Subgroups — C₃₁×#23021 #21479780

Order 7936.

874 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2933	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1946	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	C₃₁×#1753	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	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#525	C₂²	1	2	yes	—
17	C₃₁×#470	C₄	1	2	yes	complemented
18	C₃₁×#469	C₄	1	2	yes	complemented
19	C₃₁×#468	C₄	1	2	yes	complemented
20	C₃₁×#468	C₄	1	2	yes	complemented
21	C₃₁×#444	—	2	4	no	—
22	C₃₁×#436	—	2	2	no	—
23	C₃₁×#435	—	2	2	no	—
24	C₃₁×#435	—	2	2	no	—
25	C₈×D₄×C₃₁	—	2	2	no	—
26	C₃₁×#430	—	2	4	no	—
27	C₂×C₃₁×#108	C₂²	1	2	yes	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₃₁×#424	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₂×C₃₁×#104	C₂²	1	2	yes	—
34	C₃₁×#416	C₂²	1	2	yes	—
35	C₃₁×#408	C₂²	1	2	yes	—
36	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#405	C₂²	1	2	yes	—
38	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#339	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₃₁×#336	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#135	—	2	2	no	—
48	C₃₁×#135	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#131	—	2	4	no	—
53	C₃₁×#131	—	4	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	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	C₂×C₄	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#123	—	2	2	no	—
71	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
72	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
73	C₃₁×#121	C₂×C₄	1	2	yes	—
74	C₃₁×#121	C₂×C₄	1	2	yes	—
75	C₃₁×#120	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₃₁×#117	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#108	—	2	4	no	—
83	C₃₁×#108	—	2	2	no	—
84	C₃₁×#108	D₄	1	4	yes	complemented
85	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
87	C₃₁×#106	C₂×C₄	1	2	yes	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×#104	—	2	4	no	—
91	C₃₁×#104	—	4	2	no	—
92	C₃₁×#103	C₂×C₄	1	2	yes	—
93	C₃₁×#103	C₂×C₄	1	2	yes	—
94	C₃₁×#102	C₂×C₄	1	2	yes	—
95	C₃₁×#102	C₂×C₄	1	2	yes	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
99	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
100	C₃₁×#97	—	2	2	no	—
101	C₃₁×#97	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₄×C₈×C₃₁	—	2	2	no	—
104	C₃₁×#95	—	2	1	no	—
105	C₂⁴×C₃₁	#41	1	1	yes	characteristic
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
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	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₃₁	—	2	2	no	—
127	SD₁₆×C₃₁	—	2	4	no	—
128	D₈×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	2	4	no	—
131	M₁₆×C₃₁	—	2	4	no	—
132	M₁₆×C₃₁	—	2	2	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	4	4	no	—
135	M₁₆×C₃₁	—	4	4	no	—
136	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	4	4	no	—
146	C₂×C₈×C₃₁	—	4	4	no	—
147	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₃₁	#41	1	2	yes	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
158	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	4	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	—	2	1	no	—
170	#23021	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	C₄×D₄	1	2	yes	—
177	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	2	2	no	—
182	D₄×C₃₁	—	4	4	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	8	2	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	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	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
204	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	2	2	no	—
212	C₈×C₃₁	—	2	4	no	—
213	C₈×C₃₁	—	2	1	no	—
214	C₈×C₃₁	—	2	1	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₈×C₃₁	C₄×D₄	1	2	yes	—
217	C₈×C₃₁	C₄×D₄	1	2	yes	—
218	#3054	C₂×C₃₁	1	1	yes	characteristic
219	C₂×#444	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#2933	C₂×C₃₁	1	2	yes	complemented
221	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#1974	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1929	C₂×C₃₁	1	2	yes	complemented
225	#1871	C₂×C₃₁	1	2	yes	complemented
226	#1866	C₂×C₃₁	1	2	yes	complemented
227	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂²×C₃₁	—	2	1	no	—
230	C₂²×C₃₁	—	2	1	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂²×C₃₁	#548	1	1	yes	characteristic
241	C₂²×C₃₁	#553	1	1	yes	characteristic
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₃₁	#520	1	1	yes	characteristic
250	C₄×C₃₁	D₄²	1	1	yes	characteristic
251	C₄×C₃₁	#547	1	1	yes	characteristic
252	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#131	C₂²×C₃₁	1	2	yes	—
254	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
255	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	#525	C₂²×C₃₁	1	2	yes	—
257	#470	C₄×C₃₁	1	2	yes	complemented
258	#469	C₄×C₃₁	1	2	yes	complemented
259	#468	C₄×C₃₁	1	2	yes	complemented
260	#468	C₄×C₃₁	1	2	yes	complemented
261	#444	—	2	4	no	—
262	#436	—	2	2	no	—
263	#435	—	2	2	no	—
264	#435	—	2	2	no	—
265	C₈×D₄	—	2	2	no	—
266	#430	—	2	4	no	—
267	C₂×#108	C₂²×C₃₁	1	2	yes	—
268	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
270	#427	C₂²×C₃₁	1	1	yes	characteristic
271	#424	C₂²×C₃₁	1	2	yes	—
272	C₂×#104	C₂²×C₃₁	1	2	yes	—
273	C₂×#104	C₂²×C₃₁	1	2	yes	—
274	#416	C₂²×C₃₁	1	2	yes	—
275	#408	C₂²×C₃₁	1	2	yes	—
276	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
