Subgroups — C₃₁×#18773 #21475532

Order 7936.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1646	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1634	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#555	—	2	2	no	—
11	C₃₁×#555	—	2	2	no	—
12	C₃₁×#555	—	2	2	no	—
13	C₃₁×#555	C₂²	1	4	yes	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#124	—	2	2	no	—
17	C₂×C₃₁×#121	—	2	1	no	—
18	C₂×C₃₁×#121	—	2	1	no	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	—	2	1	no	—
22	C₂×C₃₁×#117	—	2	1	no	—
23	C₃₁×#494	C₂²	1	2	yes	complemented
24	C₃₁×#487	C₂²	1	2	yes	complemented
25	C₃₁×#483	—	2	1	no	—
26	C₃₁×#478	—	2	2	no	—
27	C₃₁×#476	C₂²	1	2	yes	complemented
28	C₃₁×#476	C₂²	1	2	yes	complemented
29	C₃₁×#474	C₂²	1	2	yes	complemented
30	C₃₁×#462	—	2	2	no	—
31	C₃₁×#461	C₂²	1	2	yes	complemented
32	C₃₁×#451	—	2	2	no	—
33	C₃₁×#450	—	2	1	no	—
34	C₄×SD₁₆×C₃₁	—	2	2	no	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#428	—	2	1	no	—
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₃₁×#418	—	2	1	no	—
40	C₃₁×#333	C₂²	1	4	yes	complemented
41	C₃₁×D₄⋊C₈	C₂²	1	4	yes	complemented
42	C₃₁×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	—	2	1	no	—
53	C₃₁×C₄⋊Q₈	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
55	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#124	—	2	8	no	—
60	C₃₁×#124	—	2	8	no	—
61	C₃₁×C₂²⋊Q₈	—	2	1	no	—
62	C₃₁×C₂²⋊Q₈	—	2	1	no	—
63	C₃₁×#121	—	2	4	no	—
64	C₃₁×#121	—	2	4	no	—
65	C₃₁×#121	—	2	4	no	—
66	C₃₁×#121	—	2	4	no	—
67	C₄×Q₈×C₃₁	—	2	1	no	—
68	C₄×Q₈×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	D₄	1	2	yes	—
74	C₄×D₄×C₃₁	D₄	1	2	yes	—
75	C₄×D₄×C₃₁	C₂³	1	2	yes	—
76	C₄×D₄×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#117	—	2	4	no	—
78	C₃₁×#117	—	2	4	no	—
79	C₃₁×#117	—	2	4	no	—
80	C₃₁×#117	—	2	4	no	—
81	C₃₁×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#107	—	2	2	no	—
93	C₃₁×#106	—	2	2	no	—
94	C₃₁×#106	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
99	C₃₁×#103	—	2	4	no	—
100	C₃₁×#103	—	2	2	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₄×C₈×C₃₁	—	2	1	no	—
107	C₄×C₈×C₃₁	C₂³	1	2	yes	—
108	C₂⁴×C₃₁	—	2	1	no	—
109	C₂⁴×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	8	2	no	—
114	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
115	C₂×D₄×C₃₁	—	2	8	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	4	2	no	—
121	C₂×D₄×C₃₁	—	8	4	no	—
122	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	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	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	2	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	Q₁₆×C₃₁	—	8	4	no	—
138	SD₁₆×C₃₁	—	4	4	no	—
139	M₁₆×C₃₁	—	4	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	4	1	no	—
154	C₄⋊C₄×C₃₁	—	4	1	no	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₃₁	—	2	8	no	—
158	C₂²⋊C₄×C₃₁	—	2	8	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂²⋊C₄×C₃₁	—	4	1	no	—
164	C₄²×C₃₁	—	2	8	no	—
165	C₄²×C₃₁	—	4	1	no	—
166	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
170	#18773	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	#120	1	1	yes	characteristic
180	Q₈×C₃₁	—	4	4	no	—
181	Q₈×C₃₁	—	4	2	no	—
182	Q₈×C₃₁	—	4	2	no	—
183	Q₈×C₃₁	—	8	2	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	8	8	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	4	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	4	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	4	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
