Subgroups — C₃₁×#21354 #21478113

Order 7936.

854 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1939	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	4	yes	maximal
6	C₃₁×#1768	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#431	—	2	4	no	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₄×M₁₆×C₃₁	—	2	4	no	—
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#383	C₄	1	2	yes	—
18	C₃₁×#382	C₄	1	2	yes	—
19	C₃₁×#381	C₄	1	2	yes	—
20	C₃₁×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#376	C₂²	1	2	yes	—
24	C₃₁×#336	C₂²	1	8	yes	—
25	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	2	8	no	—
30	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	4	yes	—
37	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊C₈	—	2	16	no	—
42	C₃₁×C₄⋊C₈	—	2	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	4	no	—
44	C₃₁×#97	—	2	4	no	—
45	C₄×C₈×C₃₁	—	2	4	no	—
46	C₄×C₈×C₃₁	—	2	4	no	—
47	C₃₁×#95	C₂×C₄	1	4	yes	—
48	C₃₁×#95	C₂×C₄	1	4	yes	—
49	C₃₁×#95	C₂×C₄	1	4	yes	—
50	C₃₁×#95	C₂×C₄	1	4	yes	—
51	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₃₁	—	2	16	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	8	no	—
71	C₂×C₈×C₃₁	#41	1	8	yes	—
72	C₄⋊C₄×C₃₁	—	2	8	no	—
73	C₂²⋊C₄×C₃₁	—	4	4	no	—
74	C₄²×C₃₁	—	2	8	no	—
75	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
76	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
77	C₄²×C₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
80	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
81	#21354	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	2	2	no	—
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	8	no	—
94	C₂×C₄×C₃₁	—	2	8	no	—
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	—	2	8	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₃₁	#131	1	4	yes	—
104	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
107	C₈×C₃₁	—	2	8	no	—
108	C₈×C₃₁	—	4	4	no	—
109	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
110	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
111	#2283	C₂×C₃₁	1	1	yes	characteristic
112	#1939	C₂×C₃₁	1	4	yes	complemented
113	#1851	C₂×C₃₁	1	4	yes	—
114	#1768	C₂×C₃₁	1	4	yes	complemented
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	4	4	no	—
119	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₃₁	C₄×#41	1	2	yes	—
121	C₂²×C₃₁	C₂×#131	1	2	yes	—
122	C₂²×C₃₁	#568	1	2	yes	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	4	8	no	—
128	C₄×C₃₁	#424	1	4	yes	—
129	C₄×C₃₁	#525	1	2	yes	—
130	C₄×C₃₁	#533	1	2	yes	—
131	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
132	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
133	#431	—	2	4	no	—
134	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
135	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
136	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
137	C₄×M₁₆	—	2	4	no	—
138	C₂×#97	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
141	#383	C₄×C₃₁	1	2	yes	—
142	#382	C₄×C₃₁	1	2	yes	—
143	#381	C₄×C₃₁	1	2	yes	—
144	#380	C₄×C₃₁	1	2	yes	—
145	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
147	#376	C₂²×C₃₁	1	2	yes	—
148	#336	C₂²×C₃₁	1	8	yes	—
149	C₂×C₃₁	—	4	2	no	—
150	C₂×C₃₁	#2296	1	1	yes	characteristic
151	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
152	C₂×C₃₁	#2905	1	1	yes	characteristic
153	C₂×C₃₁	#2969	1	4	yes	—
154	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	8	no	—
159	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
160	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
162	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₃₁	1	4	yes	—
166	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
167	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊C₈	—	2	16	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	#95	C₂×C₄×C₃₁	1	4	yes	—
177	#95	C₂×C₄×C₃₁	1	4	yes	—
178	#95	C₂×C₄×C₃₁	1	4	yes	—
179	#95	C₂×C₄×C₃₁	1	4	yes	—
180	C₃₁	#21354	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
186	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
196	M₁₆	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	#41×C₃₁	1	8	yes	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
206	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
207	C₄²	C₂×D₄×C₃₁	1	2	yes	—
208	C₄²	C₂×D₄×C₃₁	1	2	yes	—
209	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
210	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
215	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
232	C₂×C₄	C₃₁×#131	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
236	C₈	—	2	8	no	—
237	C₈	—	4	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂²	C₄×#41×C₃₁	1	2	yes	—
244	C₂²	C₂×C₃₁×#131	1	2	yes	—
245	C₂²	C₃₁×#568	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	C₃₁×#424	1	4	yes	—
252	C₄	C₃₁×#525	1	2	yes	—
253	C₄	C₃₁×#533	1	2	yes	—
254	C₂	—	4	2	no	—
255	C₂	C₃₁×#2296	1	1	yes	characteristic
256	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
257	C₂	C₃₁×#2905	1	1	yes	characteristic
258	C₂	C₃₁×#2969	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1939	C₂	semidirect
6. C₃₁×#1768	C₂	semidirect
81. #21354	C₃₁	direct (N × H)
112. #1939	C₂×C₃₁	semidirect
114. #1768	C₂×C₃₁	semidirect
180. C₃₁	#21354	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×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2283	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1768	2
#21354	31

Subgroups — C31×#21354 #21478113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21354 #21478113