Subgroups — C₂²×#41×D₃₁ #16860507

Order 3968.

6520 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#92495	C₂	1	3	yes	maximal, complemented
3	C₂²×#92497	C₂	1	3	yes	maximal, complemented
4	C₂×#41×D₃₁	C₂	1	48	yes	maximal, direct product factor
5	C₂²×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×D₃₁	C₂	1	3	yes	maximal, complemented
7	C₂³×C₄×D₃₁	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₃₁	C₂²	1	3	yes	—
10	C₂×#41×C₃₁	C₂²	1	24	yes	complemented
11	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₃₁	C₂²	1	3	yes	complemented
13	C₂³×C₄×C₃₁	C₂²	1	3	yes	complemented
14	C₂²×C₂²⋊D₃₁	C₂²	1	6	yes	complemented
15	C₂³×Dic₃₁	C₂²	1	3	yes	complemented
16	#41×D₃₁	C₂²	1	256	yes	direct product factor
17	C₂×#92499	C₂²	1	24	yes	complemented
18	C₂×Q₈×D₃₁	C₂²	1	24	yes	complemented
19	C₂×#92497	C₂²	1	72	yes	complemented
20	C₂×D₄×D₃₁	C₂²	1	72	yes	complemented
21	C₂×#92495	C₂²	1	72	yes	complemented
22	C₂²×D₁₂₄	C₂²	1	3	yes	complemented
23	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₃₁	C₂²	1	3	yes	—
25	C₂²×C₄×D₃₁	C₂²	1	72	yes	complemented
26	C₂²×C₄×D₃₁	C₂²	1	3	yes	complemented
27	C₂²×C₄×D₃₁	C₂²	1	3	yes	complemented
28	C₂²×Dic₆₂	C₂²	1	3	yes	complemented
29	C₂⁴×C₃₁	C₂³	1	3	yes	—
30	C₂³×D₃₁	—	2	24	no	—
31	C₂³×D₃₁	C₂³	1	3	yes	—
32	C₂³×D₃₁	C₂³	1	36	yes	—
33	C₂³×D₃₁	C₂³	1	1	yes	characteristic
34	#41×C₃₁	C₂³	1	64	yes	complemented
35	C₂×Q₈×C₃₁	C₂³	1	12	yes	complemented
36	C₂×D₄×C₃₁	C₂³	1	36	yes	complemented
37	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
38	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂³	1	36	yes	complemented
40	C₂×C₂²⋊D₃₁	C₂³	1	72	yes	complemented
41	C₂²×Dic₃₁	C₂³	1	3	yes	—
42	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃₁	C₂³	1	36	yes	complemented
44	#92499	C₂³	1	64	yes	complemented
45	Q₈×D₃₁	C₂³	1	64	yes	complemented
46	#92497	C₂³	1	192	yes	complemented
47	D₄×D₃₁	C₂³	1	192	yes	complemented
48	#92495	C₂³	1	192	yes	complemented
49	C₂×D₁₂₄	C₂³	1	36	yes	complemented
50	C₂×C₄×D₃₁	C₂³	1	12	yes	—
51	C₂×C₄×D₃₁	C₂³	1	36	yes	—
52	C₂×C₄×D₃₁	C₂³	1	192	yes	complemented
53	C₂×C₄×D₃₁	C₂³	1	36	yes	complemented
54	C₂×C₄×D₃₁	C₂³	1	36	yes	complemented
55	C₂×Dic₆₂	C₂³	1	36	yes	complemented
