Subgroups — C₃₁×#13239 #21469998

Order 7936.

502 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	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1970	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×D₁₆	—	2	2	no	—
8	C₂×C₃₁×D₁₆	—	2	2	no	—
9	C₂×C₃₁×D₁₆	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#102	—	2	2	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#390	—	2	2	no	—
16	C₃₁×#367	—	2	2	no	—
17	C₃₁×#357	—	2	4	no	—
18	C₃₁×#346	—	2	2	no	—
19	C₂²×D₄×C₃₁	—	2	2	no	—
20	C₂×D₈×C₃₁	—	2	2	no	—
21	C₂×D₈×C₃₁	—	2	2	no	—
22	C₂×D₈×C₃₁	—	2	2	no	—
23	C₂×D₈×C₃₁	—	2	2	no	—
24	C₂×D₈×C₃₁	—	2	2	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
30	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₃₁×D₁₆	—	2	8	no	—
32	C₂×C₁₆×C₃₁	—	2	1	no	—
33	C₂×C₁₆×C₃₁	—	2	1	no	—
34	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
36	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
37	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
38	C₃₁×#107	—	2	2	no	—
39	C₃₁×#107	—	2	2	no	—
40	C₃₁×#102	—	4	4	no	—
41	C₄×C₈×C₃₁	—	2	1	no	—
42	C₄×C₈×C₃₁	—	2	1	no	—
43	C₂⁴×C₃₁	—	4	2	no	—
44	C₂×D₄×C₃₁	—	2	4	no	—
45	C₂×D₄×C₃₁	—	4	2	no	—
46	C₂×D₄×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	—	4	2	no	—
49	C₂×D₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	D₈×C₃₁	—	2	4	no	—
54	D₈×C₃₁	—	2	4	no	—
55	D₈×C₃₁	—	4	2	no	—
56	D₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	2	1	no	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	D₈	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
65	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	—	8	8	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₁₆×C₃₁	—	2	2	no	—
72	C₁₆×C₃₁	C₂²⋊C₄	1	4	yes	complemented
73	#13239	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	4	4	no	—
75	C₂³×C₃₁	—	8	2	no	—
76	C₂³×C₃₁	—	8	2	no	—
77	C₂³×C₃₁	—	8	4	no	—
78	C₂³×C₃₁	—	8	4	no	—
79	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
80	D₄×C₃₁	—	4	4	no	—
81	D₄×C₃₁	—	4	4	no	—
82	D₄×C₃₁	—	8	2	no	—
83	D₄×C₃₁	—	8	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	8	2	no	—
87	C₂×C₄×C₃₁	—	8	4	no	—
88	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
90	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₈×C₃₁	#121	1	1	yes	characteristic
98	C₈×C₃₁	#124	1	1	yes	characteristic
99	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
100	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂×#357	C₂×C₃₁	1	2	yes	complemented
102	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
103	#1970	C₂×C₃₁	1	2	yes	complemented
104	C₂²×C₃₁	—	8	8	no	—
105	C₂²×C₃₁	—	8	4	no	—
106	C₂²×C₃₁	—	16	4	no	—
107	C₂²×C₃₁	—	16	2	no	—
108	C₂²×C₃₁	—	16	2	no	—
109	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
110	C₂²×C₃₁	#466	1	1	yes	characteristic
111	C₂²×C₃₁	#493	1	1	yes	characteristic
112	C₂²×C₃₁	#509	1	4	yes	—
113	C₄×C₃₁	—	4	2	no	—
114	C₄×C₃₁	—	16	2	no	—
115	C₄×C₃₁	#390	1	1	yes	characteristic
116	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
117	C₄×C₃₁	#466	1	1	yes	characteristic
118	C₄×C₃₁	#495	1	1	yes	characteristic
119	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
120	C₂×D₁₆	—	2	2	no	—
121	C₂×D₁₆	—	2	2	no	—
122	C₂×D₁₆	C₄×C₃₁	1	4	yes	complemented
123	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
124	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×#107	C₂²×C₃₁	1	2	yes	—
126	C₂×#102	—	2	2	no	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	#390	—	2	2	no	—
129	#367	—	2	2	no	—
130	#357	—	2	4	no	—
131	#346	—	2	2	no	—
132	C₂×C₃₁	—	16	4	no	—
133	C₂×C₃₁	#1970	1	1	yes	characteristic
134	C₂×C₃₁	#2177	1	2	yes	—
135	C₂×C₃₁	#2218	1	2	yes	—
136	C₂×C₃₁	#2250	1	2	yes	—
137	C₂²×D₄	—	2	2	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
144	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	4	4	no	—
148	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
149	D₁₆	—	2	8	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
153	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
154	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
155	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
156	#107	—	2	2	no	—
157	#107	—	2	2	no	—
158	#102	—	4	4	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₃₁	#13239	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	—	4	2	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	D₈×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
172	D₈	—	2	4	no	—
173	D₈	—	2	4	no	—
174	D₈	—	4	2	no	—
175	D₈	—	4	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	D₈×C₃₁	1	2	yes	—
179	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
184	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	8	8	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	1	no	—
190	C₁₆	—	2	2	no	—
191	C₁₆	C₂²⋊C₄×C₃₁	1	4	yes	complemented
192	C₂³	—	4	4	no	—
193	C₂³	—	8	2	no	—
194	C₂³	—	8	2	no	—
195	C₂³	—	8	4	no	—
196	C₂³	—	8	4	no	—
197	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	2	no	—
201	D₄	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
208	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
209	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
215	C₈	C₃₁×#121	1	1	yes	characteristic
216	C₈	C₃₁×#124	1	1	yes	characteristic
217	C₂²	—	8	8	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	4	no	—
220	C₂²	—	16	2	no	—
221	C₂²	—	16	2	no	—
222	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
223	C₂²	C₃₁×#466	1	1	yes	characteristic
224	C₂²	C₃₁×#493	1	1	yes	characteristic
225	C₂²	C₃₁×#509	1	4	yes	—
226	C₄	—	4	2	no	—
227	C₄	—	16	2	no	—
228	C₄	C₃₁×#390	1	1	yes	characteristic
229	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
230	C₄	C₃₁×#466	1	1	yes	characteristic
231	C₄	C₃₁×#495	1	1	yes	characteristic
232	C₂	—	16	4	no	—
233	C₂	C₃₁×#1970	1	1	yes	characteristic
234	C₂	C₃₁×#2177	1	2	yes	—
235	C₂	C₃₁×#2218	1	2	yes	—
236	C₂	C₃₁×#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#367	C₂	semidirect
3. C₂×C₃₁×#357	C₂	semidirect
4. C₂×C₃₁×#346	C₂	semidirect
5. C₃₁×#1970	C₂	semidirect
9. C₂×C₃₁×D₁₆	C₄	semidirect
72. C₁₆×C₃₁	C₂²⋊C₄	semidirect
73. #13239	C₃₁	direct (N × H)
100. C₂×#367	C₂×C₃₁	semidirect
101. C₂×#357	C₂×C₃₁	semidirect
102. C₂×#346	C₂×C₃₁	semidirect
103. #1970	C₂×C₃₁	semidirect
122. C₂×D₁₆	C₄×C₃₁	semidirect
161. C₃₁	#13239	direct (N × H)
191. C₁₆	C₂²⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×D₁₆	2
C₂×C₃₁×#367	2
C₂×C₃₁×#357	2
C₂×C₃₁×#346	2
C₃₁×#1970	2
#13239	31

Subgroups — C31×#13239 #21469998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13239 #21469998