Subgroups — C₄²×C₂²⋊C₄×C₃₁ #21470506

Order 7936.

3342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	6	yes	maximal
3	C₂×C₄³×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₁×#95	C₂	1	6	yes	maximal, complemented
5	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
6	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
7	C₂²×C₄²×C₃₁	—	2	8	no	—
8	C₂²×C₄²×C₃₁	C₄	1	8	yes	complemented
9	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	12	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊C₄×C₃₁	C₄	1	48	yes	direct product factor
13	C₂×C₃₁×#95	C₂²	1	12	yes	—
14	C₄³×C₃₁	—	2	8	no	—
15	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	—	2	12	no	—
17	C₂³×C₄×C₃₁	—	2	12	no	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	24	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	48	yes	—
24	C₂×C₄²×C₃₁	—	2	12	no	—
25	C₂×C₄²×C₃₁	—	2	24	no	—
26	C₂×C₄²×C₃₁	—	2	24	no	—
27	C₂×C₄²×C₃₁	—	2	32	no	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	48	yes	complemented
31	C₂×C₄²×C₃₁	D₄	1	8	yes	—
32	C₃₁×#95	C₂×C₄	1	48	yes	complemented
33	C₂⁴×C₃₁	—	2	2	no	—
34	C₂⁴×C₃₁	—	2	6	no	—
35	C₂⁴×C₃₁	C₂²×C₄	1	12	yes	—
36	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
37	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	—	2	8	no	—
39	C₂²×C₄×C₃₁	—	2	6	no	—
40	C₂²×C₄×C₃₁	—	2	24	no	—
41	C₂²×C₄×C₃₁	—	2	6	no	—
42	C₂²×C₄×C₃₁	—	2	24	no	—
43	C₂²×C₄×C₃₁	—	2	24	no	—
44	C₂²×C₄×C₃₁	—	2	96	no	—
45	C₂²×C₄×C₃₁	C₄²	1	48	yes	complemented
46	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₃₁	C₂²×C₄	1	48	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₃₁	#41	1	12	yes	—
53	C₂²⋊C₄×C₃₁	C₄²	1	64	yes	direct product factor
54	C₄²×C₃₁	—	2	24	no	—
55	C₄²×C₃₁	—	2	96	no	—
56	C₄²×C₃₁	C₂²⋊C₄	1	16	yes	direct product factor
57	C₄²×C₂²⋊C₄	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₃₁	—	2	2	no	—
59	C₂³×C₃₁	—	2	6	no	—
60	C₂³×C₃₁	—	2	24	no	—
61	C₂³×C₃₁	—	2	24	no	—
62	C₂³×C₃₁	C₂×C₄²	1	16	yes	—
63	C₂³×C₃₁	C₂×C₄²	1	12	yes	—
64	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
65	C₂³×C₃₁	C₂³×C₄	1	6	yes	—
66	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
67	C₂³×C₃₁	C₂×#41	1	6	yes	—
68	C₂×C₄×C₃₁	—	2	6	no	—
69	C₂×C₄×C₃₁	—	2	6	no	—
70	C₂×C₄×C₃₁	—	2	48	no	—
71	C₂×C₄×C₃₁	—	2	24	no	—
72	C₂×C₄×C₃₁	—	2	48	no	—
73	C₂×C₄×C₃₁	—	2	96	no	—
74	C₂×C₄×C₃₁	C₂×C₄²	1	12	yes	—
75	C₂×C₄×C₃₁	C₂×C₄²	1	12	yes	—
76	C₂×C₄×C₃₁	C₂×C₄²	1	32	yes	complemented
77	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
78	C₂×C₄×C₃₁	#117	1	12	yes	—
79	C₂×C₄×C₃₁	C₄×D₄	1	48	yes	—
80	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic
81	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	6	yes	—
