Subgroups — C₃₁×#3381 #16843314

Order 3968.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#554	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#499	C₂	1	2	yes	maximal, complemented
4	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#479	C₂	1	2	yes	maximal, complemented
6	C₃₁×#478	C₂	1	2	yes	maximal
7	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#474	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
11	C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal
12	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊Q₈	—	2	1	no	—
14	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#125	—	2	2	no	—
17	C₃₁×#125	C₂²	1	2	yes	—
18	C₃₁×#125	C₂²	1	2	yes	—
19	C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₃₁×C₂²⋊Q₈	—	2	2	no	—
21	C₄×Q₈×C₃₁	—	2	1	no	—
22	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₃₁×#107	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	2	yes	—
30	C₃₁×#106	C₂²	1	2	yes	—
31	C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₃₁×#106	C₂²	1	2	yes	—
33	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	1	yes	characteristic
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂²	1	2	yes	—
41	C₂×Q₈×C₃₁	—	2	1	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	SD₁₆×C₃₁	Q₈	1	4	yes	complemented
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₄⋊C₄×C₃₁	—	2	1	no	—
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	2	no	—
60	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	C₂³	1	2	yes	—
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₃₁	—	2	1	no	—
73	Q₈×C₃₁	—	2	2	no	—
74	Q₈×C₃₁	—	2	1	no	—
75	Q₈×C₃₁	—	4	2	no	—
76	Q₈×C₃₁	—	8	1	no	—
77	Q₈×C₃₁	C₂×Q₈	1	2	yes	—
78	D₄×C₃₁	—	2	1	no	—
79	D₄×C₃₁	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	—	8	2	no	—
87	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	2	no	—
94	C₈×C₃₁	C₂×Q₈	1	2	yes	—
95	#3381	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₃₁	—	2	2	no	—
97	C₂²×C₃₁	—	4	1	no	—
98	C₂²×C₃₁	—	4	1	no	—
99	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₃₁	—	2	2	no	—
101	C₄×C₃₁	—	2	2	no	—
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	4	1	no	—
105	C₄×C₃₁	—	4	2	no	—
106	C₄×C₃₁	—	4	1	no	—
107	C₄×C₃₁	—	4	2	no	—
108	C₄×C₃₁	#135	1	2	yes	—
109	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
111	#557	C₂×C₃₁	1	1	yes	characteristic, complemented
112	#554	C₂×C₃₁	1	1	yes	characteristic
113	#499	C₂×C₃₁	1	2	yes	complemented
114	#498	C₂×C₃₁	1	1	yes	characteristic, complemented
115	#479	C₂×C₃₁	1	2	yes	complemented
116	#478	C₂×C₃₁	1	2	yes	—
117	#477	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#474	C₂×C₃₁	1	1	yes	characteristic
119	C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
120	C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
121	C₄×SD₁₆	C₂×C₃₁	1	2	yes	—
122	C₂×C₃₁	—	4	2	no	—
123	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
124	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
125	C₂×C₃₁	#577	1	1	yes	characteristic
126	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
129	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
130	#125	—	2	2	no	—
131	#125	C₂²×C₃₁	1	2	yes	—
132	#125	C₂²×C₃₁	1	2	yes	—
133	C₂²⋊Q₈	—	2	1	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	1	no	—
136	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
138	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
139	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₃₁	1	2	yes	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	#107	C₂²×C₃₁	1	2	yes	—
143	#107	C₂²×C₃₁	1	2	yes	—
144	#106	C₂²×C₃₁	1	2	yes	—
145	#106	C₂²×C₃₁	1	1	yes	characteristic
146	#106	C₂²×C₃₁	1	2	yes	—
147	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
149	#103	C₂²×C₃₁	1	2	yes	—
150	#103	C₂²×C₃₁	1	1	yes	characteristic
151	#102	C₂²×C₃₁	1	2	yes	—
152	#102	C₂²×C₃₁	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	C₃₁	#3381	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	SD₁₆	Q₈×C₃₁	1	4	yes	complemented
164	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₃₁	1	2	yes	—
166	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
177	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
178	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
180	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	C₂³×C₃₁	1	2	yes	—
186	C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	C₂×Q₈×C₃₁	1	2	yes	—
193	D₄	—	2	1	no	—
194	D₄	C₂×Q₈×C₃₁	1	2	yes	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×Q₈×C₃₁	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	C₃₁×#135	1	2	yes	—
223	C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
224	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
225	C₂	—	4	2	no	—
226	C₂	D₄×Q₈×C₃₁	1	1	yes	characteristic
227	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
228	C₂	C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#557	C₂	semidirect
3. C₃₁×#499	C₂	semidirect
4. C₃₁×#498	C₂	semidirect
5. C₃₁×#479	C₂	semidirect
7. C₃₁×#477	C₂	semidirect
9. C₈×Q₈×C₃₁	C₂	semidirect
48. SD₁₆×C₃₁	Q₈	semidirect
95. #3381	C₃₁	direct (N × H)
111. #557	C₂×C₃₁	semidirect
113. #499	C₂×C₃₁	semidirect
114. #498	C₂×C₃₁	semidirect
115. #479	C₂×C₃₁	semidirect
117. #477	C₂×C₃₁	semidirect
119. C₈×Q₈	C₂×C₃₁	semidirect
155. C₃₁	#3381	direct (N × H)
163. SD₁₆	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#557	2
C₃₁×#554	2
C₃₁×#499	2
C₃₁×#498	2
C₃₁×#479	2
C₃₁×#478	2
C₃₁×#477	2
C₃₁×#474	2
C₈×Q₈×C₃₁	2
C₄×SD₁₆×C₃₁	2
#3381	31

Subgroups — C31×#3381 #16843314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3381 #16843314