Subgroups — C₃₁×#22804 #21479563

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1806	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#442	C₄	1	2	yes	complemented
15	C₃₁×#441	C₄	1	2	yes	complemented
16	C₃₁×#440	C₄	1	2	yes	complemented
17	C₃₁×#440	C₄	1	2	yes	complemented
18	C₃₁×#433	—	2	2	no	—
19	C₃₁×#433	—	2	2	no	—
20	C₃₁×#432	—	2	2	no	—
21	C₃₁×#431	—	2	2	no	—
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×#423	C₂²	1	2	yes	—
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#104	C₂²	1	2	yes	—
29	C₂×C₃₁×#104	C₂²	1	2	yes	—
30	C₃₁×#416	C₂²	1	2	yes	—
31	C₃₁×#408	C₂²	1	2	yes	—
32	C₃₁×#408	C₂²	1	2	yes	—
33	C₃₁×#405	C₂²	1	2	yes	—
34	C₃₁×#405	C₂²	1	2	yes	—
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#339	C₂²	1	2	yes	—
40	C₃₁×#339	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#135	—	2	2	no	—
45	C₃₁×#135	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#117	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#108	—	2	2	no	—
69	C₃₁×#108	—	2	2	no	—
70	C₃₁×#108	—	4	2	no	—
71	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
72	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
73	C₃₁×#106	C₂×C₄	1	2	yes	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×C₄⋊C₈	—	2	2	no	—
76	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
77	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
78	C₃₁×#104	—	4	2	no	—
79	C₃₁×#104	—	4	2	no	—
80	C₃₁×#103	C₂×C₄	1	2	yes	—
81	C₃₁×#103	C₂×C₄	1	2	yes	—
82	C₃₁×#102	C₂×C₄	1	2	yes	—
83	C₃₁×#102	C₂×C₄	1	2	yes	—
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	—	2	2	no	—
86	C₃₁×C₂²⋊C₈	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×#97	—	2	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#97	C₂×C₄	1	2	yes	—
92	C₃₁×#97	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	2	2	no	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	#41×C₃₁	—	2	2	no	—
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	Q₁₆×C₃₁	—	2	2	no	—
113	SD₁₆×C₃₁	—	2	4	no	—
114	D₈×C₃₁	—	2	2	no	—
115	M₁₆×C₃₁	—	2	2	no	—
116	M₁₆×C₃₁	—	2	2	no	—
117	M₁₆×C₃₁	—	2	2	no	—
118	M₁₆×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₃₁	#41	1	2	yes	—
134	C₂×C₈×C₃₁	#41	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂²⋊C₄×C₃₁	—	2	2	no	—
141	C₂²⋊C₄×C₃₁	—	2	2	no	—
142	C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
151	C₄²×C₃₁	C₂×D₄	1	2	yes	—
152	#22804	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	2	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₃₁	—	2	2	no	—
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	2	1	no	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	2	no	—
190	C₈×C₃₁	#117	1	2	yes	—
191	C₈×C₃₁	#117	1	2	yes	—
192	#2943	C₂×C₃₁	1	1	yes	characteristic
193	#2925	C₂×C₃₁	1	2	yes	complemented
194	#1975	C₂×C₃₁	1	1	yes	characteristic
195	#1974	C₂×C₃₁	1	1	yes	characteristic
196	#1929	C₂×C₃₁	1	2	yes	complemented
197	#1929	C₂×C₃₁	1	2	yes	complemented
198	#1849	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1806	C₂×C₃₁	1	2	yes	complemented
200	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₃₁	#555	1	1	yes	characteristic
210	C₂²×C₃₁	#556	1	1	yes	characteristic
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	8	1	no	—
218	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
219	C₄×C₃₁	#520	1	1	yes	characteristic
220	C₄×C₃₁	#547	1	1	yes	characteristic
221	C₄×C₃₁	#550	1	1	yes	characteristic
222	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
223	C₄×#41	C₂²×C₃₁	1	2	yes	—
224	#442	C₄×C₃₁	1	2	yes	complemented
225	#441	C₄×C₃₁	1	2	yes	complemented
226	#440	C₄×C₃₁	1	2	yes	complemented
227	#440	C₄×C₃₁	1	2	yes	complemented
228	#433	—	2	2	no	—
229	#433	—	2	2	no	—
230	#432	—	2	2	no	—
231	#431	—	2	2	no	—
232	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#108	C₂²×C₃₁	1	2	yes	—
235	#427	C₂²×C₃₁	1	1	yes	characteristic
236	#423	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#104	C₂²×C₃₁	1	2	yes	—
239	C₂×#104	C₂²×C₃₁	1	2	yes	—
240	#416	C₂²×C₃₁	1	2	yes	—
241	#408	C₂²×C₃₁	1	2	yes	—
242	#408	C₂²×C₃₁	1	2	yes	—
243	#405	C₂²×C₃₁	1	2	yes	—
244	#405	C₂²×C₃₁	1	2	yes	—
245	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	#339	C₂²×C₃₁	1	2	yes	—
250	#339	C₂²×C₃₁	1	2	yes	—
251	#336	C₂²×C₃₁	1	1	yes	characteristic
