Finite Group Databasebeta

← #34201649

Subgroups — #34201649

Order 20224.

274 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774794C211yesmaximal, characteristic
2#23765619C212yesmaximal
3#23761387C212yesmaximal
4#23761351C211yesmaximal, characteristic
5C79×#2102C211yesmaximal, characteristic
6#19099120—21no—
7#19098849—22no—
8#19098821—21no—
9#19098821C2212yes—
10#19098774—22no—
11#19098387—22no—
12#19098378—22no—
13#19098345C2212yes—
14#430×C79C2211yescharacteristic
15#423×C79C2211yescharacteristic
16#344×C79—21no—
17#339×C79—21no—
18H(3,ℤ/4)×C79C2211yescharacteristic
19C2×Dic316—21no—
20C2×Dic316D412yes—
21#14579161—21no—
22#14579161—21no—
23#14579153—41no—
24#14579140—22no—
25#14579140—41no—
26C4×Dic158—22no—
27C4×Dic158—41no—
28#14579120—41no—
29#14579101—22no—
30#14579093—21no—
31#14579093D412yes—
32#14579086—22no—
33#14579086—22no—
34C2×M16×C79—21no—
35C2×M16×C79D411yescharacteristic
36C2×M16×C79C2311yescharacteristic
37#117×C79—21no—
38C2×C42×C79D411yescharacteristic
39#108×C79—22no—
40C4⋊C8×C79—22no—
41C2×Dic158—22no—
42C2×Dic158—41no—
43Dic316—42no—
44Dic316—81no—
45C4⋊Dic79—22no—
46C4⋊Dic79—41no—
47C4⋊Dic79—41no—
48C4⋊Dic79—41no—
49Dic79⋊C4—42no—
50Dic79⋊C4—42no—
51C22⋊Dic79—42no—
52C4×Dic79—42no—
53C79⋊C16—42no—
54C22×C4×C79—21no—
55C22×C4×C79C2×D411yescharacteristic
56M16×C79—41no—
57M16×C79—41no—
58M16×C79—41no—
59M16×C79—41no—
60C2×C8×C79—21no—
61C2×C8×C79—41no—
62C2×C8×C79C2×D412yes—
63C4⋊C4×C79—41no—
64C22⋊C4×C79—41no—
65C42×C79—21no—
66C42×C79—41no—
67C42×C79SD1611yescharacteristic
68C42×C79SD1611yescharacteristic
69C23×C79AGL(1,ℤ/8)11yescharacteristic
70C2×C4×C79—21no—
71C2×C4×C79—21no—
72C2×C4×C79—21no—
73C2×C4×C79—21no—
74C2×C4×C79—42no—
75C2×C4×C79—81no—
76C2×C4×C79#12011yescharacteristic
77C2×C4×C79C2×SD1611yescharacteristic
78C2×Dic79—42no—
79C2×Dic79—42no—
80Dic158—42no—
81Dic158—42no—
82Dic158—82no—
83C8×C79—22no—
84C8×C79—41no—
85C8×C79—81no—
86C22×C79—21no—
87C22×C79—41no—
88C22×C79#45011yescharacteristic
89C4×C79—22no—
90C4×C79—21no—
91C4×C79—41no—
92C4×C79—161no—
93C4×C79#45411yescharacteristic
94C4×C79#45811yescharacteristic
95Dic79—82no—
96Dic79—82no—
97#12308—791nomaximal, Sylow
98C2×C79—21no—
99C2×C79—41no—
100C2×C79#161611yescharacteristic
101#3255—791no—
102#2239—792no—
103#2102D7911yescharacteristic
104#1340—792no—
105#1327—791no—
106C79#1230811yescharacteristic, complemented, Sylow
107#533—1581no—
108#475—1582no—
109#470—792no—
110#470—1581no—
111#462—1582no—
112#430C2×D7911yescharacteristic
113#423C2×D7911yescharacteristic
114#365—1582no—
115#362—1582no—
116#349—792no—
117#344—21no—
118#339—21no—
119H(3,ℤ/4)C2×D7911yescharacteristic
120C2×Q16—792no—
121C2×Q16—1581no—
122C2×M16—21no—
123C2×M16C22⋊D7911yescharacteristic
124C2×M16C22×D7911yescharacteristic
125C4⋊Q8—1581no—
126C4⋊Q8—1581no—
127#125—3161no—
128C22⋊Q8—3161no—
129C22⋊Q8—1582no—
130C4×Q8—3161no—
131C4×Q8—1582no—
132#117—21no—
133#117—3161no—
134C2×C42C22⋊D7911yescharacteristic
135M32—1582no—
136#108—22no—
137#106—792no—
138#106—1581no—
139C4⋊C8—22no—
140#103—1582no—
141#103—1582no—
142C2×Q8—3161no—
143C2×Q8—1582no—
144C22×C4—21no—
145C22×C4C2×C22⋊D7911yescharacteristic
146Q16—3162no—
147Q16—6321no—
148M16—41no—
149M16—41no—
150M16—41no—
151M16—41no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8D4×D7912yes—
155C4⋊C4—41no—
156C4⋊C4—3162no—
157C4⋊C4—3161no—
158C4⋊C4—3161no—
159C4⋊C4—3161no—
160C4⋊C4—3162no—
161C4⋊C4—1582no—
162C22⋊C4—41no—
163C22⋊C4—3162no—
164C42—21no—
165C42—41no—
166C42—3162no—
167C42#1192961111yescharacteristic
168C42#1192961111yescharacteristic
169C16—3162no—
170C23#1457918611yescharacteristic
171Q8—3162no—
172Q8—3162no—
173Q8—6322no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—81no—
180C2×C4—3162no—
181C2×C4—3162no—
182C2×C4#1457912911yescharacteristic
183C2×C4C2×#1192961111yescharacteristic
184C8—22no—
185C8—41no—
186C8—81no—
187C22—21no—
188C22—41no—
189C22#1909871311yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—161no—
194C4—6322no—
195C4—6322no—
196C4#1909873511yescharacteristic
197C4#1909875211yescharacteristic
198C2—21no—
199C2—41no—
200C2#2376243611yescharacteristic
subgroupG/H
centerC2#23762436
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#2102C2
socleC2×C79#1616
prime radicalC23×C79AGL(1,ℤ/8)
hypercenter#2102D79
hypocenterC79#12308
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2102

Complements

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

NH ≅ G/Ntype
106. C79#12308semidirect

Maximal subgroups

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

subgroupindex
#237747942
#237656192
#237613872
#237613512
C79×#21022
#1230879