Subgroups — C₃₁×#23114 #21479873

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1975	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1876	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1765	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	—
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#464	C₄	1	2	yes	complemented
17	C₃₁×#463	C₄	1	2	yes	complemented
18	C₃₁×#462	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#461	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#460	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#459	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#436	—	2	1	no	—
23	C₃₁×#435	—	2	2	no	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₃₁×#433	—	2	2	no	—
26	C₃₁×#432	—	2	1	no	—
27	C₃₁×#431	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	—
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#104	C₂²	1	2	yes	—
34	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#408	C₂²	1	2	yes	—
39	C₃₁×#405	C₂²	1	2	yes	—
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#97	C₂²	1	2	yes	—
42	C₄³×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#341	C₂²	1	2	yes	—
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₃₁×#339	C₂²	1	1	yes	characteristic
46	C₃₁×#336	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#135	—	4	2	no	—
50	C₃₁×#135	—	4	2	no	—
51	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
55	C₃₁×#131	—	2	4	no	—
56	C₃₁×#131	—	4	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#124	C₂×C₄	1	2	yes	—
67	C₃₁×#121	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₃₁	—	2	1	no	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₃₁×#117	—	2	1	no	—
78	C₃₁×#117	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#108	—	4	2	no	—
86	C₃₁×C₄⋊C₈	—	2	1	no	—
87	C₃₁×C₄⋊C₈	—	2	1	no	—
88	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
90	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
91	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
92	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
93	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
94	C₃₁×#104	—	2	2	no	—
95	C₃₁×#104	—	2	4	no	—
96	C₃₁×#104	—	2	2	no	—
97	C₃₁×#104	—	4	2	no	—
98	C₃₁×#103	C₂×C₄	1	2	yes	—
99	C₃₁×#103	C₂×C₄	1	2	yes	—
100	C₃₁×#102	C₂×C₄	1	2	yes	—
101	C₃₁×#102	C₂×C₄	1	2	yes	—
102	C₃₁×C₂²⋊C₈	—	2	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	2	no	—
104	C₃₁×C₂²⋊C₈	—	2	2	no	—
105	C₃₁×C₂²⋊C₈	—	2	2	no	—
106	C₃₁×#97	—	2	2	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₃₁×#97	—	4	2	no	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₄×C₈×C₃₁	—	2	1	no	—
111	#41×C₃₁	—	2	1	no	—
112	#41×C₃₁	—	2	1	no	—
113	#41×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	2	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	1	no	—
119	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
120	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	Q₁₆×C₃₁	—	4	2	no	—
137	SD₁₆×C₃₁	—	4	2	no	—
138	SD₁₆×C₃₁	—	4	2	no	—
139	D₈×C₃₁	—	4	2	no	—
140	M₁₆×C₃₁	—	2	1	no	—
141	M₁₆×C₃₁	—	2	1	no	—
142	M₁₆×C₃₁	—	2	2	no	—
143	M₁₆×C₃₁	—	2	2	no	—
144	M₁₆×C₃₁	—	2	1	no	—
145	M₁₆×C₃₁	—	4	4	no	—
146	M₁₆×C₃₁	#41	1	2	yes	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₃₁	—	2	1	no	—
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₃₁	—	2	2	no	—
169	C₂²⋊C₄×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	—	2	1	no	—
176	C₄²×C₃₁	—	4	2	no	—
177	C₄²×C₃₁	—	4	2	no	—
178	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	#41	1	2	yes	—
183	#23114	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	2	1	no	—
189	Q₈×C₃₁	—	2	1	no	—
190	Q₈×C₃₁	—	2	2	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	1	no	—
193	D₄×C₃₁	—	2	1	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	8	1	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	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	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	8	1	no	—
