Subgroups — #34181950

Order 20224.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763572	C₂	1	1	yes	maximal, characteristic
2	#23763564	C₂	1	1	yes	maximal, characteristic
3	#23761259	C₂	1	4	yes	maximal
4	C₇₉×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579097	—	2	1	no	—
6	C₂×#14579097	—	2	1	no	—
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579093	—	2	1	no	—
9	#19098587	C₂²	1	2	yes	—
10	#19098429	C₂²	1	2	yes	—
11	#431×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#322×C₇₉	C₄	1	2	yes	complemented
14	C₈²×C₇₉	C₄	1	2	yes	complemented
15	C₂×#11929608	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	#14579097	—	4	2	no	—
18	#14579097	—	4	2	no	—
19	#14579095	—	4	2	no	—
20	#14579093	—	4	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
22	C₇₉⋊#95	—	2	4	no	—
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₇₉	—	2	2	no	—
27	C₂²⋊C₈×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₉	—	4	2	no	—
34	#11929608	—	4	2	no	—
35	#11929608	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	#41	1	2	yes	—
41	C₂×C₈×C₇₉	—	2	4	no	—
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	D₈	1	2	yes	—
47	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
49	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
50	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₇₉	#41	1	2	yes	—
53	C₂³×C₇₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	4	2	no	—
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#117	1	2	yes	—
57	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#125	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₉	—	8	4	no	—
64	C₂×Dic₇₉	—	8	2	no	—
65	C₈×C₇₉	—	4	4	no	—
66	C₈×C₇₉	#106	1	2	yes	—
67	C₈×C₇₉	#106	1	2	yes	—
68	C₈×C₇₉	#107	1	2	yes	—
69	C₈×C₇₉	#107	1	2	yes	—
70	C₇₉⋊C₈	—	8	2	no	—
71	C₂²×C₇₉	—	2	2	no	—
72	C₂²×C₇₉	#382	1	1	yes	characteristic
73	C₂²×C₇₉	#486	1	1	yes	characteristic
74	C₂²×C₇₉	#487	1	1	yes	characteristic
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	2	2	no	—
77	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
78	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
79	C₄×C₇₉	#486	1	1	yes	characteristic
80	C₄×C₇₉	#487	1	1	yes	characteristic
81	Dic₇₉	—	16	2	no	—
82	#8790	—	79	1	no	maximal, Sylow
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	—	2	1	no	—
85	C₂×C₇₉	#1833	1	2	yes	—
86	C₂×C₇₉	#1834	1	1	yes	characteristic
87	#1848	—	79	1	no	—
88	#1846	—	79	1	no	—
89	#1450	D₇₉	1	1	yes	characteristic
90	#1286	—	79	4	no	—
91	C₇₉	#8790	1	1	yes	characteristic, complemented, Sylow
92	#431	C₂×D₇₉	1	2	yes	—
93	C₂×#108	—	158	1	no	—
94	C₂×#108	—	158	1	no	—
95	C₂×#107	—	158	1	no	—
96	C₂×#106	—	158	1	no	—
97	#423	—	79	2	no	—
98	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
99	#382	—	79	2	no	—
100	#322	Dic₇₉	1	2	yes	complemented
101	C₈²	Dic₇₉	1	2	yes	complemented
102	C₂×M₁₆	—	158	2	no	—
103	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
104	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	158	2	no	—
106	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
107	#108	—	316	2	no	—
108	#108	—	316	2	no	—
109	#107	—	316	2	no	—
110	#106	—	316	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	158	4	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
115	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
117	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
119	#95	—	158	4	no	—
120	C₂²×C₄	—	316	2	no	—
121	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
122	C₂²×C₄	#11929624	1	2	yes	—
123	M₁₆	—	316	2	no	—
124	M₁₆	—	316	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	316	2	no	—
127	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
131	C₂×C₈	D₆₃₂	1	2	yes	—
132	C₂×C₈	#11929613	1	2	yes	—
133	C₂×C₈	#11929613	1	2	yes	—
134	C₂×C₈	Dic₃₁₆	1	2	yes	—
135	C₄⋊C₄	—	316	2	no	—
136	C₄⋊C₄	—	316	2	no	—
137	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
139	C₄²	#11929624	1	2	yes	—
140	C₂³	#14579149	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	632	4	no	—
143	C₂×C₄	—	632	2	no	—
144	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
145	C₂×C₄	#14579120	1	2	yes	—
146	C₂×C₄	#14579149	1	1	yes	characteristic
147	C₂×C₄	#14579153	1	2	yes	—
148	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
149	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
150	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	632	2	no	—
154	C₈	#14579093	1	2	yes	—
155	C₈	#14579093	1	2	yes	—
156	C₈	#14579095	1	2	yes	—
157	C₈	#14579095	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#19098429	1	1	yes	characteristic
160	C₂²	#19098910	1	1	yes	characteristic
161	C₂²	#19098914	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	1264	2	no	—
165	C₄	C₂×#14579093	1	1	yes	characteristic
166	C₄	C₂×#14579095	1	1	yes	characteristic
167	C₄	#19098910	1	1	yes	characteristic
168	C₄	#19098914	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#23763524	1	2	yes	—
172	C₂	#23763528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098429
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1450	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#124
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1450	D₇₉
hypocenter	C₇₉	#8790
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
13. #322×C₇₉	C₄	semidirect
14. C₈²×C₇₉	C₄	semidirect
91. C₇₉	#8790	semidirect
100. #322	Dic₇₉	semidirect
101. C₈²	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763572	2
#23763564	2
#23761259	2
C₇₉×#1450	2
#8790	79