56	C₂³×C₃₁	—	2	12	no	—
57	C₂³×C₃₁	C₂⁴	1	18	yes	—
58	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₂²×D₃₁	—	2	12	no	—
60	C₂²×D₃₁	—	2	144	no	—
61	C₂²×D₃₁	#41	1	8	yes	direct product factor
62	C₂²×D₃₁	C₂⁴	1	18	yes	—
63	C₂²×D₃₁	C₂⁴	1	48	yes	—
64	C₂²×D₃₁	C₂⁴	1	6	yes	—
65	Q₈×C₃₁	C₂⁴	1	16	yes	complemented
66	D₄×C₃₁	C₂⁴	1	48	yes	complemented
67	C₂×C₄×C₃₁	C₂⁴	1	18	yes	—
68	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
69	C₂×C₄×C₃₁	C₂⁴	1	48	yes	complemented
70	C₂²⋊D₃₁	C₂⁴	1	96	yes	complemented
71	C₂×Dic₃₁	C₂⁴	1	18	yes	—
72	C₂×Dic₃₁	C₂⁴	1	6	yes	—
73	C₂×Dic₃₁	C₂⁴	1	48	yes	complemented
74	D₁₂₄	C₂⁴	1	48	yes	complemented
75	C₄×D₃₁	C₂⁴	1	16	yes	—
76	C₄×D₃₁	C₂⁴	1	48	yes	—
77	C₄×D₃₁	C₂⁴	1	48	yes	complemented
78	C₄×D₃₁	C₂⁴	1	48	yes	complemented
79	Dic₆₂	C₂⁴	1	48	yes	complemented
80	C₂³×#41	—	31	1	no	maximal, Sylow
81	C₂²×C₃₁	—	2	36	no	—
82	C₂²×C₃₁	C₂×#41	1	4	yes	complemented
83	C₂²×C₃₁	C₂⁵	1	12	yes	—
84	C₂²×C₃₁	C₂⁵	1	3	yes	—
85	C₂×D₃₁	—	2	36	no	—
86	C₂×D₃₁	—	2	96	no	—
87	C₂×D₃₁	C₂×#41	1	24	yes	direct product factor
88	C₂×D₃₁	C₂⁵	1	12	yes	—
89	C₂×D₃₁	C₂⁵	1	4	yes	—
90	C₄×C₃₁	C₂⁵	1	12	yes	—
91	C₄×C₃₁	C₂⁵	1	4	yes	—
92	Dic₃₁	C₂⁵	1	12	yes	—
93	Dic₃₁	C₂⁵	1	4	yes	—
94	C₂²×#41	—	31	48	no	—
95	C₂²×#41	—	31	1	no	—
96	C₂²×#41	—	31	3	no	—
97	C₂²×#41	—	31	3	no	—
98	C₂²×#41	D₃₁	1	1	yes	characteristic, direct product factor
99	C₂³×Q₈	—	31	1	no	—
100	C₂³×D₄	—	31	3	no	—
101	C₂⁴×C₄	—	31	3	no	—
102	C₂×C₃₁	—	2	12	no	—
103	C₂×C₃₁	C₂²×#41	1	6	yes	complemented
104	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
105	D₃₁	—	2	12	no	—
106	D₃₁	C₂²×#41	1	8	yes	direct product factor
107	C₂⁵	—	31	3	no	—
108	C₂×#41	—	31	256	no	—
109	C₂×#41	—	31	24	no	—
110	C₂×#41	—	31	72	no	—
111	C₂×#41	—	31	72	no	—
112	C₂×#41	C₂×D₃₁	1	24	yes	direct product factor
113	C₂²×Q₈	—	31	24	no	—
114	C₂²×Q₈	—	31	3	no	—
115	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
116	C₂²×D₄	—	31	6	no	—
117	C₂²×D₄	—	31	72	no	—
118	C₂²×D₄	—	31	3	no	—
119	C₂²×D₄	C₂×D₃₁	1	3	yes	complemented
120	C₂³×C₄	—	31	3	no	—
121	C₂³×C₄	—	31	3	no	—
122	C₂³×C₄	—	31	3	no	—
123	C₂³×C₄	—	31	1	no	—
124	C₂³×C₄	—	31	3	no	—
125	C₂³×C₄	—	31	72	no	—
126	C₂³×C₄	C₂×D₃₁	1	3	yes	complemented
