Subgroups — C₃₁×#21253 #21478012

Order 7936.

1562 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#421	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#104	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1980	C₂	1	8	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#537	—	2	2	no	—
10	C₃₁×#536	C₄	1	4	yes	complemented
11	C₃₁×#535	C₄	1	4	yes	complemented
12	C₃₁×#534	—	2	2	no	—
13	C₃₁×#532	C₂²	1	4	yes	—
14	C₃₁×#525	—	2	4	no	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#421	—	2	8	no	—
19	C₂×C₃₁×#104	—	2	8	no	—
20	C₂×C₃₁×#104	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#104	C₂²	1	8	yes	complemented
22	C₃₁×#405	—	2	4	no	—
23	C₄×M₁₆×C₃₁	C₂²	1	4	yes	complemented
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#385	—	2	2	no	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
27	C₂×#41×C₃₁	—	2	2	no	—
28	C₂×#41×C₃₁	—	2	2	no	—
29	C₂×#41×C₃₁	—	2	4	no	—
30	C₂×#41×C₃₁	—	2	4	no	—
31	C₂×#41×C₃₁	—	2	4	no	—
32	C₂×#41×C₃₁	—	2	4	no	—
33	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×#41×C₃₁	C₂³	1	4	yes	—
35	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	—	2	1	no	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	—	2	4	no	—
43	C₂×M₁₆×C₃₁	D₄	1	8	yes	complemented
44	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
47	C₃₁×#127	C₂×C₄	1	4	yes	complemented
48	C₃₁×#124	C₂×C₄	1	8	yes	complemented
49	C₃₁×#123	—	2	8	no	—
50	C₃₁×C₂²⋊Q₈	—	2	8	no	—
51	C₃₁×#121	—	2	8	no	—
52	C₃₁×#121	—	2	8	no	—
53	C₃₁×#120	—	2	8	no	—
54	C₄×D₄×C₃₁	—	2	8	no	—
55	C₃₁×#117	—	2	4	no	—
56	C₃₁×#117	C₂×C₄	1	4	yes	—
57	C₃₁×#117	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
63	C₂×C₄²×C₃₁	—	2	4	no	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#104	—	2	16	no	—
69	C₃₁×#104	—	2	16	no	—
70	C₃₁×#97	—	2	4	no	—
71	C₄×C₈×C₃₁	—	2	4	no	—
72	C₃₁×#95	—	2	4	no	—
73	C₂⁴×C₃₁	—	2	2	no	—
74	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
75	#41×C₃₁	—	2	8	no	—
76	#41×C₃₁	—	2	8	no	—
77	#41×C₃₁	—	2	4	no	—
78	#41×C₃₁	—	2	4	no	—
79	#41×C₃₁	—	2	4	no	—
80	#41×C₃₁	—	2	4	no	—
81	#41×C₃₁	—	4	4	no	—
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	4	4	no	—
84	#41×C₃₁	—	4	4	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	4	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	8	no	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	4	no	—
97	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	4	no	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	4	4	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₃₁	#41	1	4	yes	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	M₁₆×C₃₁	—	2	8	no	—
115	M₁₆×C₃₁	—	2	4	no	—
116	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
117	C₂×C₈×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	2	8	no	—
120	C₄⋊C₄×C₃₁	—	4	8	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
123	C₂²⋊C₄×C₃₁	—	2	16	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
126	C₄²×C₃₁	—	2	4	no	—
127	C₄²×C₃₁	—	2	4	no	—
128	C₄²×C₃₁	—	4	2	no	—
129	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
130	C₄²×C₃₁	C₂×D₄	1	4	yes	—
131	#21253	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	4	no	—
135	C₂³×C₃₁	—	8	4	no	—
136	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂³×C₃₁	#124	1	2	yes	—
138	C₂³×C₃₁	#127	1	2	yes	—
139	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₃₁	—	2	4	no	—
141	Q₈×C₃₁	—	2	4	no	—
142	Q₈×C₃₁	—	4	4	no	—
143	D₄×C₃₁	—	2	4	no	—
144	D₄×C₃₁	—	2	4	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	4	4	no	—
147	D₄×C₃₁	—	4	8	no	—
148	D₄×C₃₁	—	4	4	no	—
149	D₄×C₃₁	—	8	8	no	—
150	C₂×C₄×C₃₁	—	2	4	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	4	4	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	8	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	4	no	—
157	C₂×C₄×C₃₁	—	4	4	no	—
158	C₂×C₄×C₃₁	—	4	4	no	—
159	C₂×C₄×C₃₁	—	4	8	no	—
160	C₂×C₄×C₃₁	—	4	4	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
