Subgroups — C₃₁×#12818 #21469577

Order 7936.

474 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×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	2	yes	maximal
4	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₄×D₈×C₃₁	—	2	1	no	—
10	C₄×D₈×C₃₁	—	2	1	no	—
11	C₄×D₈×C₃₁	C₄	1	2	yes	—
12	C₄×D₈×C₃₁	C₄	1	2	yes	—
13	C₂×C₃₁×#108	—	2	1	no	—
14	C₂×C₃₁×#108	C₂²	1	2	yes	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₃₁×#423	—	2	1	no	—
17	C₂×C₃₁×#102	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×C₄⋊C₁₆	—	2	1	no	—
20	C₃₁×C₄⋊C₁₆	—	2	1	no	—
21	C₃₁×#361	—	2	4	no	—
22	C₂²×D₄×C₃₁	—	2	1	no	—
23	C₂³×C₄×C₃₁	—	4	1	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×D₈×C₃₁	D₄	1	2	yes	—
29	C₂×D₈×C₃₁	Q₈	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₄×D₄×C₃₁	—	2	2	no	—
34	C₄×D₄×C₃₁	—	2	2	no	—
35	C₄×D₄×C₃₁	—	4	1	no	—
36	C₄×D₄×C₃₁	—	4	1	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₃₁×M₃₂	—	2	2	no	—
41	C₃₁×M₃₂	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	2	2	no	—
43	C₃₁×#108	—	2	2	no	—
44	C₃₁×#108	—	2	2	no	—
45	C₃₁×#108	—	4	2	no	—
46	C₃₁×#107	—	2	1	no	—
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×C₄⋊C₈	—	4	1	no	—
49	C₃₁×C₄⋊C₈	—	4	1	no	—
50	C₃₁×#102	—	2	2	no	—
51	C₃₁×#102	—	2	2	no	—
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂⁴×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂×D₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	—	4	2	no	—
65	C₂²×C₄×C₃₁	—	4	2	no	—
66	C₂²×C₄×C₃₁	—	8	4	no	—
67	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	D₈×C₃₁	—	2	2	no	—
71	D₈×C₃₁	—	2	2	no	—
72	D₈×C₃₁	—	2	2	no	—
73	D₈×C₃₁	C₄⋊C₄	1	4	yes	—
74	M₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
86	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	4	1	no	—
89	C₂²⋊C₄×C₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₃₁	—	4	2	no	—
91	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₃₁	D₈	1	1	yes	characteristic
94	C₄²×C₃₁	D₈	1	1	yes	characteristic
95	C₁₆×C₃₁	—	4	2	no	—
96	#12818	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	4	2	no	—
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	8	2	no	—
101	C₂³×C₃₁	—	8	2	no	—
102	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
103	D₄×C₃₁	—	2	4	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	D₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	8	4	no	—
111	C₂×C₄×C₃₁	—	8	2	no	—
112	C₂×C₄×C₃₁	—	8	4	no	—
113	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	—	2	1	no	—
125	C₈×C₃₁	—	8	2	no	—
126	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₈×C₃₁	#117	1	1	yes	characteristic
128	C₈×C₃₁	#121	1	1	yes	characteristic
129	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
131	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₂×#361	C₂×C₃₁	1	2	yes	—
133	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#1387	C₂×C₃₁	1	2	yes	complemented
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	4	4	no	—
138	C₂²×C₃₁	—	8	4	no	—
139	C₂²×C₃₁	—	8	4	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
143	C₂²×C₃₁	#461	1	1	yes	characteristic
144	C₂²×C₃₁	#476	1	1	yes	characteristic
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	8	2	no	—
149	C₄×C₃₁	#361	1	2	yes	complemented
150	C₄×C₃₁	#380	1	1	yes	characteristic
151	C₄×C₃₁	#417	1	1	yes	characteristic
152	C₄×C₃₁	#459	1	1	yes	characteristic
153	C₄×C₃₁	#474	1	1	yes	characteristic
154	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄×D₄	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
157	C₄×D₈	—	2	1	no	—
158	C₄×D₈	—	2	1	no	—
159	C₄×D₈	C₄×C₃₁	1	2	yes	—
160	C₄×D₈	C₄×C₃₁	1	2	yes	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	C₂²×C₃₁	1	2	yes	—
163	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
164	#423	—	2	1	no	—
165	C₂×#102	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₄⋊C₁₆	—	2	1	no	—
168	C₄⋊C₁₆	—	2	1	no	—
169	#361	—	2	4	no	—
170	C₂×C₃₁	—	2	1	no	—
171	C₂×C₃₁	—	2	1	no	—
172	C₂×C₃₁	—	8	4	no	—
173	C₂×C₃₁	#1802	1	1	yes	characteristic
174	C₂×C₃₁	C₂×#361	1	1	yes	characteristic
175	C₂×C₃₁	#2148	1	1	yes	characteristic
176	C₂²×D₄	—	2	1	no	—
177	C₂³×C₄	—	4	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×D₈	D₄×C₃₁	1	2	yes	—
183	C₂×D₈	Q₈×C₃₁	1	2	yes	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₄×D₄	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	4	1	no	—
190	C₄×D₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₂²⋊C₄	—	4	1	no	—
193	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
194	M₃₂	—	2	2	no	—
195	M₃₂	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	#108	—	2	2	no	—
198	#108	—	2	2	no	—
199	#108	—	4	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂×C₄×C₃₁	1	2	yes	—
202	C₄⋊C₈	—	4	1	no	—
203	C₄⋊C₈	—	4	1	no	—
204	#102	—	2	2	no	—
205	#102	—	2	2	no	—
206	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
210	C₃₁	#12818	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	—	4	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	8	4	no	—
222	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	D₈	—	2	2	no	—
228	D₈	C₄⋊C₄×C₃₁	1	4	yes	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
240	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
241	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₄²	D₈×C₃₁	1	1	yes	characteristic
249	C₄²	D₈×C₃₁	1	1	yes	characteristic
250	C₁₆	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	2	no	—
256	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
269	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
273	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	8	2	no	—
280	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
281	C₈	C₃₁×#117	1	1	yes	characteristic
282	C₈	C₃₁×#121	1	1	yes	characteristic
283	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	4	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
292	C₂²	C₃₁×#461	1	1	yes	characteristic
293	C₂²	C₃₁×#476	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	2	no	—
298	C₄	C₃₁×#361	1	2	yes	complemented
299	C₄	C₃₁×#380	1	1	yes	characteristic
300	C₄	C₃₁×#417	1	1	yes	characteristic
301	C₄	C₃₁×#459	1	1	yes	characteristic
302	C₄	C₃₁×#474	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	4	no	—
306	C₂	C₃₁×#1802	1	1	yes	characteristic
307	C₂	C₂×C₃₁×#361	1	1	yes	characteristic
308	C₂	C₃₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2150	C₂	semidirect
4. C₃₁×#1848	C₂	semidirect
5. C₃₁×#1387	C₂	semidirect
96. #12818	C₃₁	direct (N × H)
131. #2150	C₂×C₃₁	semidirect
133. #1848	C₂×C₃₁	semidirect
134. #1387	C₂×C₃₁	semidirect
149. C₄×C₃₁	#361	semidirect
210. C₃₁	#12818	direct (N × H)
298. C₄	C₃₁×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₃₁	2
C₃₁×#2150	2
C₂×C₃₁×#361	2
C₃₁×#1848	2
C₃₁×#1387	2
#12818	31

Subgroups — C31×#12818 #21469577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12818 #21469577