Subgroups — C₃₁×#3356 #16843289

Order 3968.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#554	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#485	C₂	1	2	yes	maximal, complemented
5	C₃₁×#483	C₂	1	2	yes	maximal, complemented
6	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#470	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#436	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#428	C₂	1	2	yes	maximal, complemented
12	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#126	—	2	2	no	—
17	C₃₁×#125	—	2	1	no	—
18	C₃₁×#124	—	2	1	no	—
19	C₃₁×C₂²⋊Q₈	—	2	2	no	—
20	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₃₁	—	2	1	no	—
23	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₃₁	—	2	1	no	—
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×#106	C₂²	1	2	yes	—
32	C₃₁×#106	C₂²	1	2	yes	—
33	C₃₁×#106	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	1	yes	characteristic
36	C₃₁×#103	C₂²	1	1	yes	characteristic
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#103	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	1	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
48	Q₁₆×C₃₁	—	2	2	no	—
49	M₁₆×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	—	2	1	no	—
72	C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₃₁	C₂×D₄	1	2	yes	—
74	Q₈×C₃₁	—	2	1	no	—
75	Q₈×C₃₁	—	2	1	no	—
76	Q₈×C₃₁	—	2	1	no	—
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	Q₈×C₃₁	—	8	1	no	—
80	D₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	8	2	no	—
90	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	#41	1	2	yes	—
97	#3356	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₃₁	—	2	2	no	—
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	—	2	2	no	—
101	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	4	1	no	—
105	C₄×C₃₁	—	4	1	no	—
106	C₄×C₃₁	—	4	2	no	—
107	C₄×C₃₁	—	4	1	no	—
108	C₄×C₃₁	—	4	1	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	#137	1	2	yes	—
111	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
112	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
113	#556	C₂×C₃₁	1	1	yes	characteristic
114	#554	C₂×C₃₁	1	1	yes	characteristic
115	#498	C₂×C₃₁	1	1	yes	characteristic, complemented
116	#485	C₂×C₃₁	1	2	yes	complemented
117	#483	C₂×C₃₁	1	2	yes	complemented
118	#477	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#475	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#470	C₂×C₃₁	1	2	yes	complemented
121	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#436	C₂×C₃₁	1	1	yes	characteristic
123	#428	C₂×C₃₁	1	2	yes	complemented
124	C₂×C₃₁	—	4	2	no	—
125	C₂×C₃₁	#547	1	1	yes	characteristic
126	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
127	C₂×C₃₁	#577	1	1	yes	characteristic
128	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
129	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
130	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
132	#126	—	2	2	no	—
133	#125	—	2	1	no	—
134	#124	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
137	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
143	#117	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
145	#107	C₂²×C₃₁	1	1	yes	characteristic
146	#107	C₂²×C₃₁	1	2	yes	—
147	#106	C₂²×C₃₁	1	2	yes	—
148	#106	C₂²×C₃₁	1	2	yes	—
149	#106	C₂²×C₃₁	1	2	yes	—
150	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#103	C₂²×C₃₁	1	1	yes	characteristic
152	#103	C₂²×C₃₁	1	1	yes	characteristic
153	#103	C₂²×C₃₁	1	2	yes	—
154	#103	C₂²×C₃₁	1	2	yes	—
155	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
156	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₃₁	#3356	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
162	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
165	Q₁₆	—	2	2	no	—
166	M₁₆	—	2	4	no	—
167	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₃₁	1	2	yes	—
169	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
179	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
180	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
181	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₂³	C₂×D₄×C₃₁	1	2	yes	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₈	#41×C₃₁	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	C₃₁×#137	1	2	yes	—
227	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
228	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	C₃₁×#547	1	1	yes	characteristic
231	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
232	C₂	C₃₁×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#498	C₂	semidirect
4. C₃₁×#485	C₂	semidirect
5. C₃₁×#483	C₂	semidirect
6. C₃₁×#477	C₂	semidirect
7. C₃₁×#475	C₂	semidirect
8. C₃₁×#470	C₂	semidirect
9. C₄×Q₁₆×C₃₁	C₂	semidirect
11. C₃₁×#428	C₂	semidirect
97. #3356	C₃₁	direct (N × H)
115. #498	C₂×C₃₁	semidirect
116. #485	C₂×C₃₁	semidirect
117. #483	C₂×C₃₁	semidirect
118. #477	C₂×C₃₁	semidirect
119. #475	C₂×C₃₁	semidirect
120. #470	C₂×C₃₁	semidirect
121. C₄×Q₁₆	C₂×C₃₁	semidirect
123. #428	C₂×C₃₁	semidirect
157. C₃₁	#3356	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×#556	2
C₃₁×#554	2
C₃₁×#498	2
C₃₁×#485	2
C₃₁×#483	2
C₃₁×#477	2
C₃₁×#475	2
C₃₁×#470	2
C₄×Q₁₆×C₃₁	2
C₃₁×#436	2
C₃₁×#428	2
#3356	31

Subgroups — C31×#3356 #16843289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3356 #16843289