Finite Group Databasebeta

← #34201553

Subgroups — #34201553

Order 20224.

310 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 67 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#23774745C211yesmaximal, characteristic
2#23765602C212yesmaximal
3#23761411C212yesmaximal
4#23761340C211yesmaximal, characteristic
5C79×#2087C211yesmaximal, characteristic
6#19099120—21no—
7#19098859—22no—
8#19098804—21no—
9#19098804C2212yes—
10#19098774—22no—
11#19098388—22no—
12#19098366—22no—
13C22⋊(C79⋊C16)C2212yes—
14C2×#107×C79C2211yescharacteristic
15C2×C4⋊C8×C79C2211yescharacteristic
16#389×C79—21no—
17#340×C79—21no—
18#340×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—
29C2×C79⋊C16—22no—
30#14579095—21no—
31#14579095D412yes—
32#14579086—22no—
33#14579086—22no—
34C22×C8×C79D411yescharacteristic
35C22×C8×C79C2311yescharacteristic
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—41no—
39C2×C42×C79D411yescharacteristic
40#107×C79—22no—
41C4⋊C8×C79—22no—
42#95×C79—41no—
43C2×Dic158—22no—
44C2×Dic158—41no—
45Dic316—42no—
46Dic316—81no—
47C4⋊Dic79—22no—
48C4⋊Dic79—41no—
49C4⋊Dic79—41no—
50C4⋊Dic79—41no—
51Dic79⋊C4—42no—
52Dic79⋊C4—42no—
53C22⋊Dic79—42no—
54C4×Dic79—42no—
55C79⋊C16—42no—
56C22×C4×C79—41no—
57C22×C4×C79—41no—
58C22×C4×C79D811yescharacteristic
59C22×C4×C79D811yescharacteristic
60C22×C4×C79C2×D411yescharacteristic
61C2×C8×C79—21no—
62C2×C8×C79—41no—
63C2×C8×C79—41no—
64C2×C8×C79C2×D412yes—
65C4⋊C4×C79—22no—
66C4⋊C4×C79—41no—
67C4⋊C4×C79—41no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—82no—
70C42×C79—21no—
71C42×C79—21no—
72C23×C79C2×D811yescharacteristic
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—21no—
77C2×C4×C79—21no—
78C2×C4×C79—41no—
79C2×C4×C79—42no—
80C2×C4×C79—81no—
81C2×C4×C79—81no—
82C2×C4×C79—81no—
83C2×C4×C79—81no—
84C2×C4×C79—81no—
85C2×C4×C79Q3212yes—
86C2×C4×C79#12011yescharacteristic
87C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
88C2×Dic79—42no—
89C2×Dic79—42no—
90Dic158—42no—
91Dic158—42no—
92Dic158—82no—
93C8×C79—22no—
94C8×C79—41no—
95C22×C79—21no—
96C22×C79—21no—
97C22×C79#44711yescharacteristic
98C22×C79C2×Q3211yescharacteristic
99C22×C79#51011yescharacteristic
100C4×C79—21no—
101C4×C79—22no—
102C4×C79—41no—
103C4×C79—82no—
104C4×C79—161no—
105C4×C79#45611yescharacteristic
106C4×C79#45811yescharacteristic
107Dic79—82no—
108Dic79—82no—
109#12292—791nomaximal, Sylow
110C2×C79—21no—
111C2×C79—21no—
112C2×C79#161011yescharacteristic
113C2×C79#218811yescharacteristic
114C2×C79#219011yescharacteristic
115#3249—791no—
116#2236—792no—
117#2087D7911yescharacteristic
118#1348—792no—
119#1322—791no—
120C79#1229211yescharacteristic, complemented, Sylow
121#533—1581no—
122#477—1582no—
123#467—792no—
124#467—1581no—
125#462—1582no—
126C2×#107C2×D7911yescharacteristic
127C2×C4⋊C8C2×D7911yescharacteristic
128#389—21no—
129#366—1582no—
130#358—1582no—
131C22⋊C16—792no—
132#340—21no—
133#340C2×D7911yescharacteristic
134C2×Q16—792no—
135C2×Q16—1581no—
136C22×C8C22⋊D7911yescharacteristic
137C22×C8C22×D7911yescharacteristic
138C4⋊Q8—1581no—
139C4⋊Q8—1581no—
140#125—3161no—
141C22⋊Q8—3161no—
142C22⋊Q8—1582no—
143C4×Q8—3161no—
144C4×Q8—1582no—
145#117—3161no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C42C22⋊D7911yescharacteristic
150C2×C16—1582no—
151#107—22no—
152#107—792no—
153#107—1581no—
154C4⋊C8—22no—
155#103—1582no—
156#103—1582no—
157#95—41no—
158C2×Q8—3161no—
159C2×Q8—1582no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#1192960911yescharacteristic
163C22×C4#1192960911yescharacteristic
164C22×C4C2×C22⋊D7911yescharacteristic
165Q16—3162no—
166Q16—6321no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D7912yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C4⋊C4—3162no—
177C4⋊C4—3161no—
178C4⋊C4—3161no—
179C4⋊C4—3161no—
180C4⋊C4—3162no—
181C4⋊C4—1582no—
182C22⋊C4—3162no—
183C42—21no—
184C42—21no—
185C42—3162no—
186C16—3162no—
187C23C2×#1192960911yescharacteristic
188Q8—3162no—
189Q8—3162no—
190Q8—6322no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—3162no—
204C2×C4—3162no—
205C2×C4#1457910712yes—
206C2×C4#1457912911yescharacteristic
207C2×C4#1457918611yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1909869611yescharacteristic
213C22C2×#1457910711yescharacteristic
214C22#1909900311yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—161no—
220C4—6322no—
221C4—6322no—
222C4#1909874511yescharacteristic
223C4#1909875211yescharacteristic
224C2—21no—
225C2—21no—
226C2#2376240211yescharacteristic
227C2#2376533311yescharacteristic
228C2#2376534511yescharacteristic
subgroupG/H
centerC22#19098696
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#2087C2
socleC22×C79#447
prime radicalC23×C79C2×D8
hypercenter#2087D79
hypocenterC79#12292
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2087

Complements

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

NH ≅ G/Ntype
120. C79#12292semidirect

Maximal subgroups

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

subgroupindex
#237747452
#237656022
#237614112
#237613402
C79×#20872
#1229279