Subgroups — C₃₁×#9812 #21466571

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	2	no	—
11	C₃₁×#486	—	2	2	no	—
12	C₃₁×#431	C₂²	1	1	yes	characteristic
13	C₃₁×#423	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#102	C₄	1	4	yes	—
16	C₂×C₃₁×#100	—	2	2	no	—
17	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#343	—	2	2	no	—
20	C₃₁×#335	—	2	2	no	—
21	C₃₁×#332	—	2	2	no	—
22	C₃₁×#331	—	2	2	no	—
23	C₃₁×#322	—	2	2	no	—
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₃₁	—	2	1	no	—
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₃₁×C₄⋊Q₈	—	2	1	no	—
33	C₃₁×C₄⋊Q₈	—	2	1	no	—
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×#127	—	2	1	no	—
37	C₃₁×#127	—	2	1	no	—
38	C₃₁×#127	—	2	1	no	—
39	C₃₁×#127	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
48	C₃₁×#102	—	2	8	no	—
49	C₃₁×#100	—	4	4	no	—
50	C₃₁×C₂²⋊C₈	—	2	1	no	—
51	C₃₁×#97	—	2	1	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	C₄×C₈×C₃₁	D₄	1	2	yes	—
55	C₄×C₈×C₃₁	D₄	1	2	yes	—
56	C₂⁴×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	—	4	1	no	—
58	C₂×Q₈×C₃₁	—	4	1	no	—
59	C₂×Q₈×C₃₁	—	8	2	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	4	4	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	4	2	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	8	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
74	M₁₆×C₃₁	—	2	2	no	—
75	M₁₆×C₃₁	—	2	2	no	—
76	M₁₆×C₃₁	—	4	2	no	—
77	M₁₆×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	1	no	—
82	C₂×C₈×C₃₁	—	4	1	no	—
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	4	1	no	—
87	C₄⋊C₄×C₃₁	—	4	1	no	—
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₃₁	#41	1	1	yes	characteristic
92	#9812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	4	2	no	—
94	C₂³×C₃₁	—	4	4	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	—	8	2	no	—
97	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	8	2	no	—
99	Q₈×C₃₁	—	8	2	no	—
100	D₄×C₃₁	—	4	4	no	—
101	D₄×C₃₁	—	8	2	no	—
102	D₄×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	8	2	no	—
104	D₄×C₃₁	—	8	2	no	—
105	D₄×C₃₁	—	8	1	no	—
106	D₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	8	2	no	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₃₁	#137	1	2	yes	—
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	—	8	1	no	—
123	C₈×C₃₁	—	8	1	no	—
124	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
125	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#1965	C₂×C₃₁	1	1	yes	characteristic
127	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1545	C₂×C₃₁	1	1	yes	characteristic
130	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	8	2	no	—
134	C₂²×C₃₁	—	8	2	no	—
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	—	16	4	no	—
137	C₂²×C₃₁	#390	1	1	yes	characteristic
138	C₂²×C₃₁	#440	1	2	yes	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	8	2	no	—
142	C₄×C₃₁	#413	1	2	yes	—
143	C₄×C₃₁	#468	1	2	yes	—
144	C₄×C₃₁	#472	1	2	yes	—
145	C₄×C₃₁	#496	1	1	yes	characteristic
146	C₄×C₃₁	#497	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
149	#486	—	2	2	no	—
150	#486	—	2	2	no	—
151	#431	C₂²×C₃₁	1	1	yes	characteristic
152	#423	C₂²×C₃₁	1	1	yes	characteristic
153	#423	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×#102	C₄×C₃₁	1	4	yes	—
155	C₂×#100	—	2	2	no	—
156	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	#343	—	2	2	no	—
159	#335	—	2	2	no	—
160	#332	—	2	2	no	—
161	#331	—	2	2	no	—
162	#322	—	2	2	no	—
163	C₂×C₃₁	—	2	2	no	—
164	C₂×C₃₁	—	16	2	no	—
165	C₂×C₃₁	#1967	1	1	yes	characteristic
166	C₂×C₃₁	#1973	1	1	yes	characteristic
167	C₂×C₃₁	#1978	1	1	yes	characteristic
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
175	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	2	1	no	—
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	—	2	1	no	—
180	#127	—	2	1	no	—
181	#127	—	2	1	no	—
182	#127	—	2	1	no	—
183	#127	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
186	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
192	#102	—	2	8	no	—
193	#100	—	4	4	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	D₄×C₃₁	1	2	yes	—
199	C₄×C₈	D₄×C₃₁	1	2	yes	—
200	C₃₁	#9812	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	8	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	8	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₄²	#41×C₃₁	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
242	Q₈	—	8	2	no	—
243	Q₈	—	8	2	no	—
244	D₄	—	4	4	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	D₄	—	8	1	no	—
250	D₄	—	8	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
258	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
259	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
261	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
262	C₂×C₄	C₃₁×#137	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	8	1	no	—
267	C₈	—	8	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	16	4	no	—
274	C₂²	C₃₁×#390	1	1	yes	characteristic
275	C₂²	C₃₁×#440	1	2	yes	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	C₃₁×#413	1	2	yes	—
280	C₄	C₃₁×#468	1	2	yes	—
281	C₄	C₃₁×#472	1	2	yes	—
282	C₄	C₃₁×#496	1	1	yes	characteristic
283	C₄	C₃₁×#497	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	16	2	no	—
286	C₂	C₃₁×#1967	1	1	yes	characteristic
287	C₂	C₃₁×#1973	1	1	yes	characteristic
288	C₂	C₃₁×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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
2. C₃₁×#1997	C₂	semidirect
4. C₃₁×#1567	C₂	semidirect
5. C₃₁×#1549	C₂	semidirect
7. C₃₁×#1451	C₂	semidirect
92. #9812	C₃₁	direct (N × H)
125. #1997	C₂×C₃₁	semidirect
127. #1567	C₂×C₃₁	semidirect
128. #1549	C₂×C₃₁	semidirect
130. #1451	C₂×C₃₁	semidirect
200. C₃₁	#9812	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₃₁×#1997	2
C₃₁×#1965	2
C₃₁×#1567	2
C₃₁×#1549	2
C₃₁×#1545	2
C₃₁×#1451	2
#9812	31

Subgroups — C31×#9812 #21466571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9812 #21466571