Subgroups — C₃₁×#3273 #16843206

Order 3968.

520 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₃₁×#546	C₂	1	2	yes	maximal, complemented
3	C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#468	C₂	1	2	yes	maximal, complemented
5	C₃₁×#463	C₂	1	2	yes	maximal, complemented
6	C₃₁×#452	C₂	1	2	yes	maximal, complemented
7	C₃₁×#447	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₃₁	C₂²	1	2	yes	complemented
12	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
13	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
20	C₃₁×#124	C₂²	1	2	yes	complemented
21	C₃₁×#123	—	2	2	no	—
22	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	C₃₁×#121	—	2	2	no	—
24	C₃₁×#121	C₂²	1	2	yes	—
25	C₃₁×#120	—	2	2	no	—
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#103	C₂²	1	2	yes	complemented
32	C₃₁×#102	C₂²	1	2	yes	complemented
33	C₃₁×#102	C₂²	1	2	yes	complemented
34	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₂⁴×C₃₁	—	2	2	no	—
37	#41×C₃₁	—	2	4	no	—
38	#41×C₃₁	—	4	2	no	—
39	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
40	C₂×D₄×C₃₁	—	2	4	no	—
41	C₂×D₄×C₃₁	—	2	2	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	Q₁₆×C₃₁	—	4	1	no	—
52	SD₁₆×C₃₁	—	2	4	no	—
53	SD₁₆×C₃₁	—	2	4	no	—
54	D₈×C₃₁	—	2	4	no	—
55	D₈×C₃₁	—	4	1	no	—
56	D₈×C₃₁	D₄	1	4	yes	complemented
57	M₁₆×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	4	4	no	—
74	C₂³×C₃₁	—	4	4	no	—
75	C₂³×C₃₁	—	8	2	no	—
76	C₂³×C₃₁	C₂×D₄	1	2	yes	—
77	Q₈×C₃₁	—	2	2	no	—
78	Q₈×C₃₁	—	2	2	no	—
79	D₄×C₃₁	—	2	2	no	—
80	D₄×C₃₁	—	2	2	no	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	4	1	no	—
83	D₄×C₃₁	—	4	4	no	—
84	D₄×C₃₁	—	4	4	no	—
85	D₄×C₃₁	—	8	2	no	—
86	D₄×C₃₁	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	C₂×D₄	1	2	yes	—
100	#3273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	2	4	no	—
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	4	2	no	—
106	C₂²×C₃₁	—	4	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	—	4	2	no	—
110	C₂²×C₃₁	—	8	8	no	—
111	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×AGL(1,ℤ/8)	C₂×C₃₁	1	2	yes	complemented
120	#546	C₂×C₃₁	1	2	yes	complemented
121	#495	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#468	C₂×C₃₁	1	2	yes	complemented
123	#463	C₂×C₃₁	1	2	yes	complemented
124	#452	C₂×C₃₁	1	2	yes	complemented
125	#447	C₂×C₃₁	1	2	yes	complemented
126	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#436	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×C₃₁	—	4	2	no	—
129	C₂×C₃₁	—	4	4	no	—
130	C₂×C₃₁	—	8	2	no	—
131	C₂×C₃₁	D₄²	1	1	yes	characteristic
132	C₂×C₃₁	#575	1	1	yes	characteristic
133	C₂×C₃₁	#576	1	1	yes	characteristic
134	C₂×#41	C₂²×C₃₁	1	2	yes	—
135	C₂²×D₄	C₂²×C₃₁	1	2	yes	complemented
136	AGL(1,ℤ/8)	—	2	8	no	—
137	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
138	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
139	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
140	C₂×D₈	C₂²×C₃₁	1	2	yes	—
141	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×M₁₆	C₂²×C₃₁	1	2	yes	complemented
144	#124	C₂²×C₃₁	1	2	yes	complemented
145	#123	—	2	2	no	—
146	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
147	#121	—	2	2	no	—
148	#121	C₂²×C₃₁	1	2	yes	—
149	#120	—	2	2	no	—
150	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
151	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
152	C₂×C₂²⋊C₄	—	2	2	no	—
