Finite Group Databasebeta

← #34201628

Subgroups — #34201628

Order 20224.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765612C212yesmaximal, complemented
2#23764980C211yesmaximal, characteristic, complemented
3#23761391C212yesmaximal, complemented
4#23761353C211yesmaximal, characteristic, complemented
5C79×#3259C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098385—22no—
8#19098376—22no—
9#19098345C2212yes—
10#19098333—21no—
11#19098328—21no—
12C79⋊#338C2211yescharacteristic
13#540×C79—21no—
14#478×C79—22no—
15#469×C79C2212yes—
16#466×C79—21no—
17#459×C79—22no—
18#424×C79C2211yescharacteristic
19C2×#11929608—21no—
20C2×#11929608—21no—
21#14579101—22no—
22#14579097—22no—
23C2×D8×C79—21no—
24C2×D8×C79D412yes—
25C2×M16×C79C2311yescharacteristic
26C22×C8×C79D411yescharacteristic
27#127×C79—21no—
28#125×C79—21no—
29#123×C79—42no—
30#121×C79—22no—
31#121×C79—42no—
32C4×D4×C79—22no—
33#117×C79D411yescharacteristic
34#107×C79—21no—
35#106×C79D412yes—
36C4⋊C8×C79—22no—
37#102×C79—22no—
38#102×C79—22no—
39#11929608—41no—
40#11929608—41no—
41#11929608—41no—
42#11929608—41no—
43C2×C79⋊C8—41no—
44C2×C79⋊C8—41no—
45C79⋊C16—42no—
46C2×D4×C79—22no—
47C2×D4×C79—42no—
48C2×D4×C79—42no—
49C22×C4×C79—42no—
50C22×C4×C79C2×D411yescharacteristic
51D8×C79—42no—
52D8×C79—81no—
53M16×C79—41no—
54C2×C8×C79—21no—
55C2×C8×C79—41no—
56C2×C8×C79C2×D412yes—
57C4⋊C4×C79—21no—
58C4⋊C4×C79—22no—
59C4⋊C4×C79—42no—
60C22⋊C4×C79—42no—
61C22⋊C4×C79—42no—
62C22⋊C4×C79SD1612yes—
63C42×C79—21no—
64C23×C79—42no—
65C23×C79C2×SD1611yescharacteristic
66D4×C79—42no—
67D4×C79—42no—
68D4×C79—162no—
69D4×C79—162no—
70C2×C4×C79—21no—
71C2×C4×C79—21no—
72C2×C4×C79—22no—
73C2×C4×C79—42no—
74C2×C4×C79—162no—
75C2×C4×C79#12011yescharacteristic
76C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
77C8×C79—22no—
78C8×C79—41no—
79C79⋊C8—81no—
80C79⋊C8—81no—
81C22×C79—21no—
82C22×C79—41no—
83C22×C79—82no—
84C22×C79—82no—
85C22×C79—82no—
86C22×C79#45011yescharacteristic
87C4×C79—21no—
88C4×C79—42no—
89C4×C79—82no—
90C4×C79#45411yescharacteristic
91C4×C79#45811yescharacteristic
92#12304—791nomaximal, Sylow
93C2×C79—21no—
94C2×C79—41no—
95C2×C79—162no—
96C2×C79#161611yescharacteristic
97#3259D7911yescharacteristic
98#2238—792no—
99#2103—791no—
100#1341—792no—
101#1328—791no—
102C79#1230411yescharacteristic, complemented, Sylow
103#540—21no—
104#478—22no—
105#469C2×D7912yes—
106#466—21no—
107#459—22no—
108#430—791no—
109#424C2×D7911yescharacteristic
110#365—1582no—
111#361—1582no—
112#349—792no—
113#344—1581no—
114#341—1581no—
115#338—791no—
116C2×D8—21no—
117C2×D8C22⋊D7912yes—
118C2×M16—1581no—
119C2×M16—1581no—
120C2×M16C22×D7911yescharacteristic
121C22×C8C22⋊D7911yescharacteristic
122#127—21no—
123#125—21no—
124#123—42no—
125#121—22no—
126#121—42no—
127C4×D4—22no—
128#117C22⋊D7911yescharacteristic
129M32—1582no—
130#108—1582no—
131#107—21no—
132#106C22⋊D7912yes—
133C4⋊C8—22no—
134#102—22no—
135#102—22no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C22×C4—42no—
140C22×C4C2×C22⋊D7911yescharacteristic
141D8—42no—
142D8—81no—
143M16—41no—
144M16—3161no—
145M16—3161no—
146M16—3161no—
147M16—3161no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—3161no—
151C2×C8—3161no—
152C2×C8C2×C22⋊D7912yes—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4#1192961212yes—
159C42—21no—
160C16—3162no—
161C23—42no—
162C23C2×#1192961211yescharacteristic
163D4—42no—
164D4—42no—
165D4—162no—
166D4—162no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—162no—
172C2×C4#1457913011yescharacteristic
173C2×C4#1457918711yescharacteristic
174C8—22no—
175C8—41no—
176C8—6321no—
177C8—6321no—
178C22—21no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22#1909871411yescharacteristic
184C4—21no—
185C4—42no—
186C4—82no—
187C4#1909873611yescharacteristic
188C4#1909875311yescharacteristic
189C2—21no—
190C2—41no—
191C2—162no—
192C2#2376243711yescharacteristic
subgroupG/H
centerC2#23762437
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
Fitting, prime radicalC79×#3259C2
socleC2×C79#1616
hypercenter#3259D79
hypocenterC79#12304
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3259

Complements

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

NH ≅ G/Ntype
1. #23765612C2semidirect
2. #23764980C2semidirect
3. #23761391C2semidirect
4. #23761353C2semidirect
102. C79#12304semidirect

Maximal subgroups

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

subgroupindex
#237656122
#237649802
#237613912
#237613532
C79×#32592
#1230479