82	C₂×C₄³	C₂×C₃₁	1	2	yes	complemented
83	C₄×#95	C₂×C₃₁	1	6	yes	complemented
84	C₂²×C₃₁	—	2	12	no	—
85	C₂²×C₃₁	—	2	12	no	—
86	C₂²×C₃₁	—	2	32	no	—
87	C₂²×C₃₁	C₄³	1	8	yes	complemented
88	C₂²×C₃₁	C₂²×C₄²	1	3	yes	—
89	C₂²×C₃₁	C₂²×C₄²	1	4	yes	—
90	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#117	1	3	yes	—
92	C₂²×C₃₁	C₂×C₄×D₄	1	12	yes	—
93	C₂²×C₃₁	C₄×#41	1	12	yes	—
94	C₄×C₃₁	—	2	12	no	—
95	C₄×C₃₁	—	2	32	no	—
96	C₄×C₃₁	C₄×C₂²⋊C₄	1	24	yes	direct product factor
97	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
98	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
99	C₂²×C₄²	—	2	8	no	—
100	C₂²×C₄²	C₄×C₃₁	1	8	yes	complemented
101	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
102	C₂²×C₄²	C₂²×C₃₁	1	12	yes	—
103	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
104	C₄×C₂²⋊C₄	C₄×C₃₁	1	48	yes	direct product factor
105	C₂×#95	C₂²×C₃₁	1	12	yes	—
106	C₄³	—	2	8	no	—
107	C₂×C₃₁	—	2	8	no	—
108	C₂×C₃₁	C₂×C₄³	1	1	yes	characteristic
109	C₂×C₃₁	C₂×C₄×C₂²⋊C₄	1	3	yes	—
110	C₂×C₃₁	C₄×#117	1	3	yes	—
111	C₂×C₃₁	C₄²×D₄	1	8	yes	—
112	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
113	C₂³×C₄	—	2	12	no	—
114	C₂³×C₄	—	2	12	no	—
115	C₂³×C₄	C₂×C₄×C₃₁	1	24	yes	—
116	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
117	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
118	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
119	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
120	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	48	yes	—
121	C₂×C₄²	—	2	24	no	—
122	C₂×C₄²	—	2	12	no	—
123	C₂×C₄²	—	2	24	no	—
124	C₂×C₄²	—	2	32	no	—
125	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
126	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₃₁	1	48	yes	complemented
128	C₂×C₄²	D₄×C₃₁	1	8	yes	—
129	#95	C₂×C₄×C₃₁	1	48	yes	complemented
130	C₃₁	C₄²×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁴	—	2	2	no	—
132	C₂⁴	—	2	6	no	—
133	C₂⁴	C₂²×C₄×C₃₁	1	12	yes	—
134	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
135	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	6	no	—
138	C₂²×C₄	—	2	24	no	—
139	C₂²×C₄	—	2	6	no	—
140	C₂²×C₄	—	2	24	no	—
141	C₂²×C₄	—	2	24	no	—
142	C₂²×C₄	—	2	96	no	—
143	C₂²×C₄	C₄²×C₃₁	1	48	yes	complemented
144	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
145	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
146	C₂²×C₄	C₂²×C₄×C₃₁	1	48	yes	—
147	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
148	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
149	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
150	C₂²×C₄	#41×C₃₁	1	12	yes	—
151	C₂²⋊C₄	C₄²×C₃₁	1	64	yes	direct product factor
