Subgroups — C₃₁×#22687 #21479446

Order 7936.

858 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2906	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1806	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1771	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	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₄×#41×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#444	—	2	4	no	—
17	C₃₁×#443	—	2	4	no	—
18	C₃₁×#442	C₄	1	2	yes	complemented
19	C₃₁×#441	C₄	1	2	yes	complemented
20	C₃₁×#440	C₄	1	2	yes	complemented
21	C₃₁×#440	C₄	1	2	yes	complemented
22	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₃₁×#427	C₂²	1	1	yes	characteristic
26	C₃₁×#424	—	2	2	no	—
27	C₃₁×#424	—	2	2	no	—
28	C₃₁×#424	C₂²	1	2	yes	—
29	C₃₁×#424	C₂²	1	2	yes	—
30	C₃₁×#423	—	2	2	no	—
31	C₃₁×#423	C₂²	1	2	yes	—
32	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#104	C₂²	1	2	yes	—
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	2	yes	—
37	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊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	C₃₁×#135	D₄	1	4	yes	—
47	C₃₁×#135	Q₈	1	4	yes	—
48	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₃₁×#131	—	2	4	no	—
52	C₃₁×#131	—	2	4	no	—
53	C₃₁×#131	—	4	2	no	—
54	C₃₁×#131	—	4	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
66	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#108	—	2	2	no	—
79	C₃₁×#108	—	2	2	no	—
80	C₃₁×#108	—	4	2	no	—
81	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
82	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
83	C₃₁×#106	C₂×C₄	1	2	yes	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
91	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
92	C₃₁×#104	—	2	4	no	—
93	C₃₁×#104	—	2	4	no	—
94	C₃₁×#103	C₂×C₄	1	2	yes	—
95	C₃₁×#103	C₂×C₄	1	2	yes	—
96	C₃₁×#102	C₂×C₄	1	2	yes	—
97	C₃₁×#102	C₂×C₄	1	2	yes	—
98	C₃₁×#97	—	2	2	no	—
99	C₃₁×#97	C₂×C₄	1	2	yes	—
100	C₃₁×#97	C₂×C₄	1	2	yes	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₃₁×#95	—	2	1	no	—
103	#41×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	C₂×D₄	1	4	yes	—
106	#41×C₃₁	C₂×Q₈	1	4	yes	—
107	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₃₁	C₄⋊C₄	1	4	yes	complemented
119	SD₁₆×C₃₁	C₄⋊C₄	1	8	yes	complemented
120	D₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	2	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	4	4	no	—
126	M₁₆×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	4	no	—
136	C₂×C₈×C₃₁	—	4	4	no	—
137	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂²⋊C₄×C₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
161	C₄²×C₃₁	C₂×D₄	1	2	yes	—
162	#22687	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
165	Q₈×C₃₁	—	2	2	no	—
166	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
167	D₄×C₃₁	—	2	4	no	—
168	D₄×C₃₁	—	2	4	no	—
169	D₄×C₃₁	—	2	2	no	—
170	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	4	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	8	2	no	—
186	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
192	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	2	1	no	—
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	2	1	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
201	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
202	C₂×#444	C₂×C₃₁	1	1	yes	characteristic
203	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
204	#2943	C₂×C₃₁	1	1	yes	characteristic
205	#2906	C₂×C₃₁	1	2	yes	complemented
206	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1866	C₂×C₃₁	1	2	yes	complemented
209	#1866	C₂×C₃₁	1	2	yes	complemented
210	#1849	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1806	C₂×C₃₁	1	2	yes	complemented
212	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	1	no	—
214	C₂²×C₃₁	—	2	1	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₃₁	#547	1	1	yes	characteristic
220	C₂²×C₃₁	#554	1	1	yes	characteristic
221	C₄×C₃₁	—	2	4	no	—
222	C₄×C₃₁	—	2	2	no	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	2	no	—
227	C₄×C₃₁	—	8	1	no	—
228	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
229	C₄×C₃₁	#520	1	1	yes	characteristic
230	C₄×C₃₁	D₄²	1	1	yes	characteristic
231	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
232	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#131	C₂²×C₃₁	1	2	yes	—
234	C₂×#131	C₂²×C₃₁	1	2	yes	—
235	C₄×#41	C₂²×C₃₁	1	2	yes	—
236	#444	—	2	4	no	—
237	#443	—	2	4	no	—
238	#442	C₄×C₃₁	1	2	yes	complemented
239	#441	C₄×C₃₁	1	2	yes	complemented
240	#440	C₄×C₃₁	1	2	yes	complemented
241	#440	C₄×C₃₁	1	2	yes	complemented
242	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#108	C₂²×C₃₁	1	2	yes	—
245	#427	C₂²×C₃₁	1	1	yes	characteristic
246	#424	—	2	2	no	—
247	#424	—	2	2	no	—
248	#424	C₂²×C₃₁	1	2	yes	—
249	#424	C₂²×C₃₁	1	2	yes	—
250	#423	—	2	2	no	—
251	#423	C₂²×C₃₁	1	2	yes	—
252	C₂×C₄⋊C₈	—	2	2	no	—
253	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#104	C₂²×C₃₁	1	2	yes	—
255	C₂×#104	C₂²×C₃₁	1	2	yes	—
256	#416	C₂²×C₃₁	1	2	yes	—
257	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	#339	C₂²×C₃₁	1	2	yes	—
