Finite Group Databasebeta

← #34201539

Subgroups — #34201539

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#23765598C212yesmaximal
3#23764888C211yesmaximal, characteristic
4#23761413C212yesmaximal
5C79×#1322C211yesmaximal, characteristic
6#19099120—21no—
7#19098859—22no—
8#19098804—21no—
9#19098804C2212yes—
10#19098774—22no—
11C2×#14579095C2211yescharacteristic
12#19098461—21no—
13#19098391—22no—
14#19098368—22no—
15#19098326—21no—
16#19098326C2211yescharacteristic
17C2×C4⋊C8×C79C2211yescharacteristic
18C22⋊C16×C79C2212yes—
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×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×Dic79⋊C4—41no—
32#14579095—21no—
33#14579095—22no—
34#14579095D412yes—
35#14579086—22no—
36#14579086—22no—
37C79⋊#95—41no—
38C22×C8×C79D411yescharacteristic
39C22×C8×C79C2311yescharacteristic
40C2×C42×C79D411yescharacteristic
41C2×C16×C79—22no—
42C4⋊C8×C79—22no—
43C2×Dic158—22no—
44C2×Dic158—41no—
45C22×Dic79—41no—
46C22×Dic79—41no—
47Dic316—42no—
48Dic316—81no—
49C4⋊Dic79—22no—
50C4⋊Dic79—22no—
51C4⋊Dic79—41no—
52C4⋊Dic79—41no—
53C4⋊Dic79—41no—
54C4⋊Dic79—41no—
55C4⋊Dic79—41no—
56C4⋊Dic79—41no—
57Dic79⋊C4—42no—
58Dic79⋊C4—42no—
59Dic79⋊C4—82no—
60C22⋊Dic79—42no—
61C4×Dic79—42no—
62C22×C4×C79D811yescharacteristic
63C22×C4×C79D811yescharacteristic
64C22×C4×C79C2×D411yescharacteristic
65C2×C8×C79—21no—
66C2×C8×C79—41no—
67C2×C8×C79—41no—
68C2×C8×C79C2×D412yes—
69C42×C79—21no—
70C42×C79—21no—
71C16×C79—42no—
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×C79SD3212yes—
80C2×C4×C79#12011yescharacteristic
81C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
82C2×Dic79—42no—
83C2×Dic79—42no—
84C2×Dic79—42no—
85C2×Dic79—81no—
86C2×Dic79—81no—
87C2×Dic79—81no—
88C2×Dic79—81no—
89C2×Dic79—81no—
90Dic158—42no—
91Dic158—42no—
92Dic158—82no—
93C8×C79—22no—
94C8×C79—41no—
95C22×C79—21no—
96C22×C79—21no—
97C22×C79#44711yescharacteristic
98C22×C79C2×SD3211yescharacteristic
99C22×C79#51011yescharacteristic
100C4×C79—21no—
101C4×C79—22no—
102C4×C79—41no—
103C4×C79#45611yescharacteristic
104C4×C79#45811yescharacteristic
105Dic79—82no—
106Dic79—82no—
107Dic79—82no—
108Dic79—161no—
109#12289—791nomaximal, Sylow
110C2×C79—21no—
111C2×C79—21no—
112C2×C79#161011yescharacteristic
113C2×C79#218511yescharacteristic
114C2×C79#219111yescharacteristic
115#3249—791no—
116#2235—792no—
117#2086—791no—
118#1348—792no—
119#1322D7911yescharacteristic
120C79#1228911yescharacteristic, complemented, Sylow
121#533—1581no—
122#477—1582no—
123#467—792no—
124#467—1581no—
125#462—1582no—
126C2×#107—791no—
127C2×C4⋊C8C2×D7911yescharacteristic
128#389—1581no—
129#367—1582no—
130#358—1582no—
131C22⋊C16C2×D7912yes—
132#340—791no—
133#340—1581no—
134C2×Q16—792no—
135C2×Q16—1581no—
136C22×C8D31611yescharacteristic
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—3161no—
147C2×C4⋊C4—1581no—
148C2×C4⋊C4—1581no—
149C2×C42D31611yescharacteristic
150C2×C16—22no—
151#107—792no—
152#107—1582no—
153#107—1581no—
154C4⋊C8—22no—
155#103—1582no—
156#103—1582no—
157#95—3161no—
158C2×Q8—3161no—
159C2×Q8—1582no—
160C22×C4—3161no—
161C22×C4—3161no—
162C22×C4D63211yescharacteristic
163C22×C4D63211yescharacteristic
164C22×C4C2×D31611yescharacteristic
165Q16—3162no—
166Q16—6321no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D7912yes—
171C4⋊C4—3162no—
172C4⋊C4—3161no—
173C4⋊C4—3161no—
174C4⋊C4—3161no—
175C4⋊C4—3161no—
176C4⋊C4—3162no—
177C4⋊C4—3161no—
178C4⋊C4—3161no—
179C4⋊C4—6322no—
180C4⋊C4—1582no—
181C4⋊C4—1582no—
182C22⋊C4—3162no—
183C42—21no—
184C42—21no—
185C42—3162no—
186C16—42no—
187C23C2×D63211yescharacteristic
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—3162no—
198C2×C4—3162no—
199C2×C4—3162no—
200C2×C4—6321no—
201C2×C4—6321no—
202C2×C4—6321no—
203C2×C4—6321no—
204C2×C4—6321no—
205C2×C4#1457910612yes—
206C2×C4#1457913111yescharacteristic
207C2×C4#1457918911yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1909869911yescharacteristic
213C22C2×#1457910611yescharacteristic
214C22#1909900611yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—6322no—
219C4—6322no—
220C4—6322no—
221C4—12641no—
222C4#1909874811yescharacteristic
223C4#1909875411yescharacteristic
224C2—21no—
225C2—21no—
226C2#2376240511yescharacteristic
227C2#2376531911yescharacteristic
228C2#2376535311yescharacteristic
subgroupG/H
centerC22#19098699
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#1322C2
socleC22×C79#447
prime radicalC23×C79C2×D8
hypercenter#1322D79
hypocenterC79#12289
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
120. C79#12289semidirect

Maximal subgroups

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

subgroupindex
#237747452
#237655982
#237648882
#237614132
C79×#13222
#1228979