Subgroups — C₂³×#131×C₇₉ #34174762

Order 20224.

11422 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#131×C₇₉	C₂	1	56	yes	maximal, direct product factor
3	C₂³×M₁₆×C₇₉	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₈×C₇₉	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₇₉	C₄	1	28	yes	—
6	C₂²×#41×C₇₉	C₂²	1	28	yes	—
7	C₂³×Q₈×C₇₉	C₄	1	1	yes	characteristic
8	C₂³×D₄×C₇₉	C₄	1	3	yes	—
9	C₂⁴×C₄×C₇₉	C₂²	1	3	yes	—
10	C₂×#131×C₇₉	C₂²	1	448	yes	direct product factor
11	C₂²×M₁₆×C₇₉	C₂²	1	84	yes	complemented
12	C₂³×C₈×C₇₉	C₂²	1	3	yes	—
13	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	84	yes	complemented
15	C₂⁵×C₇₉	C₂×C₄	1	3	yes	—
16	C₂×#41×C₇₉	C₂×C₄	1	336	yes	—
17	C₂×#41×C₇₉	C₂³	1	112	yes	—
18	C₂²×Q₈×C₇₉	C₂×C₄	1	28	yes	—
19	C₂²×D₄×C₇₉	C₂×C₄	1	84	yes	—
20	C₂³×C₄×C₇₉	C₂×C₄	1	3	yes	—
21	C₂³×C₄×C₇₉	C₂×C₄	1	42	yes	—
22	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₇₉	C₂³	1	42	yes	—
24	#131×C₇₉	C₂³	1	512	yes	direct product factor
25	C₂×M₁₆×C₇₉	C₂³	1	336	yes	complemented
26	C₂²×C₈×C₇₉	C₂³	1	42	yes	—
27	C₂²×C₈×C₇₉	C₂³	1	14	yes	—
28	C₂²×C₈×C₇₉	C₂³	1	336	yes	complemented
29	C₂⁴×C₇₉	—	2	24	no	—
30	C₂⁴×C₇₉	C₂²×C₄	1	42	yes	—
31	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
32	#41×C₇₉	C₂²×C₄	1	448	yes	—
33	#41×C₇₉	C₂⁴	1	64	yes	—
34	C₂×Q₈×C₇₉	C₂²×C₄	1	112	yes	—
35	C₂×D₄×C₇₉	C₂²×C₄	1	336	yes	—
36	C₂²×C₄×C₇₉	C₂²×C₄	1	7	yes	—
37	C₂²×C₄×C₇₉	C₂²×C₄	1	42	yes	—
38	C₂²×C₄×C₇₉	C₂²×C₄	1	252	yes	—
39	C₂²×C₄×C₇₉	C₂⁴	1	7	yes	—
40	C₂²×C₄×C₇₉	C₂⁴	1	84	yes	—
41	M₁₆×C₇₉	C₂⁴	1	192	yes	complemented
42	C₂×C₈×C₇₉	C₂⁴	1	28	yes	—
43	C₂×C₈×C₇₉	C₂⁴	1	84	yes	—
44	C₂×C₈×C₇₉	C₂⁴	1	192	yes	complemented
45	C₂³×C₇₉	—	2	168	no	—
46	C₂³×C₇₉	#131	1	8	yes	direct product factor
47	C₂³×C₇₉	C₂³×C₄	1	84	yes	—
48	C₂³×C₇₉	C₂³×C₄	1	7	yes	—
49	Q₈×C₇₉	C₂³×C₄	1	64	yes	—
50	D₄×C₇₉	C₂³×C₄	1	192	yes	—
51	C₂×C₄×C₇₉	C₂³×C₄	1	21	yes	—
52	C₂×C₄×C₇₉	C₂³×C₄	1	84	yes	—
53	C₂×C₄×C₇₉	C₂³×C₄	1	168	yes	—
54	C₂×C₄×C₇₉	C₂⁵	1	7	yes	—
55	C₂×C₄×C₇₉	C₂⁵	1	24	yes	—
56	C₈×C₇₉	C₂⁵	1	8	yes	—
57	C₈×C₇₉	C₂⁵	1	24	yes	—