163	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
165	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
166	C₂×C₄×C₃₁	#121	1	4	yes	—
167	C₂×C₄×C₃₁	#121	1	4	yes	—
168	C₂×C₄×C₃₁	#124	1	2	yes	—
169	C₂×C₄×C₃₁	#127	1	2	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
174	C₈×C₃₁	—	2	8	no	—
175	#3449	C₂×C₃₁	1	1	yes	characteristic
176	C₂×#421	C₂×C₃₁	1	2	yes	complemented
177	C₂²×#104	C₂×C₃₁	1	2	yes	complemented
178	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1980	C₂×C₃₁	1	8	yes	complemented
181	C₂²×C₃₁	—	2	2	no	—
182	C₂²×C₃₁	—	2	4	no	—
183	C₂²×C₃₁	—	4	2	no	—
184	C₂²×C₃₁	—	4	2	no	—
185	C₂²×C₃₁	—	4	8	no	—
186	C₂²×C₃₁	—	4	4	no	—
187	C₂²×C₃₁	—	8	8	no	—
188	C₂²×C₃₁	#390	1	4	yes	—
189	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
191	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
192	C₄×C₃₁	—	2	8	no	—
193	C₄×C₃₁	—	4	4	no	—
194	C₄×C₃₁	—	4	4	no	—
195	C₄×C₃₁	—	8	2	no	—
196	C₄×C₃₁	#390	1	4	yes	—
197	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
198	C₄×C₃₁	C₂×#121	1	2	yes	—
199	C₂²×#41	C₂²×C₃₁	1	2	yes	—
200	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
201	#537	—	2	2	no	—
202	#536	C₄×C₃₁	1	4	yes	complemented
203	#535	C₄×C₃₁	1	4	yes	complemented
204	#534	—	2	2	no	—
205	#532	C₂²×C₃₁	1	4	yes	—
206	#525	—	2	4	no	—
207	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
209	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
210	#421	—	2	8	no	—
211	C₂×#104	—	2	8	no	—
212	C₂×#104	C₂²×C₃₁	1	8	yes	complemented
213	C₂×#104	C₂²×C₃₁	1	8	yes	complemented
214	#405	—	2	4	no	—
215	C₄×M₁₆	C₂²×C₃₁	1	4	yes	complemented
216	#389	—	2	2	no	—
217	#385	—	2	2	no	—
218	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
219	C₂×C₃₁	—	2	1	no	—
220	C₂×C₃₁	—	2	4	no	—
221	C₂×C₃₁	—	2	1	no	—
222	C₂×C₃₁	—	8	4	no	—
223	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
224	C₂×C₃₁	#2901	1	2	yes	—
225	C₂×#41	—	2	2	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	2	4	no	—
228	C₂×#41	—	2	4	no	—
229	C₂×#41	—	2	4	no	—
230	C₂×#41	—	2	4	no	—
231	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
232	C₂×#41	C₂³×C₃₁	1	4	yes	—
233	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
234	C₂²×D₄	—	2	2	no	—
235	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
236	C₂³×C₄	—	2	2	no	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	4	no	—
241	C₂×M₁₆	D₄×C₃₁	1	8	yes	complemented
242	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
243	C₂²×C₈	—	2	2	no	—
244	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
245	#127	C₂×C₄×C₃₁	1	4	yes	complemented
246	#124	C₂×C₄×C₃₁	1	8	yes	complemented
247	#123	—	2	8	no	—
248	C₂²⋊Q₈	—	2	8	no	—
249	#121	—	2	8	no	—
250	#121	—	2	8	no	—
251	#120	—	2	8	no	—
252	C₄×D₄	—	2	8	no	—
253	#117	—	2	4	no	—
254	#117	C₂×C₄×C₃₁	1	4	yes	—
255	#117	D₄×C₃₁	1	4	yes	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	4	no	—
258	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
259	C₂×C₂²⋊C₄	—	2	1	no	—
260	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
261	C₂×C₄²	—	2	4	no	—
262	C₂×C₄²	—	2	1	no	—
263	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
264	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
265	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
266	#104	—	2	16	no	—
267	#104	—	2	16	no	—
268	#97	—	2	4	no	—
269	C₄×C₈	—	2	4	no	—
270	#95	—	2	4	no	—
271	C₃₁	#21253	1	1	yes	characteristic, direct product factor, Sylow
272	C₂⁴	—	2	2	no	—
273	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
274	#41	—	2	8	no	—
275	#41	—	2	8	no	—
276	#41	—	2	4	no	—
277	#41	—	2	4	no	—
278	#41	—	2	4	no	—
279	#41	—	2	4	no	—
280	#41	—	4	4	no	—
281	#41	—	4	4	no	—
282	#41	—	4	4	no	—
283	#41	—	4	4	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	4	no	—
286	C₂×Q₈	—	2	4	no	—
287	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
288	C₂×D₄	—	2	4	no	—
