Subgroups — C₄²×#41×C₃₁ #21490752

Order 7936.

4086 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₃₁	C₂	1	4	yes	maximal, complemented
3	C₄²×D₄×C₃₁	C₂	1	12	yes	maximal, complemented
4	C₄×C₃₁×#117	C₂	1	9	yes	maximal, complemented
5	C₂×C₄³×C₃₁	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₄×#41×C₃₁	C₄	1	48	yes	direct product factor
8	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	18	yes	—
10	C₂×C₃₁×#117	C₂²	1	9	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	12	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	9	yes	—
13	C₄×C₄⋊C₄×C₃₁	C₂²	1	36	yes	—
14	C₄×C₂²⋊C₄×C₃₁	C₂²	1	36	yes	—
15	C₃₁×#376	C₂²	1	6	yes	—
16	C₄³×C₃₁	C₂²	1	9	yes	—
17	C₄³×C₃₁	C₂²	1	12	yes	—
18	C₄³×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×#41×C₃₁	C₂×C₄	1	24	yes	—
20	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	18	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	48	yes	complemented
25	C₄×D₄×C₃₁	C₂×C₄	1	144	yes	complemented
26	C₃₁×#117	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄×C₃₁	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂³	1	18	yes	—
29	C₂×C₄²×C₃₁	—	2	48	no	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	72	yes	complemented
31	C₂×C₄²×C₃₁	C₂³	1	36	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	9	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	18	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
36	C₃₁×#95	C₂³	1	24	yes	—
37	C₂⁴×C₃₁	C₂⁴	1	3	yes	—
38	#41×C₃₁	C₄²	1	64	yes	direct product factor
39	C₂×Q₈×C₃₁	C₂²×C₄	1	12	yes	—
40	C₂×D₄×C₃₁	C₂²×C₄	1	36	yes	—
41	C₂²×C₄×C₃₁	—	2	72	no	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	72	yes	—
43	C₂²×C₄×C₃₁	C₂²×C₄	1	36	yes	—
44	C₂²×C₄×C₃₁	C₂⁴	1	18	yes	—
45	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
47	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
48	C₄⋊C₄×C₃₁	C₂²×C₄	1	72	yes	—
49	C₂²⋊C₄×C₃₁	C₂²×C₄	1	72	yes	—
50	C₄²×C₃₁	—	2	72	no	—
51	C₄²×C₃₁	C₂²×C₄	1	12	yes	—
52	C₄²×C₃₁	C₂²×C₄	1	36	yes	—
53	C₄²×C₃₁	C₂²×C₄	1	72	yes	—
54	C₄²×C₃₁	#41	1	16	yes	direct product factor
55	C₄²×#41	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₃₁	—	2	12	no	—
57	C₂³×C₃₁	C₂³×C₄	1	18	yes	—
58	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
59	Q₈×C₃₁	C₂×C₄²	1	16	yes	complemented
60	D₄×C₃₁	C₂×C₄²	1	48	yes	complemented
61	C₂×C₄×C₃₁	—	2	36	no	—
62	C₂×C₄×C₃₁	—	2	144	no	—
63	C₂×C₄×C₃₁	C₂×C₄²	1	48	yes	complemented