58	C₂²×C₇₉	—	2	168	no	—
59	C₂²×C₇₉	C₂×#131	1	28	yes	direct product factor
60	C₂²×C₇₉	C₂⁴×C₄	1	24	yes	—
61	C₂²×C₇₉	C₂⁴×C₄	1	7	yes	—
62	C₄×C₇₉	C₂⁴×C₄	1	7	yes	—
63	C₄×C₇₉	C₂⁴×C₄	1	24	yes	—
64	C₄×C₇₉	C₂⁶	1	1	yes	characteristic
65	C₂³×#131	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₇₉	—	2	24	no	—
67	C₂×C₇₉	C₂²×#131	1	14	yes	direct product factor
68	C₂×C₇₉	C₂⁵×C₄	1	1	yes	characteristic
69	C₂³×#41	C₂×C₇₉	1	1	yes	characteristic
70	C₂²×#131	C₂×C₇₉	1	56	yes	direct product factor
71	C₂³×M₁₆	C₂×C₇₉	1	3	yes	complemented
72	C₂⁴×C₈	C₂×C₇₉	1	3	yes	complemented
73	C₇₉	C₂³×#131	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×#41	C₄×C₇₉	1	28	yes	—
75	C₂²×#41	C₂²×C₇₉	1	28	yes	—
76	C₂³×Q₈	C₄×C₇₉	1	1	yes	characteristic
77	C₂³×D₄	C₄×C₇₉	1	3	yes	—
78	C₂⁴×C₄	C₂²×C₇₉	1	3	yes	—
79	C₂×#131	C₂²×C₇₉	1	448	yes	direct product factor
80	C₂²×M₁₆	C₂²×C₇₉	1	84	yes	complemented
81	C₂³×C₈	C₂²×C₇₉	1	3	yes	—
82	C₂³×C₈	C₂²×C₇₉	1	1	yes	characteristic
83	C₂³×C₈	C₂²×C₇₉	1	84	yes	complemented
84	C₂⁵	C₂×C₄×C₇₉	1	3	yes	—
85	C₂×#41	C₂×C₄×C₇₉	1	336	yes	—
86	C₂×#41	C₂³×C₇₉	1	112	yes	—
87	C₂²×Q₈	C₂×C₄×C₇₉	1	28	yes	—
88	C₂²×D₄	C₂×C₄×C₇₉	1	84	yes	—
89	C₂³×C₄	C₂×C₄×C₇₉	1	3	yes	—
90	C₂³×C₄	C₂×C₄×C₇₉	1	42	yes	—
91	C₂³×C₄	C₂³×C₇₉	1	1	yes	characteristic
92	C₂³×C₄	C₂³×C₇₉	1	42	yes	—
93	#131	C₂³×C₇₉	1	512	yes	direct product factor
94	C₂×M₁₆	C₂³×C₇₉	1	336	yes	complemented
95	C₂²×C₈	C₂³×C₇₉	1	14	yes	—
96	C₂²×C₈	C₂³×C₇₉	1	42	yes	—
97	C₂²×C₈	C₂³×C₇₉	1	336	yes	complemented
98	C₂⁴	—	2	24	no	—
99	C₂⁴	C₂²×C₄×C₇₉	1	42	yes	—
100	C₂⁴	C₂²×C₄×C₇₉	1	1	yes	characteristic
101	#41	C₂²×C₄×C₇₉	1	448	yes	—
102	#41	C₂⁴×C₇₉	1	64	yes	—
103	C₂×Q₈	C₂²×C₄×C₇₉	1	112	yes	—
104	C₂×D₄	C₂²×C₄×C₇₉	1	336	yes	—
105	C₂²×C₄	C₂²×C₄×C₇₉	1	7	yes	—
106	C₂²×C₄	C₂²×C₄×C₇₉	1	42	yes	—
107	C₂²×C₄	C₂²×C₄×C₇₉	1	252	yes	—
108	C₂²×C₄	C₂⁴×C₇₉	1	7	yes	—
109	C₂²×C₄	C₂⁴×C₇₉	1	84	yes	—
110	M₁₆	C₂⁴×C₇₉	1	192	yes	complemented
111	C₂×C₈	C₂⁴×C₇₉	1	28	yes	—
112	C₂×C₈	C₂⁴×C₇₉	1	84	yes	—
113	C₂×C₈	C₂⁴×C₇₉	1	192	yes	complemented
114	C₂³	—	2	168	no	—
115	C₂³	#131×C₇₉	1	8	yes	direct product factor
