Subgroups — C₃₁×#14342 #21471101

Order 7936.

1998 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	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#381	C₂	1	2	yes	maximal
3	C₃₁×#1445	C₂	1	8	yes	maximal, complemented
4	C₃₁×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₃₁	—	2	8	no	—
8	C₂²×C₄⋊C₄×C₃₁	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#381	—	2	8	no	—
12	C₂×C₃₁×#95	C₂²	1	8	yes	—
13	C₂×C₃₁×#95	C₂²	1	16	yes	—
14	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	—	2	8	no	—
16	C₂³×C₄×C₃₁	—	2	8	no	—
17	C₂³×C₄×C₃₁	—	4	8	no	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₃₁	D₄	1	8	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₁	—	2	32	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
26	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
27	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
28	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
29	C₃₁×#95	—	2	16	no	—
30	C₃₁×#95	—	4	16	no	—
31	C₂⁴×C₃₁	—	2	4	no	—
32	C₂⁴×C₃₁	—	2	4	no	—
33	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
34	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
35	C₂⁴×C₃₁	#41	1	8	yes	—
36	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	—	2	16	no	—
38	C₂²×C₄×C₃₁	—	2	32	no	—
39	C₂²×C₄×C₃₁	—	2	8	no	—
40	C₂²×C₄×C₃₁	—	2	16	no	—
41	C₂²×C₄×C₃₁	—	2	8	no	—
42	C₂²×C₄×C₃₁	—	2	16	no	—
43	C₂²×C₄×C₃₁	—	2	8	no	—
44	C₂²×C₄×C₃₁	—	2	16	no	—
45	C₂²×C₄×C₃₁	—	2	16	no	—
46	C₂²×C₄×C₃₁	—	2	16	no	—
47	C₂²×C₄×C₃₁	—	4	16	no	—
48	C₂²×C₄×C₃₁	—	4	32	no	—
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	16	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
53	C₄⋊C₄×C₃₁	—	2	32	no	—
54	C₂²⋊C₄×C₃₁	—	4	16	no	—
55	#14342	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₃₁	—	2	4	no	—
57	C₂³×C₃₁	—	2	4	no	—
58	C₂³×C₃₁	—	2	8	no	—
59	C₂³×C₃₁	—	2	8	no	—
60	C₂³×C₃₁	—	2	32	no	—
61	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
62	C₂³×C₃₁	#117	1	8	yes	—
63	C₂³×C₃₁	#123	1	16	yes	—
64	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
65	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
66	C₂³×C₃₁	C₂×#41	1	4	yes	—
67	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
68	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
69	C₂×C₄×C₃₁	—	2	8	no	—
70	C₂×C₄×C₃₁	—	2	16	no	—
71	C₂×C₄×C₃₁	—	2	32	no	—
72	C₂×C₄×C₃₁	—	2	8	no	—
73	C₂×C₄×C₃₁	—	2	16	no	—
74	C₂×C₄×C₃₁	—	4	32	no	—
75	C₂×C₄×C₃₁	—	4	16	no	—
76	C₂×C₄×C₃₁	—	4	8	no	—
77	C₂×C₄×C₃₁	—	4	8	no	—
78	C₂×C₄×C₃₁	—	4	8	no	—
79	C₂×C₄×C₃₁	—	4	32	no	—
80	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
81	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
82	C₂×#381	C₂×C₃₁	1	2	yes	—
83	#1445	C₂×C₃₁	1	8	yes	complemented
84	C₂²⋊#95	C₂×C₃₁	1	4	yes	—
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	—	2	16	no	—
88	C₂²×C₃₁	—	2	32	no	—
89	C₂²×C₃₁	#381	1	8	yes	—
90	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#117	1	2	yes	—
92	C₂²×C₃₁	#518	1	2	yes	—
93	C₂²×C₃₁	#519	1	2	yes	—
94	C₂²×C₃₁	C₂×#123	1	4	yes	—
95	C₂²×C₃₁	#535	1	4	yes	—
96	C₂²×C₃₁	#536	1	4	yes	—
97	C₂²×C₃₁	#539	1	16	yes	—
98	C₄×C₃₁	—	2	8	no	—
99	C₄×C₃₁	—	4	8	no	—
100	C₄×C₃₁	—	8	8	no	—
101	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
102	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
103	C₂²×C₄⋊C₄	—	2	8	no	—
104	C₂²×C₄⋊C₄	C₄×C₃₁	1	8	yes	complemented
105	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
106	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