127	C₃₁	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
128	C₂⁴	—	31	1	no	—
129	C₂⁴	—	31	36	no	—
130	C₂⁴	—	31	3	no	—
131	C₂⁴	—	62	24	no	—
132	C₂⁴	C₂²×D₃₁	1	3	yes	—
133	#41	—	31	64	no	—
134	#41	—	31	192	no	—
135	#41	—	31	192	no	—
136	#41	C₂²×D₃₁	1	64	yes	direct product factor
137	C₂×Q₈	—	31	64	no	—
138	C₂×Q₈	—	31	36	no	—
139	C₂×Q₈	C₂²×D₃₁	1	12	yes	complemented
140	C₂×D₄	—	31	72	no	—
141	C₂×D₄	—	31	192	no	—
142	C₂×D₄	—	31	36	no	—
143	C₂×D₄	C₂²×D₃₁	1	36	yes	complemented
144	C₂²×C₄	—	31	1	no	—
145	C₂²×C₄	—	31	3	no	—
146	C₂²×C₄	—	31	36	no	—
147	C₂²×C₄	—	31	12	no	—
148	C₂²×C₄	—	31	36	no	—
149	C₂²×C₄	—	31	36	no	—
150	C₂²×C₄	—	31	36	no	—
151	C₂²×C₄	—	31	192	no	—
152	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
153	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₃₁	1	36	yes	complemented
155	C₂³	—	2	12	no	—
156	C₂³	—	31	8	no	—
157	C₂³	—	31	6	no	—
158	C₂³	—	31	48	no	—
159	C₂³	—	31	18	no	—
160	C₂³	—	62	144	no	—
161	C₂³	—	62	12	no	—
162	C₂³	C₂³×D₃₁	1	18	yes	—
163	C₂³	C₂³×D₃₁	1	1	yes	characteristic
164	Q₈	—	31	48	no	—
165	Q₈	C₂³×D₃₁	1	16	yes	complemented
166	D₄	—	31	96	no	—
167	D₄	—	31	48	no	—
168	D₄	C₂³×D₃₁	1	48	yes	complemented
169	C₂×C₄	—	31	6	no	—
170	C₂×C₄	—	31	18	no	—
171	C₂×C₄	—	31	48	no	—
172	C₂×C₄	—	31	16	no	—
173	C₂×C₄	—	31	48	no	—
174	C₂×C₄	—	31	48	no	—
175	C₂×C₄	—	31	48	no	—
176	C₂×C₄	C₂³×D₃₁	1	6	yes	—
177	C₂×C₄	C₂³×D₃₁	1	18	yes	—
178	C₂×C₄	C₂³×D₃₁	1	48	yes	complemented
179	C₂²	—	2	36	no	—
180	C₂²	—	31	24	no	—
181	C₂²	—	31	4	no	—
182	C₂²	—	31	12	no	—
183	C₂²	—	62	96	no	—
184	C₂²	—	62	36	no	—
185	C₂²	#41×D₃₁	1	4	yes	direct product factor
186	C₂²	C₂⁴×D₃₁	1	12	yes	—
187	C₂²	C₂⁴×D₃₁	1	3	yes	—
188	C₄	—	31	4	no	—
189	C₄	—	31	12	no	—
190	C₄	C₂⁴×D₃₁	1	4	yes	—
191	C₄	C₂⁴×D₃₁	1	12	yes	—
192	C₂	—	2	12	no	—
193	C₂	—	31	8	no	—
194	C₂	—	62	12	no	—
195	C₂	C₂×#41×D₃₁	1	6	yes	direct product factor
196	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator, cosocle	C₂×C₃₁	C₂⁶
Frattini	C₂	C₂⁵×D₃₁
Fitting	C₂²×#41×C₃₁	C₂
socle	C₂³×C₃₁	C₂⁴
hypercenter	C₂²×#41	D₃₁
hypocenter	C₃₁	C₂³×#41
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#92499	C₂	semidirect
2. C₂²×#92495	C₂	semidirect
3. C₂²×#92497	C₂	semidirect
4. C₂×#41×D₃₁	C₂	direct (N × H)