153	#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
154	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
155	#103	C₂²×C₃₁	1	2	yes	complemented
156	#102	C₂²×C₃₁	1	2	yes	complemented
157	#102	C₂²×C₃₁	1	2	yes	complemented
158	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
159	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
160	C₃₁	#3273	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	2	no	—
162	#41	—	2	4	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
171	C₂×D₄	C₂³×C₃₁	1	2	yes	—
172	C₂×D₄	C₂³×C₃₁	1	2	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
176	Q₁₆	—	4	1	no	—
177	SD₁₆	—	2	4	no	—
178	SD₁₆	—	2	4	no	—
179	D₈	—	2	4	no	—
180	D₈	—	4	1	no	—
181	D₈	D₄×C₃₁	1	4	yes	complemented
182	M₁₆	—	2	4	no	—
183	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₃₁	1	2	yes	—
185	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
195	C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	4	no	—
199	C₂³	—	4	4	no	—
200	C₂³	—	8	2	no	—
201	C₂³	C₂×D₄×C₃₁	1	2	yes	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	1	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	8	2	no	—
211	D₄	C₂×D₄×C₃₁	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
221	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	8	8	no	—
235	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
242	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂	—	4	2	no	—
244	C₂	—	4	4	no	—
245	C₂	—	8	2	no	—
246	C₂	D₄²×C₃₁	1	1	yes	characteristic
247	C₂	C₃₁×#575	1	1	yes	characteristic
248	C₂	C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
2. C₃₁×#546	C₂	semidirect
3. C₃₁×#495	C₂	semidirect
4. C₃₁×#468	C₂	semidirect
5. C₃₁×#463	C₂	semidirect
6. C₃₁×#452	C₂	semidirect
7. C₃₁×#447	C₂	semidirect
8. C₄×D₈×C₃₁	C₂	semidirect
9. C₃₁×#436	C₂	semidirect
11. C₂²×D₄×C₃₁	C₂²	semidirect
13. C₂×Q₁₆×C₃₁	C₂²	semidirect
14. C₂×SD₁₆×C₃₁	C₂²	semidirect
15. C₂×SD₁₆×C₃₁	C₂²	semidirect
19. C₂×M₁₆×C₃₁	C₂²	semidirect
20. C₃₁×#124	C₂²	semidirect
22. C₃₁×C₂²⋊Q₈	C₂²	semidirect
27. C₄×D₄×C₃₁	C₂²	semidirect
29. C₃₁×#107	C₂²	semidirect
30. C₃₁×C₄⋊C₈	C₂²	semidirect
31. C₃₁×#103	C₂²	semidirect
32. C₃₁×#102	C₂²	semidirect
33. C₃₁×#102	C₂²	semidirect
34. C₃₁×C₂²⋊C₈	C₂²	semidirect
35. C₄×C₈×C₃₁	C₂²	semidirect
56. D₈×C₃₁	D₄	semidirect
100. #3273	C₃₁	direct (N × H)
119. C₂×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
120. #546	C₂×C₃₁	semidirect
121. #495	C₂×C₃₁	semidirect
122. #468	C₂×C₃₁	semidirect
123. #463	C₂×C₃₁	semidirect
124. #452	C₂×C₃₁	semidirect
125. #447	C₂×C₃₁	semidirect
126. C₄×D₈	C₂×C₃₁	semidirect
127. #436	C₂×C₃₁	semidirect
135. C₂²×D₄	C₂²×C₃₁	semidirect
137. C₂×Q₁₆	C₂²×C₃₁	semidirect
138. C₂×SD₁₆	C₂²×C₃₁	semidirect
139. C₂×SD₁₆	C₂²×C₃₁	semidirect
143. C₂×M₁₆	C₂²×C₃₁	semidirect
144. #124	C₂²×C₃₁	semidirect
146. C₂²⋊Q₈	C₂²×C₃₁	semidirect
151. C₄×D₄	C₂²×C₃₁	semidirect
153. #107	C₂²×C₃₁	semidirect
154. C₄⋊C₈	C₂²×C₃₁	semidirect
155. #103	C₂²×C₃₁	semidirect
156. #102	C₂²×C₃₁	semidirect
157. #102	C₂²×C₃₁	semidirect
158. C₂²⋊C₈	C₂²×C₃₁	semidirect
159. C₄×C₈	C₂²×C₃₁	semidirect
160. C₃₁	#3273	direct (N × H)
181. D₈	D₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#546	2
C₃₁×#495	2
C₃₁×#468	2
C₃₁×#463	2
C₃₁×#452	2
C₃₁×#447	2
C₄×D₈×C₃₁	2
C₃₁×#436	2
#3273	31

Subgroups — C31×#3273 #16843206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3273 #16843206