289	C₂×D₄	—	2	4	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	4	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	4	4	no	—
295	C₂×D₄	—	4	4	no	—
296	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	4	4	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
308	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
311	C₂²×C₄	#41×C₃₁	1	4	yes	—
312	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
313	M₁₆	—	2	8	no	—
314	M₁₆	—	2	4	no	—
315	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
316	C₂×C₈	—	2	8	no	—
317	C₂×C₈	—	2	4	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	4	8	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
322	C₂²⋊C₄	—	2	16	no	—
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
325	C₄²	—	2	4	no	—
326	C₄²	—	2	4	no	—
327	C₄²	—	4	2	no	—
328	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
329	C₄²	C₂×D₄×C₃₁	1	4	yes	—
330	C₂³	—	2	4	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	8	4	no	—
334	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
335	C₂³	C₃₁×#124	1	2	yes	—
336	C₂³	C₃₁×#127	1	2	yes	—
337	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
338	Q₈	—	2	4	no	—
339	Q₈	—	2	4	no	—
340	Q₈	—	4	4	no	—
341	D₄	—	2	4	no	—
342	D₄	—	2	4	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	8	no	—
346	D₄	—	4	4	no	—
347	D₄	—	8	8	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	8	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
361	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
362	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
363	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
364	C₂×C₄	C₃₁×#121	1	4	yes	—
365	C₂×C₄	C₃₁×#121	1	4	yes	—
366	C₂×C₄	C₃₁×#124	1	2	yes	—
367	C₂×C₄	C₃₁×#127	1	2	yes	—
368	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
370	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
372	C₈	—	2	8	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	8	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	8	8	no	—
380	C₂²	C₃₁×#390	1	4	yes	—
381	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
382	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
383	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
384	C₄	—	2	8	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	4	no	—
387	C₄	—	8	2	no	—
388	C₄	C₃₁×#390	1	4	yes	—
389	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
390	C₄	C₂×C₃₁×#121	1	2	yes	—
391	C₂	—	2	1	no	—
392	C₂	—	2	4	no	—
393	C₂	—	2	1	no	—
394	C₂	—	8	4	no	—
395	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
396	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#421	C₂	semidirect
3. C₂²×C₃₁×#104	C₂	semidirect
4. C₃₁×#2873	C₂	semidirect
5. C₃₁×#2296	C₂	semidirect
6. C₃₁×#1980	C₂	semidirect
10. C₃₁×#536	C₄	semidirect
11. C₃₁×#535	C₄	semidirect
20. C₂×C₃₁×#104	C₂²	semidirect
21. C₂×C₃₁×#104	C₂²	semidirect
23. C₄×M₁₆×C₃₁	C₂²	semidirect
26. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
43. C₂×M₁₆×C₃₁	D₄	semidirect
46. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
47. C₃₁×#127	C₂×C₄	semidirect
48. C₃₁×#124	C₂×C₄	semidirect
131. #21253	C₃₁	direct (N × H)
176. C₂×#421	C₂×C₃₁	semidirect
177. C₂²×#104	C₂×C₃₁	semidirect
178. #2873	C₂×C₃₁	semidirect
179. #2296	C₂×C₃₁	semidirect
180. #1980	C₂×C₃₁	semidirect
202. #536	C₄×C₃₁	semidirect
203. #535	C₄×C₃₁	semidirect
212. C₂×#104	C₂²×C₃₁	semidirect
213. C₂×#104	C₂²×C₃₁	semidirect
215. C₄×M₁₆	C₂²×C₃₁	semidirect
218. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
241. C₂×M₁₆	D₄×C₃₁	semidirect
244. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
245. #127	C₂×C₄×C₃₁	semidirect
246. #124	C₂×C₄×C₃₁	semidirect
271. C₃₁	#21253	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×#3449	2
C₂×C₃₁×#421	2
C₂²×C₃₁×#104	2
C₃₁×#2873	2
C₃₁×#2296	2
C₃₁×#1980	2
#21253	31

Subgroups — C31×#21253 #21478012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21253 #21478012