Subgroups — C₂×C₄×C₃₁×Q₃₂ #21490205

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₄×C₃₁×Q₃₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#358	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₄	1	8	yes	direct product factor
10	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×Q₁₆×C₃₁	—	2	4	no	—
14	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	—	2	2	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#366	C₂²	1	4	yes	—
19	C₃₁×#358	C₂²	1	8	yes	—
20	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
21	C₂²×Q₈×C₃₁	—	2	2	no	—
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	4	no	—
24	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
25	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₃₁	—	2	8	no	—
29	C₄×Q₈×C₃₁	—	4	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
32	C₂×C₄²×C₃₁	—	4	2	no	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₃₁×Q₃₂	C₂×C₄	1	16	yes	direct product factor
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
38	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#107	—	2	2	no	—
40	C₃₁×#107	C₂³	1	4	yes	—
41	C₃₁×#103	—	2	8	no	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	C₄×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂×Q₈×C₃₁	—	2	8	no	—
45	C₂×Q₈×C₃₁	—	2	4	no	—
46	C₂×Q₈×C₃₁	—	4	2	no	—
47	C₂×Q₈×C₃₁	—	4	4	no	—
48	C₂²×C₄×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₃₁	—	2	8	no	—
54	Q₁₆×C₃₁	—	2	4	no	—
55	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	#41	1	2	yes	—
61	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	8	no	—
65	C₄²×C₃₁	—	4	8	no	—
66	C₄²×C₃₁	D₈	1	2	yes	—
67	C₄²×C₃₁	C₂×D₄	1	2	yes	—
68	C₁₆×C₃₁	—	2	2	no	—
69	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
70	C₂×C₄×Q₃₂	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
72	Q₈×C₃₁	—	2	8	no	—
73	Q₈×C₃₁	—	4	4	no	—
74	Q₈×C₃₁	—	4	8	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	—	4	8	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	8	4	no	—
80	C₂×C₄×C₃₁	Q₃₂	1	4	yes	direct product factor
81	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₃₁	#135	1	2	yes	—
86	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	C₄×D₄	1	2	yes	—
89	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
91	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
92	C₂×C₄×Q₁₆	C₂×C₃₁	1	2	yes	—
93	C₄×Q₃₂	C₂×C₃₁	1	8	yes	direct product factor
94	C₂×#366	C₂×C₃₁	1	1	yes	characteristic
95	C₂×#358	C₂×C₃₁	1	2	yes	—
96	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
97	C₂²×C₃₁	C₄×D₈	1	2	yes	—
98	C₂²×C₃₁	C₂×Q₃₂	1	2	yes	—
99	C₂²×C₃₁	#508	1	2	yes	—
100	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₃₁	—	2	2	no	—
102	C₄×C₃₁	—	4	8	no	—
103	C₄×C₃₁	—	8	4	no	—
104	C₄×C₃₁	C₄×D₈	1	2	yes	—
105	C₄×C₃₁	C₂×Q₃₂	1	4	yes	direct product factor
106	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
108	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
109	C₂×C₄×Q₈	—	2	2	no	—
110	C₂×Q₃₂	C₄×C₃₁	1	8	yes	direct product factor
111	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
112	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
113	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
114	C₄×Q₁₆	—	2	4	no	—
115	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
116	C₂×#107	C₂²×C₃₁	1	2	yes	—
117	C₂×#103	—	2	2	no	—
118	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
119	#366	C₂²×C₃₁	1	4	yes	—
120	#358	C₂²×C₃₁	1	8	yes	—
121	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
122	C₂×C₃₁	C₄×Q₃₂	1	4	yes	direct product factor
123	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
124	C₂×C₃₁	C₂²×Q₃₂	1	1	yes	characteristic
125	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	—
130	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
131	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
133	C₄×Q₈	—	2	8	no	—
134	C₄×Q₈	—	4	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄²	—	4	2	no	—
138	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
139	Q₃₂	C₂×C₄×C₃₁	1	16	yes	direct product factor
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
142	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
143	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
144	#107	—	2	2	no	—
145	#107	C₂³×C₃₁	1	4	yes	—
146	#103	—	2	8	no	—
147	C₄×C₈	D₄×C₃₁	1	2	yes	—
148	C₄×C₈	C₂³×C₃₁	1	2	yes	—
149	C₃₁	C₂×C₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	—	2	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
157	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₂×C₈	#41×C₃₁	1	2	yes	—
167	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄²	—	4	8	no	—
172	C₄²	D₈×C₃₁	1	2	yes	—
173	C₄²	C₂×D₄×C₃₁	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
176	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	Q₈	—	4	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	C₃₁×Q₃₂	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
187	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
190	C₂×C₄	C₃₁×#135	1	2	yes	—
191	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₄×C₃₁	1	2	yes	—
194	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
195	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
196	C₂²	C₄×D₈×C₃₁	1	2	yes	—
197	C₂²	C₂×C₃₁×Q₃₂	1	2	yes	—
198	C₂²	C₃₁×#508	1	2	yes	—
199	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	8	no	—
202	C₄	—	8	4	no	—
203	C₄	C₄×D₈×C₃₁	1	2	yes	—
204	C₄	C₂×C₃₁×Q₃₂	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
206	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
207	C₂	C₄×C₃₁×Q₃₂	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
209	C₂	C₂²×C₃₁×Q₃₂	1	1	yes	characteristic
210	C₂	C₂×C₃₁×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₃₁×Q₃₂	C₂	direct (N × H)
9. C₂×C₃₁×Q₃₂	C₄	direct (N × H)
34. C₃₁×Q₃₂	C₂×C₄	direct (N × H)
70. C₂×C₄×Q₃₂	C₃₁	direct (N × H)
80. C₂×C₄×C₃₁	Q₃₂	direct (N × H)
93. C₄×Q₃₂	C₂×C₃₁	direct (N × H)
105. C₄×C₃₁	C₂×Q₃₂	direct (N × H)
110. C₂×Q₃₂	C₄×C₃₁	direct (N × H)
122. C₂×C₃₁	C₄×Q₃₂	direct (N × H)
139. Q₃₂	C₂×C₄×C₃₁	direct (N × H)
149. C₃₁	C₂×C₄×Q₃₂	direct (N × H)
185. C₂×C₄	C₃₁×Q₃₂	direct (N × H)
204. C₄	C₂×C₃₁×Q₃₂	direct (N × H)
207. C₂	C₄×C₃₁×Q₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×Q₃₂	2
C₂×C₄×Q₁₆×C₃₁	2
C₄×C₃₁×Q₃₂	2
C₂×C₃₁×#366	2
C₂×C₃₁×#358	2
C₂×C₄×C₁₆×C₃₁	2
C₂×C₄×Q₃₂	31

Subgroups — C2×C4×C31×Q32 #21490205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₁×Q₃₂ #21490205