Subgroups — C₃₁×#12989 #21469748

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2930	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#349	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#525	C₄	1	1	yes	characteristic
9	C₃₁×#504	—	2	4	no	—
10	C₂×C₃₁×M₃₂	—	2	2	no	—
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#436	—	2	2	no	—
14	C₃₁×#435	—	2	2	no	—
15	C₃₁×#424	C₄	1	1	yes	characteristic
16	C₃₁×#408	C₄	1	1	yes	characteristic
17	C₃₁×#407	C₄	1	1	yes	characteristic
18	C₃₁×#405	C₂²	1	1	yes	characteristic
19	C₃₁×#349	—	2	2	no	—
20	C₃₁×#349	—	2	4	no	—
21	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₃₁×#131	—	2	2	no	—
24	C₃₁×#131	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#123	C₈	1	2	yes	—
36	C₃₁×C₂²⋊Q₈	—	2	1	no	—
37	C₃₁×#121	—	2	1	no	—
38	C₃₁×#120	C₈	1	2	yes	—
39	C₄×D₄×C₃₁	—	2	2	no	—
40	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
41	C₃₁×M₃₂	—	2	4	no	—
42	C₃₁×M₃₂	—	2	2	no	—
43	C₃₁×M₃₂	—	2	2	no	—
44	C₃₁×M₃₂	—	2	2	no	—
45	C₃₁×M₃₂	D₄	1	4	yes	complemented
46	C₂×C₁₆×C₃₁	—	2	4	no	—
47	C₂×C₁₆×C₃₁	—	2	2	no	—
48	C₂×C₁₆×C₃₁	—	2	2	no	—
49	C₃₁×C₄⋊C₈	—	2	2	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	2	2	no	—
52	C₃₁×#97	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
55	#41×C₃₁	—	2	2	no	—
56	#41×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₃₁	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	2	no	—
71	M₁₆×C₃₁	—	4	1	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	M₁₆×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	#41	1	2	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
89	C₄²×C₃₁	—	2	1	no	—
90	C₁₆×C₃₁	—	2	4	no	—
91	C₁₆×C₃₁	—	4	2	no	—
92	#12989	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	2	1	no	—
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	—	8	1	no	—
96	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
98	Q₈×C₃₁	—	2	2	no	—
99	D₄×C₃₁	—	2	4	no	—
100	D₄×C₃₁	—	2	2	no	—
101	D₄×C₃₁	—	8	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#131	1	2	yes	—
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	—	4	1	no	—
117	C₈×C₃₁	—	4	1	no	—
118	C₈×C₃₁	#120	1	1	yes	characteristic
119	C₈×C₃₁	#121	1	2	yes	—
120	C₈×C₃₁	#123	1	1	yes	characteristic
121	#3407	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#2930	C₂×C₃₁	1	1	yes	characteristic
123	#2115	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂×#349	C₂×C₃₁	1	2	yes	complemented
125	#1378	C₂×C₃₁	1	2	yes	complemented
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	4	1	no	—
128	C₂²×C₃₁	—	4	2	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	8	2	no	—
133	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₃₁	C₈×D₄	1	2	yes	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	4	1	no	—
138	C₄×C₃₁	#386	1	1	yes	characteristic
139	C₄×C₃₁	#408	1	1	yes	characteristic
140	C₄×C₃₁	#435	1	2	yes	—
141	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
142	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
143	#525	C₄×C₃₁	1	1	yes	characteristic
144	#504	—	2	4	no	—
145	C₂×M₃₂	—	2	2	no	—
146	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
147	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
148	#436	—	2	2	no	—
149	#435	—	2	2	no	—
150	#424	C₄×C₃₁	1	1	yes	characteristic
151	#408	C₄×C₃₁	1	1	yes	characteristic
152	#407	C₄×C₃₁	1	1	yes	characteristic
153	#405	C₂²×C₃₁	1	1	yes	characteristic
154	#349	—	2	2	no	—
155	#349	—	2	4	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	2	2	no	—
158	C₂×C₃₁	—	4	2	no	—
159	C₂×C₃₁	—	8	1	no	—
160	C₂×C₃₁	#1869	1	1	yes	characteristic
161	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
163	#131	—	2	2	no	—
164	#131	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
172	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₃₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
175	#123	C₈×C₃₁	1	2	yes	—
176	C₂²⋊Q₈	—	2	1	no	—
177	#121	—	2	1	no	—
178	#120	C₈×C₃₁	1	2	yes	—
179	C₄×D₄	—	2	2	no	—
180	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
181	M₃₂	—	2	4	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	D₄×C₃₁	1	4	yes	complemented
186	C₂×C₁₆	—	2	4	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₃₁	#12989	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	M₁₆×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
201	C₂×D₄	M₁₆×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	#41×C₃₁	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
230	C₄²	—	2	1	no	—
231	C₁₆	—	2	4	no	—
232	C₁₆	—	4	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	8	1	no	—
236	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
237	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	8	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
253	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₃₁×#131	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	C₃₁×#120	1	1	yes	characteristic
259	C₈	C₃₁×#121	1	2	yes	—
260	C₈	C₃₁×#123	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	2	no	—
268	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
269	C₂²	C₈×D₄×C₃₁	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	C₃₁×#386	1	1	yes	characteristic
274	C₄	C₃₁×#408	1	1	yes	characteristic
275	C₄	C₃₁×#435	1	2	yes	—
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	—	4	2	no	—
279	C₂	—	8	1	no	—
280	C₂	C₃₁×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#386
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#1869
prime radical	C₃₁×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3407	C₂	semidirect
3. C₃₁×#2115	C₂	semidirect
4. C₂×C₃₁×#349	C₂	semidirect
5. C₃₁×#1378	C₂	semidirect
45. C₃₁×M₃₂	D₄	semidirect
92. #12989	C₃₁	direct (N × H)
121. #3407	C₂×C₃₁	semidirect
123. #2115	C₂×C₃₁	semidirect
124. C₂×#349	C₂×C₃₁	semidirect
125. #1378	C₂×C₃₁	semidirect
185. M₃₂	D₄×C₃₁	semidirect
194. C₃₁	#12989	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3407	2
C₃₁×#2930	2
C₃₁×#2115	2
C₂×C₃₁×#349	2
C₃₁×#1378	2
#12989	31

Subgroups — C31×#12989 #21469748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12989 #21469748