Subgroups — C₃₁×#23018 #21479777

Order 7936.

892 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2933	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1871	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#525	C₂²	1	2	yes	—
17	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#465	C₄	1	2	yes	complemented
22	C₃₁×#465	C₄	1	2	yes	complemented
23	C₃₁×#443	—	2	4	no	—
24	C₃₁×#436	—	2	2	no	—
25	C₃₁×#435	—	2	2	no	—
26	C₃₁×#435	—	2	2	no	—
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	—	2	4	no	—
29	C₂×C₃₁×#108	C₂²	1	2	yes	—
30	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
32	C₃₁×#427	C₂²	1	1	yes	characteristic
33	C₃₁×#424	C₂²	1	2	yes	—
34	C₂×C₃₁×#104	C₂²	1	2	yes	—
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#339	C₂²	1	2	yes	—
44	C₃₁×#339	C₂²	1	2	yes	—
45	C₃₁×#336	C₂²	1	1	yes	characteristic
46	C₃₁×#336	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#135	—	2	2	no	—
50	C₃₁×#135	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#131	—	2	4	no	—
55	C₃₁×#131	—	4	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	4	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂×C₄	1	2	yes	—
78	C₃₁×#121	C₂×C₄	1	2	yes	—
79	C₃₁×#120	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₃₁×#117	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#108	—	2	4	no	—
87	C₃₁×#108	—	2	2	no	—
88	C₃₁×#108	D₄	1	4	yes	complemented
89	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#106	C₂×C₄	1	2	yes	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×#104	—	2	4	no	—
95	C₃₁×#104	—	4	2	no	—
96	C₃₁×#103	C₂×C₄	1	2	yes	—
97	C₃₁×#103	C₂×C₄	1	2	yes	—
98	C₃₁×#102	C₂×C₄	1	2	yes	—
99	C₃₁×#102	C₂×C₄	1	2	yes	—
100	C₃₁×C₂²⋊C₈	—	2	2	no	—
101	C₃₁×C₂²⋊C₈	—	2	2	no	—
102	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
103	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
104	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
105	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
106	C₃₁×#97	—	2	2	no	—
107	C₄×C₈×C₃₁	—	2	2	no	—
108	C₄×C₈×C₃₁	—	2	2	no	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₃₁×#95	—	2	1	no	—
111	C₂⁴×C₃₁	#41	1	1	yes	characteristic
112	#41×C₃₁	—	2	2	no	—
113	#41×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	Q₁₆×C₃₁	—	2	2	no	—
135	SD₁₆×C₃₁	—	2	4	no	—
136	D₈×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	2	2	no	—
140	M₁₆×C₃₁	—	4	4	no	—
141	M₁₆×C₃₁	—	4	4	no	—
142	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	4	4	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
166	C₂×C₈×C₃₁	#41	1	2	yes	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
170	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₂²⋊C₄×C₃₁	—	2	2	no	—
173	C₂²⋊C₄×C₃₁	—	2	2	no	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	1	no	—
179	C₄²×C₃₁	—	2	1	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	#23018	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	4	1	no	—
188	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
189	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
190	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
191	Q₈×C₃₁	—	2	2	no	—
192	Q₈×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	4	2	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	4	no	—
198	D₄×C₃₁	—	8	2	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
228	C₈×C₃₁	—	2	1	no	—
229	C₈×C₃₁	—	2	1	no	—
230	C₈×C₃₁	—	2	1	no	—
231	C₈×C₃₁	—	2	1	no	—
232	C₈×C₃₁	—	2	4	no	—
233	C₈×C₃₁	—	2	1	no	—
234	C₈×C₃₁	—	2	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	C₄×D₄	1	2	yes	—
237	C₈×C₃₁	C₄×D₄	1	2	yes	—
238	#3050	C₂×C₃₁	1	1	yes	characteristic
239	C₂×#443	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#2933	C₂×C₃₁	1	2	yes	complemented
