Subgroups — C₃₁×#33289 #21490048

Order 7936.

862 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	2	yes	maximal
2	C₃₁×#1990	C₂	1	8	yes	maximal
3	C₃₁×#1834	C₂	1	4	yes	maximal
4	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#384	—	2	2	no	—
11	C₃₁×#384	—	2	2	no	—
12	C₃₁×#384	C₂²	1	4	yes	—
13	C₃₁×#384	C₂²	1	4	yes	—
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#340	C₂²	1	16	yes	—
16	C₂²×Q₈×C₃₁	—	2	4	no	—
17	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
18	C₃₁×C₄⋊Q₈	—	2	16	no	—
19	C₃₁×C₄⋊Q₈	—	4	8	no	—
20	C₃₁×C₄⋊Q₈	—	4	8	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	D₄	1	2	yes	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
32	C₄×C₈×C₃₁	—	2	8	no	—
33	C₄×C₈×C₃₁	Q₈	1	8	yes	complemented
34	C₂×Q₈×C₃₁	—	4	8	no	—
35	C₂×Q₈×C₃₁	—	4	8	no	—
36	C₂×Q₈×C₃₁	—	4	8	no	—
37	C₂²×C₄×C₃₁	—	2	8	no	—
38	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	8	no	—
44	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
45	C₄⋊C₄×C₃₁	—	2	16	no	—
46	C₄⋊C₄×C₃₁	—	4	8	no	—
47	C₄⋊C₄×C₃₁	—	4	8	no	—
48	C₄⋊C₄×C₃₁	—	4	8	no	—
49	C₄⋊C₄×C₃₁	—	4	8	no	—
50	C₄⋊C₄×C₃₁	—	4	8	no	—
51	C₄²×C₃₁	D₈	1	4	yes	—
52	C₄²×C₃₁	SD₁₆	1	8	yes	—
53	C₄²×C₃₁	Q₁₆	1	4	yes	—
54	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
55	C₄²×C₃₁	#41	1	8	yes	—
56	#33289	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₃₁	—	8	16	no	—
59	C₂×C₄×C₃₁	—	4	8	no	—
60	C₂×C₄×C₃₁	—	4	8	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	#124	1	4	yes	—
63	C₂×C₄×C₃₁	#124	1	4	yes	—
64	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
67	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
70	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
71	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
72	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
75	C₈×C₃₁	—	2	8	no	—
76	#2868	C₂×C₃₁	1	2	yes	—
77	#1990	C₂×C₃₁	1	8	yes	—
78	#1834	C₂×C₃₁	1	4	yes	—
79	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
80	C₂²×C₃₁	C₂×#124	1	2	yes	—
81	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
82	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
83	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
84	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
85	C₄×C₃₁	—	8	8	no	—
86	C₄×C₃₁	#486	1	8	yes	—
87	C₄×C₃₁	#487	1	8	yes	—
88	C₄×C₃₁	#498	1	4	yes	—
89	C₄×C₃₁	#500	1	4	yes	—
90	C₄×C₃₁	#533	1	4	yes	—
91	C₂×C₄⋊Q₈	—	2	2	no	—
92	C₂×C₄⋊Q₈	—	2	2	no	—
93	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
94	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
95	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
96	#384	—	2	2	no	—
97	#384	—	2	2	no	—
98	#384	C₂²×C₃₁	1	4	yes	—
99	#384	C₂²×C₃₁	1	4	yes	—
100	C₄³	C₂²×C₃₁	1	1	yes	characteristic
101	#340	C₂²×C₃₁	1	16	yes	—
102	C₂×C₃₁	#2862	1	1	yes	characteristic
103	C₂×C₃₁	C₂×#486	1	2	yes	—
104	C₂×C₃₁	C₂×#487	1	2	yes	—
105	C₂×C₃₁	C₂×#498	1	1	yes	characteristic
106	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
107	C₂²×Q₈	—	2	4	no	—
108	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
109	C₄⋊Q₈	—	2	16	no	—
110	C₄⋊Q₈	—	4	8	no	—
111	C₄⋊Q₈	—	4	8	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
118	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
119	C₂×C₄²	D₄×C₃₁	1	2	yes	—
120	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
123	C₄×C₈	—	2	8	no	—
124	C₄×C₈	Q₈×C₃₁	1	8	yes	complemented
125	C₃₁	#33289	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×Q₈	—	4	8	no	—
127	C₂×Q₈	—	4	8	no	—
128	C₂×Q₈	—	4	8	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
131	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
132	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
134	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
137	C₄⋊C₄	—	2	16	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄²	D₈×C₃₁	1	4	yes	—
144	C₄²	SD₁₆×C₃₁	1	8	yes	—
145	C₄²	Q₁₆×C₃₁	1	4	yes	—
146	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
147	C₄²	#41×C₃₁	1	8	yes	—
148	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
149	Q₈	—	8	16	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	C₃₁×#124	1	4	yes	—
154	C₂×C₄	C₃₁×#124	1	4	yes	—
155	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
156	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
157	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
158	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
159	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
160	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
161	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
162	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
163	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
164	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
165	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
166	C₈	—	2	8	no	—
167	C₂²	C₂×C₃₁×#124	1	2	yes	—
168	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
169	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
170	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
171	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
172	C₄	—	8	8	no	—
173	C₄	C₃₁×#486	1	8	yes	—
174	C₄	C₃₁×#487	1	8	yes	—
175	C₄	C₃₁×#498	1	4	yes	—
176	C₄	C₃₁×#500	1	4	yes	—
177	C₄	C₃₁×#533	1	4	yes	—
178	C₂	C₃₁×#2862	1	1	yes	characteristic
179	C₂	C₂×C₃₁×#486	1	2	yes	—
180	C₂	C₂×C₃₁×#487	1	2	yes	—
181	C₂	C₂×C₃₁×#498	1	1	yes	characteristic
182	C₂	C₂×C₃₁×#500	1	1	yes	characteristic

	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
33. C₄×C₈×C₃₁	Q₈	semidirect
56. #33289	C₃₁	direct (N × H)
124. C₄×C₈	Q₈×C₃₁	semidirect
125. C₃₁	#33289	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×#2868	2
C₃₁×#1990	2
C₃₁×#1834	2
C₄²×C₈×C₃₁	2
#33289	31

Subgroups — C31×#33289 #21490048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33289 #21490048