Subgroups — C₃₁×#24804 #21481563

Order 7936.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3056	C₂	1	2	yes	maximal
3	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2052	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1901	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#525	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₃₁×#473	—	2	4	no	—
20	C₃₁×#470	—	2	2	no	—
21	C₃₁×#468	—	2	2	no	—
22	C₃₁×#467	—	2	1	no	—
23	C₃₁×#467	—	2	1	no	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#467	—	2	1	no	—
26	C₃₁×#467	—	2	2	no	—
27	C₃₁×#465	—	2	2	no	—
28	C₃₁×#435	—	2	1	no	—
29	C₃₁×#435	—	2	1	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₂×C₃₁×#108	—	2	4	no	—
33	C₂×C₃₁×#108	C₂²	1	2	yes	—
34	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
36	C₃₁×#428	C₂²	1	2	yes	—
37	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
38	C₃₁×#424	C₂²	1	1	yes	characteristic
39	C₃₁×#419	C₂²	1	2	yes	—
40	C₃₁×#417	C₂²	1	2	yes	—
41	C₂×C₃₁×#103	C₂²	1	2	yes	—
42	C₂×C₃₁×#101	C₂²	1	2	yes	—
43	C₂×C₃₁×#101	C₂²	1	2	yes	—
44	C₃₁×#408	C₂²	1	1	yes	characteristic
45	C₃₁×#407	C₂²	1	1	yes	characteristic
46	C₃₁×#405	C₂²	1	1	yes	characteristic
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#343	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#137	—	2	4	no	—
53	C₃₁×#137	—	4	2	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	1	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₃₁×#131	—	2	2	no	—
61	C₃₁×#131	—	2	2	no	—
62	C₂×M₁₆×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	2	1	no	—
64	C₂×M₁₆×C₃₁	—	2	4	no	—
65	C₂×M₁₆×C₃₁	—	4	2	no	—
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂²×C₈×C₃₁	—	2	2	no	—
71	C₂²×C₈×C₃₁	—	2	2	no	—
72	C₂²×C₈×C₃₁	—	2	2	no	—
73	C₂²×C₈×C₃₁	—	2	1	no	—
74	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#123	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	—	2	2	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	2	no	—
85	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
86	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
87	C₃₁×#121	D₄	1	1	yes	characteristic
88	C₃₁×#121	D₄	1	1	yes	characteristic
89	C₃₁×#120	—	2	1	no	—
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₃₁×#117	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₃₁×#108	—	2	4	no	—
97	C₃₁×#108	—	2	2	no	—
98	C₃₁×#108	D₄	1	4	yes	complemented
99	C₃₁×#107	—	2	1	no	—
100	C₃₁×#107	—	2	1	no	—
101	C₃₁×#107	—	2	2	no	—
102	C₃₁×#106	—	2	2	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#103	—	2	2	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#103	—	2	2	no	—
109	C₃₁×#103	—	2	2	no	—
110	C₃₁×#102	—	2	2	no	—
111	C₃₁×#101	—	2	4	no	—
112	C₃₁×#101	—	4	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	1	no	—
114	C₃₁×C₂²⋊C₈	—	2	1	no	—
115	C₃₁×C₂²⋊C₈	—	2	1	no	—
116	C₃₁×C₂²⋊C₈	—	2	1	no	—
117	C₃₁×#97	—	2	1	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₂⁴×C₃₁	#41	1	1	yes	characteristic
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	—	2	2	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	—	2	1	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₃₁	—	4	4	no	—
149	Q₁₆×C₃₁	—	4	2	no	—
150	Q₁₆×C₃₁	—	4	2	no	—
151	SD₁₆×C₃₁	—	4	4	no	—
152	M₁₆×C₃₁	—	2	2	no	—
153	M₁₆×C₃₁	—	2	1	no	—
154	M₁₆×C₃₁	—	2	2	no	—
155	M₁₆×C₃₁	—	2	1	no	—
156	M₁₆×C₃₁	—	2	1	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	1	no	—
159	M₁₆×C₃₁	—	4	4	no	—
160	M₁₆×C₃₁	—	4	2	no	—
161	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	4	4	no	—
173	C₂×C₈×C₃₁	—	4	2	no	—
174	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₃₁	#41	1	2	yes	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	4	2	no	—
185	C₄⋊C₄×C₃₁	—	4	2	no	—
186	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²⋊C₄×C₃₁	—	2	1	no	—
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	2	1	no	—
191	C₂²⋊C₄×C₃₁	—	2	1	no	—
192	C₂²⋊C₄×C₃₁	—	2	1	no	—
193	C₂²⋊C₄×C₃₁	—	4	2	no	—
194	C₂²⋊C₄×C₃₁	—	4	2	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	#24804	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	1	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	1	no	—
