Subgroups — C₃₁×#2951 #16842884

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#520	C₂	1	2	yes	maximal, complemented
3	C₃₁×#440	C₂	1	4	yes	maximal, complemented
4	C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#417	C₂	1	2	yes	maximal, complemented
8	C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#135	C₄	1	8	yes	complemented
11	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
17	C₄×D₄×C₃₁	—	2	4	no	—
18	C₄×D₄×C₃₁	C₂²	1	4	yes	—
19	C₃₁×#117	—	2	2	no	—
20	C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#107	C₂²	1	2	yes	—
24	C₃₁×#107	C₂²	1	2	yes	—
25	C₃₁×#103	C₂²	1	4	yes	—
26	C₃₁×#102	C₂²	1	4	yes	—
27	C₃₁×#97	C₂²	1	2	yes	—
28	C₄×C₈×C₃₁	C₂²	1	2	yes	—
29	#41×C₃₁	—	2	4	no	—
30	#41×C₃₁	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
32	C₂×D₄×C₃₁	—	2	2	no	—
33	C₂×D₄×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₃₁	—	2	2	no	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	—	2	4	no	—
37	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
40	D₈×C₃₁	C₂×C₄	1	4	yes	complemented
41	M₁₆×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₄⋊C₄×C₃₁	—	2	4	no	—
48	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
51	C₂²⋊C₄×C₃₁	—	2	4	no	—
52	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
53	C₄²×C₃₁	—	2	4	no	—
54	C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₃₁	—	2	2	no	—
58	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
59	D₄×C₃₁	—	2	8	no	—
60	D₄×C₃₁	—	2	2	no	—
61	D₄×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	8	no	—
64	C₂×C₄×C₃₁	—	2	4	no	—
65	C₂×C₄×C₃₁	—	8	4	no	—
66	C₂×C₄×C₃₁	—	8	2	no	—
67	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#41	1	2	yes	—
72	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₈×C₃₁	—	2	2	no	—
74	C₈×C₃₁	C₂²×C₄	1	2	yes	—
75	C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	#2951	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₃₁	—	2	1	no	—
78	C₂²×C₃₁	—	2	1	no	—
79	C₂²×C₃₁	—	2	4	no	—
80	C₂²×C₃₁	—	4	2	no	—
81	C₂²×C₃₁	—	4	2	no	—
82	C₂²×C₃₁	C₄×D₄	1	2	yes	—
83	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₃₁	—	2	4	no	—
85	C₄×C₃₁	—	2	2	no	—
86	C₄×C₃₁	—	2	2	no	—
87	C₄×C₃₁	—	4	4	no	—
88	C₄×C₃₁	C₄×D₄	1	2	yes	—
89	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
91	C₂×#135	C₂×C₃₁	1	1	yes	characteristic
92	#520	C₂×C₃₁	1	2	yes	complemented
93	#440	C₂×C₃₁	1	4	yes	complemented
94	C₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
95	C₄×D₈	C₂×C₃₁	1	2	yes	complemented
96	C₂×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
97	#417	C₂×C₃₁	1	2	yes	complemented
98	#405	C₂×C₃₁	1	1	yes	characteristic, complemented
99	C₂×C₃₁	—	2	2	no	—
100	C₂×C₃₁	—	4	4	no	—
101	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂×C₃₁	#576	1	1	yes	characteristic
103	C₂×C₃₁	#578	1	1	yes	characteristic
104	C₂×#41	C₂²×C₃₁	1	2	yes	—
105	#135	C₄×C₃₁	1	8	yes	complemented
106	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
107	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
108	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
109	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
111	C₄×Q₈	C₂²×C₃₁	1	4	yes	—
112	C₄×D₄	—	2	4	no	—
113	C₄×D₄	C₂²×C₃₁	1	4	yes	—
114	#117	—	2	2	no	—
115	#117	C₂²×C₃₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
118	#107	C₂²×C₃₁	1	2	yes	—
119	#107	C₂²×C₃₁	1	2	yes	—
120	#103	C₂²×C₃₁	1	4	yes	—
121	#102	C₂²×C₃₁	1	4	yes	—
122	#97	C₂²×C₃₁	1	2	yes	—
123	C₄×C₈	C₂²×C₃₁	1	2	yes	—
124	C₃₁	#2951	1	1	yes	characteristic, direct product factor, Sylow
125	#41	—	2	4	no	—
126	#41	C₂×C₄×C₃₁	1	8	yes	complemented
127	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	C₂³×C₃₁	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
134	Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
135	SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
136	D₈	C₂×C₄×C₃₁	1	4	yes	complemented
137	M₁₆	—	2	2	no	—
138	C₂×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
139	C₂×C₈	C₂³×C₃₁	1	2	yes	—
140	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
145	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
146	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	C₂³×C₃₁	1	2	yes	—
151	C₂³	—	2	2	no	—
152	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
153	Q₈	—	2	2	no	—
154	Q₈	C₂²×C₄×C₃₁	1	4	yes	—
155	D₄	—	2	8	no	—
156	D₄	—	2	2	no	—
157	D₄	C₂²×C₄×C₃₁	1	4	yes	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
164	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
166	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄	#41×C₃₁	1	2	yes	—
168	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂²×C₄×C₃₁	1	2	yes	—
171	C₈	C₂²×C₄×C₃₁	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	C₄×D₄×C₃₁	1	2	yes	—
178	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	4	no	—
183	C₄	C₄×D₄×C₃₁	1	2	yes	—
184	C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
185	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	4	no	—
188	C₂	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
189	C₂	C₃₁×#576	1	1	yes	characteristic
190	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#520	C₂	semidirect
3. C₃₁×#440	C₂	semidirect
4. C₄×Q₁₆×C₃₁	C₂	semidirect
5. C₄×D₈×C₃₁	C₂	semidirect
6. C₂×C₃₁×#107	C₂	semidirect
7. C₃₁×#417	C₂	semidirect
8. C₃₁×#405	C₂	semidirect
10. C₃₁×#135	C₄	semidirect
30. #41×C₃₁	C₂×C₄	semidirect
38. Q₁₆×C₃₁	C₂×C₄	semidirect
39. SD₁₆×C₃₁	C₂×C₄	semidirect
40. D₈×C₃₁	C₂×C₄	semidirect
42. C₂×C₈×C₃₁	C₂×C₄	semidirect
76. #2951	C₃₁	direct (N × H)
92. #520	C₂×C₃₁	semidirect
93. #440	C₂×C₃₁	semidirect
94. C₄×Q₁₆	C₂×C₃₁	semidirect
95. C₄×D₈	C₂×C₃₁	semidirect
96. C₂×#107	C₂×C₃₁	semidirect
97. #417	C₂×C₃₁	semidirect
98. #405	C₂×C₃₁	semidirect
105. #135	C₄×C₃₁	semidirect
124. C₃₁	#2951	direct (N × H)
126. #41	C₂×C₄×C₃₁	semidirect
134. Q₁₆	C₂×C₄×C₃₁	semidirect
135. SD₁₆	C₂×C₄×C₃₁	semidirect
136. D₈	C₂×C₄×C₃₁	semidirect
138. C₂×C₈	C₂×C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#135	2
C₃₁×#520	2
C₃₁×#440	2
C₄×Q₁₆×C₃₁	2
C₄×D₈×C₃₁	2
C₂×C₃₁×#107	2
C₃₁×#417	2
C₃₁×#405	2
#2951	31

Subgroups — C31×#2951 #16842884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2951 #16842884