208	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
209	C₂×C₄×C₃₁	#135	1	2	yes	—
210	C₂×C₄×C₃₁	#137	1	2	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₂×#555	C₂×C₃₁	1	1	yes	characteristic
218	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#3077	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1646	C₂×C₃₁	1	4	yes	complemented
222	#1634	C₂×C₃₁	1	4	yes	complemented
223	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1486	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1478	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂²×C₃₁	—	2	1	no	—
227	C₂²×C₃₁	—	2	1	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	4	2	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	8	4	no	—
235	C₂²×C₃₁	—	8	4	no	—
236	C₂²×C₃₁	#452	1	2	yes	—
237	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
238	C₄×C₃₁	—	2	1	no	—
239	C₄×C₃₁	—	2	1	no	—
240	C₄×C₃₁	—	2	1	no	—
241	C₄×C₃₁	—	4	2	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	4	no	—
244	C₄×C₃₁	—	8	1	no	—
245	C₄×C₃₁	—	8	1	no	—
246	C₄×C₃₁	#452	1	2	yes	—
247	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
248	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
249	C₄×C₃₁	#575	1	2	yes	—
250	#555	—	2	2	no	—
251	#555	—	2	2	no	—
252	#555	—	2	2	no	—
253	#555	C₂²×C₃₁	1	4	yes	—
254	#533	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#124	—	2	2	no	—
257	C₂×#121	—	2	1	no	—
258	C₂×#121	—	2	1	no	—
259	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#117	—	2	1	no	—
262	C₂×#117	—	2	1	no	—
263	#494	C₂²×C₃₁	1	2	yes	complemented
264	#487	C₂²×C₃₁	1	2	yes	complemented
265	#483	—	2	1	no	—
266	#478	—	2	2	no	—
267	#476	C₂²×C₃₁	1	2	yes	complemented
268	#476	C₂²×C₃₁	1	2	yes	complemented
269	#474	C₂²×C₃₁	1	2	yes	complemented
270	#462	—	2	2	no	—
271	#461	C₂²×C₃₁	1	2	yes	complemented
272	#451	—	2	2	no	—
273	#450	—	2	1	no	—
274	C₄×SD₁₆	—	2	2	no	—
275	#431	C₂²×C₃₁	1	1	yes	characteristic
276	#431	C₂²×C₃₁	1	1	yes	characteristic
277	#428	—	2	1	no	—
278	#423	C₂²×C₃₁	1	1	yes	characteristic
279	#418	—	2	1	no	—
280	#333	C₂²×C₃₁	1	4	yes	complemented
281	D₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
282	D₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
283	C₂×C₃₁	—	2	2	no	—
284	C₂×C₃₁	—	8	2	no	—
285	C₂×C₃₁	—	8	2	no	—
286	C₂×C₃₁	C₂×#452	1	1	yes	characteristic
287	C₂×C₃₁	#3005	1	1	yes	characteristic
288	C₂×C₃₁	#3019	1	1	yes	characteristic
289	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
290	C₂²×D₄	—	2	2	no	—
291	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
292	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
293	C₂³×C₄	—	2	1	no	—
294	C₂³×C₄	—	2	1	no	—
295	C₂×Q₁₆	—	2	4	no	—
296	C₂×SD₁₆	—	2	2	no	—
297	C₂×M₁₆	—	2	1	no	—
298	C₄⋊Q₈	—	2	1	no	—
299	C₄⋊Q₈	—	2	1	no	—
300	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
301	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
304	#125	—	2	1	no	—
305	#124	—	2	8	no	—
306	#124	—	2	8	no	—
307	C₂²⋊Q₈	—	2	1	no	—
308	C₂²⋊Q₈	—	2	1	no	—
309	#121	—	2	4	no	—
310	#121	—	2	4	no	—
311	#121	—	2	4	no	—
312	#121	—	2	4	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×Q₈	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	D₄×C₃₁	1	2	yes	—
320	C₄×D₄	D₄×C₃₁	1	2	yes	—
321	C₄×D₄	C₂³×C₃₁	1	2	yes	—
322	C₄×D₄	C₂³×C₃₁	1	2	yes	—
323	#117	—	2	4	no	—
324	#117	—	2	4	no	—
325	#117	—	2	4	no	—
326	#117	—	2	4	no	—
327	#117	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	#106	—	2	2	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
