Subgroups — C₂×C₃₁×#2255 #21490486

Order 7936.

518 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	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2255	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	2	yes	maximal
5	C₂×C₃₁×#366	C₂	1	2	yes	maximal
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₃₁×#125	—	2	1	no	—
9	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#500	—	2	2	no	—
11	C₃₁×#500	C₂²	1	4	yes	—
12	C₃₁×#499	—	2	2	no	—
13	C₃₁×#499	C₂²	1	4	yes	—
14	C₂×C₃₁×#107	—	2	1	no	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#367	C₂²	1	8	yes	—
21	C₃₁×#366	C₂²	1	8	yes	—
22	C₃₁×#346	C₂²	1	4	yes	—
23	C₂²×Q₈×C₃₁	—	4	1	no	—
24	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊Q₈	—	2	4	no	—
27	C₃₁×C₄⋊Q₈	—	4	2	no	—
28	C₃₁×#125	—	2	4	no	—
29	C₃₁×#125	—	4	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₃₁	Q₈	1	8	yes	complemented
38	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
39	C₃₁×#107	—	2	4	no	—
40	C₃₁×#107	—	2	1	no	—
41	C₃₁×#107	—	2	1	no	—
42	C₃₁×#107	—	2	2	no	—
43	C₃₁×#107	C₂³	1	2	yes	—
44	C₃₁×#107	C₂³	1	2	yes	—
45	C₃₁×#107	C₂³	1	4	yes	—
46	C₃₁×#106	—	2	4	no	—
47	C₄×C₈×C₃₁	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂×Q₈×C₃₁	—	4	4	no	—
50	C₂×Q₈×C₃₁	—	8	4	no	—
51	C₂²×C₄×C₃₁	—	4	1	no	—
52	C₂²×C₄×C₃₁	—	4	1	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	1	no	—
66	C₄⋊C₄×C₃₁	—	4	1	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄⋊C₄×C₃₁	—	4	4	no	—
71	C₄²×C₃₁	D₈	1	2	yes	—
72	C₄²×C₃₁	C₂×D₄	1	2	yes	—
73	C₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
74	C₂×#2255	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₃₁	—	8	8	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	—	8	2	no	—
83	C₂×C₄×C₃₁	—	8	4	no	—
84	C₂×C₄×C₃₁	—	8	2	no	—
85	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
91	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
92	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
93	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
94	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
95	C₂×#499	C₂×C₃₁	1	1	yes	characteristic
96	#2255	C₂×C₃₁	1	8	yes	direct product factor
97	C₂×#367	C₂×C₃₁	1	2	yes	—
98	C₂×#366	C₂×C₃₁	1	2	yes	—
99	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×C₃₁	#500	1	2	yes	—
101	C₂²×C₃₁	#509	1	2	yes	—
102	C₂²×C₃₁	#510	1	2	yes	—
103	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₃₁	—	2	2	no	—
105	C₄×C₃₁	—	2	2	no	—
106	C₄×C₃₁	—	8	2	no	—
107	C₄×C₃₁	—	8	4	no	—
108	C₄×C₃₁	—	8	2	no	—
109	C₄×C₃₁	#500	1	2	yes	—
110	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×#125	—	2	1	no	—
114	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
115	#500	—	2	2	no	—
116	#500	C₂²×C₃₁	1	4	yes	—
117	#499	—	2	2	no	—
118	#499	C₂²×C₃₁	1	4	yes	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×#107	C₂²×C₃₁	1	2	yes	—
123	C₂×#106	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	#367	C₂²×C₃₁	1	8	yes	—
126	#366	C₂²×C₃₁	1	8	yes	—
127	#346	C₂²×C₃₁	1	4	yes	—
128	C₂×C₃₁	#2255	1	4	yes	direct product factor
129	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	#125	—	2	4	no	—
138	#125	—	4	2	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
146	C₂×C₁₆	Q₈×C₃₁	1	8	yes	complemented
147	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
148	#107	—	2	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	2	2	no	—
152	#107	C₂³×C₃₁	1	2	yes	—
153	#107	C₂³×C₃₁	1	4	yes	—
154	#107	C₂³×C₃₁	1	2	yes	—
155	#106	—	2	4	no	—
156	C₄×C₈	D₄×C₃₁	1	2	yes	—
157	C₄×C₈	C₂³×C₃₁	1	2	yes	—
158	C₃₁	C₂×#2255	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	8	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
170	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
171	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄²	D₈×C₃₁	1	2	yes	—
182	C₄²	C₂×D₄×C₃₁	1	2	yes	—
183	C₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
184	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
196	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
197	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
199	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
200	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
201	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
202	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
203	C₂²	C₃₁×#500	1	2	yes	—
204	C₂²	C₃₁×#509	1	2	yes	—
205	C₂²	C₃₁×#510	1	2	yes	—
206	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₃₁×#500	1	2	yes	—
213	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
214	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
215	C₂	C₃₁×#2255	1	4	yes	direct product factor
216	C₂	C₂×C₃₁×#500	1	1	yes	characteristic
217	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
218	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
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
3. C₃₁×#2255	C₂	direct (N × H)
37. C₂×C₁₆×C₃₁	Q₈	semidirect
73. C₁₆×C₃₁	C₂×Q₈	semidirect
74. C₂×#2255	C₃₁	direct (N × H)
96. #2255	C₂×C₃₁	direct (N × H)
128. C₂×C₃₁	#2255	direct (N × H)
146. C₂×C₁₆	Q₈×C₃₁	semidirect
158. C₃₁	C₂×#2255	direct (N × H)
183. C₁₆	C₂×Q₈×C₃₁	semidirect
215. C₂	C₃₁×#2255	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₂×C₃₁×#499	2
C₃₁×#2255	2
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×#346	2
C₂×#2255	31

Subgroups — C2×C31×#2255 #21490486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2255 #21490486