Subgroups — C₃₁×#19928 #21476687

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3386	C₂	1	2	yes	maximal
3	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1694	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1649	C₂	1	2	yes	maximal
9	C₃₁×#1648	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	—	2	1	no	—
13	C₃₁×#557	C₂²	1	2	yes	—
14	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#501	—	2	2	no	—
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#494	—	2	1	no	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#494	C₂²	1	1	yes	characteristic
21	C₃₁×#478	—	2	2	no	—
22	C₃₁×#477	—	2	2	no	—
23	C₃₁×#475	C₂²	1	2	yes	—
24	C₃₁×#474	C₂²	1	2	yes	—
25	C₃₁×#467	—	2	1	no	—
26	C₃₁×#462	—	2	2	no	—
27	C₃₁×#462	C₂²	1	2	yes	—
28	C₃₁×#462	C₂²	1	2	yes	—
29	C₃₁×#451	—	2	2	no	—
30	C₃₁×#446	C₂²	1	2	yes	—
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#440	—	2	2	no	—
33	C₄×Q₁₆×C₃₁	—	2	1	no	—
34	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	1	yes	characteristic
39	C₃₁×#332	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	—	2	2	no	—
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	2	1	no	—
52	C₂×Q₁₆×C₃₁	—	4	4	no	—
53	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#125	—	2	2	no	—
61	C₃₁×#125	—	4	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	—	4	2	no	—
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#107	—	2	2	no	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	—	4	2	no	—
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×#106	D₄	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	—	2	1	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	D₄	1	1	yes	characteristic
86	C₃₁×#103	D₄	1	2	yes	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	—	4	4	no	—
99	C₂×Q₈×C₃₁	—	8	2	no	—
100	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	Q₁₆×C₃₁	—	2	2	no	—
106	Q₁₆×C₃₁	—	4	4	no	—
107	Q₁₆×C₃₁	—	4	2	no	—
108	Q₁₆×C₃₁	—	4	2	no	—
109	Q₁₆×C₃₁	—	8	1	no	—
110	Q₁₆×C₃₁	—	8	1	no	—
111	Q₁₆×C₃₁	—	8	2	no	—
112	SD₁₆×C₃₁	Q₁₆	1	4	yes	complemented
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	4	2	no	—
118	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₃₁	—	2	1	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	1	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
141	#19928	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	1	no	—
143	Q₈×C₃₁	—	2	1	no	—
144	Q₈×C₃₁	—	2	2	no	—
145	Q₈×C₃₁	—	2	2	no	—
146	Q₈×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	8	4	no	—
151	Q₈×C₃₁	C₂×Q₁₆	1	2	yes	complemented
152	D₄×C₃₁	—	2	1	no	—
153	D₄×C₃₁	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	2	1	no	—
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	4	1	no	—
171	C₈×C₃₁	—	4	1	no	—
172	C₈×C₃₁	—	4	2	no	—
173	C₈×C₃₁	C₂×Q₁₆	1	2	yes	complemented
174	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
175	#3395	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#3386	C₂×C₃₁	1	2	yes	—
177	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
178	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1694	C₂×C₃₁	1	2	yes	complemented
180	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1669	C₂×C₃₁	1	1	yes	characteristic
182	#1649	C₂×C₃₁	1	2	yes	—
183	#1648	C₂×C₃₁	1	2	yes	complemented
184	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
186	C₂²×C₃₁	—	2	2	no	—
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	D₄²	1	1	yes	characteristic
190	C₄×C₃₁	—	2	2	no	—
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	4	1	no	—
193	C₄×C₃₁	—	4	2	no	—
194	C₄×C₃₁	—	4	2	no	—
195	C₄×C₃₁	—	4	1	no	—
196	C₄×C₃₁	—	8	2	no	—
197	C₄×C₃₁	—	8	2	no	—
198	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
199	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₃₁	#576	1	1	yes	characteristic
201	C₄×C₃₁	#578	1	1	yes	characteristic
202	C₂²×Q₁₆	—	2	1	no	—
203	#557	C₂²×C₃₁	1	2	yes	—
204	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
205	#501	—	2	2	no	—
206	#500	C₂²×C₃₁	1	2	yes	—
207	#500	C₂²×C₃₁	1	2	yes	—
208	#494	—	2	1	no	—
209	#494	C₂²×C₃₁	1	1	yes	characteristic
210	#494	C₂²×C₃₁	1	1	yes	characteristic
211	#478	—	2	2	no	—
212	#477	—	2	2	no	—
213	#475	C₂²×C₃₁	1	2	yes	—
214	#474	C₂²×C₃₁	1	2	yes	—
215	#467	—	2	1	no	—
216	#462	—	2	2	no	—
217	#462	C₂²×C₃₁	1	2	yes	—
218	#462	C₂²×C₃₁	1	2	yes	—
219	#451	—	2	2	no	—
220	#446	C₂²×C₃₁	1	2	yes	—
221	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
222	#440	—	2	2	no	—
