Subgroups — C₂×C₃₁×#2248 #21490470

Order 7936.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	4	yes	maximal, complemented
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2248	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₃₁	—	2	1	no	—
7	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×D₈×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃₁×#127	—	2	1	no	—
12	C₂×C₃₁×SD₃₂	—	2	16	no	—
13	C₂×C₃₁×SD₃₂	C₂²	1	16	yes	complemented
14	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#494	—	2	2	no	—
16	C₃₁×#494	C₂²	1	4	yes	complemented
17	C₃₁×#493	—	2	2	no	—
18	C₃₁×#493	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₃₁	C₂²	1	4	yes	complemented
21	C₂²×Q₈×C₃₁	—	2	2	no	—
22	C₂²×D₄×C₃₁	—	2	2	no	—
23	C₂²×D₄×C₃₁	—	4	2	no	—
24	C₂×Q₁₆×C₃₁	—	2	8	no	—
25	C₂×Q₁₆×C₃₁	—	2	2	no	—
26	C₂×Q₁₆×C₃₁	—	2	4	no	—
27	C₂×Q₁₆×C₃₁	—	4	4	no	—
28	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
29	C₂×D₈×C₃₁	—	2	8	no	—
30	C₂×D₈×C₃₁	—	2	2	no	—
31	C₂×D₈×C₃₁	—	2	4	no	—
32	C₂×D₈×C₃₁	—	4	4	no	—
33	C₂×D₈×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	—	4	2	no	—
38	C₃₁×#127	—	2	4	no	—
39	C₃₁×#127	—	4	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₃₁×SD₃₂	—	2	32	no	—
43	C₂×C₁₆×C₃₁	D₄	1	8	yes	complemented
44	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	C₄×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂⁴×C₃₁	—	4	2	no	—
48	C₂×Q₈×C₃₁	—	2	4	no	—
49	C₂×Q₈×C₃₁	—	4	2	no	—
50	C₂×Q₈×C₃₁	—	4	4	no	—
51	C₂×Q₈×C₃₁	—	4	4	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	4	4	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	—	4	8	no	—
56	C₂×D₄×C₃₁	—	4	2	no	—
57	C₂×D₄×C₃₁	—	8	8	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	D₈	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₃₁	—	2	8	no	—
62	Q₁₆×C₃₁	—	4	4	no	—
63	Q₁₆×C₃₁	—	4	8	no	—
64	D₈×C₃₁	—	2	8	no	—
65	D₈×C₃₁	—	4	4	no	—
66	D₈×C₃₁	—	4	8	no	—
67	C₂×C₈×C₃₁	D₈	1	4	yes	—
68	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	4	8	no	—
73	C₄²×C₃₁	D₈	1	2	yes	—
74	C₄²×C₃₁	C₂×D₄	1	2	yes	—
75	C₁₆×C₃₁	C₂×D₄	1	8	yes	complemented
76	C₂×#2248	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	—	8	4	no	—
79	C₂³×C₃₁	—	8	8	no	—
80	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
81	Q₈×C₃₁	—	4	4	no	—
82	Q₈×C₃₁	—	4	4	no	—
83	Q₈×C₃₁	—	8	4	no	—
84	D₄×C₃₁	—	4	4	no	—
85	D₄×C₃₁	—	4	4	no	—
86	D₄×C₃₁	—	8	4	no	—
87	D₄×C₃₁	—	8	16	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	8	4	no	—
90	C₂×C₄×C₃₁	SD₃₂	1	8	yes	complemented
91	C₂×C₄×C₃₁	#127	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₃₁	#127	1	2	yes	—
97	C₈×C₃₁	C₂×D₈	1	4	yes	—
98	C₈×C₃₁	C₂²×D₄	1	2	yes	—
99	C₂²×SD₃₂	C₂×C₃₁	1	4	yes	complemented
100	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
102	#2248	C₂×C₃₁	1	8	yes	direct product factor
103	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
104	C₂²×C₃₁	—	8	4	no	—
105	C₂²×C₃₁	—	8	8	no	—
106	C₂²×C₃₁	—	16	8	no	—
107	C₂²×C₃₁	#493	1	2	yes	—
108	C₂²×C₃₁	C₂×SD₃₂	1	4	yes	—
109	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₃₁	—	8	4	no	—
111	C₄×C₃₁	#493	1	2	yes	—
112	C₄×C₃₁	C₂×SD₃₂	1	8	yes	complemented
113	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
114	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
115	C₂²×Q₁₆	—	2	1	no	—
116	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
117	C₂²×D₈	—	2	1	no	—
118	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×#127	—	2	1	no	—
121	C₂×SD₃₂	—	2	16	no	—
122	C₂×SD₃₂	C₂²×C₃₁	1	16	yes	complemented
123	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
124	#494	—	2	2	no	—
125	#494	C₂²×C₃₁	1	4	yes	complemented
126	#493	—	2	2	no	—
127	#493	C₂²×C₃₁	1	4	yes	—
128	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
129	C₄×C₁₆	C₂²×C₃₁	1	4	yes	complemented
130	C₂×C₃₁	—	16	4	no	—
131	C₂×C₃₁	#2248	1	4	yes	direct product factor
