Subgroups — C₃₁×#2841 #16842774

Order 3968.

396 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#400	C₂	1	2	yes	maximal, complemented
2	C₃₁×#399	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#398	C₂	1	2	yes	maximal, complemented
4	C₃₁×#398	C₂	1	2	yes	maximal, complemented
5	C₃₁×#396	C₂	1	2	yes	maximal
6	C₃₁×#395	C₂	1	2	yes	maximal, complemented
7	C₃₁×#393	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#381	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#380	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#95	C₂²	1	2	yes	—
24	C₃₁×#95	C₂²	1	2	yes	—
25	C₃₁×#95	C₂²	1	2	yes	—
26	C₃₁×#95	C₂²	1	2	yes	—
27	C₃₁×#95	C₂²	1	1	yes	characteristic
28	C₃₁×#95	C₂²	1	2	yes	—
29	C₃₁×#95	C₂²	1	2	yes	—
30	C₃₁×#95	C₂²	1	1	yes	characteristic
31	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₃₁	—	4	1	no	—
33	C₂×Q₈×C₃₁	—	4	2	no	—
34	C₂²×C₄×C₃₁	—	4	1	no	—
35	C₂²×C₄×C₃₁	—	4	1	no	—
36	C₂²×C₄×C₃₁	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₃₁	—	4	2	no	—
46	C₄⋊C₄×C₃₁	—	4	2	no	—
47	C₄⋊C₄×C₃₁	—	4	2	no	—
48	C₄⋊C₄×C₃₁	—	4	2	no	—
49	C₄⋊C₄×C₃₁	—	4	1	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₂²⋊C₄×C₃₁	—	4	2	no	—
53	C₂²⋊C₄×C₃₁	—	4	1	no	—
54	C₂²⋊C₄×C₃₁	—	4	2	no	—
55	C₂²⋊C₄×C₃₁	—	4	1	no	—
56	C₂³×C₃₁	—	2	2	no	—
57	C₂³×C₃₁	—	2	1	no	—
58	C₂³×C₃₁	—	2	1	no	—
59	C₂³×C₃₁	—	2	2	no	—
60	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
62	Q₈×C₃₁	—	4	4	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
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	—	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	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
86	#2841	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	—	2	4	no	—
89	C₂²×C₃₁	—	4	1	no	—
90	C₂²×C₃₁	—	4	2	no	—
91	C₂²×C₃₁	—	4	1	no	—
92	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
96	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
97	C₄×C₃₁	—	2	4	no	—
98	C₄×C₃₁	—	4	2	no	—
99	C₄×C₃₁	—	4	1	no	—
100	C₄×C₃₁	—	4	2	no	—
101	C₄×C₃₁	—	4	2	no	—
102	C₄×C₃₁	—	4	2	no	—
103	C₄×C₃₁	—	4	2	no	—
104	C₄×C₃₁	—	4	1	no	—
105	#400	C₂×C₃₁	1	2	yes	complemented
106	#399	C₂×C₃₁	1	1	yes	characteristic
107	#398	C₂×C₃₁	1	2	yes	complemented
108	#398	C₂×C₃₁	1	2	yes	complemented
109	#396	C₂×C₃₁	1	2	yes	—
110	#395	C₂×C₃₁	1	2	yes	complemented
111	#393	C₂×C₃₁	1	1	yes	characteristic
112	#381	C₂×C₃₁	1	1	yes	characteristic
113	#380	C₂×C₃₁	1	2	yes	—
114	C₂×C₃₁	—	4	2	no	—
115	C₂×C₃₁	#543	1	1	yes	characteristic
116	C₂×C₃₁	#544	1	2	yes	—
117	C₂×C₃₁	#560	1	1	yes	characteristic
118	C₂×C₃₁	#562	1	2	yes	—
119	C₂×C₃₁	#563	1	1	yes	characteristic
120	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
133	#95	C₂²×C₃₁	1	2	yes	—
134	#95	C₂²×C₃₁	1	2	yes	—
135	#95	C₂²×C₃₁	1	2	yes	—
136	#95	C₂²×C₃₁	1	2	yes	—
137	#95	C₂²×C₃₁	1	1	yes	characteristic
138	#95	C₂²×C₃₁	1	2	yes	—
139	#95	C₂²×C₃₁	1	2	yes	—
140	#95	C₂²×C₃₁	1	1	yes	characteristic
141	C₃₁	#2841	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	Q₈×C₃₁	1	4	yes	complemented
148	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
150	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
151	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
152	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
153	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	1	no	—
165	C₂²⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	4	1	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
172	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
173	Q₈	—	4	4	no	—
174	C₂×C₄	—	2	2	no	—
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	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
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	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×Q₈×C₃₁	1	4	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	4	1	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	1	no	—
202	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
203	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
206	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	4	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₂	—	4	2	no	—
216	C₂	C₃₁×#543	1	1	yes	characteristic
217	C₂	C₃₁×#544	1	2	yes	—
218	C₂	C₃₁×#560	1	1	yes	characteristic
219	C₂	C₃₁×#562	1	2	yes	—
220	C₂	C₃₁×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#400	C₂	semidirect
3. C₃₁×#398	C₂	semidirect
4. C₃₁×#398	C₂	semidirect
6. C₃₁×#395	C₂	semidirect
36. C₂²×C₄×C₃₁	Q₈	semidirect
86. #2841	C₃₁	direct (N × H)
105. #400	C₂×C₃₁	semidirect
107. #398	C₂×C₃₁	semidirect
108. #398	C₂×C₃₁	semidirect
110. #395	C₂×C₃₁	semidirect
141. C₃₁	#2841	direct (N × H)
147. C₂²×C₄	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#400	2
C₃₁×#399	2
C₃₁×#398	2
C₃₁×#396	2
C₃₁×#395	2
C₃₁×#393	2
C₃₁×#381	2
C₃₁×#380	2
#2841	31

Subgroups — C31×#2841 #16842774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2841 #16842774