Subgroups — C₃₁×#12930 #21469689

Order 7936.

414 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#365	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
11	C₃₁×#427	—	2	4	no	—
12	C₂×C₃₁×#107	—	2	1	no	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₃₁×#407	—	2	1	no	—
15	C₃₁×#365	—	2	4	no	—
16	C₃₁×C₂²⋊C₁₆	C₄	1	4	yes	complemented
17	C₃₁×#341	—	2	1	no	—
18	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	1	no	—
20	C₂×M₁₆×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×#117	—	2	4	no	—
28	C₃₁×#117	—	4	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
31	C₂×C₄²×C₃₁	—	4	1	no	—
32	C₃₁×M₃₂	—	2	2	no	—
33	C₃₁×M₃₂	—	2	2	no	—
34	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
35	C₃₁×#108	—	2	2	no	—
36	C₃₁×#108	—	2	2	no	—
37	C₃₁×#107	—	2	4	no	—
38	C₃₁×#106	—	2	2	no	—
39	C₃₁×#106	D₄	1	4	yes	—
40	C₃₁×C₂²⋊C₈	—	4	2	no	—
41	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	—	2	1	no	—
44	C₂²×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₄×C₃₁	—	2	1	no	—
46	C₂²×C₄×C₃₁	—	4	2	no	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₃₁	—	4	1	no	—
52	M₁₆×C₃₁	—	4	1	no	—
53	M₁₆×C₃₁	—	4	1	no	—
54	M₁₆×C₃₁	—	4	1	no	—
55	C₂×C₈×C₃₁	—	2	1	no	—
56	C₂×C₈×C₃₁	—	2	1	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	4	1	no	—
60	C₂×C₈×C₃₁	—	4	1	no	—
61	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₃₁	#41	1	2	yes	—
66	C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	—	4	4	no	—
70	C₁₆×C₃₁	—	2	4	no	—
71	#12930	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	—	2	1	no	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	4	2	no	—
75	C₂³×C₃₁	#106	1	2	yes	—
76	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	8	4	no	—
85	C₂×C₄×C₃₁	#106	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
92	C₈×C₃₁	—	2	2	no	—
93	C₈×C₃₁	—	8	1	no	—
94	C₈×C₃₁	—	8	1	no	—
95	C₈×C₃₁	C₄×D₄	1	2	yes	—
96	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
97	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#365	C₂×C₃₁	1	2	yes	complemented
99	#2112	C₂×C₃₁	1	1	yes	characteristic
100	#1810	C₂×C₃₁	1	1	yes	characteristic
101	#1383	C₂×C₃₁	1	2	yes	complemented
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	4	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
108	C₂²×C₃₁	#450	1	1	yes	characteristic
109	C₂²×C₃₁	#483	1	1	yes	characteristic
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	8	4	no	—
112	C₄×C₃₁	#385	1	1	yes	characteristic
113	C₄×C₃₁	#428	1	1	yes	characteristic
114	C₄×C₃₁	#448	1	1	yes	characteristic
115	C₄×C₃₁	#481	1	1	yes	characteristic
116	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
117	C₂×#117	—	2	1	no	—
118	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
119	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
120	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
121	#427	—	2	4	no	—
122	C₂×#107	—	2	1	no	—
123	C₂×#106	C₂²×C₃₁	1	2	yes	—
124	#407	—	2	1	no	—
125	#365	—	2	4	no	—
126	C₂²⋊C₁₆	C₄×C₃₁	1	4	yes	complemented
127	#341	—	2	1	no	—
128	C₂×C₃₁	—	2	1	no	—
129	C₂×C₃₁	—	2	1	no	—
130	C₂×C₃₁	—	4	2	no	—
131	C₂×C₃₁	#1853	1	1	yes	characteristic
132	C₂×C₃₁	#2155	1	2	yes	—
133	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
139	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
141	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
142	#117	—	2	4	no	—
143	#117	—	4	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₂²⋊C₄	—	4	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
150	#108	—	2	2	no	—
151	#108	—	2	2	no	—
152	#107	—	2	4	no	—
153	#106	—	2	2	no	—
154	#106	D₄×C₃₁	1	4	yes	—
155	C₂²⋊C₈	—	4	2	no	—
156	C₃₁	#12930	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
164	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	M₁₆	—	4	1	no	—
168	M₁₆	—	4	1	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	4	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
180	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
181	C₂×C₈	#41×C₃₁	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	4	no	—
185	C₄²	—	4	4	no	—
186	C₁₆	—	2	4	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₃₁×#106	1	2	yes	—
191	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₃₁×#106	1	2	yes	—
201	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
203	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
204	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	8	1	no	—
209	C₈	—	8	1	no	—
210	C₈	C₄×D₄×C₃₁	1	2	yes	—
211	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
218	C₂²	C₃₁×#450	1	1	yes	characteristic
219	C₂²	C₃₁×#483	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	8	4	no	—
222	C₄	C₃₁×#385	1	1	yes	characteristic
223	C₄	C₃₁×#428	1	1	yes	characteristic
224	C₄	C₃₁×#448	1	1	yes	characteristic
225	C₄	C₃₁×#481	1	1	yes	characteristic
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	4	2	no	—
229	C₂	C₃₁×#1853	1	1	yes	characteristic
230	C₂	C₃₁×#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#365	C₂	semidirect
5. C₃₁×#1383	C₂	semidirect
16. C₃₁×C₂²⋊C₁₆	C₄	semidirect
71. #12930	C₃₁	direct (N × H)
98. C₂×#365	C₂×C₃₁	semidirect
101. #1383	C₂×C₃₁	semidirect
126. C₂²⋊C₁₆	C₄×C₃₁	semidirect
156. C₃₁	#12930	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×#427	2
C₂×C₃₁×#365	2
C₃₁×#2112	2
C₃₁×#1810	2
C₃₁×#1383	2
#12930	31

Subgroups — C31×#12930 #21469689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12930 #21469689