Subgroups — C₃₁×#21323 #21478082

Order 7936.

1182 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1993	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1991	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#533	—	2	4	no	—
9	C₃₁×#533	C₄	1	8	yes	complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×#125	C₂²	1	2	yes	—
12	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
13	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₃₁	—	2	4	no	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#382	—	2	2	no	—
21	C₃₁×#382	C₂²	1	2	yes	—
22	C₃₁×#382	C₂²	1	2	yes	—
23	C₂×C₃₁×#95	—	2	2	no	—
24	C₃₁×#340	C₂²	1	16	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×C₄⋊Q₈	—	2	4	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
35	C₃₁×#125	—	2	4	no	—
36	C₃₁×#125	C₂×C₄	1	8	yes	complemented
37	C₃₁×C₂²⋊Q₈	—	2	16	no	—
38	C₄×Q₈×C₃₁	—	2	16	no	—
39	C₃₁×#117	—	2	4	no	—
40	C₃₁×#117	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	D₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#97	—	2	4	no	—
55	C₄×C₈×C₃₁	—	2	4	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	8	no	—
60	C₂×Q₈×C₃₁	—	4	8	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	4	4	no	—
64	C₂²×C₄×C₃₁	—	4	8	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₁	#41	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₃₁	—	2	16	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
79	C₄⋊C₄×C₃₁	—	2	16	no	—
80	C₄⋊C₄×C₃₁	—	2	8	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
85	C₂²⋊C₄×C₃₁	—	2	8	no	—
86	C₄²×C₃₁	—	2	8	no	—
87	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
89	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
90	C₄²×C₃₁	C₂×D₄	1	2	yes	—
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	#21323	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	2	4	no	—
94	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂³×C₃₁	#124	1	2	yes	—
96	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	4	16	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	4	8	no	—
103	C₂×C₄×C₃₁	—	4	8	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	4	8	no	—
106	C₂×C₄×C₃₁	—	8	8	no	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
111	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
112	C₂×C₄×C₃₁	#124	1	2	yes	—
113	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
118	C₈×C₃₁	—	2	8	no	—
119	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
120	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
121	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
122	#1993	C₂×C₃₁	1	4	yes	complemented
123	#1991	C₂×C₃₁	1	4	yes	complemented
124	#1736	C₂×C₃₁	1	4	yes	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	4	no	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	#391	1	4	yes	—
129	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
131	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₃₁	#575	1	4	yes	—
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	—	2	4	no	—
135	C₄×C₃₁	—	4	8	no	—
136	C₄×C₃₁	—	8	4	no	—
137	C₄×C₃₁	#391	1	4	yes	—
138	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
139	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
140	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
141	#533	—	2	4	no	—
142	#533	C₄×C₃₁	1	8	yes	complemented
143	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
144	C₂×#125	C₂²×C₃₁	1	2	yes	—
145	C₂×C₂²⋊Q₈	—	2	2	no	—
146	C₂×C₄×Q₈	—	2	2	no	—
147	C₂×#117	C₂²×C₃₁	1	2	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
149	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	C₄×M₁₆	—	2	4	no	—
151	C₂×#97	C₂²×C₃₁	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
153	#382	—	2	2	no	—
154	#382	C₂²×C₃₁	1	2	yes	—
155	#382	C₂²×C₃₁	1	2	yes	—
156	C₂×#95	—	2	2	no	—
157	#340	C₂²×C₃₁	1	16	yes	—
158	C₂×C₃₁	—	2	4	no	—
159	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
160	C₂×C₃₁	#2896	1	2	yes	—
161	C₂×C₃₁	#3140	1	2	yes	—
162	C₂×C₃₁	#3166	1	2	yes	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	D₄×C₃₁	1	2	yes	—
166	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
170	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
171	C₄⋊Q₈	—	2	4	no	—
172	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
173	#125	—	2	4	no	—
174	#125	C₂×C₄×C₃₁	1	8	yes	complemented
175	C₂²⋊Q₈	—	2	16	no	—
176	C₄×Q₈	—	2	16	no	—
177	#117	—	2	4	no	—
178	#117	C₂×C₄×C₃₁	1	8	yes	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
188	C₂×C₄²	D₄×C₃₁	1	2	yes	—
189	C₂×C₄²	D₄×C₃₁	1	4	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	#97	—	2	4	no	—
193	C₄×C₈	—	2	4	no	—
194	#95	—	2	4	no	—
195	#95	—	2	4	no	—
196	C₃₁	#21323	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	—	4	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
212	C₂²×C₄	#41×C₃₁	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
224	C₂²⋊C₄	—	2	8	no	—
225	C₄²	—	2	8	no	—
226	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
227	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
228	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
229	C₄²	C₂×D₄×C₃₁	1	2	yes	—
230	C₄²	C₂×D₄×C₃₁	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
233	C₂³	C₃₁×#124	1	2	yes	—
234	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
235	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
236	Q₈	—	4	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
248	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
249	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
250	C₂×C₄	C₃₁×#124	1	2	yes	—
251	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
256	C₈	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	C₃₁×#391	1	4	yes	—
261	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
263	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
264	C₂²	C₃₁×#575	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	8	4	no	—
269	C₄	C₃₁×#391	1	4	yes	—
270	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
271	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
274	C₂	C₃₁×#2896	1	2	yes	—
275	C₂	C₃₁×#3140	1	2	yes	—
276	C₂	C₃₁×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1993	C₂	semidirect
5. C₃₁×#1991	C₂	semidirect
9. C₃₁×#533	C₄	semidirect
31. C₂×M₁₆×C₃₁	Q₈	semidirect
34. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
36. C₃₁×#125	C₂×C₄	semidirect
92. #21323	C₃₁	direct (N × H)
122. #1993	C₂×C₃₁	semidirect
123. #1991	C₂×C₃₁	semidirect
142. #533	C₄×C₃₁	semidirect
169. C₂×M₁₆	Q₈×C₃₁	semidirect
172. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
174. #125	C₂×C₄×C₃₁	semidirect
196. C₃₁	#21323	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×#533	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#382	2
C₃₁×#1993	2
C₃₁×#1991	2
C₃₁×#1736	2
#21323	31

Subgroups — C31×#21323 #21478082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21323 #21478082