Subgroups — C₃₁×#7867 #21464626

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	2	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1276	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₃₁	—	2	2	no	—
5	C₂²×C₄²×C₃₁	C₄	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#432	—	2	2	no	—
8	C₃₁×#432	C₄	1	4	yes	complemented
9	C₃₁×#431	—	2	2	no	—
10	C₃₁×#431	C₂²	1	4	yes	—
11	C₃₁×#423	C₄	1	4	yes	complemented
12	C₃₁×#407	—	2	2	no	—
13	C₄×M₁₆×C₃₁	—	2	2	no	—
14	C₄³×C₃₁	—	2	2	no	—
15	C₄³×C₃₁	C₄	1	2	yes	—
16	C₄³×C₃₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₃₁	—	2	2	no	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	D₄	1	2	yes	—
20	C₂×M₁₆×C₃₁	—	2	4	no	—
21	C₂×C₄²×C₃₁	—	2	16	no	—
22	C₂×C₄²×C₃₁	—	2	4	no	—
23	C₂×C₄²×C₃₁	—	2	2	no	—
24	C₂×C₄²×C₃₁	—	2	4	no	—
25	C₂×C₄²×C₃₁	—	2	8	no	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
36	C₃₁×C₂²⋊C₈	—	2	4	no	—
37	C₃₁×C₂²⋊C₈	—	2	4	no	—
38	C₃₁×#97	—	2	4	no	—
39	C₄×C₈×C₃₁	—	2	4	no	—
40	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	—	2	16	no	—
42	C₂²×C₄×C₃₁	—	2	8	no	—
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	2	8	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	M₁₆×C₃₁	—	4	8	no	—
57	C₂×C₈×C₃₁	—	2	8	no	—
58	C₄²×C₃₁	—	2	16	no	—
59	C₄²×C₃₁	—	2	16	no	—
60	C₄²×C₃₁	—	2	8	no	—
61	C₄²×C₃₁	—	2	4	no	—
62	C₄²×C₃₁	C₄²	1	1	yes	characteristic
63	C₄²×C₃₁	C₄²	1	1	yes	characteristic
64	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₃₁	M₁₆	1	8	yes	complemented
70	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
73	#7867	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	#95	1	2	yes	—
77	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	—	2	16	no	—
79	C₂×C₄×C₃₁	—	2	32	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	8	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	#95	1	2	yes	—
90	C₂×C₄×C₃₁	#95	1	2	yes	—
91	C₂×C₄×C₃₁	#95	1	2	yes	—
92	C₂×C₄×C₃₁	#104	1	8	yes	—
93	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
98	C₈×C₃₁	—	4	8	no	—
99	#2920	C₂×C₃₁	1	2	yes	—
100	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
101	#1276	C₂×C₃₁	1	4	yes	complemented
102	C₂²×C₃₁	—	2	8	no	—
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	#339	1	4	yes	—
107	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
108	C₂²×C₃₁	#407	1	2	yes	—
109	C₄×C₃₁	—	2	16	no	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	#339	1	4	yes	—
114	C₄×C₃₁	C₄×M₁₆	1	2	yes	—
115	C₄×C₃₁	C₂×#104	1	4	yes	—
116	C₄×C₃₁	#423	1	2	yes	—
117	C₂²×C₄²	—	2	2	no	—
118	C₂²×C₄²	C₄×C₃₁	1	2	yes	—
119	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#432	C₄×C₃₁	1	4	yes	complemented
122	#431	—	2	2	no	—
123	#431	C₂²×C₃₁	1	4	yes	—
124	#423	C₄×C₃₁	1	4	yes	complemented
125	#407	—	2	2	no	—
126	C₄×M₁₆	—	2	2	no	—
127	C₄³	—	2	2	no	—
128	C₄³	C₄×C₃₁	1	2	yes	—
129	C₄³	C₂²×C₃₁	1	2	yes	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	—	2	4	no	—
132	C₂×C₃₁	#1728	1	1	yes	characteristic
133	C₂×C₃₁	C₂×#339	1	2	yes	—
134	C₂³×C₄	—	2	2	no	—
135	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₂³×C₄	D₄×C₃₁	1	2	yes	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×C₄²	—	2	16	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	4	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	8	no	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	D₄×C₃₁	1	2	yes	—
148	C₂×C₄²	D₄×C₃₁	1	2	yes	—
149	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₃₁	#7867	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	—	2	16	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
170	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
174	M₁₆	—	4	8	no	—
175	C₂×C₈	—	2	8	no	—
176	C₄²	—	2	16	no	—
177	C₄²	—	2	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	4	no	—
180	C₄²	C₄²×C₃₁	1	1	yes	characteristic
181	C₄²	C₄²×C₃₁	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
184	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
185	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
186	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
187	C₄²	M₁₆×C₃₁	1	8	yes	complemented
188	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
189	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
190	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₃₁×#95	1	2	yes	—
194	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	32	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	C₃₁×#95	1	2	yes	—
207	C₂×C₄	C₃₁×#95	1	2	yes	—
208	C₂×C₄	C₃₁×#95	1	2	yes	—
209	C₂×C₄	C₃₁×#104	1	8	yes	—
210	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
215	C₈	—	4	8	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	C₃₁×#339	1	4	yes	—
221	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
222	C₂²	C₃₁×#407	1	2	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₃₁×#339	1	4	yes	—
228	C₄	C₄×M₁₆×C₃₁	1	2	yes	—
229	C₄	C₂×C₃₁×#104	1	4	yes	—
230	C₄	C₃₁×#423	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	C₃₁×#1728	1	1	yes	characteristic
234	C₂	C₂×C₃₁×#339	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1276	C₂	semidirect
8. C₃₁×#432	C₄	semidirect
11. C₃₁×#423	C₄	semidirect
35. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
69. C₄²×C₃₁	M₁₆	semidirect
73. #7867	C₃₁	direct (N × H)
101. #1276	C₂×C₃₁	semidirect
121. #432	C₄×C₃₁	semidirect
124. #423	C₄×C₃₁	semidirect
152. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
157. C₃₁	#7867	direct (N × H)
187. C₄²	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂×C₄³×C₃₁	2
C₃₁×#1276	2
#7867	31

Subgroups — C31×#7867 #21464626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7867 #21464626