Subgroups — C₃₁×#33225 #21489984

Order 7936.

1006 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#487	C₂	1	4	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1956	C₂	1	8	yes	maximal
5	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃₁×#487	—	2	16	no	—
11	C₂×C₃₁×#103	C₂²	1	16	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#384	—	2	2	no	—
15	C₃₁×#384	—	2	2	no	—
16	C₃₁×#384	—	2	2	no	—
17	C₃₁×#384	C₂²	1	4	yes	—
18	C₄³×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₃₁	—	2	2	no	—
20	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
21	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
22	C₃₁×C₄⋊Q₈	—	2	8	no	—
23	C₃₁×C₄⋊Q₈	—	2	8	no	—
24	C₃₁×C₄⋊Q₈	—	2	16	no	—
25	C₃₁×C₄⋊Q₈	—	4	8	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₃₁×#103	—	2	32	no	—
37	C₄×C₈×C₃₁	—	2	8	no	—
38	C₄×C₈×C₃₁	D₄	1	8	yes	—
39	C₂×Q₈×C₃₁	—	2	8	no	—
40	C₂×Q₈×C₃₁	—	2	16	no	—
41	C₂×Q₈×C₃₁	—	4	4	no	—
42	C₂×Q₈×C₃₁	—	4	8	no	—
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₃₁	—	2	8	no	—
50	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
51	C₄⋊C₄×C₃₁	—	2	8	no	—
52	C₄⋊C₄×C₃₁	—	4	16	no	—
53	C₄⋊C₄×C₃₁	—	4	4	no	—
54	C₄⋊C₄×C₃₁	—	4	4	no	—
55	C₄⋊C₄×C₃₁	—	4	4	no	—
56	C₄⋊C₄×C₃₁	—	4	16	no	—
57	C₄²×C₃₁	SD₁₆	1	8	yes	—
58	C₄²×C₃₁	Q₁₆	1	8	yes	—
59	C₄²×C₃₁	C₂×D₄	1	4	yes	—
60	C₄²×C₃₁	#41	1	8	yes	—
61	#33225	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₃₁	—	4	16	no	—
64	Q₈×C₃₁	—	8	8	no	—
65	C₂×C₄×C₃₁	—	4	8	no	—
66	C₂×C₄×C₃₁	—	4	4	no	—
67	C₂×C₄×C₃₁	—	4	4	no	—
68	C₂×C₄×C₃₁	—	4	8	no	—
69	C₂×C₄×C₃₁	#124	1	8	yes	—
70	C₂×C₄×C₃₁	#127	1	2	yes	—
71	C₂×C₄×C₃₁	#127	1	2	yes	—
72	C₂×C₄×C₃₁	C₂×SD₁₆	1	8	yes	—
73	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
74	C₂×C₄×C₃₁	C₂×Q₁₆	1	8	yes	—
75	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
76	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
78	C₈×C₃₁	—	2	8	no	—
79	C₂×#487	C₂×C₃₁	1	4	yes	—
80	#2868	C₂×C₃₁	1	1	yes	characteristic
81	#2868	C₂×C₃₁	1	1	yes	characteristic
82	#1956	C₂×C₃₁	1	8	yes	—
83	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
84	C₂²×C₃₁	C₂×#124	1	2	yes	—
85	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
86	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
87	C₂²×C₃₁	C₂²×Q₁₆	1	2	yes	—
88	C₄×C₃₁	—	8	4	no	—
89	C₄×C₃₁	—	8	4	no	—
90	C₄×C₃₁	#487	1	16	yes	complemented
91	C₄×C₃₁	#492	1	4	yes	—
92	C₄×C₃₁	#494	1	4	yes	—
93	C₄×C₃₁	#532	1	4	yes	—
94	C₂×C₄⋊Q₈	—	2	2	no	—
95	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
96	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
97	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
98	#487	—	2	16	no	—
99	C₂×#103	C₂²×C₃₁	1	16	yes	—
100	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
101	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
102	#384	—	2	2	no	—
103	#384	—	2	2	no	—
104	#384	—	2	2	no	—
105	#384	C₂²×C₃₁	1	4	yes	—
106	C₄³	C₂²×C₃₁	1	1	yes	characteristic
107	C₂×C₃₁	#2858	1	1	yes	characteristic
108	C₂×C₃₁	C₂×#487	1	4	yes	—
109	C₂×C₃₁	C₂×#492	1	1	yes	characteristic
110	C₂×C₃₁	C₂×#494	1	1	yes	characteristic
111	C₂²×Q₈	—	2	2	no	—
112	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
113	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
114	C₄⋊Q₈	—	2	8	no	—
115	C₄⋊Q₈	—	2	8	no	—
116	C₄⋊Q₈	—	2	16	no	—
117	C₄⋊Q₈	—	4	8	no	—
118	C₂×C₄⋊C₄	—	2	8	no	—
119	C₂×C₄⋊C₄	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
124	C₂×C₄²	D₄×C₃₁	1	2	yes	—
125	C₂×C₄²	D₄×C₃₁	1	2	yes	—
126	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
128	#103	—	2	32	no	—
129	C₄×C₈	—	2	8	no	—
130	C₄×C₈	D₄×C₃₁	1	8	yes	—
131	C₃₁	#33225	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	16	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	4	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	16	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	16	no	—
150	C₄²	SD₁₆×C₃₁	1	8	yes	—
151	C₄²	Q₁₆×C₃₁	1	8	yes	—
152	C₄²	C₂×D₄×C₃₁	1	4	yes	—
153	C₄²	#41×C₃₁	1	8	yes	—
154	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
155	Q₈	—	4	16	no	—
156	Q₈	—	8	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	8	no	—
161	C₂×C₄	C₃₁×#124	1	8	yes	—
162	C₂×C₄	C₃₁×#127	1	2	yes	—
163	C₂×C₄	C₃₁×#127	1	2	yes	—
164	C₂×C₄	C₂×SD₁₆×C₃₁	1	8	yes	—
165	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
166	C₂×C₄	C₂×Q₁₆×C₃₁	1	8	yes	—
167	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
168	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
169	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	C₂×C₃₁×#124	1	2	yes	—
172	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
173	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
174	C₂²	C₂²×Q₁₆×C₃₁	1	2	yes	—
175	C₄	—	8	4	no	—
176	C₄	—	8	4	no	—
177	C₄	C₃₁×#487	1	16	yes	complemented
178	C₄	C₃₁×#492	1	4	yes	—
179	C₄	C₃₁×#494	1	4	yes	—
180	C₄	C₃₁×#532	1	4	yes	—
181	C₂	C₃₁×#2858	1	1	yes	characteristic
182	C₂	C₂×C₃₁×#487	1	4	yes	—
183	C₂	C₂×C₃₁×#492	1	1	yes	characteristic
184	C₂	C₂×C₃₁×#494	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
61. #33225	C₃₁	direct (N × H)
90. C₄×C₃₁	#487	semidirect
131. C₃₁	#33225	direct (N × H)
177. C₄	C₃₁×#487	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#487	2
C₃₁×#2868	2
C₃₁×#1956	2
C₄²×C₈×C₃₁	2
#33225	31

Subgroups — C31×#33225 #21489984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33225 #21489984