107	#381	—	2	8	no	—
108	C₂×#95	C₂²×C₃₁	1	8	yes	—
109	C₂×#95	C₂²×C₃₁	1	16	yes	—
110	C₂×C₃₁	—	2	8	no	—
111	C₂×C₃₁	C₂×#381	1	1	yes	characteristic
112	C₂×C₃₁	#2309	1	1	yes	characteristic
113	C₂×C₃₁	#2310	1	1	yes	characteristic
114	C₂×C₃₁	#2313	1	4	yes	—
115	C₂×C₃₁	#2623	1	8	yes	—
116	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	—	2	8	no	—
119	C₂³×C₄	—	4	8	no	—
120	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
121	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
122	C₂³×C₄	D₄×C₃₁	1	8	yes	—
123	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
124	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
125	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
126	C₂×C₄⋊C₄	—	2	32	no	—
127	C₂×C₄⋊C₄	—	2	16	no	—
128	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
129	C₂×C₂²⋊C₄	—	2	8	no	—
130	C₂×C₂²⋊C₄	—	2	16	no	—
131	#95	—	2	16	no	—
132	#95	—	4	16	no	—
133	C₃₁	#14342	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	—	2	4	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
137	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
138	C₂⁴	#41×C₃₁	1	8	yes	—
139	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	—	2	16	no	—
141	C₂²×C₄	—	2	32	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	16	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	16	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	16	no	—
148	C₂²×C₄	—	2	16	no	—
149	C₂²×C₄	—	2	16	no	—
150	C₂²×C₄	—	4	16	no	—
151	C₂²×C₄	—	4	32	no	—
152	C₂²×C₄	C₂²⋊C₄×C₃₁	1	16	yes	—
153	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
154	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
155	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
156	C₄⋊C₄	—	2	32	no	—
157	C₂²⋊C₄	—	4	16	no	—
158	C₂³	—	2	4	no	—
159	C₂³	—	2	4	no	—
160	C₂³	—	2	8	no	—
161	C₂³	—	2	8	no	—
162	C₂³	—	2	32	no	—
163	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
164	C₂³	C₃₁×#117	1	8	yes	—
165	C₂³	C₃₁×#123	1	16	yes	—
166	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
167	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
168	C₂³	C₂×#41×C₃₁	1	4	yes	—
169	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
170	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	—	2	32	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	16	no	—
176	C₂×C₄	—	4	32	no	—
177	C₂×C₄	—	4	16	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	4	32	no	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	16	no	—
186	C₂²	—	2	32	no	—
187	C₂²	C₃₁×#381	1	8	yes	—
188	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂²	C₂×C₃₁×#117	1	2	yes	—
190	C₂²	C₃₁×#518	1	2	yes	—
191	C₂²	C₃₁×#519	1	2	yes	—
192	C₂²	C₂×C₃₁×#123	1	4	yes	—
193	C₂²	C₃₁×#535	1	4	yes	—
194	C₂²	C₃₁×#536	1	4	yes	—
195	C₂²	C₃₁×#539	1	16	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	—	8	8	no	—
199	C₂	—	2	8	no	—
200	C₂	C₂×C₃₁×#381	1	1	yes	characteristic
201	C₂	C₃₁×#2309	1	1	yes	characteristic
202	C₂	C₃₁×#2310	1	1	yes	characteristic
203	C₂	C₃₁×#2313	1	4	yes	—
204	C₂	C₃₁×#2623	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1445	C₂	semidirect
8. C₂²×C₄⋊C₄×C₃₁	C₄	semidirect
55. #14342	C₃₁	direct (N × H)
83. #1445	C₂×C₃₁	semidirect
104. C₂²×C₄⋊C₄	C₄×C₃₁	semidirect
133. C₃₁	#14342	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#381	2
C₃₁×#1445	2
C₃₁×C₂²⋊#95	2
#14342	31

Subgroups — C31×#14342 #21471101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14342 #21471101