Subgroups — C₃₁×#19255 #21476014

Order 7936.

798 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×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3036	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1681	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
10	C₂³×C₈×C₃₁	—	2	1	no	—
11	C₃₁×#532	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#486	C₂²	1	2	yes	complemented
16	C₃₁×#466	—	2	1	no	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#463	—	2	4	no	—
19	C₃₁×#461	C₂²	1	4	yes	—
20	C₃₁×#459	C₂²	1	4	yes	—
21	C₄×D₈×C₃₁	—	2	1	no	—
22	C₄×D₈×C₃₁	—	2	1	no	—
23	C₈×D₄×C₃₁	—	2	1	no	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₈×D₄×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#423	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃₁×#417	—	2	2	no	—
30	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	2	yes	complemented
33	C₃₁×#335	C₂²	1	2	yes	complemented
34	C₃₁×#332	C₂²	1	4	yes	complemented
35	C₃₁×#331	C₂²	1	4	yes	complemented
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	—	2	1	no	—
39	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
40	C₂×SD₁₆×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	4	no	—
42	C₂×D₈×C₃₁	—	2	1	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	4	4	no	—
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₃₁×#127	C₂³	1	2	yes	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	D₄	1	2	yes	—
61	C₄×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×C₄⋊C₈	—	2	1	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
70	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
71	C₃₁×#103	—	2	4	no	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂⁴×C₃₁	—	2	1	no	—
82	#41×C₃₁	—	4	4	no	—
83	#41×C₃₁	—	4	4	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	D₈	1	4	yes	complemented
92	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
102	SD₁₆×C₃₁	—	4	8	no	—
103	D₈×C₃₁	—	4	2	no	—
104	D₈×C₃₁	—	4	8	no	—
105	D₈×C₃₁	—	4	2	no	—
106	M₁₆×C₃₁	—	4	4	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	4	no	—
110	C₂×C₈×C₃₁	—	4	1	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₂×C₈×C₃₁	—	4	2	no	—
113	C₂×C₈×C₃₁	—	4	4	no	—
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	—	4	2	no	—
126	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₁	#41	1	1	yes	characteristic
129	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
130	#19255	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	2	4	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	4	2	no	—
136	C₂³×C₃₁	#121	1	1	yes	characteristic
137	Q₈×C₃₁	—	4	2	no	—
138	Q₈×C₃₁	—	4	2	no	—
139	D₄×C₃₁	—	2	2	no	—
140	D₄×C₃₁	—	2	4	no	—
141	D₄×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	8	2	no	—
146	D₄×C₃₁	—	8	4	no	—
147	D₄×C₃₁	C₂×D₈	1	4	yes	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	4	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	8	4	no	—
158	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
159	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
160	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₃₁	#135	1	2	yes	—
163	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
166	C₈×C₃₁	—	2	1	no	—
167	C₈×C₃₁	—	2	1	no	—
168	C₈×C₃₁	—	2	1	no	—
169	C₈×C₃₁	—	4	4	no	—
170	C₈×C₃₁	—	4	2	no	—
171	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
172	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#3036	C₂×C₃₁	1	2	yes	complemented
174	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1681	C₂×C₃₁	1	4	yes	complemented
176	#1673	C₂×C₃₁	1	2	yes	complemented
177	#1667	C₂×C₃₁	1	2	yes	complemented
178	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1547	C₂×C₃₁	1	2	yes	complemented
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	2	1	no	—
182	C₂²×C₃₁	—	2	4	no	—
183	C₂²×C₃₁	—	4	8	no	—
184	C₂²×C₃₁	—	4	2	no	—
185	C₂²×C₃₁	—	4	2	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	#466	1	2	yes	—
190	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	4	2	no	—
195	C₄×C₃₁	—	8	2	no	—
196	C₄×C₃₁	#466	1	2	yes	—
197	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
199	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₃₁	#575	1	1	yes	characteristic
201	C₂×AGL(1,ℤ/8)	—	2	2	no	—
202	C₂³×C₈	—	2	1	no	—
203	#532	C₂²×C₃₁	1	2	yes	—
204	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
205	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
206	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
207	#486	C₂²×C₃₁	1	2	yes	complemented
208	#466	—	2	1	no	—
209	#466	—	2	1	no	—
210	#463	—	2	4	no	—
211	#461	C₂²×C₃₁	1	4	yes	—
212	#459	C₂²×C₃₁	1	4	yes	—
213	C₄×D₈	—	2	1	no	—
214	C₄×D₈	—	2	1	no	—
215	C₈×D₄	—	2	1	no	—
216	C₈×D₄	—	2	1	no	—
217	C₈×D₄	C₂²×C₃₁	1	2	yes	—
218	C₈×D₄	C₂²×C₃₁	1	2	yes	—
219	#423	C₂²×C₃₁	1	2	yes	—
220	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
221	#417	—	2	2	no	—
