Subgroups — #23778113

Order 10112.

1424 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099399	C₂	1	1	yes	maximal, characteristic
2	#19099329	C₂	1	6	yes	maximal, complemented
3	#19099328	C₂	1	8	yes	maximal, complemented
4	C₂×#14579169	C₂	1	6	yes	maximal, complemented
5	C₂×#14579168	C₂	1	1	yes	maximal, characteristic, complemented
6	#131×D₇₉	C₂	1	8	yes	maximal, complemented
7	#568×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14579209	C₄	1	2	yes	—
9	#14579208	C₄	1	4	yes	—
10	#14579207	C₄	1	4	yes	—
11	#14579206	C₄	1	6	yes	—
12	C₂×#11929624	C₂²	1	6	yes	—
13	#41×D₇₉	C₂²	1	8	yes	—
14	#14579169	C₂²	1	12	yes	complemented
15	#14579169	C₂²	1	4	yes	complemented
16	#14579169	C₂²	1	12	yes	complemented
17	#14579169	C₂²	1	12	yes	complemented
18	#14579168	C₂²	1	4	yes	complemented
19	#14579168	C₂²	1	4	yes	complemented
20	#14579167	C₂²	1	24	yes	complemented
21	C₂×#11929608	C₂²	1	3	yes	complemented
22	C₂×#11929607	C₂²	1	12	yes	complemented
23	M₁₆×D₇₉	C₂²	1	24	yes	complemented
24	C₂²×C₇₉⋊C₈	C₂²	1	3	yes	complemented
25	C₂×C₈×D₇₉	C₂²	1	12	yes	complemented
26	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
27	#131×C₇₉	C₂²	1	8	yes	complemented
28	C₂×M₁₆×C₇₉	C₂²	1	6	yes	complemented
29	#11929625	C₂×C₄	1	8	yes	—
30	#11929624	C₂×C₄	1	12	yes	—
31	#11929624	C₂×C₄	1	12	yes	—
32	#11929624	C₂³	1	12	yes	—
33	#11929624	C₂³	1	4	yes	—
34	#11929623	C₂×C₄	1	24	yes	—
35	C₂×Dic₁₅₈	C₂×C₄	1	6	yes	—
36	Q₈×D₇₉	C₂×C₄	1	8	yes	—
37	C₂×D₃₁₆	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊D₇₉	C₂×C₄	1	12	yes	—
39	D₄×D₇₉	C₂×C₄	1	24	yes	—
40	C₂×C₄×D₇₉	C₂³	1	12	yes	—
41	#11929608	C₂³	1	12	yes	—
42	#11929607	C₂³	1	24	yes	—
43	#11929607	C₂³	1	8	yes	—
44	C₂×C₇₉⋊C₈	C₂³	1	6	yes	—
45	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₇₉⋊C₈	C₂³	1	6	yes	—
47	C₂×C₇₉⋊C₈	C₂³	1	3	yes	—
48	C₈×D₇₉	C₂³	1	8	yes	—
49	C₈×D₇₉	C₂³	1	24	yes	—
50	#41×C₇₉	C₂×C₄	1	4	yes	—
51	#41×C₇₉	C₂³	1	4	yes	—
52	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×D₄×C₇₉	C₂×C₄	1	3	yes	—
54	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
55	M₁₆×C₇₉	C₂³	1	4	yes	—
56	M₁₆×C₇₉	C₂³	1	12	yes	—
57	C₂×C₈×C₇₉	C₂³	1	12	yes	—
58	C₂³×C₇₉	C₂²×C₄	1	3	yes	—
59	C₂²×D₇₉	C₂²×C₄	1	12	yes	—
60	Q₈×C₇₉	C₂²×C₄	1	4	yes	—
61	D₄×C₇₉	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₇₉	C₂²×C₄	1	6	yes	—
63	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
64	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂⁴	1	6	yes	—
66	C₂²⋊D₇₉	C₂²×C₄	1	24	yes	—
67	C₂²⋊D₇₉	C₂²×C₄	1	8	yes	—
68	C₂×Dic₇₉	C₂²×C₄	1	12	yes	—
69	D₃₁₆	C₂²×C₄	1	12	yes	—
70	D₃₁₆	C₂²×C₄	1	4	yes	—
71	C₄×D₇₉	C₂²×C₄	1	24	yes	—
72	C₄×D₇₉	C₂⁴	1	8	yes	—
73	Dic₁₅₈	C₂²×C₄	1	12	yes	—
74	Dic₁₅₈	C₂²×C₄	1	4	yes	—
75	C₈×C₇₉	C₂⁴	1	8	yes	—
76	C₇₉⋊C₈	C₂⁴	1	1	yes	characteristic
77	C₇₉⋊C₈	C₂⁴	1	6	yes	—
78	C₇₉⋊C₈	C₂⁴	1	1	yes	characteristic
79	C₂²×C₇₉	—	4	3	no	—
80	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
81	C₂²×C₇₉	C₂³×C₄	1	6	yes	—
82	C₂×D₇₉	—	4	12	no	—
83	C₂×D₇₉	C₂³×C₄	1	8	yes	—
84	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₇₉	C₂³×C₄	1	6	yes	—
86	C₄×C₇₉	C₂⁵	1	1	yes	characteristic
87	Dic₇₉	C₂³×C₄	1	8	yes	—
88	C₂×C₇₉	—	2	6	no	—
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
91	D₇₉	—	2	8	no	—
92	#3573	—	79	1	no	maximal, Sylow
93	C₇₉	#3573	1	1	yes	characteristic, complemented, Sylow
