Subgroups — C₃₁×#12830 #21469589

Order 7936.

378 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×#2957	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2145	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#359	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₃₁×#568	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#444	C₄	1	4	yes	complemented
13	C₃₁×#443	—	2	2	no	—
14	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₃₁×#427	C₂²	1	1	yes	characteristic
17	C₃₁×#421	—	2	1	no	—
18	C₃₁×#405	C₂²	1	1	yes	characteristic
19	C₃₁×C₄⋊C₁₆	—	2	2	no	—
20	C₃₁×#360	—	2	2	no	—
21	C₃₁×#359	—	2	2	no	—
22	C₃₁×#344	—	2	1	no	—
23	C₃₁×H(3,ℤ/4)	—	2	1	no	—
24	C₂×#41×C₃₁	—	2	1	no	—
25	C₃₁×#135	—	2	1	no	—
26	C₃₁×#135	—	2	1	no	—
27	C₃₁×#135	C₂×C₄	1	2	yes	—
28	C₃₁×#135	D₄	1	2	yes	—
29	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	—	4	2	no	—
34	C₂×M₁₆×C₃₁	—	2	1	no	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₂×M₁₆×C₃₁	—	4	2	no	—
37	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#117	—	2	1	no	—
40	C₃₁×#117	D₄	1	1	yes	characteristic
41	C₃₁×M₃₂	—	2	1	no	—
42	C₃₁×M₃₂	—	2	1	no	—
43	C₂×C₁₆×C₃₁	—	2	1	no	—
44	C₂×C₁₆×C₃₁	—	2	1	no	—
45	C₂×C₁₆×C₃₁	—	2	1	no	—
46	C₂×C₁₆×C₃₁	—	2	1	no	—
47	C₃₁×#108	—	2	1	no	—
48	C₃₁×#108	—	2	1	no	—
49	C₃₁×#108	—	2	1	no	—
50	C₃₁×#108	C₂×C₄	1	2	yes	—
51	C₃₁×#107	—	2	1	no	—
52	C₃₁×#106	—	2	1	no	—
53	C₃₁×#104	—	2	4	no	—
54	C₃₁×#97	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₃₁	D₄	1	2	yes	—
56	#41×C₃₁	—	2	2	no	—
57	#41×C₃₁	—	2	2	no	—
58	#41×C₃₁	—	4	1	no	—
59	#41×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂×D₄×C₃₁	—	2	1	no	—
62	C₂×D₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₃₁	—	2	1	no	—
66	Q₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
67	SD₁₆×C₃₁	—	2	2	no	—
68	SD₁₆×C₃₁	—	2	2	no	—
69	D₈×C₃₁	—	2	1	no	—
70	D₈×C₃₁	C₄⋊C₄	1	2	yes	—
71	M₁₆×C₃₁	—	2	1	no	—
72	M₁₆×C₃₁	—	2	1	no	—
73	M₁₆×C₃₁	—	2	2	no	—
74	M₁₆×C₃₁	—	4	4	no	—
75	M₁₆×C₃₁	—	4	1	no	—
76	M₁₆×C₃₁	—	4	1	no	—
77	M₁₆×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	—	4	4	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
87	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₄⋊C₄×C₃₁	—	4	1	no	—
90	C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂²⋊C₄×C₃₁	—	4	1	no	—
92	C₄²×C₃₁	—	4	1	no	—
93	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₃₁	—	4	1	no	—
95	C₁₆×C₃₁	—	4	1	no	—
96	#12830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	4	1	no	—
98	C₂³×C₃₁	#135	1	1	yes	characteristic
99	Q₈×C₃₁	—	2	2	no	—
100	Q₈×C₃₁	—	4	1	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	1	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	8	1	no	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
114	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₈×C₃₁	—	2	2	no	—
117	C₈×C₃₁	—	4	2	no	—
118	C₈×C₃₁	—	8	1	no	—
119	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₃₁	#117	1	1	yes	characteristic
121	C₈×C₃₁	#121	1	1	yes	characteristic
122	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
123	#2957	C₂×C₃₁	1	1	yes	characteristic
124	#2151	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#2145	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#359	C₂×C₃₁	1	1	yes	characteristic
127	#1833	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#1388	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1382	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	2	no	—
132	C₂²×C₃₁	—	16	1	no	—
133	C₂²×C₃₁	#416	1	1	yes	characteristic
134	C₂²×C₃₁	#463	1	1	yes	characteristic
135	C₂²×C₃₁	#478	1	1	yes	characteristic
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	4	2	no	—
138	C₄×C₃₁	—	16	1	no	—
139	C₄×C₃₁	#380	1	1	yes	characteristic
