Subgroups — C₃₁×#3045 #16842978

Order 3968.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#464	C₂	1	2	yes	maximal, complemented
6	C₃₁×#463	C₂	1	2	yes	maximal, complemented
7	C₃₁×#460	C₂	1	2	yes	maximal, complemented
8	C₃₁×#459	C₂	1	2	yes	maximal, complemented
9	C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
16	C₃₁×#135	—	2	4	no	—
17	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₂×D₈×C₃₁	C₂²	1	2	yes	—
22	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#127	C₂²	1	2	yes	—
26	C₃₁×#124	C₂²	1	2	yes	—
27	C₃₁×#121	—	2	1	no	—
28	C₃₁×#121	—	2	1	no	—
29	C₃₁×#120	—	2	2	no	—
30	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
33	C₃₁×#117	—	2	1	no	—
34	C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
38	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
39	C₃₁×#103	C₂²	1	2	yes	complemented
40	C₃₁×#102	C₂²	1	2	yes	—
41	C₃₁×#102	C₂²	1	2	yes	—
42	C₃₁×#102	C₂²	1	2	yes	complemented
43	C₂⁴×C₃₁	—	2	1	no	—
44	#41×C₃₁	—	2	2	no	—
45	#41×C₃₁	—	2	2	no	—
46	#41×C₃₁	—	2	2	no	—
47	#41×C₃₁	D₄	1	4	yes	complemented
48	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	—	2	1	no	—
52	C₂×D₄×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	—	2	1	no	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	C₂³	1	2	yes	—
58	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	Q₁₆×C₃₁	—	2	2	no	—
66	SD₁₆×C₃₁	—	2	4	no	—
67	SD₁₆×C₃₁	—	2	2	no	—
68	SD₁₆×C₃₁	—	2	2	no	—
69	D₈×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	2	2	no	—
71	M₁₆×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂³	1	2	yes	—
76	C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	C₂³	1	2	yes	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	2	no	—
89	C₂³×C₃₁	—	2	1	no	—
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	—	8	1	no	—
92	C₂³×C₃₁	—	8	1	no	—
93	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₃₁	—	2	1	no	—
95	Q₈×C₃₁	—	2	1	no	—
96	Q₈×C₃₁	—	2	1	no	—
97	Q₈×C₃₁	C₂×D₄	1	2	yes	—
98	D₄×C₃₁	—	2	4	no	—
99	D₄×C₃₁	—	2	2	no	—
100	D₄×C₃₁	—	2	1	no	—
101	D₄×C₃₁	—	2	1	no	—
102	D₄×C₃₁	—	2	2	no	—
103	D₄×C₃₁	—	2	1	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	D₄×C₃₁	—	8	2	no	—
107	D₄×C₃₁	—	8	2	no	—
108	D₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	8	2	no	—
115	C₂×C₄×C₃₁	—	8	1	no	—
116	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	#41	1	2	yes	—
121	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	—	2	2	no	—
125	#3045	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₃₁	—	2	1	no	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	8	4	no	—
138	C₂²×C₃₁	#121	1	2	yes	—
139	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	1	no	—
144	C₄×C₃₁	—	4	2	no	—
145	C₄×C₃₁	#121	1	2	yes	—
146	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₂×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#535	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#520	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#464	C₂×C₃₁	1	2	yes	complemented
153	#463	C₂×C₃₁	1	2	yes	complemented
154	#460	C₂×C₃₁	1	2	yes	complemented
155	#459	C₂×C₃₁	1	2	yes	complemented
156	#424	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#417	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂×C₃₁	—	2	2	no	—
160	C₂×C₃₁	—	4	2	no	—
161	C₂×C₃₁	—	8	1	no	—
162	C₂×C₃₁	—	8	2	no	—
163	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
164	C₂×C₃₁	#576	1	1	yes	characteristic
165	C₂×C₃₁	#577	1	1	yes	characteristic
166	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
168	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
169	AGL(1,ℤ/8)	—	2	4	no	—
170	#135	—	2	4	no	—
171	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
172	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
173	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