219	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂³×C₄	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	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₈×C₃₁	—	2	2	no	—
230	C₈×C₃₁	—	2	2	no	—
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	#3035	C₂×C₃₁	1	1	yes	characteristic
237	#2935	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1975	C₂×C₃₁	1	2	yes	complemented
240	#1962	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1929	C₂×C₃₁	1	2	yes	complemented
243	#1876	C₂×C₃₁	1	2	yes	complemented
244	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1765	C₂×C₃₁	1	2	yes	complemented
246	C₄×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	1	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂²×C₃₁	#553	1	1	yes	characteristic
257	C₂²×C₃₁	#555	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	1	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	1	no	—
266	C₄×C₃₁	—	4	1	no	—
267	C₄×C₃₁	#444	1	2	yes	—
268	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
269	C₄×C₃₁	#518	1	1	yes	characteristic
270	C₄×C₃₁	#546	1	1	yes	characteristic
271	C₄×C₃₁	#550	1	1	yes	characteristic
272	C₂×#135	C₂²×C₃₁	1	2	yes	—
273	C₂×#131	C₂²×C₃₁	1	2	yes	—
274	#532	C₂²×C₃₁	1	1	yes	characteristic
275	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
276	#464	C₄×C₃₁	1	2	yes	complemented
277	#463	C₄×C₃₁	1	2	yes	complemented
278	#462	C₄×C₃₁	1	1	yes	characteristic, complemented
279	#461	C₄×C₃₁	1	1	yes	characteristic, complemented
280	#460	C₄×C₃₁	1	1	yes	characteristic, complemented
281	#459	C₄×C₃₁	1	1	yes	characteristic, complemented
282	#436	—	2	1	no	—
283	#435	—	2	2	no	—
284	C₈×D₄	—	2	1	no	—
285	#433	—	2	2	no	—
286	#432	—	2	1	no	—
287	#431	—	2	1	no	—
288	C₂×#108	C₂²×C₃₁	1	2	yes	—
289	#423	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#104	C₂²×C₃₁	1	2	yes	—
294	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#104	C₂²×C₃₁	1	2	yes	—
296	#416	C₂²×C₃₁	1	2	yes	—
297	#408	C₂²×C₃₁	1	2	yes	—
298	#408	C₂²×C₃₁	1	2	yes	—
299	#405	C₂²×C₃₁	1	2	yes	—
300	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#97	C₂²×C₃₁	1	2	yes	—
302	C₄³	C₂²×C₃₁	1	1	yes	characteristic
303	#341	C₂²×C₃₁	1	2	yes	—
304	#339	C₂²×C₃₁	1	1	yes	characteristic
305	#339	C₂²×C₃₁	1	1	yes	characteristic
306	#336	C₂²×C₃₁	1	2	yes	—
307	C₂×C₃₁	—	2	1	no	—
308	C₂×C₃₁	—	2	1	no	—
309	C₂×C₃₁	—	8	1	no	—
310	C₂×C₃₁	—	8	2	no	—
311	C₂×C₃₁	#2363	1	1	yes	characteristic
312	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
313	C₂×C₃₁	#2956	1	1	yes	characteristic
314	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×#41	C₂³×C₃₁	1	2	yes	—
316	#135	—	4	2	no	—
317	#135	—	4	2	no	—
318	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
319	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
320	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
321	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
322	#131	—	2	4	no	—
323	#131	—	4	2	no	—
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
327	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	—	2	2	no	—
329	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
330	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
331	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
332	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
333	#124	C₂×C₄×C₃₁	1	2	yes	—
334	#121	—	2	1	no	—
335	#121	—	2	1	no	—
336	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
337	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	—	2	1	no	—
340	C₄×D₄	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
344	#117	—	2	1	no	—
345	#117	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#108	—	4	2	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
358	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
361	#104	—	2	2	no	—
362	#104	—	2	4	no	—
363	#104	—	2	2	no	—
364	#104	—	4	2	no	—
365	#103	C₂×C₄×C₃₁	1	2	yes	—