344	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
345	#103	—	2	4	no	—
346	#103	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	C₂³×C₃₁	1	2	yes	—
354	C₃₁	#18773	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁴	—	2	1	no	—
356	C₂⁴	—	2	1	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	8	2	no	—
361	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	4	2	no	—
368	C₂×D₄	—	8	4	no	—
369	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	4	4	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	Q₁₆	—	8	4	no	—
385	SD₁₆	—	4	4	no	—
386	M₁₆	—	4	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	4	1	no	—
401	C₄⋊C₄	—	4	1	no	—
402	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
403	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	4	1	no	—
411	C₄²	—	2	8	no	—
412	C₄²	—	4	1	no	—
413	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	C₃₁×#120	1	1	yes	characteristic
426	Q₈	—	4	4	no	—
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	8	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	8	8	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
453	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
454	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
455	C₂×C₄	C₃₁×#135	1	2	yes	—
456	C₂×C₄	C₃₁×#137	1	2	yes	—
457	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₈	—	4	2	no	—
461	C₈	—	4	2	no	—
462	C₈	—	4	2	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	8	4	no	—
472	C₂²	—	8	4	no	—
473	C₂²	C₃₁×#452	1	2	yes	—
474	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	4	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	1	no	—
483	C₄	C₃₁×#452	1	2	yes	—
484	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
485	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
486	C₄	C₃₁×#575	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	8	2	no	—
489	C₂	—	8	2	no	—
490	C₂	C₂×C₃₁×#452	1	1	yes	characteristic
491	C₂	C₃₁×#3005	1	1	yes	characteristic
492	C₂	C₃₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#555	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
2. C₃₁×#3209	C₂	semidirect
3. C₃₁×#3101	C₂	semidirect
4. C₃₁×#3077	C₂	semidirect
5. C₃₁×#1646	C₂	semidirect
6. C₃₁×#1634	C₂	semidirect
7. C₃₁×#1548	C₂	semidirect
8. C₃₁×#1486	C₂	semidirect
9. C₃₁×#1478	C₂	semidirect
23. C₃₁×#494	C₂²	semidirect
24. C₃₁×#487	C₂²	semidirect
27. C₃₁×#476	C₂²	semidirect
28. C₃₁×#476	C₂²	semidirect
29. C₃₁×#474	C₂²	semidirect
31. C₃₁×#461	C₂²	semidirect
40. C₃₁×#333	C₂²	semidirect
41. C₃₁×D₄⋊C₈	C₂²	semidirect
42. C₃₁×D₄⋊C₈	C₂²	semidirect
170. #18773	C₃₁	direct (N × H)
218. #3209	C₂×C₃₁	semidirect
219. #3101	C₂×C₃₁	semidirect
220. #3077	C₂×C₃₁	semidirect
221. #1646	C₂×C₃₁	semidirect
222. #1634	C₂×C₃₁	semidirect
223. #1548	C₂×C₃₁	semidirect
224. #1486	C₂×C₃₁	semidirect
225. #1478	C₂×C₃₁	semidirect
263. #494	C₂²×C₃₁	semidirect
264. #487	C₂²×C₃₁	semidirect
267. #476	C₂²×C₃₁	semidirect
268. #476	C₂²×C₃₁	semidirect
269. #474	C₂²×C₃₁	semidirect
271. #461	C₂²×C₃₁	semidirect
280. #333	C₂²×C₃₁	semidirect
281. D₄⋊C₈	C₂²×C₃₁	semidirect
282. D₄⋊C₈	C₂²×C₃₁	semidirect
354. C₃₁	#18773	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×#555	2
C₃₁×#3209	2
C₃₁×#3101	2
C₃₁×#3077	2
C₃₁×#1646	2
C₃₁×#1634	2
C₃₁×#1548	2
C₃₁×#1486	2
C₃₁×#1478	2
#18773	31

Subgroups — C31×#18773 #21475532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18773 #21475532