152	C₄²	—	2	24	no	—
153	C₄²	—	2	96	no	—
154	C₄²	C₂²⋊C₄×C₃₁	1	16	yes	direct product factor
155	C₂³	—	2	2	no	—
156	C₂³	—	2	6	no	—
157	C₂³	—	2	24	no	—
158	C₂³	—	2	24	no	—
159	C₂³	C₂×C₄²×C₃₁	1	16	yes	—
160	C₂³	C₂×C₄²×C₃₁	1	12	yes	—
161	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
162	C₂³	C₂³×C₄×C₃₁	1	6	yes	—
163	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
164	C₂³	C₂×#41×C₃₁	1	6	yes	—
165	C₂×C₄	—	2	6	no	—
166	C₂×C₄	—	2	6	no	—
167	C₂×C₄	—	2	48	no	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	2	48	no	—
170	C₂×C₄	—	2	96	no	—
171	C₂×C₄	C₂×C₄²×C₃₁	1	12	yes	—
172	C₂×C₄	C₂×C₄²×C₃₁	1	12	yes	—
173	C₂×C₄	C₂×C₄²×C₃₁	1	32	yes	complemented
174	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
175	C₂×C₄	C₃₁×#117	1	12	yes	—
176	C₂×C₄	C₄×D₄×C₃₁	1	48	yes	—
177	C₂²	—	2	12	no	—
178	C₂²	—	2	12	no	—
179	C₂²	—	2	32	no	—
180	C₂²	C₄³×C₃₁	1	8	yes	complemented
181	C₂²	C₂²×C₄²×C₃₁	1	3	yes	—
182	C₂²	C₂²×C₄²×C₃₁	1	4	yes	—
183	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
184	C₂²	C₂×C₃₁×#117	1	3	yes	—
185	C₂²	C₂×C₄×D₄×C₃₁	1	12	yes	—
186	C₂²	C₄×#41×C₃₁	1	12	yes	—
187	C₄	—	2	12	no	—
188	C₄	—	2	32	no	—
189	C₄	C₄×C₂²⋊C₄×C₃₁	1	24	yes	direct product factor
190	C₂	—	2	8	no	—
191	C₂	C₂×C₄³×C₃₁	1	1	yes	characteristic
192	C₂	C₂×C₄×C₂²⋊C₄×C₃₁	1	3	yes	—
193	C₂	C₄×C₃₁×#117	1	3	yes	—
194	C₂	C₄²×D₄×C₃₁	1	8	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₃₁	C₂²
commutator	C₂	C₂×C₄³×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
socle	C₂⁴×C₃₁	C₂⁴
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄³×C₃₁	C₂	semidirect
4. C₄×C₃₁×#95	C₂	semidirect
8. C₂²×C₄²×C₃₁	C₄	semidirect
12. C₄×C₂²⋊C₄×C₃₁	C₄	direct (N × H)
30. C₂×C₄²×C₃₁	C₂×C₄	semidirect
32. C₃₁×#95	C₂×C₄	semidirect
45. C₂²×C₄×C₃₁	C₄²	semidirect
53. C₂²⋊C₄×C₃₁	C₄²	direct (N × H)
56. C₄²×C₃₁	C₂²⋊C₄	direct (N × H)
57. C₄²×C₂²⋊C₄	C₃₁	direct (N × H)
76. C₂×C₄×C₃₁	C₂×C₄²	semidirect
82. C₂×C₄³	C₂×C₃₁	semidirect
83. C₄×#95	C₂×C₃₁	semidirect
87. C₂²×C₃₁	C₄³	semidirect
96. C₄×C₃₁	C₄×C₂²⋊C₄	direct (N × H)
100. C₂²×C₄²	C₄×C₃₁	semidirect
104. C₄×C₂²⋊C₄	C₄×C₃₁	direct (N × H)
127. C₂×C₄²	C₂×C₄×C₃₁	semidirect
129. #95	C₂×C₄×C₃₁	semidirect
130. C₃₁	C₄²×C₂²⋊C₄	direct (N × H)
143. C₂²×C₄	C₄²×C₃₁	semidirect
151. C₂²⋊C₄	C₄²×C₃₁	direct (N × H)
154. C₄²	C₂²⋊C₄×C₃₁	direct (N × H)
173. C₂×C₄	C₂×C₄²×C₃₁	semidirect
180. C₂²	C₄³×C₃₁	semidirect
189. C₄	C₄×C₂²⋊C₄×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₃₁	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×C₄³×C₃₁	2
C₄×C₃₁×#95	2
C₄²×C₂²⋊C₄	31

Subgroups — C42×C22⋊C4×C31 #21470506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂²⋊C₄×C₃₁ #21470506