116	C₂³	C₂³×C₄×C₇₉	1	84	yes	—
117	C₂³	C₂³×C₄×C₇₉	1	7	yes	—
118	Q₈	C₂³×C₄×C₇₉	1	64	yes	—
119	D₄	C₂³×C₄×C₇₉	1	192	yes	—
120	C₂×C₄	C₂³×C₄×C₇₉	1	21	yes	—
121	C₂×C₄	C₂³×C₄×C₇₉	1	84	yes	—
122	C₂×C₄	C₂³×C₄×C₇₉	1	168	yes	—
123	C₂×C₄	C₂⁵×C₇₉	1	7	yes	—
124	C₂×C₄	C₂⁵×C₇₉	1	24	yes	—
125	C₈	C₂⁵×C₇₉	1	8	yes	—
126	C₈	C₂⁵×C₇₉	1	24	yes	—
127	C₂²	—	2	168	no	—
128	C₂²	C₂×#131×C₇₉	1	28	yes	direct product factor
129	C₂²	C₂⁴×C₄×C₇₉	1	24	yes	—
130	C₂²	C₂⁴×C₄×C₇₉	1	7	yes	—
131	C₄	C₂⁴×C₄×C₇₉	1	7	yes	—
132	C₄	C₂⁴×C₄×C₇₉	1	24	yes	—
133	C₄	C₂⁶×C₇₉	1	1	yes	characteristic
134	C₂	—	2	24	no	—
135	C₂	C₂²×#131×C₇₉	1	14	yes	direct product factor
136	C₂	C₂⁵×C₄×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈×C₇₉	C₂²
commutator	C₂	C₂⁵×C₄×C₇₉
Frattini, cosocle	C₄	C₂⁶×C₇₉
socle	C₂⁴×C₇₉	C₂²×C₄
prime radical	C₂³×#41×C₇₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#131×C₇₉	C₂	direct (N × H)
3. C₂³×M₁₆×C₇₉	C₂	semidirect
4. C₂⁴×C₈×C₇₉	C₂	semidirect
10. C₂×#131×C₇₉	C₂²	direct (N × H)
11. C₂²×M₁₆×C₇₉	C₂²	semidirect
14. C₂³×C₈×C₇₉	C₂²	semidirect
24. #131×C₇₉	C₂³	direct (N × H)
25. C₂×M₁₆×C₇₉	C₂³	semidirect
28. C₂²×C₈×C₇₉	C₂³	semidirect
41. M₁₆×C₇₉	C₂⁴	semidirect
44. C₂×C₈×C₇₉	C₂⁴	semidirect
46. C₂³×C₇₉	#131	direct (N × H)
59. C₂²×C₇₉	C₂×#131	direct (N × H)
65. C₂³×#131	C₇₉	direct (N × H)
67. C₂×C₇₉	C₂²×#131	direct (N × H)
70. C₂²×#131	C₂×C₇₉	direct (N × H)
71. C₂³×M₁₆	C₂×C₇₉	semidirect
72. C₂⁴×C₈	C₂×C₇₉	semidirect
73. C₇₉	C₂³×#131	direct (N × H)
79. C₂×#131	C₂²×C₇₉	direct (N × H)
80. C₂²×M₁₆	C₂²×C₇₉	semidirect
83. C₂³×C₈	C₂²×C₇₉	semidirect
93. #131	C₂³×C₇₉	direct (N × H)
94. C₂×M₁₆	C₂³×C₇₉	semidirect
97. C₂²×C₈	C₂³×C₇₉	semidirect
110. M₁₆	C₂⁴×C₇₉	semidirect
113. C₂×C₈	C₂⁴×C₇₉	semidirect
115. C₂³	#131×C₇₉	direct (N × H)
128. C₂²	C₂×#131×C₇₉	direct (N × H)
135. C₂	C₂²×#131×C₇₉	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×#41×C₇₉	2
C₂²×#131×C₇₉	2
C₂³×M₁₆×C₇₉	2
C₂⁴×C₈×C₇₉	2
C₂³×#131	79

Subgroups — C23×#131×C79 #34174762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#131×C₇₉ #34174762