5. C₂²×Q₈×D₃₁	C₂	semidirect
6. C₂²×D₄×D₃₁	C₂	semidirect
7. C₂³×C₄×D₃₁	C₂	semidirect
8. C₂²×#41×C₃₁	C₂	semidirect
10. C₂×#41×C₃₁	C₂²	semidirect
11. C₂²×Q₈×C₃₁	C₂²	semidirect
12. C₂²×D₄×C₃₁	C₂²	semidirect
13. C₂³×C₄×C₃₁	C₂²	semidirect
14. C₂²×C₂²⋊D₃₁	C₂²	semidirect
15. C₂³×Dic₃₁	C₂²	semidirect
16. #41×D₃₁	C₂²	direct (N × H)
17. C₂×#92499	C₂²	semidirect
18. C₂×Q₈×D₃₁	C₂²	semidirect
19. C₂×#92497	C₂²	semidirect
20. C₂×D₄×D₃₁	C₂²	semidirect
21. C₂×#92495	C₂²	semidirect
22. C₂²×D₁₂₄	C₂²	semidirect
25. C₂²×C₄×D₃₁	C₂²	semidirect
26. C₂²×C₄×D₃₁	C₂²	semidirect
27. C₂²×C₄×D₃₁	C₂²	semidirect
28. C₂²×Dic₆₂	C₂²	semidirect
34. #41×C₃₁	C₂³	semidirect
35. C₂×Q₈×C₃₁	C₂³	semidirect
36. C₂×D₄×C₃₁	C₂³	semidirect
39. C₂²×C₄×C₃₁	C₂³	semidirect
40. C₂×C₂²⋊D₃₁	C₂³	semidirect
43. C₂²×Dic₃₁	C₂³	semidirect
44. #92499	C₂³	semidirect
45. Q₈×D₃₁	C₂³	semidirect
46. #92497	C₂³	semidirect
47. D₄×D₃₁	C₂³	semidirect
48. #92495	C₂³	semidirect
49. C₂×D₁₂₄	C₂³	semidirect
52. C₂×C₄×D₃₁	C₂³	semidirect
53. C₂×C₄×D₃₁	C₂³	semidirect
54. C₂×C₄×D₃₁	C₂³	semidirect
55. C₂×Dic₆₂	C₂³	semidirect
61. C₂²×D₃₁	#41	direct (N × H)
65. Q₈×C₃₁	C₂⁴	semidirect
66. D₄×C₃₁	C₂⁴	semidirect
69. C₂×C₄×C₃₁	C₂⁴	semidirect
70. C₂²⋊D₃₁	C₂⁴	semidirect
73. C₂×Dic₃₁	C₂⁴	semidirect
74. D₁₂₄	C₂⁴	semidirect
77. C₄×D₃₁	C₂⁴	semidirect
78. C₄×D₃₁	C₂⁴	semidirect
79. Dic₆₂	C₂⁴	semidirect
82. C₂²×C₃₁	C₂×#41	semidirect
87. C₂×D₃₁	C₂×#41	direct (N × H)
98. C₂²×#41	D₃₁	direct (N × H)
103. C₂×C₃₁	C₂²×#41	semidirect
106. D₃₁	C₂²×#41	direct (N × H)
112. C₂×#41	C₂×D₃₁	direct (N × H)
115. C₂²×Q₈	C₂×D₃₁	semidirect
119. C₂²×D₄	C₂×D₃₁	semidirect
126. C₂³×C₄	C₂×D₃₁	semidirect
127. C₃₁	C₂³×#41	semidirect
136. #41	C₂²×D₃₁	direct (N × H)
139. C₂×Q₈	C₂²×D₃₁	semidirect
143. C₂×D₄	C₂²×D₃₁	semidirect
154. C₂²×C₄	C₂²×D₃₁	semidirect
165. Q₈	C₂³×D₃₁	semidirect
168. D₄	C₂³×D₃₁	semidirect
178. C₂×C₄	C₂³×D₃₁	semidirect
185. C₂²	#41×D₃₁	direct (N × H)
195. C₂	C₂×#41×D₃₁	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×#92499	2
C₂²×#92495	2
C₂²×#92497	2
C₂×#41×D₃₁	2
C₂²×Q₈×D₃₁	2
C₂²×D₄×D₃₁	2
C₂³×C₄×D₃₁	2
C₂²×#41×C₃₁	2
C₂³×#41	31

Subgroups — C22×#41×D31 #16860507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×D₃₁ #16860507