201	C₂³×C₃₁	—	4	1	no	—
202	C₂³×C₃₁	#121	1	1	yes	characteristic
203	C₂³×C₃₁	#121	1	1	yes	characteristic
204	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	1	no	—
206	Q₈×C₃₁	—	2	1	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	Q₈×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	2	1	no	—
212	D₄×C₃₁	—	2	1	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	1	no	—
215	D₄×C₃₁	—	4	1	no	—
216	D₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	8	2	no	—
237	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
238	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
245	C₈×C₃₁	—	2	1	no	—
246	C₈×C₃₁	—	2	1	no	—
247	C₈×C₃₁	—	2	4	no	—
248	C₈×C₃₁	—	4	1	no	—
249	C₈×C₃₁	—	4	1	no	—
250	C₈×C₃₁	—	4	2	no	—
251	C₈×C₃₁	#121	1	2	yes	—
252	C₈×C₃₁	#121	1	2	yes	—
253	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3056	C₂×C₃₁	1	2	yes	—
255	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#2929	C₂×C₃₁	1	1	yes	characteristic
257	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#2064	C₂×C₃₁	1	2	yes	complemented
259	#2052	C₂×C₃₁	1	2	yes	complemented
260	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1901	C₂×C₃₁	1	2	yes	complemented
263	#1816	C₂×C₃₁	1	1	yes	characteristic, complemented
264	C₂²×C₃₁	—	2	1	no	—
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	2	no	—
273	C₂²×C₃₁	—	4	1	no	—
274	C₂²×C₃₁	—	4	1	no	—
275	C₂²×C₃₁	—	4	1	no	—
276	C₂²×C₃₁	#473	1	2	yes	complemented
277	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
278	C₂²×C₃₁	#548	1	1	yes	characteristic
279	C₂²×C₃₁	#553	1	1	yes	characteristic
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	4	1	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	8	2	no	—
286	C₄×C₃₁	—	8	2	no	—
287	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
288	C₄×C₃₁	#537	1	1	yes	characteristic
289	C₄×C₃₁	D₄²	1	1	yes	characteristic
290	C₄×C₃₁	#547	1	1	yes	characteristic
291	C₂×#137	C₂²×C₃₁	1	2	yes	—
292	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
294	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
295	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
296	#525	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
298	#473	—	2	4	no	—
299	#470	—	2	2	no	—
300	#468	—	2	2	no	—
301	#467	—	2	1	no	—
302	#467	—	2	1	no	—
303	#467	—	2	1	no	—
304	#467	—	2	1	no	—
305	#467	—	2	2	no	—
306	#465	—	2	2	no	—
307	#435	—	2	1	no	—
308	#435	—	2	1	no	—
309	C₈×D₄	—	2	1	no	—
310	C₈×D₄	—	2	1	no	—
311	C₂×#108	—	2	4	no	—
312	C₂×#108	C₂²×C₃₁	1	2	yes	—
313	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
315	#428	C₂²×C₃₁	1	2	yes	—
316	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
317	#424	C₂²×C₃₁	1	1	yes	characteristic
318	#419	C₂²×C₃₁	1	2	yes	—
319	#417	C₂²×C₃₁	1	2	yes	—
320	C₂×#103	C₂²×C₃₁	1	2	yes	—
321	C₂×#101	C₂²×C₃₁	1	2	yes	—
322	C₂×#101	C₂²×C₃₁	1	2	yes	—
323	#408	C₂²×C₃₁	1	1	yes	characteristic
324	#407	C₂²×C₃₁	1	1	yes	characteristic
325	#405	C₂²×C₃₁	1	1	yes	characteristic
326	#343	C₂²×C₃₁	1	2	yes	—
327	#343	C₂²×C₃₁	1	2	yes	—
328	C₂×C₃₁	—	2	1	no	—
329	C₂×C₃₁	—	2	1	no	—
330	C₂×C₃₁	—	2	2	no	—
331	C₂×C₃₁	—	4	1	no	—
332	C₂×C₃₁	—	8	1	no	—
333	C₂×C₃₁	#2416	1	1	yes	characteristic
334	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
335	C₂×C₃₁	#3067	1	1	yes	characteristic
336	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
338	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
339	#137	—	2	4	no	—
340	#137	—	4	2	no	—
341	C₂×Q₁₆	—	2	2	no	—
342	C₂×Q₁₆	—	2	1	no	—
343	C₂×Q₁₆	—	2	1	no	—
344	C₂×Q₁₆	—	2	2	no	—
345	C₂×Q₁₆	—	2	2	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	#131	—	2	2	no	—
348	#131	—	2	2	no	—
349	C₂×M₁₆	—	2	2	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	4	no	—
352	C₂×M₁₆	—	4	2	no	—
353	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
354	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
356	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
357	C₂²×C₈	—	2	2	no	—
358	C₂²×C₈	—	2	2	no	—
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	1	no	—
361	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