241	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1974	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1929	C₂×C₃₁	1	2	yes	complemented
245	#1871	C₂×C₃₁	1	2	yes	complemented
246	#1866	C₂×C₃₁	1	2	yes	complemented
247	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	#443	1	2	yes	complemented
262	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
263	C₂²×C₃₁	#548	1	1	yes	characteristic
264	C₂²×C₃₁	#553	1	1	yes	characteristic
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
273	C₄×C₃₁	#520	1	1	yes	characteristic
274	C₄×C₃₁	D₄²	1	1	yes	characteristic
275	C₄×C₃₁	#547	1	1	yes	characteristic
276	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#131	C₂²×C₃₁	1	2	yes	—
278	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
279	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
280	#525	C₂²×C₃₁	1	2	yes	—
281	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
282	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
283	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
284	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
285	#465	C₄×C₃₁	1	2	yes	complemented
286	#465	C₄×C₃₁	1	2	yes	complemented
287	#443	—	2	4	no	—
288	#436	—	2	2	no	—
289	#435	—	2	2	no	—
290	#435	—	2	2	no	—
291	C₈×D₄	—	2	2	no	—
292	C₂×#108	—	2	4	no	—
293	C₂×#108	C₂²×C₃₁	1	2	yes	—
294	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
296	#427	C₂²×C₃₁	1	1	yes	characteristic
297	#424	C₂²×C₃₁	1	2	yes	—
298	C₂×#104	C₂²×C₃₁	1	2	yes	—
299	C₂×#104	C₂²×C₃₁	1	2	yes	—
300	#416	C₂²×C₃₁	1	2	yes	—
301	#408	C₂²×C₃₁	1	2	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
303	#405	C₂²×C₃₁	1	2	yes	—
304	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
306	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
307	#339	C₂²×C₃₁	1	2	yes	—
308	#339	C₂²×C₃₁	1	2	yes	—
309	#336	C₂²×C₃₁	1	1	yes	characteristic
310	#336	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₃₁	—	2	1	no	—
312	C₂×C₃₁	—	2	1	no	—
313	C₂×C₃₁	—	2	2	no	—
314	C₂×C₃₁	—	4	1	no	—
315	C₂×C₃₁	—	8	2	no	—
316	C₂×C₃₁	#2361	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
318	C₂×C₃₁	#2957	1	1	yes	characteristic
319	C₂×#41	C₂³×C₃₁	1	2	yes	—
320	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
321	#135	—	2	2	no	—
322	#135	—	2	2	no	—
323	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
325	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
326	#131	—	2	4	no	—
327	#131	—	4	2	no	—
328	C₂×M₁₆	—	2	2	no	—
329	C₂×M₁₆	—	2	2	no	—
330	C₂×M₁₆	—	2	4	no	—
331	C₂×M₁₆	—	4	2	no	—
332	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
333	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
334	C₂²×C₈	—	2	2	no	—
335	C₂²×C₈	—	2	2	no	—
336	C₂²×C₈	—	2	2	no	—
337	C₂²×C₈	—	2	2	no	—
338	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
346	#123	—	2	2	no	—
347	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
348	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
349	#121	C₂×C₄×C₃₁	1	2	yes	—
350	#121	C₂×C₄×C₃₁	1	2	yes	—
351	#120	—	2	2	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	—	2	2	no	—
354	#117	C₂³×C₃₁	1	2	yes	—
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	#108	—	2	4	no	—
359	#108	—	2	2	no	—
360	#108	D₄×C₃₁	1	4	yes	complemented
361	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
362	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
363	#106	C₂×C₄×C₃₁	1	2	yes	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	#104	—	2	4	no	—
367	#104	—	4	2	no	—
368	#103	C₂×C₄×C₃₁	1	2	yes	—
369	#103	C₂×C₄×C₃₁	1	2	yes	—
370	#102	C₂×C₄×C₃₁	1	2	yes	—
371	#102	C₂×C₄×C₃₁	1	2	yes	—
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
