Subgroups — C₃₁×#33060 #21489819

Order 7936.

822 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×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1931	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1907	C₂	1	4	yes	maximal, complemented
9	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
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	—	2	2	no	—
18	C₃₁×#477	—	2	1	no	—
19	C₃₁×#476	—	2	1	no	—
20	C₃₁×#475	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#460	—	2	1	no	—
23	C₃₁×#460	—	2	1	no	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	—	2	1	no	—
26	C₃₁×#428	C₂²	1	2	yes	—
27	C₃₁×#423	C₂²	1	1	yes	characteristic
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#418	C₂²	1	2	yes	—
30	C₃₁×#417	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	4	yes	—
32	C₂×C₃₁×#100	C₂²	1	4	yes	—
33	C₃₁×#405	—	2	2	no	—
34	C₃₁×#405	C₂²	1	2	yes	—
35	C₄×M₁₆×C₃₁	—	2	1	no	—
36	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	—	2	1	no	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#343	C₂²	1	4	yes	—
40	C₃₁×#341	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	C₂³	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	C₂³	1	2	yes	—
46	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₃₁×#131	—	2	4	no	—
50	C₃₁×#131	—	2	4	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
70	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
71	C₃₁×#117	—	2	1	no	—
72	C₃₁×#117	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#100	—	2	8	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#97	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	—	2	1	no	—
94	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
95	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
96	C₄×C₈×C₃₁	D₄	1	2	yes	—
97	C₂⁴×C₃₁	—	2	2	no	—
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	#41	1	4	yes	—
100	C₂×Q₈×C₃₁	—	4	1	no	—
101	C₂×Q₈×C₃₁	—	4	1	no	—
102	C₂×Q₈×C₃₁	—	8	2	no	—
103	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	4	no	—
109	C₂×D₄×C₃₁	—	4	1	no	—
110	C₂×D₄×C₃₁	—	4	1	no	—
111	C₂×D₄×C₃₁	—	4	1	no	—
112	C₂×D₄×C₃₁	—	4	1	no	—
113	C₂×D₄×C₃₁	—	8	2	no	—
114	C₂×D₄×C₃₁	—	8	2	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	SD₁₆×C₃₁	—	4	4	no	—
124	D₈×C₃₁	—	4	4	no	—
125	M₁₆×C₃₁	—	2	8	no	—
126	M₁₆×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	—	2	4	no	—
128	M₁₆×C₃₁	—	2	4	no	—
129	M₁₆×C₃₁	#41	1	4	yes	—
130	C₂×C₈×C₃₁	—	2	8	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	4	1	no	—
143	C₄⋊C₄×C₃₁	—	4	1	no	—
144	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₂²⋊C₄×C₃₁	—	2	1	no	—
148	C₂²⋊C₄×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	#33060	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	4	no	—
160	C₂³×C₃₁	—	8	2	no	—
161	C₂³×C₃₁	—	8	2	no	—
162	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
163	Q₈×C₃₁	—	2	1	no	—
164	Q₈×C₃₁	—	8	2	no	—
165	Q₈×C₃₁	—	8	2	no	—
166	Q₈×C₃₁	#124	1	2	yes	—
167	D₄×C₃₁	—	2	4	no	—
168	D₄×C₃₁	—	2	1	no	—
169	D₄×C₃₁	—	4	4	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	8	2	no	—
173	D₄×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	D₄×C₃₁	#124	1	2	yes	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	8	2	no	—
187	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
188	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
189	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	4	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#3038	C₂×C₃₁	1	1	yes	characteristic
203	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1931	C₂×C₃₁	1	4	yes	complemented
207	#1907	C₂×C₃₁	1	4	yes	complemented
208	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	2	1	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	16	4	no	—
218	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
219	C₂²×C₃₁	#529	1	2	yes	—
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	4	1	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	#471	1	2	yes	—
227	C₄×C₃₁	#472	1	2	yes	—
228	C₄×C₃₁	#525	1	2	yes	—
229	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
230	C₄×C₃₁	#532	1	1	yes	characteristic
231	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
232	C₂×#131	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
235	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
236	#486	—	2	1	no	—
237	#486	—	2	1	no	—
238	#486	—	2	2	no	—
239	#477	—	2	1	no	—
240	#476	—	2	1	no	—
241	#475	—	2	1	no	—
242	#474	—	2	1	no	—
243	#460	—	2	1	no	—
244	#460	—	2	1	no	—
245	#459	—	2	1	no	—
246	#459	—	2	1	no	—
247	#428	C₂²×C₃₁	1	2	yes	—
248	#423	C₂²×C₃₁	1	1	yes	characteristic
249	#423	C₂²×C₃₁	1	1	yes	characteristic
250	#418	C₂²×C₃₁	1	2	yes	—
251	#417	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	4	yes	—
253	C₂×#100	C₂²×C₃₁	1	4	yes	—
