Subgroups — C₃₁×#19276 #21476035

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1682	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1681	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	—	2	1	no	—
11	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#486	—	2	1	no	—
16	C₃₁×#486	—	2	1	no	—
17	C₃₁×#486	C₂²	1	4	yes	—
18	C₃₁×#462	—	2	1	no	—
19	C₃₁×#462	C₂²	1	2	yes	complemented
20	C₃₁×#461	—	2	1	no	—
21	C₃₁×#461	C₂²	1	2	yes	complemented
22	C₃₁×#460	—	2	1	no	—
23	C₃₁×#460	C₂²	1	2	yes	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₃₁×#419	—	2	1	no	—
34	C₃₁×#418	—	2	1	no	—
35	C₂×C₃₁×#102	—	2	2	no	—
36	C₂×C₄×C₈×C₃₁	—	2	1	no	—
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	4	yes	complemented
39	C₃₁×#332	C₂²	1	4	yes	complemented
40	C₃₁×#331	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₃₁×#137	—	4	4	no	—
47	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	D₄	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	—	2	1	no	—
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	1	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	1	no	—
89	C₃₁×C₂²⋊C₈	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
93	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	2	yes	—
95	C₂⁴×C₃₁	—	4	1	no	—
96	#41×C₃₁	—	2	2	no	—
97	#41×C₃₁	D₈	1	4	yes	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	4	1	no	—
100	C₂×Q₈×C₃₁	—	4	2	no	—
101	C₂×Q₈×C₃₁	—	4	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	4	4	no	—
108	C₂×D₄×C₃₁	—	4	1	no	—
109	C₂×D₄×C₃₁	—	4	1	no	—
110	C₂×D₄×C₃₁	—	4	1	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
122	Q₁₆×C₃₁	—	4	4	no	—
123	SD₁₆×C₃₁	—	4	4	no	—
124	SD₁₆×C₃₁	—	4	4	no	—
125	D₈×C₃₁	—	4	4	no	—
126	M₁₆×C₃₁	—	4	2	no	—
127	M₁₆×C₃₁	—	4	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	4	4	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	4	1	no	—
140	C₄⋊C₄×C₃₁	—	4	1	no	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	#41	1	1	yes	characteristic
152	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
153	#19276	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	8	2	no	—
157	C₂³×C₃₁	—	8	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	#121	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	1	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	Q₈×C₃₁	—	4	2	no	—
163	Q₈×C₃₁	—	8	2	no	—
164	Q₈×C₃₁	C₂×D₈	1	2	yes	—
165	D₄×C₃₁	—	2	4	no	—
166	D₄×C₃₁	—	2	1	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	4	no	—
170	D₄×C₃₁	—	8	2	no	—
171	D₄×C₃₁	—	8	2	no	—
172	D₄×C₃₁	—	8	2	no	—
173	D₄×C₃₁	—	8	2	no	—
174	D₄×C₃₁	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	8	2	no	—
185	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
186	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
189	C₂×C₄×C₃₁	#135	1	2	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	4	2	no	—
197	C₈×C₃₁	—	4	2	no	—
198	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#3038	C₂×C₃₁	1	1	yes	characteristic
201	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1682	C₂×C₃₁	1	4	yes	complemented
203	#1681	C₂×C₃₁	1	4	yes	complemented
204	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	16	4	no	—
216	C₂²×C₃₁	#466	1	2	yes	—
217	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	#466	1	2	yes	—
225	C₄×C₃₁	#471	1	2	yes	—
226	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
227	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
228	C₄×C₃₁	#575	1	2	yes	—
229	C₂×#137	—	2	1	no	—
230	C₂×AGL(1,ℤ/8)	—	2	1	no	—
231	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
234	#486	—	2	1	no	—
235	#486	—	2	1	no	—
236	#486	C₂²×C₃₁	1	4	yes	—
237	#462	—	2	1	no	—
238	#462	C₂²×C₃₁	1	2	yes	complemented
239	#461	—	2	1	no	—
240	#461	C₂²×C₃₁	1	2	yes	complemented
241	#460	—	2	1	no	—
242	#460	C₂²×C₃₁	1	2	yes	—
243	#459	—	2	1	no	—
244	#459	C₂²×C₃₁	1	2	yes	—
245	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
246	C₈×D₄	—	2	2	no	—
247	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
248	#431	—	2	1	no	—
249	#431	C₂²×C₃₁	1	1	yes	characteristic
250	#423	C₂²×C₃₁	1	1	yes	characteristic
251	#423	C₂²×C₃₁	1	1	yes	characteristic
252	#419	—	2	1	no	—
253	#418	—	2	1	no	—
254	C₂×#102	—	2	2	no	—
255	C₂×C₄×C₈	—	2	1	no	—
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#335	C₂²×C₃₁	1	4	yes	complemented
258	#332	C₂²×C₃₁	1	4	yes	complemented
259	#331	C₂²×C₃₁	1	4	yes	—
260	C₂×C₃₁	—	2	2	no	—
261	C₂×C₃₁	—	4	2	no	—
262	C₂×C₃₁	—	16	2	no	—
263	C₂×C₃₁	#3040	1	1	yes	characteristic
264	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
265	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
266	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
