Subgroups — C₃₁×#20668 #21477427

Order 7936.

1322 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2879	C₂	1	2	yes	maximal
4	C₃₁×#1811	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1810	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1738	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#131	—	2	8	no	—
9	C₂×C₃₁×#131	C₄	1	8	yes	—
10	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	—	2	8	no	—
16	C₂×C₃₁×#108	C₂²	1	4	yes	—
17	C₂×C₃₁×#108	C₂²	1	4	yes	—
18	C₃₁×#408	—	2	8	no	—
19	C₃₁×#407	C₂²	1	8	yes	—
20	C₃₁×#341	C₂²	1	8	yes	—
21	C₂×#41×C₃₁	—	2	4	no	—
22	C₂×#41×C₃₁	—	2	4	no	—
23	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×#41×C₃₁	D₄	1	4	yes	—
25	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
26	C₂²×D₄×C₃₁	Q₈	1	2	yes	—
27	C₂²×D₄×C₃₁	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₃₁×#131	—	2	32	no	—
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	—	2	8	no	—
34	C₂×M₁₆×C₃₁	—	2	8	no	—
35	C₂×M₁₆×C₃₁	—	2	8	no	—
36	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
38	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
39	C₂²×C₈×C₃₁	—	2	8	no	—
40	C₂²×C₈×C₃₁	—	2	4	no	—
41	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₃₁	D₄	1	8	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#108	—	2	8	no	—
48	C₃₁×#108	—	2	8	no	—
49	C₃₁×C₂²⋊C₈	—	2	16	no	—
50	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	#41×C₃₁	—	2	8	no	—
53	#41×C₃₁	—	2	8	no	—
54	#41×C₃₁	—	2	8	no	—
55	#41×C₃₁	—	2	8	no	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
58	C₂×D₄×C₃₁	—	2	8	no	—
59	C₂×D₄×C₃₁	—	2	4	no	—
60	C₂×D₄×C₃₁	C₄⋊C₄	1	8	yes	—
61	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	#41	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	8	no	—
74	M₁₆×C₃₁	—	2	16	no	—
75	M₁₆×C₃₁	—	2	8	no	—
76	M₁₆×C₃₁	—	2	8	no	—
77	M₁₆×C₃₁	—	4	8	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	16	no	—
80	C₂×C₈×C₃₁	—	2	8	no	—
81	C₂×C₈×C₃₁	—	4	8	no	—
82	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	—
83	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
88	#20668	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	2	4	no	—
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	—	4	4	no	—
92	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
95	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
96	Q₈×C₃₁	—	2	8	no	—
97	D₄×C₃₁	—	2	16	no	—
98	D₄×C₃₁	—	2	8	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	8	no	—
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₃₁	#117	1	4	yes	—
107	C₂×C₄×C₃₁	#121	1	4	yes	—
108	C₂×C₄×C₃₁	#121	1	4	yes	—
109	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
114	C₈×C₃₁	—	2	4	no	—
115	C₈×C₃₁	—	4	8	no	—
116	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
117	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
119	C₂²×#108	C₂×C₃₁	1	2	yes	complemented
120	#2879	C₂×C₃₁	1	2	yes	—
121	#1811	C₂×C₃₁	1	4	yes	complemented
122	#1810	C₂×C₃₁	1	4	yes	complemented
123	#1738	C₂×C₃₁	1	2	yes	complemented
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	8	no	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	—	4	8	no	—
130	C₂²×C₃₁	#380	1	4	yes	—
131	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	#380	1	4	yes	—
135	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
137	C₄×C₃₁	C₂×#121	1	2	yes	—
138	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×#131	—	2	8	no	—
140	C₂×#131	C₄×C₃₁	1	8	yes	—
141	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
142	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
143	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
145	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#108	—	2	8	no	—
147	C₂×#108	C₂²×C₃₁	1	4	yes	—
148	C₂×#108	C₂²×C₃₁	1	4	yes	—
149	#408	—	2	8	no	—
150	#407	C₂²×C₃₁	1	8	yes	—
151	#341	C₂²×C₃₁	1	8	yes	—
152	C₂×C₃₁	—	2	2	no	—
153	C₂×C₃₁	—	2	4	no	—
154	C₂×C₃₁	—	4	4	no	—
155	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
156	C₂×C₃₁	#2916	1	2	yes	—
157	C₂×#41	—	2	4	no	—
158	C₂×#41	—	2	4	no	—
159	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
160	C₂×#41	D₄×C₃₁	1	4	yes	—
161	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
162	C₂²×D₄	Q₈×C₃₁	1	2	yes	—
163	C₂²×D₄	Q₈×C₃₁	1	1	yes	characteristic
164	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
165	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	#131	—	2	32	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	—	2	8	no	—
172	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
173	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
174	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
178	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
179	C₂²×C₈	D₄×C₃₁	1	8	yes	—
180	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
181	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
183	#108	—	2	8	no	—
184	#108	—	2	8	no	—
185	C₂²⋊C₈	—	2	16	no	—
186	C₃₁	#20668	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
188	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	#41	—	2	8	no	—
192	#41	—	2	8	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	C₄⋊C₄×C₃₁	1	8	yes	—
198	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	#41×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
210	M₁₆	—	2	8	no	—
211	M₁₆	—	2	16	no	—
212	M₁₆	—	2	8	no	—
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	8	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	16	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	4	8	no	—
219	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	—
220	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
221	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
224	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
229	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
230	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
231	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
232	Q₈	—	2	8	no	—
233	D₄	—	2	16	no	—
234	D₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
242	C₂×C₄	C₃₁×#117	1	4	yes	—
243	C₂×C₄	C₃₁×#121	1	4	yes	—
244	C₂×C₄	C₃₁×#121	1	4	yes	—
245	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
248	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	4	8	no	—
252	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
253	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	C₃₁×#380	1	4	yes	—
261	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
263	C₄	—	2	4	no	—
264	C₄	C₃₁×#380	1	4	yes	—
265	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
266	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
267	C₄	C₂×C₃₁×#121	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	2	4	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
272	C₂	C₃₁×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#108	C₂	semidirect
4. C₃₁×#1811	C₂	semidirect
5. C₃₁×#1810	C₂	semidirect
6. C₃₁×#1738	C₂	semidirect
88. #20668	C₃₁	direct (N × H)
119. C₂²×#108	C₂×C₃₁	semidirect
121. #1811	C₂×C₃₁	semidirect
122. #1810	C₂×C₃₁	semidirect
123. #1738	C₂×C₃₁	semidirect
186. C₃₁	#20668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#131	2
C₂²×C₃₁×#108	2
C₃₁×#2879	2
C₃₁×#1811	2
C₃₁×#1810	2
C₃₁×#1738	2
#20668	31

Subgroups — C31×#20668 #21477427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20668 #21477427