64	C₂×C₄×C₃₁	C₂³×C₄	1	36	yes	—
65	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
66	C₂×C₄×C₃₁	C₂³×C₄	1	6	yes	—
67	C₂×C₄×C₃₁	C₂³×C₄	1	18	yes	—
68	C₂×C₄×C₃₁	C₂×#41	1	24	yes	—
69	C₂×C₄×#41	C₂×C₃₁	1	3	yes	—
70	C₄²×Q₈	C₂×C₃₁	1	4	yes	complemented
71	C₄²×D₄	C₂×C₃₁	1	12	yes	complemented
72	C₄×#117	C₂×C₃₁	1	9	yes	complemented
73	C₂×C₄³	C₂×C₃₁	1	3	yes	complemented
74	C₂²×C₃₁	—	2	36	no	—
75	C₂²×C₃₁	C₂²×C₄²	1	12	yes	—
76	C₂²×C₃₁	C₂⁴×C₄	1	3	yes	—
77	C₂²×C₃₁	C₂²×#41	1	4	yes	—
78	C₄×C₃₁	—	2	36	no	—
79	C₄×C₃₁	C₂²×C₄²	1	4	yes	—
80	C₄×C₃₁	C₂²×C₄²	1	12	yes	—
81	C₄×C₃₁	C₄×#41	1	24	yes	direct product factor
82	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
83	C₄×#41	C₄×C₃₁	1	48	yes	direct product factor
84	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
85	C₂×C₄×D₄	C₂²×C₃₁	1	18	yes	—
86	C₂×#117	C₂²×C₃₁	1	9	yes	—
87	C₂²×C₄²	C₂²×C₃₁	1	12	yes	—
88	C₂²×C₄²	C₂²×C₃₁	1	9	yes	—
89	C₄×C₄⋊C₄	C₂²×C₃₁	1	36	yes	—
90	C₄×C₂²⋊C₄	C₂²×C₃₁	1	36	yes	—
91	#376	C₂²×C₃₁	1	6	yes	—
92	C₄³	C₂²×C₃₁	1	1	yes	characteristic
93	C₄³	C₂²×C₃₁	1	9	yes	—
94	C₄³	C₂²×C₃₁	1	12	yes	—
95	C₂×C₃₁	—	2	12	no	—
96	C₂×C₃₁	C₂³×C₄²	1	1	yes	characteristic
97	C₂×C₃₁	C₂×C₄×#41	1	6	yes	—
98	C₂×#41	C₂×C₄×C₃₁	1	24	yes	—
99	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
100	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
101	C₂³×C₄	C₂³×C₃₁	1	18	yes	—
102	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
103	C₄×Q₈	C₂×C₄×C₃₁	1	48	yes	complemented
104	C₄×D₄	C₂×C₄×C₃₁	1	144	yes	complemented
105	#117	C₂×C₄×C₃₁	1	72	yes	complemented
106	C₂×C₄⋊C₄	C₂³×C₃₁	1	18	yes	—
107	C₂×C₂²⋊C₄	C₂³×C₃₁	1	18	yes	—
108	C₂×C₄²	—	2	48	no	—
109	C₂×C₄²	C₂×C₄×C₃₁	1	72	yes	complemented
110	C₂×C₄²	C₂³×C₃₁	1	36	yes	—
111	C₂×C₄²	C₂³×C₃₁	1	9	yes	—
112	C₂×C₄²	C₂³×C₃₁	1	18	yes	—
113	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
114	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
115	#95	C₂³×C₃₁	1	24	yes	—
116	C₃₁	C₄²×#41	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	C₂⁴×C₃₁	1	3	yes	—
118	#41	C₄²×C₃₁	1	64	yes	direct product factor
119	C₂×Q₈	C₂²×C₄×C₃₁	1	12	yes	—
120	C₂×D₄	C₂²×C₄×C₃₁	1	36	yes	—
121	C₂²×C₄	—	2	72	no	—
122	C₂²×C₄	C₂²×C₄×C₃₁	1	72	yes	—
123	C₂²×C₄	C₂²×C₄×C₃₁	1	36	yes	—
124	C₂²×C₄	C₂⁴×C₃₁	1	18	yes	—
125	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
127	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
