Subgroups — C₃₁×#3123 #16843056

Order 3968.

394 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#518	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#485	C₂	1	2	yes	maximal, complemented
4	C₃₁×#478	C₂	1	2	yes	maximal, complemented
5	C₃₁×#461	C₂	1	2	yes	maximal, complemented
6	C₃₁×#450	C₂	1	2	yes	maximal
7	C₃₁×#440	C₂	1	2	yes	maximal, complemented
8	C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#126	—	2	2	no	—
15	C₃₁×#125	—	2	2	no	—
16	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
19	C₄×D₄×C₃₁	—	2	2	no	—
20	C₄×D₄×C₃₁	C₂²	1	2	yes	—
21	C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₃₁×#117	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
25	C₃₁×#107	C₂²	1	2	yes	complemented
26	C₃₁×#106	C₂²	1	2	yes	complemented
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₃₁×#103	C₂²	1	2	yes	complemented
29	C₃₁×#103	C₂²	1	2	yes	complemented
30	C₃₁×#102	C₂²	1	2	yes	—
31	C₃₁×#102	C₂²	1	2	yes	—
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
34	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₂⁴×C₃₁	—	2	1	no	—
36	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂×D₄×C₃₁	—	2	4	no	—
38	C₂×D₄×C₃₁	—	2	1	no	—
39	C₂×D₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	SD₁₆×C₃₁	—	2	4	no	—
44	SD₁₆×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂×C₈×C₃₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
52	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₃₁	—	2	1	no	—
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	D₄	1	2	yes	—
59	C₄²×C₃₁	—	2	1	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₃₁	—	2	4	no	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	—	8	2	no	—
66	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₃₁	—	2	2	no	—
68	Q₈×C₃₁	—	2	2	no	—
69	D₄×C₃₁	—	2	4	no	—
70	D₄×C₃₁	—	2	2	no	—
71	D₄×C₃₁	#41	1	4	yes	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	8	2	no	—
77	C₂×C₄×C₃₁	—	8	2	no	—
78	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₈×C₃₁	—	2	2	no	—
83	C₈×C₃₁	—	2	2	no	—
84	#3123	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	—	4	2	no	—
89	C₂²×C₃₁	—	4	2	no	—
90	C₂²×C₃₁	—	8	4	no	—
91	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	2	no	—
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	4	2	no	—
96	C₄×C₃₁	—	4	2	no	—
97	C₄×C₃₁	—	4	2	no	—
98	C₄×C₃₁	C₂×#41	1	2	yes	—
99	#541	C₂×C₃₁	1	1	yes	characteristic, complemented
100	#518	C₂×C₃₁	1	1	yes	characteristic
101	#485	C₂×C₃₁	1	2	yes	complemented
102	#478	C₂×C₃₁	1	2	yes	complemented
103	#461	C₂×C₃₁	1	2	yes	complemented
104	#450	C₂×C₃₁	1	2	yes	—
105	#440	C₂×C₃₁	1	2	yes	complemented
106	C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
107	#433	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×C₃₁	—	4	1	no	—
109	C₂×C₃₁	—	4	4	no	—
110	C₂×C₃₁	#525	1	1	yes	characteristic
111	C₂×C₃₁	#577	1	2	yes	—
112	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
113	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
114	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
115	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
116	#126	—	2	2	no	—
117	#125	—	2	2	no	—
118	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
119	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
120	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
121	C₄×D₄	—	2	2	no	—
122	C₄×D₄	C₂²×C₃₁	1	2	yes	—
123	C₄×D₄	C₂²×C₃₁	1	2	yes	—
124	#117	C₂²×C₃₁	1	1	yes	characteristic
125	#117	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	2	no	—
127	#107	C₂²×C₃₁	1	2	yes	complemented
128	#106	C₂²×C₃₁	1	2	yes	complemented
129	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
130	#103	C₂²×C₃₁	1	2	yes	complemented
131	#103	C₂²×C₃₁	1	2	yes	complemented
132	#102	C₂²×C₃₁	1	2	yes	—
133	#102	C₂²×C₃₁	1	2	yes	—
134	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
135	#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
136	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
137	C₃₁	#3123	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	—	2	1	no	—
139	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	C₂³×C₃₁	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	2	4	no	—
148	C₂×C₈	C₂³×C₃₁	1	2	yes	—
149	C₂×C₈	C₂³×C₃₁	1	2	yes	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
155	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
156	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
157	C₂²⋊C₄	—	2	1	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	C₄²	C₂³×C₃₁	1	1	yes	characteristic
166	C₂³	—	2	4	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	8	2	no	—
169	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	2	no	—
174	D₄	#41×C₃₁	1	4	yes	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
184	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×#41×C₃₁	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	4	4	no	—
203	C₂	C₃₁×#525	1	1	yes	characteristic
204	C₂	C₃₁×#577	1	2	yes	—

	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 ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#541	C₂	semidirect
3. C₃₁×#485	C₂	semidirect
4. C₃₁×#478	C₂	semidirect
5. C₃₁×#461	C₂	semidirect
7. C₃₁×#440	C₂	semidirect
8. C₄×SD₁₆×C₃₁	C₂	semidirect
9. C₃₁×#433	C₂	semidirect
13. C₃₁×C₄⋊Q₈	C₂²	semidirect
16. C₃₁×#125	C₂²	semidirect
18. C₄×Q₈×C₃₁	C₂²	semidirect
25. C₃₁×#107	C₂²	semidirect
26. C₃₁×#106	C₂²	semidirect
27. C₃₁×C₄⋊C₈	C₂²	semidirect
28. C₃₁×#103	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
33. C₃₁×#97	C₂²	semidirect
34. C₄×C₈×C₃₁	C₂²	semidirect
84. #3123	C₃₁	direct (N × H)
99. #541	C₂×C₃₁	semidirect
101. #485	C₂×C₃₁	semidirect
102. #478	C₂×C₃₁	semidirect
103. #461	C₂×C₃₁	semidirect
105. #440	C₂×C₃₁	semidirect
106. C₄×SD₁₆	C₂×C₃₁	semidirect
107. #433	C₂×C₃₁	semidirect
115. C₄⋊Q₈	C₂²×C₃₁	semidirect
118. #125	C₂²×C₃₁	semidirect
120. C₄×Q₈	C₂²×C₃₁	semidirect
127. #107	C₂²×C₃₁	semidirect
128. #106	C₂²×C₃₁	semidirect
129. C₄⋊C₈	C₂²×C₃₁	semidirect
130. #103	C₂²×C₃₁	semidirect
131. #103	C₂²×C₃₁	semidirect
135. #97	C₂²×C₃₁	semidirect
136. C₄×C₈	C₂²×C₃₁	semidirect
137. C₃₁	#3123	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×#541	2
C₃₁×#518	2
C₃₁×#485	2
C₃₁×#478	2
C₃₁×#461	2
C₃₁×#450	2
C₃₁×#440	2
C₄×SD₁₆×C₃₁	2
C₃₁×#433	2
#3123	31

Subgroups — C31×#3123 #16843056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3123 #16843056