Subgroups — C₃₁×#3353 #16843286

Order 3968.

374 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#547	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₃₁×#482	C₂	1	2	yes	maximal, complemented
6	C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#468	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#428	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₃₁	—	2	1	no	—
13	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×#126	—	2	2	no	—
18	C₃₁×#124	—	2	1	no	—
19	C₃₁×#123	—	2	2	no	—
20	C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃₁×#121	C₂²	1	2	yes	—
23	C₄×Q₈×C₃₁	—	2	1	no	—
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	2	yes	—
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×#106	C₂²	1	1	yes	characteristic
33	C₃₁×#106	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	1	yes	characteristic
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#102	C₂²	1	1	yes	characteristic
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	#41×C₃₁	—	4	4	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	—	2	1	no	—
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	SD₁₆×C₃₁	—	2	2	no	—
52	M₁₆×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	1	no	—
73	C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	C₂×D₄	1	2	yes	—
76	Q₈×C₃₁	—	2	1	no	—
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	Q₈×C₃₁	—	4	2	no	—
80	D₄×C₃₁	—	2	2	no	—
81	D₄×C₃₁	—	2	1	no	—
82	D₄×C₃₁	—	2	1	no	—
83	D₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	#41	1	2	yes	complemented
100	#3353	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	4	1	no	—
111	C₄×C₃₁	—	4	1	no	—
112	C₄×C₃₁	—	4	2	no	—
113	C₄×C₃₁	—	4	2	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
116	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
118	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#547	C₂×C₃₁	1	1	yes	characteristic
120	#500	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#484	C₂×C₃₁	1	2	yes	complemented
122	#482	C₂×C₃₁	1	2	yes	complemented
123	#475	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#474	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#468	C₂×C₃₁	1	2	yes	complemented
126	C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#436	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#428	C₂×C₃₁	1	2	yes	complemented
129	C₂×C₃₁	—	4	2	no	—
130	C₂×C₃₁	—	8	1	no	—
131	C₂×C₃₁	#547	1	1	yes	characteristic
132	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₃₁	#578	1	1	yes	characteristic
134	C₂×#41	—	2	1	no	—
135	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
139	#126	—	2	2	no	—
140	#124	—	2	1	no	—
141	#123	—	2	2	no	—
142	C₂²⋊Q₈	—	2	1	no	—
143	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
144	#121	C₂²×C₃₁	1	2	yes	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
149	#117	C₂²×C₃₁	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
151	#107	C₂²×C₃₁	1	2	yes	—
152	#107	C₂²×C₃₁	1	2	yes	—
153	#107	C₂²×C₃₁	1	2	yes	—
154	#106	C₂²×C₃₁	1	1	yes	characteristic
155	#106	C₂²×C₃₁	1	2	yes	—
156	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
157	#103	C₂²×C₃₁	1	1	yes	characteristic
158	#103	C₂²×C₃₁	1	2	yes	—
159	#102	C₂²×C₃₁	1	1	yes	characteristic
160	#102	C₂²×C₃₁	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
162	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₃₁	#3353	1	1	yes	characteristic, direct product factor, Sylow
164	#41	—	4	4	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
174	SD₁₆	—	2	2	no	—
175	M₁₆	—	2	4	no	—
176	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃₁	1	2	yes	—
178	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
187	C₄⋊C₄	C₂³×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	1	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₃₁	1	1	yes	characteristic
197	C₂³	—	2	1	no	—
198	C₂³	C₂×D₄×C₃₁	1	2	yes	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	#41×C₃₁	1	2	yes	complemented
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
238	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
239	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₃₁×#547	1	1	yes	characteristic
243	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#547	C₂
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. C₃₁×#556	C₂	semidirect
3. C₃₁×#500	C₂	semidirect
4. C₃₁×#484	C₂	semidirect
5. C₃₁×#482	C₂	semidirect
6. C₃₁×#475	C₂	semidirect
7. C₃₁×#474	C₂	semidirect
8. C₃₁×#468	C₂	semidirect
9. C₄×SD₁₆×C₃₁	C₂	semidirect
10. C₃₁×#436	C₂	semidirect
11. C₃₁×#428	C₂	semidirect
99. C₈×C₃₁	#41	semidirect
100. #3353	C₃₁	direct (N × H)
118. #556	C₂×C₃₁	semidirect
120. #500	C₂×C₃₁	semidirect
121. #484	C₂×C₃₁	semidirect
122. #482	C₂×C₃₁	semidirect
123. #475	C₂×C₃₁	semidirect
124. #474	C₂×C₃₁	semidirect
125. #468	C₂×C₃₁	semidirect
126. C₄×SD₁₆	C₂×C₃₁	semidirect
127. #436	C₂×C₃₁	semidirect
128. #428	C₂×C₃₁	semidirect
163. C₃₁	#3353	direct (N × H)
222. C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#556	2
C₃₁×#547	2
C₃₁×#500	2
C₃₁×#484	2
C₃₁×#482	2
C₃₁×#475	2
C₃₁×#474	2
C₃₁×#468	2
C₄×SD₁₆×C₃₁	2
C₃₁×#436	2
C₃₁×#428	2
#3353	31

Subgroups — C31×#3353 #16843286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3353 #16843286