132	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
133	C₂×C₃₁	C₂²×SD₃₂	1	2	yes	—
134	C₂²×Q₈	—	2	2	no	—
135	C₂²×D₄	—	2	2	no	—
136	C₂²×D₄	—	4	2	no	—
137	C₂×Q₁₆	—	2	8	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	4	4	no	—
141	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
142	C₂×D₈	—	2	8	no	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	4	no	—
146	C₂×D₈	C₂³×C₃₁	1	4	yes	—
147	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	C₂×C₄⋊C₄	—	4	2	no	—
154	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
155	SD₃₂	—	2	32	no	—
156	C₂×C₁₆	D₄×C₃₁	1	8	yes	complemented
157	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
158	C₄×C₈	D₄×C₃₁	1	2	yes	—
159	C₄×C₈	C₂³×C₃₁	1	2	yes	—
160	C₃₁	C₂×#2248	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	4	2	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	8	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	8	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	D₈×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
175	Q₁₆	—	2	8	no	—
176	Q₁₆	—	4	4	no	—
177	Q₁₆	—	4	8	no	—
178	D₈	—	2	8	no	—
179	D₈	—	4	4	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	D₈×C₃₁	1	4	yes	—
182	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
184	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
185	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
186	C₄⋊C₄	—	4	8	no	—
187	C₄²	D₈×C₃₁	1	2	yes	—
188	C₄²	C₂×D₄×C₃₁	1	2	yes	—
189	C₁₆	C₂×D₄×C₃₁	1	8	yes	complemented
190	C₂³	—	4	4	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	8	no	—
193	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	8	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	16	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	4	no	—
203	C₂×C₄	C₃₁×SD₃₂	1	8	yes	complemented
204	C₂×C₄	C₃₁×#127	1	2	yes	—
205	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
207	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
208	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
209	C₈	C₃₁×#127	1	2	yes	—
210	C₈	C₂×D₈×C₃₁	1	4	yes	—
211	C₈	C₂²×D₄×C₃₁	1	2	yes	—
212	C₂²	—	8	4	no	—
213	C₂²	—	8	8	no	—
214	C₂²	—	16	8	no	—
215	C₂²	C₃₁×#493	1	2	yes	—
216	C₂²	C₂×C₃₁×SD₃₂	1	4	yes	—
217	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	C₃₁×#493	1	2	yes	—
220	C₄	C₂×C₃₁×SD₃₂	1	8	yes	complemented
221	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
222	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
223	C₂	—	16	4	no	—
224	C₂	C₃₁×#2248	1	4	yes	direct product factor
225	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
226	C₂	C₂²×C₃₁×SD₃₂	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	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
1. C₂²×C₃₁×SD₃₂	C₂	semidirect
2. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#2248	C₂	direct (N × H)
5. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
13. C₂×C₃₁×SD₃₂	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
20. C₄×C₁₆×C₃₁	C₂²	semidirect
43. C₂×C₁₆×C₃₁	D₄	semidirect
75. C₁₆×C₃₁	C₂×D₄	semidirect
76. C₂×#2248	C₃₁	direct (N × H)
90. C₂×C₄×C₃₁	SD₃₂	semidirect
99. C₂²×SD₃₂	C₂×C₃₁	semidirect
100. C₂×#494	C₂×C₃₁	semidirect
102. #2248	C₂×C₃₁	direct (N × H)
103. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
112. C₄×C₃₁	C₂×SD₃₂	semidirect
122. C₂×SD₃₂	C₂²×C₃₁	semidirect
125. #494	C₂²×C₃₁	semidirect
129. C₄×C₁₆	C₂²×C₃₁	semidirect
131. C₂×C₃₁	#2248	direct (N × H)
156. C₂×C₁₆	D₄×C₃₁	semidirect
160. C₃₁	C₂×#2248	direct (N × H)
189. C₁₆	C₂×D₄×C₃₁	semidirect
203. C₂×C₄	C₃₁×SD₃₂	semidirect
220. C₄	C₂×C₃₁×SD₃₂	semidirect
224. C₂	C₃₁×#2248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₂×C₃₁×#494	2
C₂×C₃₁×#493	2
C₃₁×#2248	2
C₂×C₄×C₁₆×C₃₁	2
C₂×#2248	31

Subgroups — C2×C31×#2248 #21490470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2248 #21490470