Subgroups — C₃₁×#8807 #21465566

Order 7936.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1831	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	2	yes	maximal
4	C₂×C₃₁×#334	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄³×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#336	—	2	4	no	—
14	C₃₁×#335	—	2	4	no	—
15	C₃₁×#334	—	2	4	no	—
16	C₂²×C₈×C₃₁	—	2	4	no	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₃₁	D₄	1	2	yes	—
24	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	—	2	4	no	—
27	C₃₁×C₄⋊C₈	—	2	4	no	—
28	C₄×C₈×C₃₁	—	2	2	no	—
29	C₄×C₈×C₃₁	—	2	2	no	—
30	C₄×C₈×C₃₁	—	2	2	no	—
31	C₄×C₈×C₃₁	C₈	1	4	yes	complemented
32	C₄×C₈×C₃₁	D₄	1	2	yes	—
33	C₄×C₈×C₃₁	D₄	1	2	yes	—
34	C₄×C₈×C₃₁	Q₈	1	2	yes	—
35	C₄×C₈×C₃₁	Q₈	1	2	yes	—
36	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	1	no	—
42	C₂×C₈×C₃₁	—	2	1	no	—
43	C₂×C₈×C₃₁	—	2	1	no	—
44	C₂×C₈×C₃₁	—	2	1	no	—
45	C₂×C₈×C₃₁	—	4	4	no	—
46	C₂×C₈×C₃₁	—	4	8	no	—
47	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
54	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
55	C₄²×C₃₁	—	2	4	no	—
56	C₄²×C₃₁	—	2	1	no	—
57	C₄²×C₃₁	—	2	1	no	—
58	C₄²×C₃₁	—	2	4	no	—
59	C₄²×C₃₁	C₂×C₈	1	2	yes	—
60	C₄²×C₃₁	M₁₆	1	2	yes	—
61	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
65	#8807	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
73	C₂×C₄×C₃₁	#108	1	4	yes	—
74	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#135	1	4	yes	—
85	C₈×C₃₁	—	2	2	no	—
86	C₈×C₃₁	—	2	2	no	—
87	C₈×C₃₁	—	8	4	no	—
88	C₈×C₃₁	C₄⋊C₈	1	4	yes	—
89	C₈×C₃₁	C₄⋊C₈	1	4	yes	—
90	#1831	C₂×C₃₁	1	2	yes	—
91	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
92	C₂×#335	C₂×C₃₁	1	2	yes	—
93	C₂×#334	C₂×C₃₁	1	2	yes	—
94	C₂²×C₃₁	#384	1	1	yes	characteristic
95	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₃₁	#423	1	1	yes	characteristic
97	C₂²×C₃₁	#427	1	2	yes	—
98	C₂²×C₃₁	C₂×#108	1	2	yes	—
99	C₄×C₃₁	—	2	4	no	—
100	C₄×C₃₁	—	2	4	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₃₁	#423	1	1	yes	characteristic
106	C₄×C₃₁	#423	1	1	yes	characteristic
107	C₄×C₃₁	#495	1	2	yes	—
108	C₄×C₃₁	#499	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
110	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
111	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
112	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
113	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
116	C₄³	C₂²×C₃₁	1	1	yes	characteristic
117	#336	—	2	4	no	—
118	#335	—	2	4	no	—
119	#334	—	2	4	no	—
120	C₂×C₃₁	#1564	1	2	yes	—
121	C₂×C₃₁	#1568	1	2	yes	—
122	C₂×C₃₁	#1842	1	1	yes	characteristic
123	C₂×C₃₁	#1846	1	1	yes	characteristic
124	C₂×C₃₁	#1848	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
129	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
132	C₂×C₄²	D₄×C₃₁	1	2	yes	—
133	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
134	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₈×C₃₁	1	4	yes	complemented
141	C₄×C₈	D₄×C₃₁	1	2	yes	—
142	C₄×C₈	D₄×C₃₁	1	2	yes	—
143	C₄×C₈	Q₈×C₃₁	1	2	yes	—
144	C₄×C₈	Q₈×C₃₁	1	2	yes	—
145	C₃₁	#8807	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
147	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	—	4	8	no	—
157	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
158	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
159	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
160	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
161	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
162	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
163	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
164	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₂×C₈×C₃₁	1	2	yes	—
170	C₄²	M₁₆×C₃₁	1	2	yes	—
171	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
174	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
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	2	no	—
181	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
182	C₂×C₄	C₃₁×#108	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
186	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
188	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
191	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
192	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
193	C₂×C₄	C₃₁×#135	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	8	4	no	—
197	C₈	C₃₁×C₄⋊C₈	1	4	yes	—
198	C₈	C₃₁×C₄⋊C₈	1	4	yes	—
199	C₂²	C₃₁×#384	1	1	yes	characteristic
200	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
201	C₂²	C₃₁×#423	1	1	yes	characteristic
202	C₂²	C₃₁×#427	1	2	yes	—
203	C₂²	C₂×C₃₁×#108	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
209	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
210	C₄	C₃₁×#423	1	1	yes	characteristic
211	C₄	C₃₁×#423	1	1	yes	characteristic
212	C₄	C₃₁×#495	1	2	yes	—
213	C₄	C₃₁×#499	1	2	yes	—
214	C₂	C₃₁×#1564	1	2	yes	—
215	C₂	C₃₁×#1568	1	2	yes	—
216	C₂	C₃₁×#1842	1	1	yes	characteristic
217	C₂	C₃₁×#1846	1	1	yes	characteristic
218	C₂	C₃₁×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
31. C₄×C₈×C₃₁	C₈	semidirect
65. #8807	C₃₁	direct (N × H)
140. C₄×C₈	C₈×C₃₁	semidirect
145. C₃₁	#8807	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×#1831	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×#335	2
C₂×C₃₁×#334	2
#8807	31

Subgroups — C31×#8807 #21465566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8807 #21465566