267	C₂²×Q₈	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	#137	—	4	4	no	—
272	AGL(1,ℤ/8)	—	4	4	no	—
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×SD₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×D₈	—	2	2	no	—
277	C₂×M₁₆	—	2	1	no	—
278	C₂×M₁₆	—	2	1	no	—
279	C₂²×C₈	—	2	2	no	—
280	C₂²×C₈	D₄×C₃₁	1	2	yes	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
285	#127	—	2	1	no	—
286	#127	—	2	1	no	—
287	#127	—	2	1	no	—
288	#127	C₂³×C₃₁	1	2	yes	—
289	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
290	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
294	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
295	#117	—	2	1	no	—
296	#117	D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
305	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
306	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
307	#103	—	2	2	no	—
308	#103	—	2	2	no	—
309	#102	—	2	4	no	—
310	#102	—	2	4	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	2	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	2	1	no	—
317	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
318	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₃₁	1	2	yes	—
320	C₃₁	#19276	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	4	1	no	—
322	#41	—	2	2	no	—
323	#41	D₈×C₃₁	1	4	yes	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	4	1	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	1	no	—
336	C₂×D₄	—	4	1	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	8	2	no	—
340	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
348	Q₁₆	—	4	4	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	—	4	4	no	—
351	D₈	—	4	4	no	—
352	M₁₆	—	4	2	no	—
353	M₁₆	—	4	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	—	4	4	no	—
359	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	4	1	no	—
366	C₄⋊C₄	—	4	1	no	—
367	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	1	no	—
375	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₄²	#41×C₃₁	1	1	yes	characteristic
378	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₂³	—	2	1	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	8	2	no	—
382	C₂³	—	8	2	no	—
383	C₂³	—	8	2	no	—
384	C₂³	C₃₁×#121	1	1	yes	characteristic
385	Q₈	—	2	1	no	—
386	Q₈	—	4	2	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	8	2	no	—
389	Q₈	C₂×D₈×C₃₁	1	2	yes	—
390	D₄	—	2	4	no	—
391	D₄	—	2	1	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	4	no	—
395	D₄	—	8	2	no	—
396	D₄	—	8	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	8	2	no	—
399	D₄	C₂×D₈×C₃₁	1	2	yes	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
411	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
412	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
413	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
414	C₂×C₄	C₃₁×#135	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₈	—	4	2	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	16	4	no	—
432	C₂²	C₃₁×#466	1	2	yes	—
433	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
434	C₄	—	2	1	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	8	2	no	—
440	C₄	C₃₁×#466	1	2	yes	—
441	C₄	C₃₁×#471	1	2	yes	—
442	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
443	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
444	C₄	C₃₁×#575	1	2	yes	—
445	C₂	—	2	2	no	—
446	C₂	—	4	2	no	—
447	C₂	—	16	2	no	—
448	C₂	C₃₁×#3040	1	1	yes	characteristic
449	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
450	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#486	C₂	semidirect
2. C₃₁×#3039	C₂	semidirect
4. C₈×#41×C₃₁	C₂	semidirect
5. C₃₁×#1682	C₂	semidirect
6. C₃₁×#1681	C₂	semidirect
7. C₃₁×#1567	C₂	semidirect
8. C₃₁×#1546	C₂	semidirect
9. C₃₁×#1545	C₂	semidirect
19. C₃₁×#462	C₂²	semidirect
21. C₃₁×#461	C₂²	semidirect
26. C₈×Q₈×C₃₁	C₂²	semidirect
28. C₈×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#335	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
153. #19276	C₃₁	direct (N × H)
198. C₂×#486	C₂×C₃₁	semidirect
199. #3039	C₂×C₃₁	semidirect
201. C₈×#41	C₂×C₃₁	semidirect
202. #1682	C₂×C₃₁	semidirect
203. #1681	C₂×C₃₁	semidirect
204. #1567	C₂×C₃₁	semidirect
205. #1546	C₂×C₃₁	semidirect
206. #1545	C₂×C₃₁	semidirect
238. #462	C₂²×C₃₁	semidirect
240. #461	C₂²×C₃₁	semidirect
245. C₈×Q₈	C₂²×C₃₁	semidirect
247. C₈×D₄	C₂²×C₃₁	semidirect
257. #335	C₂²×C₃₁	semidirect
258. #332	C₂²×C₃₁	semidirect
320. C₃₁	#19276	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₃₁×#3039	2
C₃₁×#3038	2
C₈×#41×C₃₁	2
C₃₁×#1682	2
C₃₁×#1681	2
C₃₁×#1567	2
C₃₁×#1546	2
C₃₁×#1545	2
#19276	31

Subgroups — C31×#19276 #21476035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19276 #21476035