Subgroups — C₃₁×#29143 #21485902

Order 7936.

890 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×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2025	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1957	C₂	1	2	yes	maximal
5	C₃₁×#1834	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1804	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#125	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#102	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#390	—	2	2	no	—
16	C₃₁×#390	C₂²	1	2	yes	—
17	C₃₁×#389	—	2	2	no	—
18	C₃₁×#389	—	2	2	no	—
19	C₃₁×#384	C₂²	1	1	yes	characteristic
20	C₃₁×#384	C₂²	1	2	yes	—
21	C₃₁×#382	C₂²	1	2	yes	—
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#340	C₂²	1	8	yes	—
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₂²×D₄×C₃₁	—	2	2	no	—
26	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
29	C₃₁×#127	—	2	2	no	—
30	C₃₁×#127	Q₈	1	4	yes	—
31	C₃₁×#125	—	4	4	no	—
32	C₄×D₄×C₃₁	—	2	8	no	—
33	C₄×D₄×C₃₁	—	4	8	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊C₈	—	2	4	no	—
48	C₃₁×#102	—	4	8	no	—
49	C₄×C₈×C₃₁	—	2	4	no	—
50	C₃₁×#95	—	2	2	no	—
51	C₃₁×#95	—	2	2	no	—
52	C₂⁴×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	—	2	8	no	—
54	C₂×D₄×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	2	4	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂×D₄×C₃₁	—	4	4	no	—
59	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₄⋊C₄×C₃₁	—	4	8	no	—
75	C₄⋊C₄×C₃₁	—	4	4	no	—
76	C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₄⋊C₄×C₃₁	—	4	4	no	—
78	C₂²⋊C₄×C₃₁	—	4	8	no	—
79	C₂²⋊C₄×C₃₁	—	4	4	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	—	2	2	no	—
83	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₃₁	#41	1	2	yes	—
85	C₄²×C₃₁	#41	1	4	yes	—
86	#29143	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	4	no	—
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	4	2	no	—
90	C₂³×C₃₁	—	4	4	no	—
91	C₂³×C₃₁	—	4	4	no	—
92	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
93	D₄×C₃₁	—	2	8	no	—
94	D₄×C₃₁	—	4	4	no	—
95	D₄×C₃₁	—	4	16	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	8	8	no	—
107	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₃₁	#124	1	2	yes	—
109	C₂×C₄×C₃₁	#124	1	2	yes	—
110	C₂×C₄×C₃₁	#135	1	4	yes	—
111	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
116	C₈×C₃₁	—	4	4	no	—
117	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
119	#2025	C₂×C₃₁	1	4	yes	complemented
120	#1957	C₂×C₃₁	1	2	yes	—
121	#1834	C₂×C₃₁	1	2	yes	complemented
122	#1804	C₂×C₃₁	1	4	yes	complemented
123	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂²×C₃₁	—	4	8	no	—
125	C₂²×C₃₁	—	4	4	no	—
126	C₂²×C₃₁	—	8	2	no	—
127	C₂²×C₃₁	—	8	2	no	—
128	C₂²×C₃₁	—	8	4	no	—
129	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
131	C₂²×C₃₁	#535	1	1	yes	characteristic
132	C₂²×C₃₁	C₂×#135	1	2	yes	—
133	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	4	2	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	—	8	4	no	—
139	C₄×C₃₁	#478	1	8	yes	—
140	C₄×C₃₁	#554	1	2	yes	—
141	C₄×C₃₁	#555	1	2	yes	—
142	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×#125	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄×D₄	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
147	C₂×#102	C₂²×C₃₁	1	8	yes	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
149	#390	—	2	2	no	—
150	#390	C₂²×C₃₁	1	2	yes	—
151	#389	—	2	2	no	—
152	#389	—	2	2	no	—
153	#384	C₂²×C₃₁	1	1	yes	characteristic
154	#384	C₂²×C₃₁	1	2	yes	—
155	#382	C₂²×C₃₁	1	2	yes	—
156	C₄³	C₂²×C₃₁	1	1	yes	characteristic
157	#340	C₂²×C₃₁	1	8	yes	—
158	#336	C₂²×C₃₁	1	2	yes	—
159	C₂×C₃₁	—	8	4	no	—
160	C₂×C₃₁	#2563	1	1	yes	characteristic
161	C₂×C₃₁	C₂×#478	1	2	yes	—
162	C₂×C₃₁	#3131	1	1	yes	characteristic
163	C₂×C₃₁	#3138	1	1	yes	characteristic
164	C₂×C₃₁	#3207	1	2	yes	—
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
169	#127	—	2	2	no	—
170	#127	Q₈×C₃₁	1	4	yes	—
171	#125	—	4	4	no	—
172	C₄×D₄	—	2	8	no	—
173	C₄×D₄	—	4	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	D₄×C₃₁	1	2	yes	—
183	C₂×C₄²	D₄×C₃₁	1	2	yes	—
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	#102	—	4	8	no	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₃₁	#29143	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	2	2	no	—
194	C₂×D₄	—	2	8	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	2	no	—
224	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
225	C₄²	#41×C₃₁	1	2	yes	—
226	C₄²	#41×C₃₁	1	4	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
233	D₄	—	2	8	no	—
234	D₄	—	4	16	no	—
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	8	no	—
247	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₃₁×#124	1	2	yes	—
249	C₂×C₄	C₃₁×#124	1	2	yes	—
250	C₂×C₄	C₃₁×#135	1	4	yes	—
251	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
252	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	4	no	—
262	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
264	C₂²	C₃₁×#535	1	1	yes	characteristic
265	C₂²	C₂×C₃₁×#135	1	2	yes	—
266	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	4	no	—
272	C₄	C₃₁×#478	1	8	yes	—
273	C₄	C₃₁×#554	1	2	yes	—
274	C₄	C₃₁×#555	1	2	yes	—
275	C₂	—	8	4	no	—
276	C₂	C₃₁×#2563	1	1	yes	characteristic
277	C₂	C₂×C₃₁×#478	1	2	yes	—
278	C₂	C₃₁×#3131	1	1	yes	characteristic
279	C₂	C₃₁×#3138	1	1	yes	characteristic
280	C₂	C₃₁×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2859	C₂	semidirect
3. C₃₁×#2025	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
6. C₃₁×#1804	C₂	semidirect
7. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
86. #29143	C₃₁	direct (N × H)
117. #2859	C₂×C₃₁	semidirect
119. #2025	C₂×C₃₁	semidirect
121. #1834	C₂×C₃₁	semidirect
122. #1804	C₂×C₃₁	semidirect
123. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
192. C₃₁	#29143	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×#2859	2
C₄×C₃₁×#127	2
C₃₁×#2025	2
C₃₁×#1957	2
C₃₁×#1834	2
C₃₁×#1804	2
C₄×C₃₁×C₄⋊C₈	2
#29143	31

Subgroups — C31×#29143 #21485902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29143 #21485902