Subgroups — C₃₁×#20648 #21477407

Order 7936.

1598 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#417	C₂	1	2	yes	maximal
4	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1803	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1802	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	—	2	4	no	—
11	C₄×#41×C₃₁	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	—	2	2	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₄²×C₃₁	—	2	1	no	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#423	—	2	4	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
23	C₃₁×#417	—	2	8	no	—
24	C₂×C₃₁×#103	C₂²	1	4	yes	—
25	C₂×C₃₁×#102	C₂²	1	4	yes	—
26	C₃₁×#384	—	2	1	no	—
27	C₃₁×#384	—	2	1	no	—
28	C₃₁×#384	C₂²	1	4	yes	—
29	C₃₁×#340	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×#41×C₃₁	D₄	1	4	yes	—
35	C₂×#41×C₃₁	Q₈	1	4	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	—	2	1	no	—
40	C₂³×C₄×C₃₁	—	2	2	no	—
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂³×C₄×C₃₁	D₄	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	4	no	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
48	C₄×Q₈×C₃₁	—	2	4	no	—
49	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₄×D₄×C₃₁	—	2	16	no	—
51	C₄×D₄×C₃₁	—	2	4	no	—
52	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	8	no	—
55	C₃₁×#117	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	8	no	—
70	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	4	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₃₁×C₄⋊C₈	—	2	8	no	—
76	C₃₁×#103	—	2	8	no	—
77	C₃₁×#102	—	2	8	no	—
78	C₂⁴×C₃₁	—	2	1	no	—
79	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
80	#41×C₃₁	—	2	8	no	—
81	#41×C₃₁	—	2	8	no	—
82	#41×C₃₁	—	2	8	no	—
83	#41×C₃₁	C₄⋊C₄	1	16	yes	complemented
84	C₂×Q₈×C₃₁	—	2	1	no	—
85	C₂×Q₈×C₃₁	—	2	1	no	—
86	C₂×Q₈×C₃₁	—	2	1	no	—
87	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	4	no	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	8	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	4	4	no	—
109	C₂²×C₄×C₃₁	—	4	4	no	—
110	C₂²×C₄×C₃₁	—	4	8	no	—
111	C₂²×C₄×C₃₁	—	4	4	no	—
112	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
113	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
120	C₂²×C₄×C₃₁	#41	1	2	yes	—
121	C₂²×C₄×C₃₁	#41	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₃₁	—	4	8	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	4	no	—
126	C₂×C₈×C₃₁	—	2	4	no	—
127	C₄⋊C₄×C₃₁	—	2	8	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	4	8	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	8	no	—
134	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₄²×C₃₁	—	2	4	no	—
138	C₄²×C₃₁	—	2	8	no	—
139	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
140	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
141	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
142	C₄²×C₃₁	C₂×D₄	1	4	yes	—
143	#20648	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×C₃₁	#121	1	2	yes	—
153	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
154	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	2	2	no	—
157	Q₈×C₃₁	—	2	2	no	—
158	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
159	D₄×C₃₁	—	2	16	no	—
160	D₄×C₃₁	—	2	2	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	16	no	—
170	C₂×C₄×C₃₁	—	2	16	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	16	no	—
175	C₂×C₄×C₃₁	—	4	4	no	—
176	C₂×C₄×C₃₁	—	4	8	no	—
177	C₂×C₄×C₃₁	—	8	8	no	—
178	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
179	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
181	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
182	C₂×C₄×C₃₁	#117	1	4	yes	—
183	C₂×C₄×C₃₁	#121	1	4	yes	—
184	C₂×C₄×C₃₁	#121	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
186	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
187	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
188	C₂×C₄×C₃₁	#137	1	4	yes	—
189	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₈×C₃₁	—	4	4	no	—
196	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
197	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×#417	C₂×C₃₁	1	2	yes	—
199	C₂×#384	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1803	C₂×C₃₁	1	4	yes	complemented
201	#1802	C₂×C₃₁	1	4	yes	complemented
202	#1736	C₂×C₃₁	1	2	yes	complemented
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	4	4	no	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	—	4	2	no	—
212	C₂²×C₃₁	#380	1	4	yes	—
213	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
217	C₂²×C₃₁	C₂×#137	1	2	yes	—
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	—	4	4	no	—
221	C₄×C₃₁	—	8	4	no	—
222	C₄×C₃₁	#380	1	4	yes	—
223	C₄×C₃₁	#417	1	8	yes	complemented
224	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
225	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
226	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
227	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
228	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
229	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
230	C₄×#41	—	2	4	no	—
231	C₄×#41	C₄×C₃₁	1	8	yes	complemented
232	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄×D₄	—	2	2	no	—
234	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
235	C₂×#117	—	2	1	no	—
236	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
237	C₂²×C₄⋊C₄	—	2	2	no	—
238	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₂²×C₄²	—	2	1	no	—
240	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
241	#423	—	2	4	no	—
242	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
243	#417	—	2	8	no	—
244	C₂×#103	C₂²×C₃₁	1	4	yes	—
245	C₂×#102	C₂²×C₃₁	1	4	yes	—
246	#384	—	2	1	no	—
247	#384	—	2	1	no	—
248	#384	C₂²×C₃₁	1	4	yes	—
249	#340	C₂²×C₃₁	1	8	yes	—
250	C₂×C₃₁	—	2	4	no	—
251	C₂×C₃₁	—	4	4	no	—
252	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