252	#336	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₃₁	—	2	1	no	—
254	C₂×C₃₁	—	2	1	no	—
255	C₂×C₃₁	—	8	2	no	—
256	C₂×C₃₁	#2354	1	1	yes	characteristic
257	C₂×C₃₁	#2955	1	1	yes	characteristic
258	C₂×C₃₁	#2956	1	1	yes	characteristic
259	C₂×#41	C₂³×C₃₁	1	2	yes	—
260	#135	—	2	2	no	—
261	#135	—	2	2	no	—
262	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
264	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
272	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
273	C₄×D₄	—	2	2	no	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
276	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
277	#117	—	2	2	no	—
278	#117	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	#108	—	2	2	no	—
285	#108	—	2	2	no	—
286	#108	—	4	2	no	—
287	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
288	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
289	#106	C₂×C₄×C₃₁	1	2	yes	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
293	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
294	#104	—	4	2	no	—
295	#104	—	4	2	no	—
296	#103	C₂×C₄×C₃₁	1	2	yes	—
297	#103	C₂×C₄×C₃₁	1	2	yes	—
298	#102	C₂×C₄×C₃₁	1	2	yes	—
299	#102	C₂×C₄×C₃₁	1	2	yes	—
300	C₂²⋊C₈	—	2	2	no	—
301	C₂²⋊C₈	—	2	2	no	—
302	C₂²⋊C₈	—	2	2	no	—
303	C₂²⋊C₈	—	2	2	no	—
304	#97	—	2	2	no	—
305	#97	—	2	2	no	—
306	#97	—	2	2	no	—
307	#97	C₂×C₄×C₃₁	1	2	yes	—
308	#97	C₂×C₄×C₃₁	1	2	yes	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	2	no	—
311	C₄×C₈	—	2	2	no	—
312	#95	—	2	1	no	—
313	C₃₁	#22804	1	1	yes	characteristic, direct product factor, Sylow
314	#41	—	2	2	no	—
315	#41	—	2	2	no	—
316	#41	—	2	2	no	—
317	#41	—	2	2	no	—
318	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
329	Q₁₆	—	2	2	no	—
330	SD₁₆	—	2	4	no	—
331	D₈	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	—	2	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
349	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₈	#41×C₃₁	1	2	yes	—
351	C₂×C₈	#41×C₃₁	1	2	yes	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
356	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	1	no	—
367	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
368	C₄²	C₂×D₄×C₃₁	1	2	yes	—
369	C₂³	—	2	2	no	—
370	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
371	Q₈	—	2	2	no	—
372	Q₈	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	2	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	4	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
394	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
399	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
400	C₈	—	2	2	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	1	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	2	no	—
405	C₈	—	4	2	no	—
406	C₈	C₃₁×#117	1	2	yes	—
407	C₈	C₃₁×#117	1	2	yes	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
414	C₂²	C₃₁×#555	1	1	yes	characteristic
415	C₂²	C₃₁×#556	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
424	C₄	C₃₁×#520	1	1	yes	characteristic
425	C₄	C₃₁×#547	1	1	yes	characteristic
426	C₄	C₃₁×#550	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	8	2	no	—
430	C₂	C₃₁×#2354	1	1	yes	characteristic
431	C₂	C₃₁×#2955	1	1	yes	characteristic
432	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
2. C₃₁×#2925	C₂	semidirect
5. C₃₁×#1929	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1849	C₂	semidirect
8. C₃₁×#1806	C₂	semidirect
9. C₃₁×#1773	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
11. C₃₁×#1771	C₂	semidirect
14. C₃₁×#442	C₄	semidirect
15. C₃₁×#441	C₄	semidirect
16. C₃₁×#440	C₄	semidirect
17. C₃₁×#440	C₄	semidirect
152. #22804	C₃₁	direct (N × H)
193. #2925	C₂×C₃₁	semidirect
196. #1929	C₂×C₃₁	semidirect
197. #1929	C₂×C₃₁	semidirect
198. #1849	C₂×C₃₁	semidirect
199. #1806	C₂×C₃₁	semidirect
200. #1773	C₂×C₃₁	semidirect
201. #1772	C₂×C₃₁	semidirect
202. #1771	C₂×C₃₁	semidirect
224. #442	C₄×C₃₁	semidirect
225. #441	C₄×C₃₁	semidirect
226. #440	C₄×C₃₁	semidirect
227. #440	C₄×C₃₁	semidirect
313. C₃₁	#22804	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×#2943	2
C₃₁×#2925	2
C₃₁×#1975	2
C₃₁×#1974	2
C₃₁×#1929	2
C₃₁×#1849	2
C₃₁×#1806	2
C₃₁×#1773	2
C₃₁×#1772	2
C₃₁×#1771	2
#22804	31

Subgroups — C31×#22804 #21479563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22804 #21479563