Subgroups — C₃₁×#19955 #21476714

Order 7936.

636 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3353	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1693	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, complemented
8	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1647	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₃₁×#556	C₂²	1	2	yes	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#484	—	2	2	no	—
21	C₃₁×#482	—	2	2	no	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#468	—	2	2	no	—
25	C₃₁×#467	—	2	1	no	—
26	C₃₁×#464	—	2	2	no	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#461	C₂²	1	2	yes	—
29	C₃₁×#452	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆×C₃₁	—	2	1	no	—
32	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#436	C₂²	1	2	yes	—
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#428	—	2	2	no	—
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×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₃₁×#135	—	4	4	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#126	—	4	2	no	—
60	C₃₁×#124	—	2	2	no	—
61	C₃₁×#123	—	4	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#117	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	—	4	2	no	—
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×#106	D₄	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#102	—	2	2	no	—
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	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	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	8	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	Q₁₆×C₃₁	—	4	2	no	—
107	Q₁₆×C₃₁	—	4	2	no	—
108	Q₁₆×C₃₁	—	8	1	no	—
109	Q₁₆×C₃₁	—	8	1	no	—
110	SD₁₆×C₃₁	—	2	2	no	—
111	SD₁₆×C₃₁	—	4	4	no	—
112	SD₁₆×C₃₁	—	8	2	no	—
113	D₈×C₃₁	—	2	2	no	—
114	M₁₆×C₃₁	—	4	4	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	4	2	no	—
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	—	2	1	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	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄×C₃₁	—	2	1	no	—
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	—	2	1	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
142	#19955	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	Q₈×C₃₁	—	2	1	no	—
146	Q₈×C₃₁	—	2	1	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	2	1	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	2	1	no	—
155	D₄×C₃₁	—	8	4	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
170	C₈×C₃₁	—	2	2	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	—	4	1	no	—
176	C₈×C₃₁	#135	1	2	yes	complemented
177	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
178	#3393	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#3353	C₂×C₃₁	1	2	yes	complemented
180	#3283	C₂×C₃₁	1	1	yes	characteristic
181	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1693	C₂×C₃₁	1	2	yes	complemented
183	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1650	C₂×C₃₁	1	2	yes	complemented
186	#1647	C₂×C₃₁	1	2	yes	complemented
187	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	D₄²	1	1	yes	characteristic
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	—	4	1	no	—
200	C₄×C₃₁	—	4	1	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	—	8	2	no	—
203	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
204	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₄×C₃₁	#576	1	1	yes	characteristic
206	C₄×C₃₁	#578	1	1	yes	characteristic
207	C₂×#135	—	2	1	no	—
208	#556	C₂²×C₃₁	1	2	yes	—
209	#547	C₂²×C₃₁	1	2	yes	—
210	#500	C₂²×C₃₁	1	2	yes	—
211	#500	C₂²×C₃₁	1	2	yes	—
212	#495	—	2	1	no	—
213	#494	C₂²×C₃₁	1	1	yes	characteristic
214	#494	C₂²×C₃₁	1	1	yes	characteristic
215	#484	—	2	2	no	—
216	#482	—	2	2	no	—
217	#475	C₂²×C₃₁	1	2	yes	—
218	#474	C₂²×C₃₁	1	2	yes	—
219	#468	—	2	2	no	—
220	#467	—	2	1	no	—
221	#464	—	2	2	no	—
222	#461	C₂²×C₃₁	1	2	yes	—
223	#461	C₂²×C₃₁	1	2	yes	—
224	#452	—	2	2	no	—
225	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₄×Q₁₆	—	2	1	no	—
227	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
229	#436	C₂²×C₃₁	1	2	yes	—
230	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
231	#428	—	2	2	no	—
232	#335	C₂²×C₃₁	1	1	yes	characteristic