277	#405	C₂²×C₃₁	1	2	yes	—
278	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	#339	C₂²×C₃₁	1	2	yes	—
282	#339	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	1	yes	characteristic
284	#336	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₃₁	—	2	1	no	—
286	C₂×C₃₁	—	2	1	no	—
287	C₂×C₃₁	—	4	2	no	—
288	C₂×C₃₁	—	8	2	no	—
289	C₂×C₃₁	#2361	1	1	yes	characteristic
290	C₂×C₃₁	#2955	1	1	yes	characteristic
291	C₂×C₃₁	#2956	1	1	yes	characteristic
292	C₂×#41	C₂³×C₃₁	1	2	yes	—
293	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
294	#135	—	2	2	no	—
295	#135	—	2	2	no	—
296	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
298	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
299	#131	—	2	4	no	—
300	#131	—	4	2	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	2	4	no	—
306	C₂×M₁₆	—	4	2	no	—
307	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
308	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
309	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
310	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	—	2	2	no	—
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	#123	—	2	2	no	—
318	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
319	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
320	#121	C₂×C₄×C₃₁	1	2	yes	—
321	#121	C₂×C₄×C₃₁	1	2	yes	—
322	#120	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	#117	C₂³×C₃₁	1	2	yes	—
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
329	#108	—	2	4	no	—
330	#108	—	2	2	no	—
331	#108	D₄×C₃₁	1	4	yes	complemented
332	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
333	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
334	#106	C₂×C₄×C₃₁	1	2	yes	—
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	—	2	2	no	—
337	#104	—	2	4	no	—
338	#104	—	4	2	no	—
339	#103	C₂×C₄×C₃₁	1	2	yes	—
340	#103	C₂×C₄×C₃₁	1	2	yes	—
341	#102	C₂×C₄×C₃₁	1	2	yes	—
342	#102	C₂×C₄×C₃₁	1	2	yes	—
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
346	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
347	#97	—	2	2	no	—
348	#97	—	2	2	no	—
349	C₄×C₈	—	2	2	no	—
350	C₄×C₈	—	2	2	no	—
351	#95	—	2	1	no	—
352	C₃₁	#23021	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	#41×C₃₁	1	1	yes	characteristic
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
372	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
374	Q₁₆	—	2	2	no	—
375	SD₁₆	—	2	4	no	—
376	D₈	—	2	2	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	4	no	—
379	M₁₆	—	2	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	4	4	no	—
383	M₁₆	—	4	4	no	—
384	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	4	4	no	—
394	C₂×C₈	—	4	4	no	—
395	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
396	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
402	C₂×C₈	#41×C₃₁	1	2	yes	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
406	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	4	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	C₄×D₄×C₃₁	1	2	yes	—
424	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
425	Q₈	—	2	2	no	—
426	Q₈	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	2	2	no	—
429	D₄	—	4	4	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	8	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
457	C₈	—	2	2	no	—
458	C₈	—	2	2	no	—
459	C₈	—	2	4	no	—
460	C₈	—	2	1	no	—
461	C₈	—	2	1	no	—
462	C₈	—	4	2	no	—
463	C₈	C₄×D₄×C₃₁	1	2	yes	—
464	C₈	C₄×D₄×C₃₁	1	2	yes	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	1	no	—
474	C₂²	—	8	1	no	—
475	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
476	C₂²	C₃₁×#548	1	1	yes	characteristic
477	C₂²	C₃₁×#553	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	8	1	no	—
484	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
485	C₄	C₃₁×#520	1	1	yes	characteristic
486	C₄	D₄²×C₃₁	1	1	yes	characteristic
487	C₄	C₃₁×#547	1	1	yes	characteristic
488	C₂	—	2	1	no	—
489	C₂	—	2	1	no	—
490	C₂	—	4	2	no	—
491	C₂	—	8	2	no	—
492	C₂	C₃₁×#2361	1	1	yes	characteristic
493	C₂	C₃₁×#2955	1	1	yes	characteristic
494	C₂	C₃₁×#2956	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×#3054	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×#444	C₂	semidirect
3. C₃₁×#2933	C₂	semidirect
4. C₂×C₃₁×#430	C₂	semidirect
5. C₃₁×#1974	C₂	semidirect
6. C₃₁×#1946	C₂	semidirect
7. C₃₁×#1929	C₂	semidirect
8. C₃₁×#1871	C₂	semidirect
9. C₃₁×#1866	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
11. C₃₁×#1753	C₂	semidirect
17. C₃₁×#470	C₄	semidirect
18. C₃₁×#469	C₄	semidirect
19. C₃₁×#468	C₄	semidirect
20. C₃₁×#468	C₄	semidirect
84. C₃₁×#108	D₄	semidirect
170. #23021	C₃₁	direct (N × H)
219. C₂×#444	C₂×C₃₁	semidirect
220. #2933	C₂×C₃₁	semidirect
221. C₂×#430	C₂×C₃₁	semidirect
222. #1974	C₂×C₃₁	semidirect
223. #1946	C₂×C₃₁	semidirect
224. #1929	C₂×C₃₁	semidirect
225. #1871	C₂×C₃₁	semidirect
226. #1866	C₂×C₃₁	semidirect
227. #1772	C₂×C₃₁	semidirect
228. #1753	C₂×C₃₁	semidirect
257. #470	C₄×C₃₁	semidirect
258. #469	C₄×C₃₁	semidirect
259. #468	C₄×C₃₁	semidirect
260. #468	C₄×C₃₁	semidirect
331. #108	D₄×C₃₁	semidirect
352. C₃₁	#23021	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3054	2
C₂×C₃₁×#444	2
C₃₁×#2933	2
C₂×C₃₁×#430	2
C₃₁×#1974	2
C₃₁×#1946	2
C₃₁×#1929	2
C₃₁×#1871	2
C₃₁×#1866	2
C₃₁×#1772	2
C₃₁×#1753	2
#23021	31

Subgroups — C31×#23021 #21479780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23021 #21479780