Subgroups — #34197252

Order 20224.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579100	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761682	C₂	1	1	yes	maximal, characteristic
4	#23761316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579100	C₄	1	2	yes	—
9	C₂×C₁₆×D₇₉	C₄	1	2	yes	—
10	#19098614	C₄	1	2	yes	—
11	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
12	#19098338	C₄	1	2	yes	—
13	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₆×Dic₇₉	C₄	1	2	yes	—
16	#19098285	C₄	1	2	yes	—
17	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	C₈²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
24	#14579100	C₈	1	8	yes	complemented
25	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₁₆×D₇₉	C₈	1	8	yes	complemented
28	Dic₇₉⋊C₈	C₈	1	4	yes	—
29	(C₂×D₇₉)⋊C₈	C₈	1	4	yes	—
30	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
39	C₂×C₄×D₇₉	C₂×C₈	1	2	yes	—
40	C₂×C₇₉⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₇₉⋊C₈	C₄²	1	1	yes	characteristic
42	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
43	C₈×D₇₉	C₂×C₈	1	4	yes	—
44	C₈×D₇₉	C₂×C₈	1	4	yes	—
45	C₄×Dic₇₉	C₄²	1	1	yes	characteristic
46	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
47	C₇₉⋊C₁₆	C₂×C₈	1	4	yes	—
48	C₇₉⋊C₁₆	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₇₉	C₂×C₈	1	4	yes	—
54	C₁₆×C₇₉	C₂×C₈	1	4	yes	—
55	C₂²×D₇₉	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
58	C₂×Dic₇₉	C₄×C₈	1	2	yes	—
59	C₂×Dic₇₉	C₄×C₈	1	1	yes	characteristic
60	C₄×D₇₉	—	2	4	no	—
61	C₄×D₇₉	C₄×C₈	1	2	yes	—
62	C₄×D₇₉	C₄×C₈	1	2	yes	—
63	C₈×C₇₉	—	2	4	no	—
64	C₈×C₇₉	C₂²×C₈	1	2	yes	—
65	C₈×C₇₉	C₂²×C₈	1	2	yes	—
66	C₇₉⋊C₈	—	2	4	no	—
67	C₇₉⋊C₈	C₄×C₈	1	2	yes	—
68	C₇₉⋊C₈	C₄×C₈	1	2	yes	—
69	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
70	C₂×D₇₉	—	2	2	no	—
71	C₂×D₇₉	C₈²	1	2	yes	—
72	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₇₉	#504	1	4	yes	—
75	Dic₇₉	—	2	2	no	—
76	Dic₇₉	C₈²	1	2	yes	—
77	#11399	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₇₉	#2108	1	2	yes	—
80	D₇₉	—	2	2	no	—
81	C₄×M₃₂	—	79	1	no	—
82	C₂×C₄×C₁₆	—	79	1	no	—
83	#1450	—	79	1	no	—
84	#1311	—	79	1	no	—
85	#1311	D₇₉	1	1	yes	characteristic, complemented
86	C₈×C₁₆	—	79	1	no	—
87	C₈×C₁₆	—	79	1	no	—
88	C₇₉	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	79	2	no	—
90	C₂²×C₁₆	—	79	2	no	—
91	#431	—	79	2	no	—
92	C₂×C₄×C₈	—	79	1	no	—
93	#346	—	79	2	no	—
94	C₄×C₁₆	—	79	2	no	—
95	C₄×C₁₆	—	79	1	no	—
96	C₄×C₁₆	—	79	1	no	—
97	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
98	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
99	#322	—	79	2	no	—
100	C₈²	—	79	1	no	—
101	C₈²	C₂×D₇₉	1	1	yes	characteristic
102	C₂²×C₈	—	79	1	no	—
103	C₂²×C₈	—	79	1	no	—
104	C₂×C₄²	—	79	1	no	—
105	M₃₂	—	79	8	no	—
106	C₂×C₁₆	—	79	2	no	—
107	C₂×C₁₆	—	79	2	no	—
108	C₂×C₁₆	—	79	8	no	—
109	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
110	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
111	C₄⋊C₈	—	79	4	no	—
112	C₂²⋊C₈	—	79	4	no	—
113	C₄×C₈	—	79	1	no	—
114	C₄×C₈	—	79	1	no	—
115	C₄×C₈	—	79	2	no	—
116	C₄×C₈	—	79	1	no	—
117	C₄×C₈	C₄×D₇₉	1	2	yes	—
118	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₂²×C₄	—	79	1	no	—
120	C₂²×C₄	—	79	2	no	—
121	C₂×C₈	—	79	1	no	—
122	C₂×C₈	—	79	1	no	—
123	C₂×C₈	—	79	4	no	—
124	C₂×C₈	—	79	4	no	—
125	C₂×C₈	—	79	4	no	—
126	C₂×C₈	C₈×D₇₉	1	4	yes	—
127	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
129	C₄²	—	79	1	no	—
130	C₄²	—	79	2	no	—
131	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
132	C₁₆	—	79	4	no	—
133	C₁₆	—	79	4	no	—
134	C₁₆	C₈×D₇₉	1	4	yes	—
135	C₁₆	C₈×D₇₉	1	4	yes	—
136	C₂³	—	79	1	no	—
137	C₂×C₄	—	79	1	no	—
138	C₂×C₄	—	79	2	no	—
139	C₂×C₄	—	79	2	no	—
140	C₂×C₄	—	79	2	no	—
141	C₂×C₄	—	158	4	no	—
142	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
143	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	79	2	no	—
146	C₈	—	79	2	no	—
147	C₈	—	158	4	no	—
148	C₈	C₂×C₈×D₇₉	1	2	yes	—
149	C₈	C₂×C₈×D₇₉	1	2	yes	—
150	C₂²	—	79	2	no	—
151	C₂²	—	158	2	no	—
152	C₂²	C₄×C₈×D₇₉	1	1	yes	characteristic
153	C₄	—	79	2	no	—
154	C₄	—	158	2	no	—
155	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
156	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
157	C₄	#19098981	1	4	yes	—
158	C₂	—	158	2	no	—
159	C₂	C₈²×D₇₉	1	1	yes	characteristic
160	C₂	#23764998	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇₉
commutator	C₂×C₇₉	C₂×C₈²
Fitting	C₇₉×#1311	C₂
socle	C₂²×C₇₉	C₂×C₄×C₈
prime radical	C₂²×D₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#11399
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ #1311

Complements

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

N	H ≅ G/N	type
4. #23761316	C₂	semidirect
5. C₈×C₇₉⋊C₁₆	C₂	semidirect
6. C₁₆×C₇₉⋊C₈	C₂	semidirect
7. C₇₉×#1311	C₂	semidirect
24. #14579100	C₈	semidirect
27. C₁₆×D₇₉	C₈	semidirect
85. #1311	D₇₉	semidirect
88. C₇₉	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579100	2
C₄×C₁₆×D₇₉	2
#23761682	2
#23761316	2
C₈×C₇₉⋊C₁₆	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#1311	2
#11399	79