140	C₄×C₃₁	#418	1	1	yes	characteristic
141	C₄×C₃₁	#463	1	1	yes	characteristic
142	C₄×C₃₁	#478	1	1	yes	characteristic
143	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
144	#568	—	2	1	no	—
145	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
146	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
147	#444	C₄×C₃₁	1	4	yes	complemented
148	#443	—	2	2	no	—
149	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
151	#427	C₂²×C₃₁	1	1	yes	characteristic
152	#421	—	2	1	no	—
153	#405	C₂²×C₃₁	1	1	yes	characteristic
154	C₄⋊C₁₆	—	2	2	no	—
155	#360	—	2	2	no	—
156	#359	—	2	2	no	—
157	#344	—	2	1	no	—
158	H(3,ℤ/4)	—	2	1	no	—
159	C₂×C₃₁	—	2	1	no	—
160	C₂×C₃₁	—	2	1	no	—
161	C₂×C₃₁	—	2	1	no	—
162	C₂×C₃₁	—	8	2	no	—
163	C₂×C₃₁	#1804	1	1	yes	characteristic
164	C₂×#41	—	2	1	no	—
165	#135	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	C₂×C₄×C₃₁	1	2	yes	—
168	#135	D₄×C₃₁	1	2	yes	—
169	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic
170	C₂×SD₁₆	—	2	1	no	—
171	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic
172	#131	—	2	4	no	—
173	#131	—	4	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	4	2	no	—
177	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
179	#117	—	2	1	no	—
180	#117	D₄×C₃₁	1	1	yes	characteristic
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	#108	—	2	1	no	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	C₂×C₄×C₃₁	1	2	yes	—
191	#107	—	2	1	no	—
192	#106	—	2	1	no	—
193	#104	—	2	4	no	—
194	#97	C₂×C₄×C₃₁	1	2	yes	—
195	C₄×C₈	D₄×C₃₁	1	2	yes	—
196	C₃₁	#12830	1	1	yes	characteristic, direct product factor, Sylow
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	4	1	no	—
200	#41	—	4	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	Q₁₆	—	2	1	no	—
207	Q₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
208	SD₁₆	—	2	2	no	—
209	SD₁₆	—	2	2	no	—
210	D₈	—	2	1	no	—
211	D₈	C₄⋊C₄×C₃₁	1	2	yes	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
227	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
228	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
235	C₁₆	—	4	1	no	—
236	C₁₆	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₃₁×#135	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	4	1	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
252	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
254	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	8	1	no	—
259	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₈	C₃₁×#117	1	1	yes	characteristic
261	C₈	C₃₁×#121	1	1	yes	characteristic
262	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	C₃₁×#416	1	1	yes	characteristic
267	C₂²	C₃₁×#463	1	1	yes	characteristic
268	C₂²	C₃₁×#478	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	16	1	no	—
272	C₄	C₃₁×#380	1	1	yes	characteristic
273	C₄	C₃₁×#418	1	1	yes	characteristic
274	C₄	C₃₁×#463	1	1	yes	characteristic
275	C₄	C₃₁×#478	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₃₁×#1804	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#418
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#1804
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2151	C₂	semidirect
5. C₃₁×#1833	C₂	semidirect
6. C₃₁×#1388	C₂	semidirect
7. C₃₁×#1382	C₂	semidirect
12. C₃₁×#444	C₄	semidirect
96. #12830	C₃₁	direct (N × H)
124. #2151	C₂×C₃₁	semidirect
127. #1833	C₂×C₃₁	semidirect
128. #1388	C₂×C₃₁	semidirect
129. #1382	C₂×C₃₁	semidirect
147. #444	C₄×C₃₁	semidirect
196. C₃₁	#12830	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×#2957	2
C₃₁×#2151	2
C₃₁×#2145	2
C₂×C₃₁×#359	2
C₃₁×#1833	2
C₃₁×#1388	2
C₃₁×#1382	2
#12830	31

Subgroups — C31×#12830 #21469589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12830 #21469589