366	#103	C₂×C₄×C₃₁	1	2	yes	—
367	#102	C₂×C₄×C₃₁	1	2	yes	—
368	#102	C₂×C₄×C₃₁	1	2	yes	—
369	C₂²⋊C₈	—	2	2	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	#97	—	2	2	no	—
374	#97	—	2	1	no	—
375	#97	—	4	2	no	—
376	C₄×C₈	—	2	2	no	—
377	C₄×C₈	—	2	1	no	—
378	C₃₁	#23114	1	1	yes	characteristic, direct product factor, Sylow
379	#41	—	2	1	no	—
380	#41	—	2	1	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	—	2	1	no	—
386	#41	—	2	1	no	—
387	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
388	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
394	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	2	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	Q₁₆	—	4	2	no	—
405	SD₁₆	—	4	2	no	—
406	SD₁₆	—	4	2	no	—
407	D₈	—	4	2	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	4	4	no	—
414	M₁₆	#41×C₃₁	1	2	yes	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	4	4	no	—
423	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
424	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
425	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
426	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	4	2	no	—
445	C₄²	—	4	2	no	—
446	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
448	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₄²	#41×C₃₁	1	2	yes	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	2	no	—
453	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
454	Q₈	—	2	2	no	—
455	Q₈	—	2	1	no	—
456	Q₈	—	2	1	no	—
457	Q₈	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	1	no	—
461	D₄	—	2	2	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	2	no	—
467	D₄	—	8	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	8	1	no	—
486	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
495	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	2	2	no	—
498	C₈	—	2	2	no	—
499	C₈	—	2	2	no	—
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	4	2	no	—
511	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
512	C₂²	C₃₁×#553	1	1	yes	characteristic
513	C₂²	C₃₁×#555	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	1	no	—
523	C₄	C₃₁×#444	1	2	yes	—
524	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
525	C₄	C₃₁×#518	1	1	yes	characteristic
526	C₄	C₃₁×#546	1	1	yes	characteristic
527	C₄	C₃₁×#550	1	1	yes	characteristic
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	8	1	no	—
531	C₂	—	8	2	no	—
532	C₂	C₃₁×#2363	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#444	1	1	yes	characteristic
534	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×#3035	C₂
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
2. C₃₁×#2935	C₂	semidirect
3. C₃₁×#2925	C₂	semidirect
4. C₃₁×#1975	C₂	semidirect
5. C₃₁×#1962	C₂	semidirect
6. C₃₁×#1950	C₂	semidirect
7. C₃₁×#1929	C₂	semidirect
8. C₃₁×#1876	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
10. C₃₁×#1765	C₂	semidirect
11. C₄×C₃₁×#104	C₂	semidirect
16. C₃₁×#464	C₄	semidirect
17. C₃₁×#463	C₄	semidirect
18. C₃₁×#462	C₄	semidirect
19. C₃₁×#461	C₄	semidirect
20. C₃₁×#460	C₄	semidirect
21. C₃₁×#459	C₄	semidirect
183. #23114	C₃₁	direct (N × H)
237. #2935	C₂×C₃₁	semidirect
238. #2925	C₂×C₃₁	semidirect
239. #1975	C₂×C₃₁	semidirect
240. #1962	C₂×C₃₁	semidirect
241. #1950	C₂×C₃₁	semidirect
242. #1929	C₂×C₃₁	semidirect
243. #1876	C₂×C₃₁	semidirect
244. #1767	C₂×C₃₁	semidirect
245. #1765	C₂×C₃₁	semidirect
246. C₄×#104	C₂×C₃₁	semidirect
276. #464	C₄×C₃₁	semidirect
277. #463	C₄×C₃₁	semidirect
278. #462	C₄×C₃₁	semidirect
279. #461	C₄×C₃₁	semidirect
280. #460	C₄×C₃₁	semidirect
281. #459	C₄×C₃₁	semidirect
378. C₃₁	#23114	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×#3035	2
C₃₁×#2935	2
C₃₁×#2925	2
C₃₁×#1975	2
C₃₁×#1962	2
C₃₁×#1950	2
C₃₁×#1929	2
C₃₁×#1876	2
C₃₁×#1767	2
C₃₁×#1765	2
C₄×C₃₁×#104	2
#23114	31

Subgroups — C31×#23114 #21479873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23114 #21479873