375	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
376	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
377	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
378	#97	—	2	2	no	—
379	C₄×C₈	—	2	2	no	—
380	C₄×C₈	—	2	2	no	—
381	C₄×C₈	—	2	2	no	—
382	#95	—	2	1	no	—
383	C₃₁	#23018	1	1	yes	characteristic, direct product factor, Sylow
384	C₂⁴	#41×C₃₁	1	1	yes	characteristic
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	2	2	no	—
388	#41	—	2	2	no	—
389	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	Q₁₆	—	2	2	no	—
408	SD₁₆	—	2	4	no	—
409	D₈	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	4	4	no	—
414	M₁₆	—	4	4	no	—
415	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	4	4	no	—
429	C₂×C₈	—	4	4	no	—
430	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
439	C₂×C₈	#41×C₃₁	1	2	yes	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
443	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₄²	—	2	2	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	4	1	no	—
460	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
461	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
462	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	D₄	—	2	2	no	—
466	D₄	—	2	2	no	—
467	D₄	—	4	4	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	2	no	—
470	D₄	—	8	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
500	C₈	—	2	1	no	—
501	C₈	—	2	1	no	—
502	C₈	—	2	1	no	—
503	C₈	—	2	1	no	—
504	C₈	—	2	4	no	—
505	C₈	—	2	1	no	—
506	C₈	—	2	1	no	—
507	C₈	—	4	2	no	—
508	C₈	C₄×D₄×C₃₁	1	2	yes	—
509	C₈	C₄×D₄×C₃₁	1	2	yes	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	C₃₁×#443	1	2	yes	complemented
523	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
524	C₂²	C₃₁×#548	1	1	yes	characteristic
525	C₂²	C₃₁×#553	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	2	no	—
533	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
534	C₄	C₃₁×#520	1	1	yes	characteristic
535	C₄	D₄²×C₃₁	1	1	yes	characteristic
536	C₄	C₃₁×#547	1	1	yes	characteristic
537	C₂	—	2	1	no	—
538	C₂	—	2	1	no	—
539	C₂	—	2	2	no	—
540	C₂	—	4	1	no	—
541	C₂	—	8	2	no	—
542	C₂	C₃₁×#2361	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
544	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#443	C₂	semidirect
3. C₃₁×#2933	C₂	semidirect
4. C₂²×C₃₁×#108	C₂	semidirect
5. C₃₁×#1974	C₂	semidirect
6. C₃₁×#1945	C₂	semidirect
7. C₃₁×#1929	C₂	semidirect
8. C₃₁×#1871	C₂	semidirect
9. C₃₁×#1866	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₃₁×#467	C₄	semidirect
18. C₃₁×#467	C₄	semidirect
19. C₃₁×#466	C₄	semidirect
20. C₃₁×#466	C₄	semidirect
21. C₃₁×#465	C₄	semidirect
22. C₃₁×#465	C₄	semidirect
88. C₃₁×#108	D₄	semidirect
182. #23018	C₃₁	direct (N × H)
239. C₂×#443	C₂×C₃₁	semidirect
240. #2933	C₂×C₃₁	semidirect
241. C₂²×#108	C₂×C₃₁	semidirect
242. #1974	C₂×C₃₁	semidirect
243. #1945	C₂×C₃₁	semidirect
244. #1929	C₂×C₃₁	semidirect
245. #1871	C₂×C₃₁	semidirect
246. #1866	C₂×C₃₁	semidirect
247. #1772	C₂×C₃₁	semidirect
248. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
261. C₂²×C₃₁	#443	semidirect
281. #467	C₄×C₃₁	semidirect
282. #467	C₄×C₃₁	semidirect
283. #466	C₄×C₃₁	semidirect
284. #466	C₄×C₃₁	semidirect
285. #465	C₄×C₃₁	semidirect
286. #465	C₄×C₃₁	semidirect
360. #108	D₄×C₃₁	semidirect
383. C₃₁	#23018	direct (N × H)
522. C₂²	C₃₁×#443	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3050	2
C₂×C₃₁×#443	2
C₃₁×#2933	2
C₂²×C₃₁×#108	2
C₃₁×#1974	2
C₃₁×#1945	2
C₃₁×#1929	2
C₃₁×#1871	2
C₃₁×#1866	2
C₃₁×#1772	2
C₈×C₂²⋊C₄×C₃₁	2
#23018	31

Subgroups — C31×#23018 #21479777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23018 #21479777