233	#334	C₂²×C₃₁	1	1	yes	characteristic
234	#332	C₂²×C₃₁	1	2	yes	—
235	#332	C₂²×C₃₁	1	2	yes	—
236	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
237	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
239	C₈²	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₃₁	—	8	1	no	—
241	C₂×C₃₁	—	8	2	no	—
242	C₂×C₃₁	#3274	1	1	yes	characteristic
243	C₂×C₃₁	#3283	1	1	yes	characteristic
244	C₂×C₃₁	#3293	1	1	yes	characteristic
245	C₂×#41	—	2	2	no	—
246	#135	—	4	4	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×SD₁₆	—	2	2	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
254	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂×M₁₆	—	2	2	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
258	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
259	#126	—	4	2	no	—
260	#124	—	2	2	no	—
261	#123	—	4	2	no	—
262	C₂²⋊Q₈	—	2	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	#121	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₄×D₄	C₂³×C₃₁	1	2	yes	—
269	#117	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	#107	—	2	2	no	—
272	#107	—	2	2	no	—
273	#107	—	2	2	no	—
274	#107	—	4	2	no	—
275	#107	D₄×C₃₁	1	1	yes	characteristic
276	#106	—	2	2	no	—
277	#106	D₄×C₃₁	1	1	yes	characteristic
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
281	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
282	#103	—	2	2	no	—
283	#103	—	2	2	no	—
284	#103	—	2	2	no	—
285	#103	D₄×C₃₁	1	1	yes	characteristic
286	#102	—	2	2	no	—
287	#102	D₄×C₃₁	1	2	yes	—
288	#102	D₄×C₃₁	1	1	yes	characteristic
289	C₂²⋊C₈	—	2	1	no	—
290	C₂²⋊C₈	—	2	2	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₃₁	#19955	1	1	yes	characteristic, direct product factor, Sylow
296	#41	—	4	4	no	—
297	#41	—	8	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂×D₄	—	4	2	no	—
302	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	4	2	no	—
307	Q₁₆	—	4	2	no	—
308	Q₁₆	—	4	2	no	—
309	Q₁₆	—	8	1	no	—
310	Q₁₆	—	8	1	no	—
311	SD₁₆	—	2	2	no	—
312	SD₁₆	—	4	4	no	—
313	SD₁₆	—	8	2	no	—
314	D₈	—	2	2	no	—
315	M₁₆	—	4	4	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
333	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²⋊C₄	—	2	1	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	4	2	no	—
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	4	2	no	—
342	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	—	2	1	no	—
345	Q₈	—	2	1	no	—
346	Q₈	—	2	1	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	D₄	—	2	1	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	8	4	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	1	no	—
376	C₈	C₃₁×#135	1	2	yes	complemented
377	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	D₄²×C₃₁	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	2	no	—
388	C₄	—	4	1	no	—
389	C₄	—	4	1	no	—
390	C₄	—	8	2	no	—
391	C₄	—	8	2	no	—
392	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
393	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
394	C₄	C₃₁×#576	1	1	yes	characteristic
395	C₄	C₃₁×#578	1	1	yes	characteristic
396	C₂	—	8	1	no	—
397	C₂	—	8	2	no	—
398	C₂	C₃₁×#3274	1	1	yes	characteristic
399	C₂	C₃₁×#3283	1	1	yes	characteristic
400	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3283	C₂
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×#3393	C₂	semidirect
2. C₃₁×#3353	C₂	semidirect
4. C₃₁×#1713	C₂	semidirect
5. C₃₁×#1693	C₂	semidirect
6. C₃₁×#1670	C₂	semidirect
7. C₃₁×#1669	C₂	semidirect
8. C₃₁×#1650	C₂	semidirect
9. C₃₁×#1647	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×SD₁₆×C₃₁	C₂	semidirect
142. #19955	C₃₁	direct (N × H)
176. C₈×C₃₁	#135	semidirect
178. #3393	C₂×C₃₁	semidirect
179. #3353	C₂×C₃₁	semidirect
181. #1713	C₂×C₃₁	semidirect
182. #1693	C₂×C₃₁	semidirect
183. #1670	C₂×C₃₁	semidirect
184. #1669	C₂×C₃₁	semidirect
185. #1650	C₂×C₃₁	semidirect
186. #1647	C₂×C₃₁	semidirect
187. #1589	C₂×C₃₁	semidirect
188. C₈×SD₁₆	C₂×C₃₁	semidirect
295. C₃₁	#19955	direct (N × H)
376. C₈	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3393	2
C₃₁×#3353	2
C₃₁×#3283	2
C₃₁×#1713	2
C₃₁×#1693	2
C₃₁×#1670	2
C₃₁×#1669	2
C₃₁×#1650	2
C₃₁×#1647	2
C₃₁×#1589	2
C₈×SD₁₆×C₃₁	2
#19955	31

Subgroups — C31×#19955 #21476714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19955 #21476714