂²×C₃₁	semidirect
325. C₄×Q₈	C₂×C₄×C₃₁	semidirect
329. C₄×D₄	C₂×C₄×C₃₁	semidirect
364. C₃₁	#20655	direct (N × H)
374. #41	C₄⋊C₄×C₃₁	semidirect
379. C₂×Q₈	C₄⋊C₄×C₃₁	semidirect
387. C₂×D₄	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3428	2
C₃₁×#2905	2
C₂×C₃₁×#421	2
C₂²×C₃₁×#104	2
C₃₁×#2296	2
C₃₁×#1806	2
C₃₁×#1733	2
C₂×C₃₁×#339	2
#20655	31

Subgroups — C31×#20655 #21477414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20655 #21477414