Finite Group Databasebeta

← #34200714

Subgroups — #34200714

Order 20224.

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

normal subgroups only

This group has 390 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, complemented
2#23765329C212yesmaximal
3#23761411C212yesmaximal, complemented
4#23761340C211yesmaximal, characteristic, complemented
5C79×#2054C211yesmaximal, characteristic
6#19099120—21no—
7C2×#14579105—22no—
8#19098859—22no—
9#19098804—21no—
10#19098804C2212yes—
11#19098774—22no—
12#19098364—22no—
13C22⋊(C79⋊C16)C2212yes—
14C22×D8×C79C2211yescharacteristic
15C2×C4⋊C8×C79C2211yescharacteristic
16C2×#102×C79—21no—
17#390×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—
29#14579105—44no—
30C2×C79⋊C16—22no—
31#14579095—21no—
32#14579095D412yes—
33#14579086—22no—
34#14579086—22no—
35C22×D4×C79—21no—
36C2×D8×C79—22no—
37C2×D8×C79—42no—
38C22×C8×C79D411yescharacteristic
39C22×C8×C79C2311yescharacteristic
40C2×C4⋊C4×C79—21no—
41C2×C22⋊C4×C79—41no—
42C2×C22⋊C4×C79—41no—
43C2×C42×C79D411yescharacteristic
44C4⋊C8×C79—22no—
45#102×C79—42no—
46C2×Dic158—22no—
47C2×Dic158—41no—
48Dic316—42no—
49Dic316—81no—
50C4⋊Dic79—22no—
51C4⋊Dic79—41no—
52C4⋊Dic79—41no—
53C4⋊Dic79—41no—
54Dic79⋊C4—42no—
55Dic79⋊C4—42no—
56C22⋊Dic79—42no—
57C4×Dic79—42no—
58C79⋊C16—42no—
59C24×C79—41no—
60C2×D4×C79—22no—
61C2×D4×C79—41no—
62C2×D4×C79—41no—
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79—41no—
66C22×C4×C79—41no—
67C22×C4×C79D811yescharacteristic
68C22×C4×C79D811yescharacteristic
69C22×C4×C79C2×D411yescharacteristic
70D8×C79—24no—
71D8×C79—42no—
72C2×C8×C79—21no—
73C2×C8×C79—41no—
74C2×C8×C79—41no—
75C2×C8×C79C2×D412yes—
76C4⋊C4×C79—41no—
77C4⋊C4×C79—41no—
78C22⋊C4×C79—82no—
79C22⋊C4×C79—82no—
80C42×C79—21no—
81C42×C79—21no—
82C23×C79—42no—
83C23×C79—81no—
84C23×C79—81no—
85C23×C79—81no—
86C23×C79—81no—
87C23×C79—81no—
88C23×C79—81no—
89C23×C79C2×D811yescharacteristic
90D4×C79—42no—
91D4×C79—42no—
92D4×C79—81no—
93D4×C79—81no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—41no—
100C2×C4×C79—81no—
101C2×C4×C79—81no—
102C2×C4×C79—81no—
103C2×C4×C79SD3212yescomplemented
104C2×C4×C79#12011yescharacteristic
105C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
106C2×Dic79—42no—
107C2×Dic79—42no—
108Dic158—42no—
109Dic158—42no—
110Dic158—82no—
111C8×C79—22no—
112C8×C79—41no—
113C22×C79—21no—
114C22×C79—21no—
115C22×C79—82no—
116C22×C79—82no—
117C22×C79—82no—
118C22×C79—161no—
119C22×C79—161no—
120C22×C79—161no—
121C22×C79—161no—
122C22×C79#44711yescharacteristic
123C22×C79C2×SD3211yescharacteristic
124C22×C79#51011yescharacteristic
125C4×C79—21no—
126C4×C79—22no—
127C4×C79—41no—
128C4×C79—161no—
129C4×C79#45411yescharacteristic
130C4×C79B(4,𝔽2)11yescharacteristic
131Dic79—82no—
132Dic79—82no—
133#12159—791nomaximal, Sylow
134C2×C79—21no—
135C2×C79—21no—
136C2×C79—162no—
137C2×C79#159811yescharacteristic
138C2×C79#218511yescharacteristic
139C2×C79#219011yescharacteristic
140#3249—791no—
141#2187—792no—
142#2054D7911yescharacteristic
143#1348—792no—
144#1322—791no—
145C79#1215911yescharacteristic, complemented, Sylow
146C22×D8C2×D7911yescharacteristic
147#533—1581no—
148C2×SD32—1582no—
149#477—1582no—
150#467—792no—
151#467—1581no—
152#462—1582no—
153C2×C4⋊C8C2×D7911yescharacteristic
154C2×#102—21no—
155#390—21no—
156#357—1582no—
157C22⋊C16—792no—
158#340C2×D7911yescharacteristic
159C22×D4—21no—
160C2×Q16—792no—
161C2×Q16—1581no—
162C2×D8—22no—
163C2×D8—42no—
164C22×C8C22⋊D7911yescharacteristic
165C22×C8C22×D7911yescharacteristic
166C4⋊Q8—1581no—
167C4⋊Q8—1581no—
168#125—3161no—
169C22⋊Q8—3161no—
170C22⋊Q8—1582no—
171C4×Q8—3161no—
172C4×Q8—1582no—
173#117—3161no—
174C2×C4⋊C4—21no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42C22⋊D7911yescharacteristic
178SD32—3164no—
179C2×C16—1582no—
180#107—792no—
181#107—1581no—
182C4⋊C8—22no—
183#103—1582no—
184#103—1582no—
185#102—42no—
186C24—41no—
187C2×Q8—3161no—
188C2×Q8—1582no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C22×C4—41no—
196C22×C4#1192960911yescharacteristic
197C22×C4#1192960911yescharacteristic
198C22×C4C2×C22⋊D7911yescharacteristic
199Q16—3162no—
200Q16—6321no—
201D8—24no—
202D8—42no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8D4×D7912yes—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—3162no—
210C4⋊C4—3161no—
211C4⋊C4—3161no—
212C4⋊C4—3161no—
213C4⋊C4—3162no—
214C4⋊C4—1582no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—3162no—
218C42—21no—
219C42—21no—
220C42—3162no—
221C16—3162no—
222C23—42no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×#1192960911yescharacteristic
230Q8—3162no—
231Q8—3162no—
232Q8—6322no—
233D4—42no—
234D4—42no—
235D4—81no—
236D4—81no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—3162no—
247C2×C4—3162no—
248C2×C4#1457910512yescomplemented
249C2×C4#1457912911yescharacteristic
250C2×C4#1457918611yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—161no—
259C22—161no—
260C22—161no—
261C22—161no—
262C22#1909869611yescharacteristic
263C22C2×#1457910511yescharacteristic
264C22#1909900311yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—161no—
269C4—6322no—
270C4—6322no—
271C4#1909873511yescharacteristic
272C4#1909874911yescharacteristic
273C2—21no—
274C2—21no—
275C2—162no—
276C2#2376233411yescharacteristic
277C2#2376531611yescharacteristic
278C2#2376534511yescharacteristic
subgroupG/H
centerC22#19098696
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#2054C2
socleC22×C79#447
prime radicalC22×D8×C79C22
hypercenter#2054D79
hypocenterC79#12159
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #23774745C2semidirect
3. #23761411C2semidirect
4. #23761340C2semidirect
103. C2×C4×C79SD32semidirect
145. C79#12159semidirect
248. C2×C4#14579105semidirect

Maximal subgroups

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

subgroupindex
#237747452
#237653292
#237614112
#237613402
C79×#20542
#1215979