367	#123	—	2	1	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	2	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
373	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
374	#121	D₄×C₃₁	1	1	yes	characteristic
375	#121	D₄×C₃₁	1	1	yes	characteristic
376	#120	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	#117	C₂³×C₃₁	1	1	yes	characteristic
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	#108	—	2	4	no	—
384	#108	—	2	2	no	—
385	#108	D₄×C₃₁	1	4	yes	complemented
386	#107	—	2	1	no	—
387	#107	—	2	1	no	—
388	#107	—	2	2	no	—
389	#106	—	2	2	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	#103	—	2	2	no	—
393	#103	—	2	2	no	—
394	#103	—	2	2	no	—
395	#103	—	2	2	no	—
396	#103	—	2	2	no	—
397	#102	—	2	2	no	—
398	#101	—	2	4	no	—
399	#101	—	4	2	no	—
400	C₂²⋊C₈	—	2	1	no	—
401	C₂²⋊C₈	—	2	1	no	—
402	C₂²⋊C₈	—	2	1	no	—
403	C₂²⋊C₈	—	2	1	no	—
404	#97	—	2	1	no	—
405	C₄×C₈	—	2	1	no	—
406	C₃₁	#24804	1	1	yes	characteristic, direct product factor, Sylow
407	C₂⁴	#41×C₃₁	1	1	yes	characteristic
408	#41	—	2	1	no	—
409	#41	—	2	1	no	—
410	#41	—	2	1	no	—
411	#41	—	2	1	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
436	Q₁₆	—	4	4	no	—
437	Q₁₆	—	4	2	no	—
438	Q₁₆	—	4	2	no	—
439	SD₁₆	—	4	4	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	1	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	2	1	no	—
444	M₁₆	—	2	1	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	1	no	—
447	M₁₆	—	4	4	no	—
448	M₁₆	—	4	2	no	—
449	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	4	4	no	—
461	C₂×C₈	—	4	2	no	—
462	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
467	C₂×C₈	#41×C₃₁	1	2	yes	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	4	2	no	—
482	C₂²⋊C₄	—	4	2	no	—
483	C₄²	—	2	1	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	C₃₁×#121	1	1	yes	characteristic
490	C₂³	C₃₁×#121	1	1	yes	characteristic
491	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	2	no	—
498	D₄	—	2	1	no	—
499	D₄	—	2	1	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	1	no	—
503	D₄	—	4	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
525	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
531	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	2	4	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	2	no	—
538	C₈	C₃₁×#121	1	2	yes	—
539	C₈	C₃₁×#121	1	2	yes	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	C₃₁×#473	1	2	yes	complemented
553	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
554	C₂²	C₃₁×#548	1	1	yes	characteristic
555	C₂²	C₃₁×#553	1	1	yes	characteristic
556	C₄	—	2	1	no	—
557	C₄	—	2	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	8	2	no	—
562	C₄	—	8	2	no	—
563	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
564	C₄	C₃₁×#537	1	1	yes	characteristic
565	C₄	D₄²×C₃₁	1	1	yes	characteristic
566	C₄	C₃₁×#547	1	1	yes	characteristic
567	C₂	—	2	2	no	—
568	C₂	—	2	1	no	—
569	C₂	—	2	1	no	—
570	C₂	—	4	1	no	—
571	C₂	—	8	1	no	—
572	C₂	C₃₁×#2416	1	1	yes	characteristic
573	C₂	C₂×C₃₁×#473	1	1	yes	characteristic
574	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#473	C₂	semidirect
3. C₂×C₃₁×#467	C₂	semidirect
5. C₂²×C₃₁×#108	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#2052	C₂	semidirect
8. C₃₁×#1979	C₂	semidirect
9. C₃₁×#1952	C₂	semidirect
10. C₃₁×#1901	C₂	semidirect
11. C₃₁×#1816	C₂	semidirect
98. C₃₁×#108	D₄	semidirect
196. #24804	C₃₁	direct (N × H)
253. C₂×#473	C₂×C₃₁	semidirect
255. C₂×#467	C₂×C₃₁	semidirect
257. C₂²×#108	C₂×C₃₁	semidirect
258. #2064	C₂×C₃₁	semidirect
259. #2052	C₂×C₃₁	semidirect
260. #1979	C₂×C₃₁	semidirect
261. #1952	C₂×C₃₁	semidirect
262. #1901	C₂×C₃₁	semidirect
263. #1816	C₂×C₃₁	semidirect
276. C₂²×C₃₁	#473	semidirect
385. #108	D₄×C₃₁	semidirect
406. C₃₁	#24804	direct (N × H)
552. C₂²	C₃₁×#473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#473	2
C₃₁×#3056	2
C₂×C₃₁×#467	2
C₃₁×#2929	2
C₂²×C₃₁×#108	2
C₃₁×#2064	2
C₃₁×#2052	2
C₃₁×#1979	2
C₃₁×#1952	2
C₃₁×#1901	2
C₃₁×#1816	2
#24804	31

Subgroups — C31×#24804 #21481563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24804 #21481563