Subgroups — #34204655

Order 20224.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763564	C₂	1	1	yes	maximal, characteristic
2	#23763560	C₂	1	1	yes	maximal, characteristic
3	#23761492	C₂	1	4	yes	maximal
4	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579095	—	2	1	no	—
6	C₂×#14579095	C₂²	1	2	yes	—
7	C₂×#14579095	C₂²	1	2	yes	—
8	C₂×#14579093	—	2	1	no	—
9	#19098433	—	2	1	no	—
10	#19098429	—	2	1	no	—
11	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#346×C₇₉	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₇₉	—	2	2	no	—
15	C₂×C₄⋊Dic₇₉	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	4	2	no	—
17	#14579095	—	2	4	no	—
18	#14579095	—	2	4	no	—
19	#14579095	—	4	2	no	—
20	#14579093	—	4	2	no	—
21	C₇₉⋊#95	—	4	2	no	—
22	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	—	2	2	no	—
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₉	Q₈	1	2	yes	—
30	C₂²×Dic₇₉	—	4	2	no	—
31	C₂²×Dic₇₉	—	4	2	no	—
32	C₄⋊Dic₇₉	—	4	2	no	—
33	C₄⋊Dic₇₉	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	8	4	no	—
37	C₂²×C₄×C₇₉	D₈	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₇₉	#41	1	2	yes	—
45	C₂×C₈×C₇₉	#41	1	2	yes	—
46	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₇₉	Q₁₆	1	2	yes	—
48	C₁₆×C₇₉	—	2	4	no	—
49	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	#107	1	4	yes	—
53	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₇₉	—	8	2	no	—
60	C₂×Dic₇₉	—	8	2	no	—
61	C₂×Dic₇₉	—	8	4	no	—
62	C₂×Dic₇₉	—	8	2	no	—
63	C₂×Dic₇₉	—	8	2	no	—
64	C₈×C₇₉	#106	1	4	yes	—
65	C₈×C₇₉	#117	1	2	yes	—
66	C₈×C₇₉	#125	1	2	yes	—
67	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
68	C₂²×C₇₉	#486	1	1	yes	characteristic
69	C₂²×C₇₉	#486	1	1	yes	characteristic
70	C₂²×C₇₉	#509	1	2	yes	—
71	C₂²×C₇₉	#510	1	2	yes	—
72	C₄×C₇₉	—	2	4	no	—
73	C₄×C₇₉	#382	1	1	yes	characteristic
74	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
75	C₄×C₇₉	#487	1	1	yes	characteristic
76	C₄×C₇₉	#487	1	1	yes	characteristic
77	Dic₇₉	—	16	2	no	—
78	Dic₇₉	—	16	2	no	—
79	#12890	—	79	1	no	maximal, Sylow
80	C₂×C₇₉	#1834	1	1	yes	characteristic
81	C₂×C₇₉	#2159	1	2	yes	—
82	C₂×C₇₉	#2243	1	2	yes	—
83	C₂×C₇₉	#2244	1	1	yes	characteristic
84	C₂×C₇₉	#2245	1	1	yes	characteristic
85	C₂×#346	D₇₉	1	1	yes	characteristic
86	#1846	—	79	1	no	—
87	#1845	—	79	1	no	—
88	#1380	—	79	4	no	—
89	C₇₉	#12890	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
91	C₂×#107	—	79	2	no	—
92	C₂×#107	—	79	2	no	—
93	C₂×#107	—	158	1	no	—
94	C₂×#106	—	158	1	no	—
95	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
96	#384	—	158	1	no	—
97	#382	—	158	1	no	—
98	#346	Dic₇₉	1	4	yes	complemented
99	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	316	2	no	—
102	C₂×C₄⋊C₄	—	158	2	no	—
103	C₂×C₄⋊C₄	—	158	2	no	—
104	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
108	#107	—	316	2	no	—
109	#107	—	158	4	no	—
110	#107	—	158	4	no	—
111	#106	—	316	2	no	—
112	C₄×C₈	Dic₁₅₈	1	2	yes	—
113	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
114	#95	—	316	2	no	—
115	C₂²×C₄	—	316	2	no	—
116	C₂²×C₄	—	316	2	no	—
117	C₂²×C₄	D₆₃₂	1	2	yes	—
118	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
120	C₂×C₈	#11929613	1	2	yes	—
121	C₂×C₈	#11929613	1	2	yes	—
122	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
124	C₂×C₈	#11929624	1	2	yes	—
125	C₂×C₈	#11929624	1	2	yes	—
126	C₄⋊C₄	—	316	2	no	—
127	C₄⋊C₄	—	316	2	no	—
128	C₄⋊C₄	—	316	2	no	—
129	C₄⋊C₄	—	316	2	no	—
130	C₄⋊C₄	—	632	4	no	—
131	C₄²	C₄⋊Dic₇₉	1	2	yes	—
132	C₄²	Dic₃₁₆	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	632	2	no	—
138	C₂×C₄	—	632	2	no	—
139	C₂×C₄	—	632	4	no	—
140	C₂×C₄	—	632	2	no	—
141	C₂×C₄	—	632	2	no	—
142	C₂×C₄	#14579095	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
144	C₂×C₄	#14579149	1	1	yes	characteristic
145	C₂×C₄	#14579149	1	1	yes	characteristic
146	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
147	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
149	C₈	#14579093	1	4	yes	—
150	C₈	#14579120	1	2	yes	—
151	C₈	#14579153	1	2	yes	—
152	C₂²	C₂×#14579095	1	1	yes	characteristic
153	C₂²	#19098910	1	1	yes	characteristic
154	C₂²	#19098910	1	1	yes	characteristic
155	C₂²	#19099001	1	2	yes	—
156	C₂²	#19099006	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	1264	2	no	—
159	C₄	—	1264	2	no	—
160	C₄	#19098429	1	1	yes	characteristic
161	C₄	C₂×#14579093	1	1	yes	characteristic
162	C₄	#19098914	1	1	yes	characteristic
163	C₄	#19098914	1	1	yes	characteristic
164	C₂	#23763528	1	1	yes	characteristic
165	C₂	#23765188	1	2	yes	—
166	C₂	#23765637	1	2	yes	—
167	C₂	#23765642	1	1	yes	characteristic
168	C₂	#23765646	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#12890
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #346×C₇₉	C₄	semidirect
89. C₇₉	#12890	semidirect
98. #346	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763564	2
#23763560	2
#23761492	2
C₂×#346×C₇₉	2
#12890	79