Finite Group Databasebeta

← #34200766

Subgroups — #34200766

Order 20224.

330 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 63 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#23765333C212yesmaximal
2#23764668C211yesmaximal, characteristic
3#23761418C212yesmaximal
4#23761342C211yesmaximal, characteristic
5C79×#3261C211yesmaximal, characteristic
6C22×Dic316C2211yescharacteristic
7C2×#14579107—22no—
8#19098571—21no—
9C2×#14579079—21no—
10#19098366—22no—
11C22⋊(C79⋊C16)C2212yes—
12#19098328C2211yescharacteristic
13#541×C79—21no—
14#479×C79—22no—
15#470×C79—21no—
16#467×C79C2212yes—
17#462×C79—22no—
18#424×C79C2211yescharacteristic
19C22×Dic158—21no—
20C2×Dic316—22no—
21C2×Dic316—42no—
22C2×#11929608—21no—
23#14579107—44no—
24C2×C79⋊C16—22no—
25#14579084—42no—
26#14579079—42no—
27C2×Q16×C79—21no—
28C2×Q16×C79D412yes—
29C2×M16×C79D411yescharacteristic
30C22×C8×C79C2311yescharacteristic
31C4⋊Q8×C79—21no—
32#126×C79—42no—
33#125×C79—21no—
34#125×C79—42no—
35C22⋊Q8×C79—22no—
36C4×Q8×C79—22no—
37#117×C79D411yescharacteristic
38#107×C79D412yes—
39#106×C79—21no—
40C4⋊C8×C79—22no—
41#103×C79—22no—
42#103×C79—22no—
43C2×Dic158—22no—
44C2×Dic158—41no—
45C2×Dic158—41no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C22×Dic79—41no—
50Dic316—24no—
51Dic316—42no—
52#11929608—41no—
53#11929608—41no—
54C2×C79⋊C8—41no—
55C79⋊C16—42no—
56C2×Q8×C79—22no—
57C22×C4×C79C2×D411yescharacteristic
58Q16×C79—42no—
59Q16×C79—81no—
60M16×C79—41no—
61C2×C8×C79—21no—
62C2×C8×C79—41no—
63C2×C8×C79C2×D412yes—
64C4⋊C4×C79—22no—
65C4⋊C4×C79—42no—
66C4⋊C4×C79—42no—
67C4⋊C4×C79—42no—
68C4⋊C4×C79—42no—
69C4⋊C4×C79D812yes—
70C22⋊C4×C79—21no—
71C22⋊C4×C79—42no—
72C42×C79—21no—
73C42×C79—42no—
74C23×C79AGL(1,ℤ/8)11yescharacteristic
75Q8×C79—42no—
76Q8×C79—42no—
77C2×C4×C79—21no—
78C2×C4×C79—21no—
79C2×C4×C79—22no—
80C2×C4×C79—42no—
81C2×C4×C79—42no—
82C2×C4×C79#12011yescharacteristic
83C2×C4×C79C2×D811yescharacteristic
84C2×Dic79—42no—
85C2×Dic79—81no—
86C2×Dic79—81no—
87Dic158—42no—
88Dic158—42no—
89Dic158—81no—
90Dic158—81no—
91C8×C79—22no—
92C8×C79—41no—
93C79⋊C8—81no—
94C22×C79—21no—
95C22×C79—22no—
96C22×C79#44711yescharacteristic
97C4×C79—21no—
98C4×C79—42no—
99C4×C79—82no—
100C4×C79—82no—
101C4×C79#45411yescharacteristic
102C4×C79B(4,𝔽2)11yescharacteristic
103Dic79—82no—
104#12168—791nomaximal, Sylow
105C2×C79—41no—
106C2×C79#159811yescharacteristic
107C2×C79#219512yes—
108#3261D7911yescharacteristic
109#2188—792no—
110#2047—791no—
111#1350—792no—
112#1323—791no—
113C79#1216811yescharacteristic, complemented, Sylow
114C22×Q16—791no—
115#541—21no—
116C2×Q32—1582no—
117#479—22no—
118#470—21no—
119#467C2×D7912yes—
120#462—22no—
121#424C2×D7911yescharacteristic
122#419—1581no—
123C2×#101—1581no—
124#358—1582no—
125C22⋊C16—792no—
126#341—791no—
127C22×Q8—1581no—
128C2×Q16—21no—
129C2×Q16—3162no—
130C2×Q16—1582no—
131C2×Q16C22⋊D7912yes—
132C2×M16—1581no—
133C2×M16C22⋊D7911yescharacteristic
134C22×C8C22×D7911yescharacteristic
135C4⋊Q8—21no—
136#126—42no—
137#125—21no—
138#125—42no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D7911yescharacteristic
142Q32—3164no—
143C2×C16—1582no—
144#107C22⋊D7912yes—
145#106—21no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—3162no—
150#101—3162no—
151C2×Q8—22no—
152C2×Q8—3161no—
153C2×Q8—3161no—
154C2×Q8—3161no—
155C2×Q8—3161no—
156C2×Q8—3161no—
157C2×Q8—1582no—
158C22×C4—3161no—
159C22×C4C2×C22⋊D7911yescharacteristic
160Q16—42no—
161Q16—81no—
162Q16—3162no—
163Q16—1584no—
164M16—41no—
165M16—3161no—
166M16—3161no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—3161no—
170C2×C8C2×C22⋊D7912yes—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4#1192960912yes—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C42—21no—
180C42—42no—
181C16—3162no—
182C23#1457918711yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—3162no—
186Q8—3162no—
187Q8—6321no—
188Q8—6321no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3162no—
195C2×C4—6321no—
196C2×C4—6321no—
197C2×C4#1457913011yescharacteristic
198C2×C4C2×#1192960911yescharacteristic
199C8—22no—
200C8—41no—
201C8—6321no—
202C22—22no—
203C22—21no—
204C22#1909869711yescharacteristic
205C4—21no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4—6322no—
210C4#1909873611yescharacteristic
211C4#1909875011yescharacteristic
212C2—41no—
213C2#2376233511yescharacteristic
214C2#2376537812yes—
subgroupG/H
centerC22#19098697
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#3261C2
socleC22×C79#447
prime radicalC23×C79AGL(1,ℤ/8)
hypercenter#3261D79
hypocenterC79#12168
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #3261

Complements

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

NH ≅ G/Ntype
113. C79#12168semidirect

Maximal subgroups

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

subgroupindex
#237653332
#237646682
#237614182
#237613422
C79×#32612
#1216879