Subgroups — C₃₁×#3215 #16843148

Order 3968.

400 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#497	C₂	1	2	yes	maximal, complemented
4	C₃₁×#489	C₂	1	2	yes	maximal, complemented
5	C₃₁×#464	C₂	1	4	yes	maximal, complemented
6	C₃₁×#451	C₂	1	2	yes	maximal, complemented
7	C₃₁×#450	C₂	1	2	yes	maximal
8	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
12	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	—
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#125	—	2	1	no	—
16	C₃₁×#124	—	2	1	no	—
17	C₃₁×#124	—	2	1	no	—
18	C₃₁×#124	C₂²	1	2	yes	—
19	C₃₁×#124	C₂²	1	2	yes	—
20	C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₃₁	—	2	1	no	—
23	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#117	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
26	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
27	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₃₁×#103	C₂²	1	4	yes	complemented
30	C₃₁×#102	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×#97	C₂²	1	2	yes	complemented
33	C₂⁴×C₃₁	—	2	1	no	—
34	C₂×Q₈×C₃₁	—	2	1	no	—
35	C₂×Q₈×C₃₁	—	2	1	no	—
36	C₂×Q₈×C₃₁	—	4	2	no	—
37	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
38	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
39	C₂×D₄×C₃₁	—	2	1	no	—
40	C₂×D₄×C₃₁	—	4	2	no	—
41	C₂×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	Q₁₆×C₃₁	—	4	4	no	—
47	SD₁₆×C₃₁	—	4	4	no	—
48	C₂×C₈×C₃₁	C₂³	1	4	yes	—
49	C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	4	2	no	—
67	C₂³×C₃₁	—	4	2	no	—
68	C₂³×C₃₁	—	4	2	no	—
69	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₃₁	—	2	4	no	—
71	Q₈×C₃₁	—	2	4	no	—
72	Q₈×C₃₁	—	4	2	no	—
73	Q₈×C₃₁	—	4	2	no	—
74	D₄×C₃₁	—	2	4	no	—
75	D₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	4	1	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₈×C₃₁	—	2	4	no	—
90	#3215	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₃₁	—	2	1	no	—
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	—	4	2	no	—
94	C₂²×C₃₁	—	4	2	no	—
95	C₂²×C₃₁	—	4	2	no	—
96	C₂²×C₃₁	—	8	4	no	—
97	C₂²×C₃₁	#137	1	2	yes	—
98	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₃₁	—	2	1	no	—
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	4	2	no	—
105	C₄×C₃₁	—	4	2	no	—
106	C₄×C₃₁	—	4	2	no	—
107	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
108	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂×#124	C₂×C₃₁	1	1	yes	characteristic
110	#497	C₂×C₃₁	1	2	yes	complemented
111	#489	C₂×C₃₁	1	2	yes	complemented
112	#464	C₂×C₃₁	1	4	yes	complemented
113	#451	C₂×C₃₁	1	2	yes	complemented
114	#450	C₂×C₃₁	1	2	yes	—
115	#432	C₂×C₃₁	1	1	yes	characteristic, complemented
116	C₂×C₃₁	—	2	2	no	—
117	C₂×C₃₁	—	8	2	no	—
118	C₂×C₃₁	#535	1	1	yes	characteristic
119	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
120	C₂×C₃₁	#575	1	1	yes	characteristic
121	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
124	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	—
125	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
126	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
127	#125	—	2	1	no	—
128	#124	—	2	1	no	—
129	#124	—	2	1	no	—
130	#124	C₂²×C₃₁	1	2	yes	—
131	#124	C₂²×C₃₁	1	2	yes	—
132	C₂²⋊Q₈	—	2	1	no	—
133	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
136	#117	—	2	1	no	—
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₂²⋊C₄	—	2	1	no	—
139	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
140	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
141	#103	C₂²×C₃₁	1	4	yes	complemented
142	#102	C₂²×C₃₁	1	4	yes	—
143	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
144	#97	C₂²×C₃₁	1	2	yes	complemented
145	C₃₁	#3215	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
151	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	C₂³×C₃₁	1	2	yes	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	4	no	—
161	C₂×C₈	C₂³×C₃₁	1	4	yes	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	D₄×C₃₁	1	1	yes	characteristic
175	C₄²	D₄×C₃₁	1	1	yes	characteristic
176	C₄²	C₂³×C₃₁	1	1	yes	characteristic
177	C₄²	C₂³×C₃₁	1	1	yes	characteristic
178	C₂³	—	2	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
183	Q₈	—	2	4	no	—
184	Q₈	—	2	4	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	D₄	—	2	4	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
202	C₈	—	2	4	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	8	4	no	—
209	C₂²	C₃₁×#137	1	2	yes	—
210	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
220	C₂	—	2	2	no	—
221	C₂	—	8	2	no	—
222	C₂	C₃₁×#535	1	1	yes	characteristic
223	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
224	C₂	C₃₁×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#533	C₂	semidirect
3. C₃₁×#497	C₂	semidirect
4. C₃₁×#489	C₂	semidirect
5. C₃₁×#464	C₂	semidirect
6. C₃₁×#451	C₂	semidirect
8. C₃₁×#432	C₂	semidirect
11. C₂×Q₁₆×C₃₁	C₂²	semidirect
13. C₃₁×C₄⋊Q₈	C₂²	semidirect
14. C₃₁×C₄⋊Q₈	C₂²	semidirect
23. C₄×Q₈×C₃₁	C₂²	semidirect
28. C₃₁×C₄⋊C₈	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
32. C₃₁×#97	C₂²	semidirect
90. #3215	C₃₁	direct (N × H)
108. #533	C₂×C₃₁	semidirect
110. #497	C₂×C₃₁	semidirect
111. #489	C₂×C₃₁	semidirect
112. #464	C₂×C₃₁	semidirect
113. #451	C₂×C₃₁	semidirect
115. #432	C₂×C₃₁	semidirect
123. C₂×Q₁₆	C₂²×C₃₁	semidirect
125. C₄⋊Q₈	C₂²×C₃₁	semidirect
126. C₄⋊Q₈	C₂²×C₃₁	semidirect
135. C₄×Q₈	C₂²×C₃₁	semidirect
140. C₄⋊C₈	C₂²×C₃₁	semidirect
141. #103	C₂²×C₃₁	semidirect
144. #97	C₂²×C₃₁	semidirect
145. C₃₁	#3215	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×#533	2
C₂×C₃₁×#124	2
C₃₁×#497	2
C₃₁×#489	2
C₃₁×#464	2
C₃₁×#451	2
C₃₁×#450	2
C₃₁×#432	2
#3215	31

Subgroups — C31×#3215 #16843148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3215 #16843148