174	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
175	C₂×D₈	C₂²×C₃₁	1	2	yes	—
176	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
177	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
179	#127	C₂²×C₃₁	1	2	yes	—
180	#124	C₂²×C₃₁	1	2	yes	—
181	#121	—	2	1	no	—
182	#121	—	2	1	no	—
183	#120	—	2	2	no	—
184	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
187	#117	—	2	1	no	—
188	#117	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
191	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
192	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
193	#103	C₂²×C₃₁	1	2	yes	complemented
194	#102	C₂²×C₃₁	1	2	yes	—
195	#102	C₂²×C₃₁	1	2	yes	—
196	#102	C₂²×C₃₁	1	2	yes	complemented
197	C₃₁	#3045	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	—	2	1	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	D₄×C₃₁	1	4	yes	complemented
203	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂³×C₃₁	1	2	yes	—
213	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	SD₁₆	—	2	4	no	—
222	SD₁₆	—	2	2	no	—
223	SD₁₆	—	2	2	no	—
224	D₈	—	2	4	no	—
225	D₈	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂³×C₃₁	1	2	yes	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
234	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
235	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂³×C₃₁	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	—	8	1	no	—
248	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	Q₈	C₂×D₄×C₃₁	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	—	2	1	no	—
256	D₄	—	2	1	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	1	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	8	2	no	—
262	D₄	—	8	2	no	—
263	D₄	C₂×D₄×C₃₁	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
273	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
275	C₂×C₄	#41×C₃₁	1	2	yes	—
276	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	C₃₁×#121	1	2	yes	—
293	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	C₃₁×#121	1	2	yes	—
300	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
307	C₂	C₃₁×#576	1	1	yes	characteristic
308	C₂	C₃₁×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₃₁×#135	C₂	semidirect
3. C₃₁×#535	C₂	semidirect
4. C₃₁×#520	C₂	semidirect
5. C₃₁×#464	C₂	semidirect
6. C₃₁×#463	C₂	semidirect
7. C₃₁×#460	C₂	semidirect
8. C₃₁×#459	C₂	semidirect
9. C₃₁×#424	C₂	semidirect
10. C₃₁×#417	C₂	semidirect
11. C₂×C₃₁×#102	C₂	semidirect
17. C₂×Q₁₆×C₃₁	C₂²	semidirect
19. C₂×SD₁₆×C₃₁	C₂²	semidirect
20. C₂×SD₁₆×C₃₁	C₂²	semidirect
22. C₂×D₈×C₃₁	C₂²	semidirect
30. C₄×Q₈×C₃₁	C₂²	semidirect
32. C₄×D₄×C₃₁	C₂²	semidirect
37. C₃₁×C₄⋊C₈	C₂²	semidirect
38. C₃₁×C₄⋊C₈	C₂²	semidirect
39. C₃₁×#103	C₂²	semidirect
42. C₃₁×#102	C₂²	semidirect
47. #41×C₃₁	D₄	semidirect
125. #3045	C₃₁	direct (N × H)
148. C₂×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
149. C₂×#135	C₂×C₃₁	semidirect
150. #535	C₂×C₃₁	semidirect
151. #520	C₂×C₃₁	semidirect
152. #464	C₂×C₃₁	semidirect
153. #463	C₂×C₃₁	semidirect
154. #460	C₂×C₃₁	semidirect
155. #459	C₂×C₃₁	semidirect
156. #424	C₂×C₃₁	semidirect
157. #417	C₂×C₃₁	semidirect
158. C₂×#102	C₂×C₃₁	semidirect
171. C₂×Q₁₆	C₂²×C₃₁	semidirect
173. C₂×SD₁₆	C₂²×C₃₁	semidirect
174. C₂×SD₁₆	C₂²×C₃₁	semidirect
176. C₂×D₈	C₂²×C₃₁	semidirect
184. C₄×Q₈	C₂²×C₃₁	semidirect
186. C₄×D₄	C₂²×C₃₁	semidirect
191. C₄⋊C₈	C₂²×C₃₁	semidirect
192. C₄⋊C₈	C₂²×C₃₁	semidirect
193. #103	C₂²×C₃₁	semidirect
196. #102	C₂²×C₃₁	semidirect
197. C₃₁	#3045	direct (N × H)
202. #41	D₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×AGL(1,ℤ/8)	2
C₂×C₃₁×#135	2
C₃₁×#535	2
C₃₁×#520	2
C₃₁×#464	2
C₃₁×#463	2
C₃₁×#460	2
C₃₁×#459	2
C₃₁×#424	2
C₃₁×#417	2
C₂×C₃₁×#102	2
#3045	31

Subgroups — C31×#3045 #16842978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3045 #16842978