94	#585	—	79	1	no	—
95	#568	—	79	8	no	—
96	#568	—	79	6	no	—
97	#568	D₇₉	1	1	yes	characteristic, complemented
98	C₂×#131	—	79	1	no	—
99	C₂×#131	—	79	8	no	—
100	C₂×#131	—	79	6	no	—
101	2₋¹⁺⁴	—	79	4	no	—
102	2₋¹⁺⁴	—	79	2	no	—
103	H(5,𝔽₂)	—	79	4	no	—
104	H(5,𝔽₂)	—	79	6	no	—
105	C₂×#41	—	79	8	no	—
106	C₂×#41	—	79	6	no	—
107	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
108	#131	—	79	4	no	—
109	#131	—	79	4	no	—
110	#131	—	79	12	no	—
111	#131	—	79	4	no	—
112	#131	—	79	12	no	—
113	#131	—	79	12	no	—
114	#131	—	79	24	no	—
115	#131	C₂×D₇₉	1	8	yes	complemented
116	C₂×M₁₆	—	79	3	no	—
117	C₂×M₁₆	—	79	24	no	—
118	C₂×M₁₆	—	79	12	no	—
119	C₂×M₁₆	C₂×D₇₉	1	6	yes	complemented
120	C₂²×C₈	—	79	3	no	—
121	C₂²×C₈	—	79	12	no	—
122	#41	—	79	8	no	—
123	#41	—	79	24	no	—
124	#41	—	79	12	no	—
125	#41	—	79	12	no	—
126	#41	—	79	4	no	—
127	#41	—	79	12	no	—
128	#41	C₄×D₇₉	1	4	yes	—
129	#41	C₂²×D₇₉	1	4	yes	—
130	C₂×Q₈	—	79	8	no	—
131	C₂×Q₈	—	79	6	no	—
132	C₂×Q₈	C₄×D₇₉	1	1	yes	characteristic
133	C₂×D₄	—	79	12	no	—
134	C₂×D₄	—	79	24	no	—
135	C₂×D₄	—	79	6	no	—
136	C₂×D₄	C₄×D₇₉	1	3	yes	—
137	C₂²×C₄	—	79	12	no	—
138	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
139	M₁₆	—	79	12	no	—
140	M₁₆	—	79	24	no	—
141	M₁₆	—	79	8	no	—
142	M₁₆	C₂²×D₇₉	1	12	yes	—
143	M₁₆	C₂²×D₇₉	1	4	yes	—
144	C₂×C₈	—	79	1	no	—
145	C₂×C₈	—	79	6	no	—
146	C₂×C₈	—	79	3	no	—
147	C₂×C₈	—	79	6	no	—
148	C₂×C₈	—	79	8	no	—
149	C₂×C₈	—	79	24	no	—
150	C₂×C₈	C₂²×D₇₉	1	12	yes	—
151	C₂³	—	79	12	no	—
152	C₂³	C₂×C₄×D₇₉	1	3	yes	—
153	Q₈	—	79	12	no	—
154	Q₈	—	79	4	no	—
155	Q₈	C₂×C₄×D₇₉	1	4	yes	—
156	D₄	—	79	24	no	—
157	D₄	—	79	8	no	—
158	D₄	—	79	12	no	—
159	D₄	—	79	4	no	—
160	D₄	C₂×C₄×D₇₉	1	12	yes	—
161	C₂×C₄	—	79	12	no	—
162	C₂×C₄	—	79	8	no	—
163	C₂×C₄	—	79	24	no	—
164	C₂×C₄	C₂×C₄×D₇₉	1	6	yes	—
165	C₂×C₄	C₂×C₄×D₇₉	1	3	yes	—
166	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
167	C₂×C₄	C₂³×D₇₉	1	6	yes	—
168	C₈	—	79	1	no	—
169	C₈	—	79	1	no	—
170	C₈	—	79	6	no	—
171	C₈	C₂³×D₇₉	1	8	yes	—
172	C₂²	—	4	3	no	—
173	C₂²	—	316	12	no	—
174	C₂²	—	79	8	no	—
175	C₂²	C₂²×C₄×D₇₉	1	6	yes	—
176	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
177	C₄	—	79	8	no	—
178	C₄	C₂²×C₄×D₇₉	1	6	yes	—
179	C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
180	C₄	C₂⁴×D₇₉	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	6	no	—
183	C₂	—	158	8	no	—
184	C₂	C₂³×C₄×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₇₉
commutator, socle	C₂×C₇₉	C₂⁴×C₄
Fitting	#568×C₇₉	C₂
prime radical	#19099399	C₂
cosocle	C₄×C₇₉	C₂⁵
hypercenter	#568	D₇₉
hypocenter	C₇₉	#3573
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #19099329	C₂	semidirect
3. #19099328	C₂	semidirect
4. C₂×#14579169	C₂	semidirect
5. C₂×#14579168	C₂	semidirect
6. #131×D₇₉	C₂	semidirect
7. #568×C₇₉	C₂	semidirect
14. #14579169	C₂²	semidirect
15. #14579169	C₂²	semidirect
16. #14579169	C₂²	semidirect
17. #14579169	C₂²	semidirect
18. #14579168	C₂²	semidirect
19. #14579168	C₂²	semidirect
20. #14579167	C₂²	semidirect
21. C₂×#11929608	C₂²	semidirect
22. C₂×#11929607	C₂²	semidirect
23. M₁₆×D₇₉	C₂²	semidirect
24. C₂²×C₇₉⋊C₈	C₂²	semidirect
25. C₂×C₈×D₇₉	C₂²	semidirect
27. #131×C₇₉	C₂²	semidirect
28. C₂×M₁₆×C₇₉	C₂²	semidirect
93. C₇₉	#3573	semidirect
97. #568	D₇₉	semidirect
115. #131	C₂×D₇₉	semidirect
119. C₂×M₁₆	C₂×D₇₉	semidirect