262	#339	C₂²×C₃₁	1	2	yes	—
263	#336	C₂²×C₃₁	1	1	yes	characteristic
264	#336	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₃₁	—	2	1	no	—
266	C₂×C₃₁	—	2	1	no	—
267	C₂×C₃₁	—	4	4	no	—
268	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₃₁	#2955	1	1	yes	characteristic
270	C₂×C₃₁	#2956	1	1	yes	characteristic
271	C₂×#41	C₂³×C₃₁	1	2	yes	—
272	#135	D₄×C₃₁	1	4	yes	—
273	#135	Q₈×C₃₁	1	4	yes	—
274	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
276	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
277	#131	—	2	4	no	—
278	#131	—	2	4	no	—
279	#131	—	4	2	no	—
280	#131	—	4	2	no	—
281	C₂×M₁₆	—	2	2	no	—
282	C₂×M₁₆	—	2	2	no	—
283	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
284	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
285	C₂²×C₈	—	2	2	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
292	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
296	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
297	#117	—	2	2	no	—
298	#117	C₂³×C₃₁	1	2	yes	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	#108	—	2	2	no	—
305	#108	—	2	2	no	—
306	#108	—	4	2	no	—
307	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
308	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
309	#106	C₂×C₄×C₃₁	1	2	yes	—
310	C₄⋊C₈	—	2	2	no	—
311	C₄⋊C₈	—	2	2	no	—
312	C₄⋊C₈	—	2	2	no	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	—	2	2	no	—
316	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
317	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
318	#104	—	2	4	no	—
319	#104	—	2	4	no	—
320	#103	C₂×C₄×C₃₁	1	2	yes	—
321	#103	C₂×C₄×C₃₁	1	2	yes	—
322	#102	C₂×C₄×C₃₁	1	2	yes	—
323	#102	C₂×C₄×C₃₁	1	2	yes	—
324	#97	—	2	2	no	—
325	#97	C₂×C₄×C₃₁	1	2	yes	—
326	#97	C₂×C₄×C₃₁	1	2	yes	—
327	C₄×C₈	—	2	2	no	—
328	#95	—	2	1	no	—
329	C₃₁	#22687	1	1	yes	characteristic, direct product factor, Sylow
330	#41	—	2	2	no	—
331	#41	—	2	2	no	—
332	#41	C₂×D₄×C₃₁	1	4	yes	—
333	#41	C₂×Q₈×C₃₁	1	4	yes	—
334	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
345	Q₁₆	C₄⋊C₄×C₃₁	1	4	yes	complemented
346	SD₁₆	C₄⋊C₄×C₃₁	1	8	yes	complemented
347	D₈	C₄⋊C₄×C₃₁	1	4	yes	complemented
348	M₁₆	—	2	2	no	—
349	M₁₆	—	2	2	no	—
350	M₁₆	—	2	2	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	4	4	no	—
353	M₁₆	—	4	4	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	—	4	4	no	—
364	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
365	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
367	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
371	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
376	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
388	C₄²	C₂×D₄×C₃₁	1	2	yes	—
389	C₂³	—	2	2	no	—
390	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
393	D₄	—	2	4	no	—
394	D₄	—	2	4	no	—
395	D₄	—	2	2	no	—
396	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	8	2	no	—
412	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
413	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
414	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
420	C₈	—	2	2	no	—
421	C₈	—	2	1	no	—
422	C₈	—	2	2	no	—
423	C₈	—	2	1	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	2	no	—
426	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
427	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
434	C₂²	C₃₁×#547	1	1	yes	characteristic
435	C₂²	C₃₁×#554	1	1	yes	characteristic
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	1	no	—
443	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
444	C₄	C₃₁×#520	1	1	yes	characteristic
445	C₄	D₄²×C₃₁	1	1	yes	characteristic
446	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	4	4	no	—
450	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
451	C₂	C₃₁×#2955	1	1	yes	characteristic
452	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	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2906	C₂	semidirect
5. C₃₁×#1946	C₂	semidirect
6. C₃₁×#1945	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1866	C₂	semidirect
9. C₃₁×#1849	C₂	semidirect
10. C₃₁×#1806	C₂	semidirect
11. C₃₁×#1771	C₂	semidirect
18. C₃₁×#442	C₄	semidirect
19. C₃₁×#441	C₄	semidirect
20. C₃₁×#440	C₄	semidirect
21. C₃₁×#440	C₄	semidirect
118. Q₁₆×C₃₁	C₄⋊C₄	semidirect
119. SD₁₆×C₃₁	C₄⋊C₄	semidirect
120. D₈×C₃₁	C₄⋊C₄	semidirect
162. #22687	C₃₁	direct (N × H)
205. #2906	C₂×C₃₁	semidirect
206. #1946	C₂×C₃₁	semidirect
207. #1945	C₂×C₃₁	semidirect
208. #1866	C₂×C₃₁	semidirect
209. #1866	C₂×C₃₁	semidirect
210. #1849	C₂×C₃₁	semidirect
211. #1806	C₂×C₃₁	semidirect
212. #1771	C₂×C₃₁	semidirect
238. #442	C₄×C₃₁	semidirect
239. #441	C₄×C₃₁	semidirect
240. #440	C₄×C₃₁	semidirect
241. #440	C₄×C₃₁	semidirect
329. C₃₁	#22687	direct (N × H)
345. Q₁₆	C₄⋊C₄×C₃₁	semidirect
346. SD₁₆	C₄⋊C₄×C₃₁	semidirect
347. D₈	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#444	2
C₂×C₃₁×#443	2
C₃₁×#2943	2
C₃₁×#2906	2
C₃₁×#1946	2
C₃₁×#1945	2
C₃₁×#1866	2
C₃₁×#1849	2
C₃₁×#1806	2
C₃₁×#1771	2
#22687	31

Subgroups — C31×#22687 #21479446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22687 #21479446