222	C₂×C₂²⋊C₈	—	2	1	no	—
223	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
224	#335	C₂²×C₃₁	1	2	yes	complemented
225	#335	C₂²×C₃₁	1	2	yes	complemented
226	#332	C₂²×C₃₁	1	4	yes	complemented
227	#331	C₂²×C₃₁	1	4	yes	complemented
228	C₂×C₃₁	—	2	2	no	—
229	C₂×C₃₁	—	4	4	no	—
230	C₂×C₃₁	—	16	2	no	—
231	C₂×C₃₁	#3036	1	1	yes	characteristic
232	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
233	C₂×C₃₁	#3067	1	1	yes	characteristic
234	C₂×#41	—	2	2	no	—
235	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
236	C₂³×C₄	—	2	1	no	—
237	AGL(1,ℤ/8)	—	4	8	no	—
238	C₂×SD₁₆	—	2	4	no	—
239	C₂×D₈	—	2	4	no	—
240	C₂×D₈	—	2	1	no	—
241	C₂×D₈	—	2	1	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	—	4	4	no	—
247	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
249	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
250	#127	C₂³×C₃₁	1	2	yes	—
251	#124	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	D₄×C₃₁	1	2	yes	—
259	C₄×D₄	C₂³×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊C₄	—	2	1	no	—
262	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
263	#107	—	2	1	no	—
264	#107	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
268	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
269	#103	—	2	4	no	—
270	#102	—	2	2	no	—
271	#102	—	2	2	no	—
272	#102	—	2	4	no	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
276	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₃₁	#19255	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	—	2	1	no	—
281	#41	—	4	4	no	—
282	#41	—	4	4	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	D₈×C₃₁	1	4	yes	complemented
291	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
301	SD₁₆	—	4	8	no	—
302	D₈	—	4	2	no	—
303	D₈	—	4	8	no	—
304	D₈	—	4	2	no	—
305	M₁₆	—	4	4	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	4	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	4	no	—
311	C₂×C₈	—	4	2	no	—
312	C₂×C₈	—	4	4	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	1	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	—	4	2	no	—
325	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₄²	#41×C₃₁	1	1	yes	characteristic
328	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
329	C₂³	—	2	4	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	C₃₁×#121	1	1	yes	characteristic
335	Q₈	—	4	2	no	—
336	Q₈	—	4	2	no	—
337	D₄	—	2	4	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	8	2	no	—
344	D₄	—	8	4	no	—
345	D₄	C₂×D₈×C₃₁	1	4	yes	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	4	no	—
356	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
357	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
358	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
359	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
360	C₂×C₄	C₃₁×#135	1	2	yes	—
361	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	4	4	no	—
368	C₈	—	4	2	no	—
369	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	4	8	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	C₃₁×#466	1	2	yes	—
380	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	C₃₁×#466	1	2	yes	—
387	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
388	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
389	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₄	C₃₁×#575	1	1	yes	characteristic
391	C₂	—	2	2	no	—
392	C₂	—	4	4	no	—
393	C₂	—	16	2	no	—
394	C₂	C₃₁×#3036	1	1	yes	characteristic
395	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
396	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#3036	C₂	semidirect
3. C₂×C₈×D₄×C₃₁	C₂	semidirect
4. C₃₁×#1681	C₂	semidirect
5. C₃₁×#1673	C₂	semidirect
6. C₃₁×#1667	C₂	semidirect
7. C₃₁×#1567	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
13. C₃₁×#500	C₂²	semidirect
14. C₃₁×#493	C₂²	semidirect
15. C₃₁×#486	C₂²	semidirect
32. C₃₁×#335	C₂²	semidirect
33. C₃₁×#335	C₂²	semidirect
34. C₃₁×#332	C₂²	semidirect
35. C₃₁×#331	C₂²	semidirect
91. C₂×D₄×C₃₁	D₈	semidirect
130. #19255	C₃₁	direct (N × H)
171. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
172. #3169	C₂×C₃₁	semidirect
173. #3036	C₂×C₃₁	semidirect
174. C₂×C₈×D₄	C₂×C₃₁	semidirect
175. #1681	C₂×C₃₁	semidirect
176. #1673	C₂×C₃₁	semidirect
177. #1667	C₂×C₃₁	semidirect
178. #1567	C₂×C₃₁	semidirect
179. #1547	C₂×C₃₁	semidirect
205. #500	C₂²×C₃₁	semidirect
206. #493	C₂²×C₃₁	semidirect
207. #486	C₂²×C₃₁	semidirect
224. #335	C₂²×C₃₁	semidirect
225. #335	C₂²×C₃₁	semidirect
226. #332	C₂²×C₃₁	semidirect
227. #331	C₂²×C₃₁	semidirect
279. C₃₁	#19255	direct (N × H)
290. C₂×D₄	D₈×C₃₁	semidirect
369. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₃₁×#3036	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1681	2
C₃₁×#1673	2
C₃₁×#1667	2
C₃₁×#1567	2
C₃₁×#1547	2
#19255	31

Subgroups — C31×#19255 #21476014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19255 #21476014