Subgroups — C₃₁×#3374 #16843307

Order 3968.

326 conjugacy classes of subgroups.

The subgroup lattice has 224 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
2	C₃₁×#557	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#491	C₂	1	2	yes	maximal
4	C₃₁×#487	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#479	C₂	1	2	yes	maximal
6	C₃₁×#475	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#462	C₂	1	2	yes	maximal
8	C₃₁×#462	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#446	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#441	C₂	1	2	yes	maximal
11	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×C₄⋊Q₈	—	2	1	no	—
15	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#125	—	2	2	no	—
19	C₃₁×#125	—	2	1	no	—
20	C₃₁×#125	—	2	2	no	—
21	C₃₁×#125	C₂²	1	2	yes	—
22	C₄×Q₈×C₃₁	—	2	1	no	—
23	C₄×Q₈×C₃₁	—	2	1	no	—
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
27	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#106	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₃₁×#103	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#103	C₂²	1	2	yes	—
39	C₃₁×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	1	no	—
42	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	Q₁₆×C₃₁	—	2	2	no	—
47	Q₁₆×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	2	yes	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₄²×C₃₁	C₂³	1	2	yes	—
73	Q₈×C₃₁	—	2	2	no	—
74	Q₈×C₃₁	—	2	1	no	—
75	Q₈×C₃₁	—	2	1	no	—
76	Q₈×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	—	2	1	no	—
78	Q₈×C₃₁	—	2	2	no	—
79	Q₈×C₃₁	—	4	2	no	—
80	Q₈×C₃₁	#41	1	2	yes	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	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	—	2	1	no	—
95	#3374	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₃₁	—	2	2	no	—
98	C₄×C₃₁	—	2	2	no	—
99	C₄×C₃₁	—	2	2	no	—
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	4	2	no	—
102	C₄×C₃₁	—	4	1	no	—
103	C₄×C₃₁	—	4	2	no	—
104	C₄×C₃₁	—	4	1	no	—
105	C₄×C₃₁	—	4	1	no	—
106	C₄×C₃₁	#137	1	2	yes	—
107	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
108	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
109	#557	C₂×C₃₁	1	1	yes	characteristic
110	#557	C₂×C₃₁	1	1	yes	characteristic
111	#491	C₂×C₃₁	1	2	yes	—
112	#487	C₂×C₃₁	1	1	yes	characteristic
113	#479	C₂×C₃₁	1	2	yes	—
114	#475	C₂×C₃₁	1	1	yes	characteristic
115	#462	C₂×C₃₁	1	2	yes	—
116	#462	C₂×C₃₁	1	1	yes	characteristic
117	#446	C₂×C₃₁	1	1	yes	characteristic
118	#441	C₂×C₃₁	1	2	yes	—
119	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
120	C₂×C₃₁	#548	1	1	yes	characteristic
121	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
122	C₂×C₃₁	#578	1	1	yes	characteristic
123	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
129	#125	—	2	2	no	—
130	#125	—	2	1	no	—
131	#125	—	2	2	no	—
132	#125	C₂²×C₃₁	1	2	yes	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
136	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
138	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
139	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
140	#107	C₂²×C₃₁	1	1	yes	characteristic
141	#106	C₂²×C₃₁	1	2	yes	—
142	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
144	#103	C₂²×C₃₁	1	1	yes	characteristic
145	#103	C₂²×C₃₁	1	1	yes	characteristic
146	#103	C₂²×C₃₁	1	2	yes	—
147	#103	C₂²×C₃₁	1	2	yes	—
148	#103	C₂²×C₃₁	1	2	yes	—
149	#103	C₂²×C₃₁	1	2	yes	—
150	#97	C₂²×C₃₁	1	2	yes	—
151	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₃₁	#3374	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
155	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
157	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
158	Q₁₆	—	2	2	no	—
159	Q₁₆	—	2	4	no	—
160	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
162	C₂×C₈	C₂³×C₃₁	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
175	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
176	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
177	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₃₁	1	1	yes	characteristic
184	C₄²	C₂³×C₃₁	1	2	yes	—
185	Q₈	—	2	2	no	—
186	Q₈	—	2	1	no	—
187	Q₈	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	#41×C₃₁	1	2	yes	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
200	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	4	1	no	—
217	C₄	C₃₁×#137	1	2	yes	—
218	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
219	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂	C₃₁×#548	1	1	yes	characteristic
221	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
222	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
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
95. #3374	C₃₁	direct (N × H)
152. C₃₁	#3374	direct (N × H)

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×#491	2
C₃₁×#487	2
C₃₁×#479	2
C₃₁×#475	2
C₃₁×#462	2
C₃₁×#446	2
C₃₁×#441	2
C₄×Q₁₆×C₃₁	2
#3374	31

Subgroups — C31×#3374 #16843307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3374 #16843307