223	C₄×Q₁₆	—	2	1	no	—
224	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
225	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
226	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	#335	C₂²×C₃₁	1	1	yes	characteristic
228	#334	C₂²×C₃₁	1	1	yes	characteristic
229	#332	C₂²×C₃₁	1	2	yes	—
230	#332	C₂²×C₃₁	1	2	yes	—
231	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
233	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₈²	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₃₁	—	4	2	no	—
236	C₂×C₃₁	#3274	1	1	yes	characteristic
237	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
238	C₂×C₃₁	#3293	1	1	yes	characteristic
239	C₂²×Q₈	—	2	2	no	—
240	C₂×Q₁₆	—	2	2	no	—
241	C₂×Q₁₆	—	2	2	no	—
242	C₂×Q₁₆	—	2	1	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	2	1	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	4	4	no	—
247	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
248	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	2	no	—
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
253	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
254	#125	—	2	2	no	—
255	#125	—	4	2	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	C₄×Q₈	—	2	2	no	—
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
261	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
264	#107	—	2	2	no	—
265	#107	—	2	2	no	—
266	#107	—	2	2	no	—
267	#107	—	4	2	no	—
268	#107	D₄×C₃₁	1	1	yes	characteristic
269	#106	—	2	2	no	—
270	#106	D₄×C₃₁	1	1	yes	characteristic
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#103	—	2	2	no	—
279	#103	D₄×C₃₁	1	1	yes	characteristic
280	#103	D₄×C₃₁	1	2	yes	—
281	#102	—	2	2	no	—
282	#102	D₄×C₃₁	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	#97	—	2	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₃₁	#19928	1	1	yes	characteristic, direct product factor, Sylow
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	4	4	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
296	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	4	2	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	4	4	no	—
302	Q₁₆	—	4	2	no	—
303	Q₁₆	—	4	2	no	—
304	Q₁₆	—	8	1	no	—
305	Q₁₆	—	8	1	no	—
306	Q₁₆	—	8	2	no	—
307	SD₁₆	Q₁₆×C₃₁	1	4	yes	complemented
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	4	2	no	—
313	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
329	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²⋊C₄	—	2	1	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	4	2	no	—
335	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
336	C₂³	—	2	1	no	—
337	Q₈	—	2	1	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	4	no	—
345	Q₈	C₂×Q₁₆×C₃₁	1	2	yes	complemented
346	D₄	—	2	1	no	—
347	D₄	C₂×Q₁₆×C₃₁	1	2	yes	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	4	no	—
358	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	4	1	no	—
365	C₈	—	4	1	no	—
366	C₈	—	4	2	no	—
367	C₈	C₂×Q₁₆×C₃₁	1	2	yes	complemented
368	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
369	C₂²	—	2	2	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	4	1	no	—
372	C₂²	D₄²×C₃₁	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	2	no	—
381	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
382	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
383	C₄	C₃₁×#576	1	1	yes	characteristic
384	C₄	C₃₁×#578	1	1	yes	characteristic
385	C₂	—	4	2	no	—
386	C₂	C₃₁×#3274	1	1	yes	characteristic
387	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
388	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3395	C₂	semidirect
4. C₃₁×#1713	C₂	semidirect
5. C₃₁×#1694	C₂	semidirect
6. C₃₁×#1670	C₂	semidirect
9. C₃₁×#1648	C₂	semidirect
10. C₃₁×#1591	C₂	semidirect
112. SD₁₆×C₃₁	Q₁₆	semidirect
141. #19928	C₃₁	direct (N × H)
151. Q₈×C₃₁	C₂×Q₁₆	semidirect
173. C₈×C₃₁	C₂×Q₁₆	semidirect
175. #3395	C₂×C₃₁	semidirect
178. #1713	C₂×C₃₁	semidirect
179. #1694	C₂×C₃₁	semidirect
180. #1670	C₂×C₃₁	semidirect
183. #1648	C₂×C₃₁	semidirect
184. #1591	C₂×C₃₁	semidirect
289. C₃₁	#19928	direct (N × H)
307. SD₁₆	Q₁₆×C₃₁	semidirect
345. Q₈	C₂×Q₁₆×C₃₁	semidirect
367. C₈	C₂×Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3395	2
C₃₁×#3386	2
D₄×Q₁₆×C₃₁	2
C₃₁×#1713	2
C₃₁×#1694	2
C₃₁×#1670	2
C₃₁×#1669	2
C₃₁×#1649	2
C₃₁×#1648	2
C₃₁×#1591	2
C₈×SD₁₆×C₃₁	2
#19928	31

Subgroups — C31×#19928 #21476687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19928 #21476687