253	C₂×C₃₁	C₂×#417	1	2	yes	—
254	C₂×C₃₁	#3038	1	1	yes	characteristic
255	C₂×C₃₁	#3039	1	1	yes	characteristic
256	C₂×C₃₁	#3079	1	2	yes	—
257	C₂×#41	—	2	2	no	—
258	C₂×#41	—	2	2	no	—
259	C₂×#41	—	2	2	no	—
260	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
261	C₂×#41	D₄×C₃₁	1	4	yes	—
262	C₂×#41	Q₈×C₃₁	1	4	yes	—
263	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
266	C₂³×C₄	—	2	1	no	—
267	C₂³×C₄	—	2	2	no	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	D₄×C₃₁	1	2	yes	—
270	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×M₁₆	—	2	4	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂×M₁₆	—	2	4	no	—
274	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
275	C₄×Q₈	—	2	4	no	—
276	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
277	C₄×D₄	—	2	16	no	—
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
280	#117	—	2	2	no	—
281	#117	—	2	8	no	—
282	#117	C₂×C₄×C₃₁	1	4	yes	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	—	2	4	no	—
286	C₂×C₄⋊C₄	—	2	4	no	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	4	8	no	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	—	2	8	no	—
297	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
298	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	D₄×C₃₁	1	4	yes	—
301	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
302	C₄⋊C₈	—	2	8	no	—
303	#103	—	2	8	no	—
304	#102	—	2	8	no	—
305	C₃₁	#20648	1	1	yes	characteristic, direct product factor, Sylow
306	C₂⁴	—	2	1	no	—
307	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
308	#41	—	2	8	no	—
309	#41	—	2	8	no	—
310	#41	—	2	8	no	—
311	#41	C₄⋊C₄×C₃₁	1	16	yes	complemented
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
317	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	8	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	—	4	8	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
341	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
348	C₂²×C₄	#41×C₃₁	1	2	yes	—
349	C₂²×C₄	#41×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	M₁₆	—	4	8	no	—
352	C₂×C₈	—	2	4	no	—
353	C₂×C₈	—	2	4	no	—
354	C₂×C₈	—	2	4	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	4	8	no	—
360	C₄⋊C₄	—	4	4	no	—
361	C₄⋊C₄	—	4	8	no	—
362	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₄²	—	2	4	no	—
366	C₄²	—	2	8	no	—
367	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
368	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
369	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
370	C₄²	C₂×D₄×C₃₁	1	4	yes	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	2	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
379	C₂³	C₃₁×#121	1	2	yes	—
380	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
381	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	2	2	no	—
385	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	complemented
386	D₄	—	2	16	no	—
387	D₄	—	2	2	no	—
388	D₄	—	2	2	no	—
389	D₄	—	2	2	no	—
390	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	complemented
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	16	no	—
397	C₂×C₄	—	2	16	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	16	no	—
402	C₂×C₄	—	4	4	no	—
403	C₂×C₄	—	4	8	no	—
404	C₂×C₄	—	8	8	no	—
405	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
406	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
407	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
408	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
409	C₂×C₄	C₃₁×#117	1	4	yes	—
410	C₂×C₄	C₃₁×#121	1	4	yes	—
411	C₂×C₄	C₃₁×#121	1	2	yes	—
412	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
413	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
414	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
415	C₂×C₄	C₃₁×#137	1	4	yes	—
416	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
422	C₈	—	4	4	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	C₃₁×#380	1	4	yes	—
433	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
435	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
436	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
437	C₂²	C₂×C₃₁×#137	1	2	yes	—
438	C₄	—	2	4	no	—
439	C₄	—	2	4	no	—
440	C₄	—	4	4	no	—
441	C₄	—	8	4	no	—
442	C₄	C₃₁×#380	1	4	yes	—
443	C₄	C₃₁×#417	1	8	yes	complemented
444	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
445	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
446	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
447	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
448	C₂	—	2	4	no	—
449	C₂	—	4	4	no	—
450	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
451	C₂	C₂×C₃₁×#417	1	2	yes	—
452	C₂	C₃₁×#3038	1	1	yes	characteristic
453	C₂	C₃₁×#3039	1	1	yes	characteristic
454	C₂	C₃₁×#3079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×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
2. C₂×C₃₁×#423	C₂	semidirect
4. C₂×C₃₁×#384	C₂	semidirect
5. C₃₁×#1803	C₂	semidirect
6. C₃₁×#1802	C₂	semidirect
7. C₃₁×#1736	C₂	semidirect
11. C₄×#41×C₃₁	C₄	semidirect
49. C₄×Q₈×C₃₁	C₂×C₄	semidirect
52. C₄×D₄×C₃₁	C₂×C₄	semidirect
83. #41×C₃₁	C₄⋊C₄	semidirect
143. #20648	C₃₁	direct (N × H)
158. Q₈×C₃₁	C₂×C₄⋊C₄	semidirect
163. D₄×C₃₁	C₂×C₄⋊C₄	semidirect
197. C₂×#423	C₂×C₃₁	semidirect
199. C₂×#384	C₂×C₃₁	semidirect
200. #1803	C₂×C₃₁	semidirect
201. #1802	C₂×C₃₁	semidirect
202. #1736	C₂×C₃₁	semidirect
223. C₄×C₃₁	#417	semidirect
231. C₄×#41	C₄×C₃₁	semidirect
276. C₄×Q₈	C₂×C₄×C₃₁	semidirect
279. C₄×D₄	C₂×C₄×C₃₁	semidirect
305. C₃₁	#20648	direct (N × H)
311. #41	C₄⋊C₄×C₃₁	semidirect
385. Q₈	C₂×C₄⋊C₄×C₃₁	semidirect
390. D₄	C₂×C₄⋊C₄×C₃₁	semidirect
443. C₄	C₃₁×#417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₂×C₃₁×#423	2
C₂×C₃₁×#417	2
C₂×C₃₁×#384	2
C₃₁×#1803	2
C₃₁×#1802	2
C₃₁×#1736	2
#20648	31

Subgroups — C31×#20648 #21477407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20648 #21477407