Subgroups — C₃₁×#21923 #21478682

Order 7936.

668 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2298	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1923	C₂	1	4	yes	maximal
3	C₃₁×#1916	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1798	C₂	1	2	yes	maximal
7	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
14	C₃₁×#388	C₄	1	4	yes	—
15	C₃₁×#387	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#336	C₂²	1	4	yes	—
21	C₃₁×#336	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
26	C₃₁×#117	—	2	2	no	—
27	C₃₁×#117	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	—	2	4	no	—
39	C₃₁×C₄⋊C₈	—	4	4	no	—
40	C₃₁×C₂²⋊C₈	—	4	4	no	—
41	C₄×C₈×C₃₁	—	4	4	no	—
42	C₃₁×#95	C₂×C₄	1	4	yes	—
43	C₃₁×#95	C₂×C₄	1	4	yes	—
44	C₃₁×#95	C₂×C₄	1	4	yes	—
45	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	2	8	no	—
64	C₂²⋊C₄×C₃₁	—	2	4	no	—
65	C₂²⋊C₄×C₃₁	—	2	8	no	—
66	C₄²×C₃₁	—	2	8	no	—
67	C₄²×C₃₁	—	2	1	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	#41	1	2	yes	—
74	C₄²×C₃₁	#41	1	2	yes	—
75	#21923	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	—	2	2	no	—
78	C₂³×C₃₁	—	2	4	no	—
79	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	4	1	no	—
91	C₂×C₄×C₃₁	—	4	1	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	#117	1	4	yes	—
94	C₂×C₄×C₃₁	#131	1	8	yes	—
95	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
98	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
99	C₈×C₃₁	—	4	4	no	—
100	C₈×C₃₁	—	4	4	no	—
101	#2298	C₂×C₃₁	1	1	yes	characteristic
102	#1923	C₂×C₃₁	1	4	yes	—
103	#1916	C₂×C₃₁	1	4	yes	complemented
104	#1843	C₂×C₃₁	1	1	yes	characteristic, complemented
105	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
106	#1798	C₂×C₃₁	1	2	yes	—
107	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	2	no	—
110	C₂²×C₃₁	—	4	4	no	—
111	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
112	C₂²×C₃₁	#520	1	2	yes	—
113	C₂²×C₃₁	C₂×#131	1	4	yes	—
114	C₄×C₃₁	—	2	4	no	—
115	C₄×C₃₁	—	2	8	no	—
116	C₄×C₃₁	—	4	1	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	4	no	—
119	C₄×C₃₁	#540	1	1	yes	characteristic
120	C₄×C₃₁	#541	1	1	yes	characteristic
121	C₄×C₃₁	#542	1	2	yes	—
122	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
125	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
126	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
128	#388	C₄×C₃₁	1	4	yes	—
129	#387	C₄×C₃₁	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
131	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
132	#376	C₂²×C₃₁	1	1	yes	characteristic
133	C₄³	C₂²×C₃₁	1	1	yes	characteristic
134	#336	C₂²×C₃₁	1	4	yes	—
135	#336	C₂²×C₃₁	1	4	yes	—
136	#336	C₂²×C₃₁	1	4	yes	—
137	C₂×C₃₁	—	8	1	no	—
138	C₂×C₃₁	#2314	1	1	yes	characteristic
139	C₂×C₃₁	#2922	1	2	yes	—
140	C₂×C₃₁	#2994	1	4	yes	—
141	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
143	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
144	#117	—	2	2	no	—
145	#117	—	2	2	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
151	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
153	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
154	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	4	4	no	—
158	C₂²⋊C₈	—	4	4	no	—
159	C₄×C₈	—	4	4	no	—
160	#95	C₂×C₄×C₃₁	1	4	yes	—
161	#95	C₂×C₄×C₃₁	1	4	yes	—
162	#95	C₂×C₄×C₃₁	1	4	yes	—
163	C₃₁	#21923	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
174	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	8	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	8	no	—
185	C₄²	—	2	8	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	#41×C₃₁	1	2	yes	—
193	C₄²	#41×C₃₁	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₃₁×#117	1	4	yes	—
212	C₂×C₄	C₃₁×#131	1	8	yes	—
213	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
216	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	4	4	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	4	no	—
222	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
223	C₂²	C₃₁×#520	1	2	yes	—
224	C₂²	C₂×C₃₁×#131	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	8	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	4	no	—
230	C₄	C₃₁×#540	1	1	yes	characteristic
231	C₄	C₃₁×#541	1	1	yes	characteristic
232	C₄	C₃₁×#542	1	2	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₃₁×#2314	1	1	yes	characteristic
235	C₂	C₃₁×#2922	1	2	yes	—
236	C₂	C₃₁×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1916	C₂	semidirect
4. C₃₁×#1843	C₂	semidirect
5. C₃₁×#1832	C₂	semidirect
7. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
75. #21923	C₃₁	direct (N × H)
103. #1916	C₂×C₃₁	semidirect
104. #1843	C₂×C₃₁	semidirect
105. #1832	C₂×C₃₁	semidirect
107. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
163. C₃₁	#21923	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×#2298	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1843	2
C₃₁×#1832	2
C₃₁×#1798	2
C₄×C₃₁×C₄⋊C₈	2
#21923	31

Subgroups — C31×#21923 #21478682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21923 #21478682