Subgroups — C₃₁×#33083 #21489842

Order 7936.

732 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×#2863	C₂	1	1	yes	maximal, characteristic
2	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1928	C₂	1	3	yes	maximal
4	C₃₁×#1921	C₂	1	3	yes	maximal
5	C₃₁×#1878	C₂	1	3	yes	maximal
6	C₃₁×#1863	C₂	1	3	yes	maximal
7	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
10	C₂×C₃₁×#107	C₂²	1	3	yes	—
11	C₂×C₃₁×#103	C₂²	1	3	yes	—
12	C₂×C₃₁×#103	C₂²	1	3	yes	—
13	C₃₁×#391	C₂²	1	3	yes	—
14	C₃₁×#389	C₂²	1	3	yes	—
15	C₃₁×#387	—	2	3	no	—
16	C₃₁×#383	—	2	1	no	—
17	C₄×C₄⋊C₄×C₃₁	—	2	3	no	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
19	C₄³×C₃₁	—	2	1	no	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	C₂²	1	3	yes	—
22	C₃₁×#340	C₂²	1	3	yes	—
23	C₃₁×#336	C₂²	1	3	yes	—
24	C₃₁×#336	C₂²	1	3	yes	—
25	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₃₁	—	2	6	no	—
28	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₃₁	—	2	6	no	—
31	C₄×Q₈×C₃₁	—	2	6	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
34	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
35	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
36	C₂×C₄²×C₃₁	—	2	3	no	—
37	C₂×C₄²×C₃₁	—	2	3	no	—
38	C₂×C₄²×C₃₁	D₄	1	3	yes	—
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
41	C₃₁×#107	—	2	6	no	—
42	C₃₁×#103	—	2	6	no	—
43	C₃₁×#103	—	2	6	no	—
44	C₃₁×#95	—	2	3	no	—
45	C₃₁×#95	—	2	3	no	—
46	C₃₁×#95	—	2	3	no	—
47	C₃₁×#95	—	2	3	no	—
48	C₂×Q₈×C₃₁	—	2	3	no	—
49	C₂×Q₈×C₃₁	—	2	3	no	—
50	C₂×Q₈×C₃₁	—	2	3	no	—
51	C₂×Q₈×C₃₁	#41	1	6	yes	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	3	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	3	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₃₁	—	4	4	no	—
60	C₂×C₈×C₃₁	—	2	3	no	—
61	C₂×C₈×C₃₁	—	2	3	no	—
62	C₂×C₈×C₃₁	—	2	3	no	—
63	C₂×C₈×C₃₁	—	2	3	no	—
64	C₄⋊C₄×C₃₁	—	2	12	no	—
65	C₄⋊C₄×C₃₁	—	2	3	no	—
66	C₄⋊C₄×C₃₁	—	2	3	no	—
67	C₄⋊C₄×C₃₁	—	2	3	no	—
68	C₄⋊C₄×C₃₁	—	4	6	no	—
69	C₄⋊C₄×C₃₁	#41	1	6	yes	—
70	C₄²×C₃₁	—	2	12	no	—
71	C₄²×C₃₁	—	2	3	no	—
72	C₄²×C₃₁	—	2	3	no	—
73	C₄²×C₃₁	—	4	3	no	—
74	C₄²×C₃₁	—	4	6	no	—
75	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
76	#33083	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	6	no	—
79	Q₈×C₃₁	—	4	1	no	—
80	Q₈×C₃₁	—	4	3	no	—
81	Q₈×C₃₁	#126	1	4	yes	—
82	C₂×C₄×C₃₁	—	2	3	no	—
83	C₂×C₄×C₃₁	—	2	6	no	—
84	C₂×C₄×C₃₁	—	2	3	no	—
85	C₂×C₄×C₃₁	—	2	3	no	—
86	C₂×C₄×C₃₁	—	2	6	no	—
87	C₂×C₄×C₃₁	—	2	3	no	—
88	C₂×C₄×C₃₁	—	2	3	no	—
89	C₂×C₄×C₃₁	—	4	6	no	—
90	C₂×C₄×C₃₁	—	4	3	no	—
91	C₂×C₄×C₃₁	—	4	3	no	—
92	C₂×C₄×C₃₁	—	4	6	no	—
93	C₂×C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
94	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
95	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
96	C₈×C₃₁	—	4	1	no	—
97	C₈×C₃₁	—	4	3	no	—
98	#2863	C₂×C₃₁	1	1	yes	characteristic
99	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
100	#1928	C₂×C₃₁	1	3	yes	—
101	#1921	C₂×C₃₁	1	3	yes	—
102	#1878	C₂×C₃₁	1	3	yes	—
103	#1863	C₂×C₃₁	1	3	yes	—
104	#1839	C₂×C₃₁	1	1	yes	characteristic
105	C₂²×C₃₁	#525	1	3	yes	—
106	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
107	C₂²×C₃₁	#575	1	3	yes	—
108	C₄×C₃₁	—	2	6	no	—
109	C₄×C₃₁	—	2	4	no	—
110	C₄×C₃₁	—	4	3	no	—
111	C₄×C₃₁	—	4	6	no	—
112	C₄×C₃₁	—	8	1	no	—
113	C₄×C₃₁	—	8	3	no	—