254	#405	—	2	2	no	—
255	#405	C₂²×C₃₁	1	2	yes	—
256	C₄×M₁₆	—	2	1	no	—
257	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×C₈	—	2	1	no	—
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#343	C₂²×C₃₁	1	4	yes	—
261	#341	C₂²×C₃₁	1	4	yes	—
262	C₂×C₃₁	—	2	2	no	—
263	C₂×C₃₁	—	4	2	no	—
264	C₂×C₃₁	—	16	2	no	—
265	C₂×C₃₁	#2850	1	1	yes	characteristic
266	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
267	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
268	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	1	no	—
272	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
273	AGL(1,ℤ/8)	—	2	8	no	—
274	C₂×SD₁₆	—	2	2	no	—
275	C₂×D₈	—	2	2	no	—
276	#131	—	2	4	no	—
277	#131	—	2	4	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
280	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
281	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	2	1	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	#127	—	2	1	no	—
289	#127	—	2	1	no	—
290	#127	—	2	1	no	—
291	#127	—	2	1	no	—
292	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
293	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	—	2	1	no	—
296	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
297	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
298	#117	—	2	1	no	—
299	#117	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	#107	—	2	2	no	—
305	#106	—	2	2	no	—
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	—	2	1	no	—
308	C₄⋊C₈	—	2	1	no	—
309	C₄⋊C₈	—	2	1	no	—
310	#103	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#100	—	2	8	no	—
317	#97	—	2	2	no	—
318	#97	—	2	1	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	—	2	1	no	—
321	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
322	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
323	C₄×C₈	D₄×C₃₁	1	2	yes	—
324	C₃₁	#33060	1	1	yes	characteristic, direct product factor, Sylow
325	C₂⁴	—	2	2	no	—
326	#41	—	2	2	no	—
327	#41	#41×C₃₁	1	4	yes	—
328	C₂×Q₈	—	4	1	no	—
329	C₂×Q₈	—	4	1	no	—
330	C₂×Q₈	—	8	2	no	—
331	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	4	1	no	—
338	C₂×D₄	—	4	1	no	—
339	C₂×D₄	—	4	1	no	—
340	C₂×D₄	—	4	1	no	—
341	C₂×D₄	—	8	2	no	—
342	C₂×D₄	—	8	2	no	—
343	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	SD₁₆	—	4	4	no	—
352	D₈	—	4	4	no	—
353	M₁₆	—	2	8	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	2	4	no	—
356	M₁₆	—	2	4	no	—
357	M₁₆	#41×C₃₁	1	4	yes	—
358	C₂×C₈	—	2	8	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
365	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²⋊C₄	—	2	1	no	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂³	—	2	1	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	4	no	—
387	C₂³	—	8	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
390	Q₈	—	2	1	no	—
391	Q₈	—	8	2	no	—
392	Q₈	—	8	2	no	—
393	Q₈	C₃₁×#124	1	2	yes	—
394	D₄	—	2	4	no	—
395	D₄	—	2	1	no	—
396	D₄	—	4	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	2	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	D₄	C₃₁×#124	1	2	yes	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
415	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
416	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
417	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
421	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
422	C₈	—	2	2	no	—
423	C₈	—	2	4	no	—
424	C₈	—	2	2	no	—
425	C₈	—	4	2	no	—
426	C₈	—	4	2	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	16	4	no	—
436	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
437	C₂²	C₃₁×#529	1	2	yes	—
438	C₄	—	2	2	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₃₁×#471	1	2	yes	—
445	C₄	C₃₁×#472	1	2	yes	—
446	C₄	C₃₁×#525	1	2	yes	—
447	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
448	C₄	C₃₁×#532	1	1	yes	characteristic
449	C₂	—	2	2	no	—
450	C₂	—	4	2	no	—
451	C₂	—	16	2	no	—
452	C₂	C₃₁×#2850	1	1	yes	characteristic
453	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
454	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#3079	C₂	semidirect
4. C₄×C₃₁×#131	C₂	semidirect
5. C₃₁×#1997	C₂	semidirect
6. C₃₁×#1965	C₂	semidirect
7. C₃₁×#1931	C₂	semidirect
8. C₃₁×#1907	C₂	semidirect
9. C₃₁×#1837	C₂	semidirect
156. #33060	C₃₁	direct (N × H)
200. C₂×#486	C₂×C₃₁	semidirect
201. #3079	C₂×C₃₁	semidirect
203. C₄×#131	C₂×C₃₁	semidirect
204. #1997	C₂×C₃₁	semidirect
205. #1965	C₂×C₃₁	semidirect
206. #1931	C₂×C₃₁	semidirect
207. #1907	C₂×C₃₁	semidirect
208. #1837	C₂×C₃₁	semidirect
324. C₃₁	#33060	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×#3079	2
C₃₁×#3038	2
C₄×C₃₁×#131	2
C₃₁×#1997	2
C₃₁×#1965	2
C₃₁×#1931	2
C₃₁×#1907	2
C₃₁×#1837	2
#33060	31

Subgroups — C31×#33060 #21489819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33060 #21489819