Subgroups — #34204645

Order 20224.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 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	#23761499	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579097	—	2	1	no	—
6	C₂×#14579097	C₂²	1	2	yes	—
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579093	C₂²	1	2	yes	—
9	#19098587	—	2	1	no	—
10	#19098429	—	2	1	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#346×C₇₉	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₇₉	C₄	1	2	yes	complemented
15	C₂×#11929608	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	#14579097	—	2	4	no	—
18	#14579097	—	4	2	no	—
19	#14579095	—	4	2	no	—
20	#14579093	—	2	4	no	—
21	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
22	C₇₉⋊#95	—	4	2	no	—
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
26	M₃₂×C₇₉	—	2	2	no	—
27	M₃₂×C₇₉	—	2	2	no	—
28	C₂×C₁₆×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	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	D₈	1	2	yes	—
44	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
45	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	#41	1	2	yes	—
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₉	SD₁₆	1	2	yes	—
52	C₁₆×C₇₉	—	2	4	no	—
53	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	4	2	no	—
55	C₂×C₄×C₇₉	#106	1	2	yes	—
56	C₂×C₄×C₇₉	#106	1	2	yes	—
57	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×D₈	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	#107	1	2	yes	—
66	C₈×C₇₉	#107	1	2	yes	—
67	C₈×C₇₉	#117	1	2	yes	—
68	C₈×C₇₉	#125	1	2	yes	—
69	C₇₉⋊C₈	—	8	2	no	—
70	C₂²×C₇₉	—	2	2	no	—
71	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
72	C₂²×C₇₉	#486	1	1	yes	characteristic
73	C₂²×C₇₉	#487	1	1	yes	characteristic
74	C₄×C₇₉	—	2	2	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	#382	1	1	yes	characteristic
77	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₉	#486	1	1	yes	characteristic
79	C₄×C₇₉	#487	1	1	yes	characteristic
80	Dic₇₉	—	16	2	no	—
81	#12887	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	#1834	1	1	yes	characteristic
85	C₂×C₇₉	#2155	1	2	yes	—
86	C₄×M₃₂	D₇₉	1	1	yes	characteristic
87	#1848	—	79	1	no	—
88	#1846	—	79	1	no	—
89	#1383	—	79	4	no	—
90	C₇₉	#12887	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
92	C₂×#108	—	79	2	no	—
93	C₂×#108	—	158	1	no	—
94	C₂×#107	—	158	1	no	—
95	C₂×#106	—	79	2	no	—
96	#423	—	158	1	no	—
97	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
98	#382	—	158	1	no	—
99	#346	Dic₇₉	1	2	yes	complemented
100	C₄×C₁₆	Dic₇₉	1	2	yes	complemented
101	C₂×M₁₆	—	158	2	no	—
102	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	158	2	no	—
105	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
109	#108	—	316	2	no	—
110	#108	—	158	4	no	—
111	#107	—	316	2	no	—
112	#106	—	158	4	no	—
113	C₄⋊C₈	—	316	2	no	—
114	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
115	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
118	#95	—	316	2	no	—
119	C₂²×C₄	—	316	2	no	—
120	C₂²×C₄	#11929613	1	2	yes	—
121	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
122	M₁₆	—	316	2	no	—
123	M₁₆	—	316	2	no	—
124	C₂×C₈	—	316	2	no	—
125	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
127	C₂×C₈	D₆₃₂	1	2	yes	—
128	C₂×C₈	Dic₃₁₆	1	2	yes	—
129	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
131	C₂×C₈	#11929624	1	2	yes	—
132	C₂×C₈	#11929624	1	2	yes	—
133	C₄⋊C₄	—	316	2	no	—
134	C₄⋊C₄	—	316	2	no	—
135	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
137	C₄²	#11929613	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#11929613	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	632	4	no	—
142	C₂×C₄	—	632	2	no	—
143	C₂×C₄	#14579093	1	2	yes	—
144	C₂×C₄	#14579093	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
146	C₂×C₄	#14579149	1	1	yes	characteristic
147	C₂×C₄	#14579149	1	1	yes	characteristic
148	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
149	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
151	C₈	—	632	2	no	—
152	C₈	#14579095	1	2	yes	—
153	C₈	#14579095	1	2	yes	—
154	C₈	#14579120	1	2	yes	—
155	C₈	#14579153	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#14579093	1	1	yes	characteristic
158	C₂²	#19098910	1	1	yes	characteristic
159	C₂²	#19098914	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1264	2	no	—
163	C₄	#19098429	1	1	yes	characteristic
164	C₄	C₂×#14579095	1	1	yes	characteristic
165	C₄	#19098910	1	1	yes	characteristic
166	C₄	#19098914	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#23763528	1	1	yes	characteristic
170	C₂	#23765173	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14579093
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#12887
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #346×C₇₉	C₄	semidirect
14. C₄×C₁₆×C₇₉	C₄	semidirect
90. C₇₉	#12887	semidirect
99. #346	Dic₇₉	semidirect
100. C₄×C₁₆	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
#23761499	2
C₄×M₃₂×C₇₉	2
#12887	79