128	C₄⋊C₄	C₂²×C₄×C₃₁	1	72	yes	—
129	C₂²⋊C₄	C₂²×C₄×C₃₁	1	72	yes	—
130	C₄²	—	2	72	no	—
131	C₄²	C₂²×C₄×C₃₁	1	12	yes	—
132	C₄²	C₂²×C₄×C₃₁	1	36	yes	—
133	C₄²	C₂²×C₄×C₃₁	1	72	yes	—
134	C₄²	#41×C₃₁	1	16	yes	direct product factor
135	C₂³	—	2	12	no	—
136	C₂³	C₂³×C₄×C₃₁	1	18	yes	—
137	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
138	Q₈	C₂×C₄²×C₃₁	1	16	yes	complemented
139	D₄	C₂×C₄²×C₃₁	1	48	yes	complemented
140	C₂×C₄	—	2	36	no	—
141	C₂×C₄	—	2	144	no	—
142	C₂×C₄	C₂×C₄²×C₃₁	1	48	yes	complemented
143	C₂×C₄	C₂³×C₄×C₃₁	1	36	yes	—
144	C₂×C₄	C₂³×C₄×C₃₁	1	6	yes	—
145	C₂×C₄	C₂³×C₄×C₃₁	1	18	yes	—
146	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
147	C₂×C₄	C₂×#41×C₃₁	1	24	yes	—
148	C₂²	—	2	36	no	—
149	C₂²	C₂²×C₄²×C₃₁	1	12	yes	—
150	C₂²	C₂⁴×C₄×C₃₁	1	3	yes	—
151	C₂²	C₂²×#41×C₃₁	1	4	yes	—
152	C₄	—	2	36	no	—
153	C₄	C₂²×C₄²×C₃₁	1	4	yes	—
154	C₄	C₂²×C₄²×C₃₁	1	12	yes	—
155	C₄	C₄×#41×C₃₁	1	24	yes	direct product factor
156	C₂	—	2	12	no	—
157	C₂	C₂³×C₄²×C₃₁	1	1	yes	characteristic
158	C₂	C₂×C₄×#41×C₃₁	1	6	yes	—

	subgroup	G/H
center	C₄³×C₃₁	C₂²
commutator	C₂	C₂³×C₄²×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈×C₃₁	C₂	semidirect
3. C₄²×D₄×C₃₁	C₂	semidirect
4. C₄×C₃₁×#117	C₂	semidirect
5. C₂×C₄³×C₃₁	C₂	semidirect
7. C₄×#41×C₃₁	C₄	direct (N × H)
24. C₄×Q₈×C₃₁	C₂×C₄	semidirect
25. C₄×D₄×C₃₁	C₂×C₄	semidirect
26. C₃₁×#117	C₂×C₄	semidirect
30. C₂×C₄²×C₃₁	C₂×C₄	semidirect
38. #41×C₃₁	C₄²	direct (N × H)
54. C₄²×C₃₁	#41	direct (N × H)
55. C₄²×#41	C₃₁	direct (N × H)
59. Q₈×C₃₁	C₂×C₄²	semidirect
60. D₄×C₃₁	C₂×C₄²	semidirect
63. C₂×C₄×C₃₁	C₂×C₄²	semidirect
70. C₄²×Q₈	C₂×C₃₁	semidirect
71. C₄²×D₄	C₂×C₃₁	semidirect
72. C₄×#117	C₂×C₃₁	semidirect
73. C₂×C₄³	C₂×C₃₁	semidirect
81. C₄×C₃₁	C₄×#41	direct (N × H)
83. C₄×#41	C₄×C₃₁	direct (N × H)
103. C₄×Q₈	C₂×C₄×C₃₁	semidirect
104. C₄×D₄	C₂×C₄×C₃₁	semidirect
105. #117	C₂×C₄×C₃₁	semidirect
109. C₂×C₄²	C₂×C₄×C₃₁	semidirect
116. C₃₁	C₄²×#41	direct (N × H)
118. #41	C₄²×C₃₁	direct (N × H)
134. C₄²	#41×C₃₁	direct (N × H)
138. Q₈	C₂×C₄²×C₃₁	semidirect
139. D₄	C₂×C₄²×C₃₁	semidirect
142. C₂×C₄	C₂×C₄²×C₃₁	semidirect
155. C₄	C₄×#41×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
C₂×C₄×#41×C₃₁	2
C₄²×Q₈×C₃₁	2
C₄²×D₄×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₄³×C₃₁	2
C₄²×#41	31

Subgroups — C42×#41×C31 #21490752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#41×C₃₁ #21490752