114	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
115	C₄×C₃₁	#529	1	3	yes	—
116	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
117	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
118	C₂×#107	C₂²×C₃₁	1	3	yes	—
119	C₂×#103	C₂²×C₃₁	1	3	yes	—
120	C₂×#103	C₂²×C₃₁	1	3	yes	—
121	#391	C₂²×C₃₁	1	3	yes	—
122	#389	C₂²×C₃₁	1	3	yes	—
123	#387	—	2	3	no	—
124	#383	—	2	1	no	—
125	C₄×C₄⋊C₄	—	2	3	no	—
126	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
127	C₄³	—	2	1	no	—
128	C₄³	C₂²×C₃₁	1	1	yes	characteristic
129	#340	C₂²×C₃₁	1	3	yes	—
130	#340	C₂²×C₃₁	1	3	yes	—
131	#336	C₂²×C₃₁	1	3	yes	—
132	#336	C₂²×C₃₁	1	3	yes	—
133	C₂×C₃₁	#2852	1	1	yes	characteristic
134	C₂×C₃₁	#3104	1	3	yes	—
135	C₂×C₃₁	#3111	1	3	yes	—
136	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
137	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
138	C₂×Q₁₆	—	2	6	no	—
139	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
141	C₄×Q₈	—	2	6	no	—
142	C₄×Q₈	—	2	6	no	—
143	C₂×C₄⋊C₄	—	2	3	no	—
144	C₂×C₄⋊C₄	—	2	3	no	—
145	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
146	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
147	C₂×C₄²	—	2	3	no	—
148	C₂×C₄²	—	2	3	no	—
149	C₂×C₄²	D₄×C₃₁	1	3	yes	—
150	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
152	#107	—	2	6	no	—
153	#103	—	2	6	no	—
154	#103	—	2	6	no	—
155	#95	—	2	3	no	—
156	#95	—	2	3	no	—
157	#95	—	2	3	no	—
158	#95	—	2	3	no	—
159	C₃₁	#33083	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	3	no	—
161	C₂×Q₈	—	2	3	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	#41×C₃₁	1	6	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
171	Q₁₆	—	4	4	no	—
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₄⋊C₄	—	2	12	no	—
177	C₄⋊C₄	—	2	3	no	—
178	C₄⋊C₄	—	2	3	no	—
179	C₄⋊C₄	—	2	3	no	—
180	C₄⋊C₄	—	4	6	no	—
181	C₄⋊C₄	#41×C₃₁	1	6	yes	—
182	C₄²	—	2	12	no	—
183	C₄²	—	2	3	no	—
184	C₄²	—	2	3	no	—
185	C₄²	—	4	3	no	—
186	C₄²	—	4	6	no	—
187	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
188	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
189	Q₈	—	2	6	no	—
190	Q₈	—	4	1	no	—
191	Q₈	—	4	3	no	—
192	Q₈	C₃₁×#126	1	4	yes	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	4	6	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	3	no	—
203	C₂×C₄	—	4	6	no	—
204	C₂×C₄	C₂×Q₁₆×C₃₁	1	6	yes	—
205	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
206	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
207	C₈	—	4	1	no	—
208	C₈	—	4	3	no	—
209	C₂²	C₃₁×#525	1	3	yes	—
210	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
211	C₂²	C₃₁×#575	1	3	yes	—
212	C₄	—	2	6	no	—
213	C₄	—	2	4	no	—
214	C₄	—	4	3	no	—
215	C₄	—	4	6	no	—
216	C₄	—	8	1	no	—
217	C₄	—	8	3	no	—
218	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
219	C₄	C₃₁×#529	1	3	yes	—
220	C₂	C₃₁×#2852	1	1	yes	characteristic
221	C₂	C₃₁×#3104	1	3	yes	—
222	C₂	C₃₁×#3111	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
75. C₄²×C₃₁	Q₁₆	semidirect
76. #33083	C₃₁	direct (N × H)
159. C₃₁	#33083	direct (N × H)
187. C₄²	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2863	2
C₄²×Q₈×C₃₁	2
C₃₁×#1928	2
C₃₁×#1921	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1839	2
#33083	31

Subgroups — C31×#33083 #21489842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33083 #21489842