Subgroups — C₃₁×#3101 #16843034

Order 3968.

448 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×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#483	C₂	1	2	yes	maximal, complemented
4	C₃₁×#476	C₂	1	2	yes	maximal, complemented
5	C₃₁×#461	C₂	1	2	yes	maximal, complemented
6	C₃₁×#450	C₂	1	2	yes	maximal
7	C₄×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
8	C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	—	2	1	no	—
11	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	—
12	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#125	—	2	1	no	—
15	C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₄×Q₈×C₃₁	—	2	1	no	—
18	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
19	C₄×D₄×C₃₁	—	2	1	no	—
20	C₄×D₄×C₃₁	—	2	1	no	—
21	C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#117	—	2	1	no	—
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
26	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#106	C₂²	1	4	yes	complemented
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	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
33	C₂⁴×C₃₁	—	2	1	no	—
34	C₂×Q₈×C₃₁	—	2	1	no	—
35	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
36	C₂×D₄×C₃₁	—	2	4	no	—
37	C₂×D₄×C₃₁	—	2	1	no	—
38	C₂×D₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	—	2	1	no	—
40	C₂²×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	4	4	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	SD₁₆×C₃₁	—	2	8	no	—
47	C₂×C₈×C₃₁	C₂³	1	4	yes	—
48	C₄⋊C₄×C₃₁	—	2	1	no	—
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	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₄²×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	D₄	1	1	yes	characteristic
61	C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₃₁	—	2	4	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	4	2	no	—
67	C₂³×C₃₁	—	4	2	no	—
68	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₃₁	—	2	4	no	—
70	Q₈×C₃₁	—	4	2	no	—
71	D₄×C₃₁	—	2	4	no	—
72	D₄×C₃₁	—	2	2	no	—
73	D₄×C₃₁	#41	1	4	yes	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	4	1	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	SD₁₆	1	4	yes	complemented
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	#3101	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	—	2	4	no	—
94	C₂²×C₃₁	—	4	8	no	—
95	C₂²×C₃₁	—	4	2	no	—
96	C₂²×C₃₁	—	4	2	no	—
97	C₂²×C₃₁	C₂×SD₁₆	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	—	4	2	no	—
103	C₄×C₃₁	—	4	2	no	—
104	C₄×C₃₁	—	4	2	no	—
105	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
106	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
107	C₄×C₃₁	C₂×#41	1	2	yes	—
108	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
110	#483	C₂×C₃₁	1	2	yes	complemented
111	#476	C₂×C₃₁	1	2	yes	complemented
112	#461	C₂×C₃₁	1	2	yes	complemented
113	#450	C₂×C₃₁	1	2	yes	—
114	C₄×SD₁₆	C₂×C₃₁	1	4	yes	complemented
115	#431	C₂×C₃₁	1	1	yes	characteristic, complemented
116	C₂×C₃₁	—	2	2	no	—
117	C₂×C₃₁	—	4	4	no	—
118	C₂×C₃₁	#525	1	1	yes	characteristic
119	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
120	C₂×C₃₁	#575	1	1	yes	characteristic
121	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
122	C₂³×C₄	—	2	1	no	—
123	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	—
124	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
125	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
126	#125	—	2	1	no	—
127	C₂²⋊Q₈	—	2	1	no	—
128	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
131	C₄×D₄	—	2	1	no	—
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	C₂²×C₃₁	1	2	yes	—
134	C₄×D₄	C₂²×C₃₁	1	2	yes	—
135	#117	—	2	1	no	—
136	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
139	#106	C₂²×C₃₁	1	4	yes	complemented
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	C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
145	C₃₁	#3101	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	C₂³×C₃₁	1	2	yes	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	4	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	SD₁₆	—	2	8	no	—
160	C₂×C₈	C₂³×C₃₁	1	4	yes	—
161	C₄⋊C₄	—	2	1	no	—
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	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	D₄×C₃₁	1	1	yes	characteristic
174	C₄²	D₄×C₃₁	1	1	yes	characteristic
175	C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	C₄²	C₂³×C₃₁	1	1	yes	characteristic
177	C₂³	—	2	4	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
182	Q₈	—	2	4	no	—
183	Q₈	—	4	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	D₄	#41×C₃₁	1	4	yes	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	SD₁₆×C₃₁	1	4	yes	complemented
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	—	2	4	no	—
206	C₂²	—	4	8	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂×SD₁₆×C₃₁	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	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
218	C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
219	C₄	C₂×#41×C₃₁	1	2	yes	—
220	C₂	—	2	2	no	—
221	C₂	—	4	4	no	—
222	C₂	C₃₁×#525	1	1	yes	characteristic
223	C₂	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
224	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#483	C₂	semidirect
4. C₃₁×#476	C₂	semidirect
5. C₃₁×#461	C₂	semidirect
7. C₄×SD₁₆×C₃₁	C₂	semidirect
8. C₃₁×#431	C₂	semidirect
12. C₃₁×C₄⋊Q₈	C₂²	semidirect
13. C₃₁×C₄⋊Q₈	C₂²	semidirect
18. C₄×Q₈×C₃₁	C₂²	semidirect
27. C₃₁×#106	C₂²	semidirect
28. C₃₁×C₄⋊C₈	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
32. C₄×C₈×C₃₁	C₂²	semidirect
82. C₂×C₄×C₃₁	SD₁₆	semidirect
90. #3101	C₃₁	direct (N × H)
108. #533	C₂×C₃₁	semidirect
110. #483	C₂×C₃₁	semidirect
111. #476	C₂×C₃₁	semidirect
112. #461	C₂×C₃₁	semidirect
114. C₄×SD₁₆	C₂×C₃₁	semidirect
115. #431	C₂×C₃₁	semidirect
124. C₄⋊Q₈	C₂²×C₃₁	semidirect
125. C₄⋊Q₈	C₂²×C₃₁	semidirect
130. C₄×Q₈	C₂²×C₃₁	semidirect
139. #106	C₂²×C₃₁	semidirect
140. C₄⋊C₈	C₂²×C₃₁	semidirect
141. #103	C₂²×C₃₁	semidirect
144. C₄×C₈	C₂²×C₃₁	semidirect
145. C₃₁	#3101	direct (N × H)
195. C₂×C₄	SD₁₆×C₃₁	semidirect

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×C₄×D₄×C₃₁	2
C₃₁×#483	2
C₃₁×#476	2
C₃₁×#461	2
C₃₁×#450	2
C₄×SD₁₆×C₃₁	2
C₃₁×#431	2
#3101	31

Subgroups — C31×#3101 #16843034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3101 #16843034