Finite Group Databasebeta

← #34186128

Subgroups — #34186128

Order 20224.

518 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 115 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#23768754C211yesmaximal, characteristic
2#23763392C211yesmaximal, characteristic
3#23761830C212yesmaximal, complemented
4#23761259C212yesmaximal, complemented
5C79×#2975C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7#19098587C2212yes—
8C2×#14579078—21no—
9#19098499C412yes—
10#19098475C412yes—
11#19098466—21no—
12#19098459—21no—
13#19098436—22no—
14#19098429C2211yescharacteristic
15#19098332—21no—
16#19098295—24no—
17C2×C4×D4×C79C2211yescharacteristic
18#436×C79—24no—
19#431×C79C2212yes—
20#407×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22C2×#11929608—22no—
23C4×Dic158—22no—
24C4×Dic158—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—22no—
31C2×C4×Dic79—21no—
32C4⋊(C79⋊C8)—22no—
33C4⋊(C79⋊C8)—22no—
34#14579078—42no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C79⋊#95C2×C412yes—
38C22×D4×C79D411yescharacteristic
39C23×C4×C79D412yes—
40C2×M16×C79—24no—
41C4×D4×C79—22no—
42C4×D4×C79—22no—
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79C2×C412yes—
45C2×C42×C79C2311yescharacteristic
46C4⋊C8×C79—22no—
47C22⋊C8×C79—24no—
48C22⋊C8×C79—22no—
49C4×C8×C79D414yes—
50C2×Dic158—22no—
51C2×Dic158—42no—
52C22×Dic79—21no—
53C22×Dic79—22no—
54C22×Dic79—21no—
55#11929608—42no—
56#11929608—42no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59C4⋊Dic79—41no—
60C4⋊Dic79—41no—
61Dic79⋊C4—41no—
62Dic79⋊C4—41no—
63Dic79⋊C4—82no—
64C22⋊Dic79—44no—
65C22⋊Dic79—82no—
66C4×Dic79—42no—
67C24×C79C22⋊C412yes—
68C2×D4×C79—22no—
69C2×D4×C79—21no—
70C2×D4×C79—22no—
71C2×D4×C79—21no—
72C22×C4×C79—22no—
73C22×C4×C79—22no—
74C22×C4×C79—82no—
75C22×C4×C79C22⋊C412yes—
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79M16×C79—84no—
80C2×C8×C79—24no—
81C4⋊C4×C79—22no—
82C22⋊C4×C79—24no—
83C42×C79C2×D412yes—
84C42×C79#4112yes—
85C23×C79—22no—
86C23×C79—22no—
87C23×C79—22no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79C2×C22⋊C411yescharacteristic
91D4×C79—42no—
92D4×C79—42no—
93C2×C4×C79—22no—
94C2×C4×C79—41no—
95C2×C4×C79—41no—
96C2×C4×C79—44no—
97C2×C4×C79—42no—
98C2×C4×C79—82no—
99C2×C4×C79C4×D412yes—
100C2×C4×C79#12011yescharacteristic
101C2×C4×C79#12112yes—
102C2×C4×C79#12311yescharacteristic
103C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C79#13712yes—
105C2×C4×C79#13711yescharacteristic
106C2×Dic79—22no—
107C2×Dic79—41no—
108C2×Dic79—41no—
109C2×Dic79—42no—
110C2×Dic79—41no—
111C2×Dic79—42no—
112C2×Dic79—41no—
113C2×Dic79—41no—
114C2×Dic79—42no—
115Dic158—24no—
116Dic158—42no—
117C8×C79—44no—
118C79⋊C8—82no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79—44no—
122C22×C79—42no—
123C22×C79—42no—
124C22×C79—42no—
125C22×C79—82no—
126C22×C79#38611yescharacteristic
127C22×C79#41711yescharacteristic
128C22×C79#41911yescharacteristic
129C4×C79—22no—
130C4×C79—21no—
131C4×C79—21no—
132C4×C79—42no—
133C4×C79#46812yes—
134C4×C79#47012yes—
135Dic79—42no—
136Dic79—82no—
137Dic79—81no—
138#9487—791nomaximal, Sylow
139C2×C79—21no—
140C2×C79—21no—
141C2×C79—82no—
142C2×C79#189111yescharacteristic
143C2×C79#189411yescharacteristic
144C2×C79#191111yescharacteristic
145#2975D7911yescharacteristic
146#2580—791no—
147#1800—791no—
148#1487—792no—
149#1286—792no—
150C79#948711yescharacteristic, complemented, Sylow
151C2×C4×Q8—791no—
152C2×C4×D4C2×D7911yescharacteristic
153#436—24no—
154#431C2×D7912yes—
155#423—792no—
156C2×#100—1581no—
157#407—22no—
158#399—792no—
159#393—792no—
160#390—1581no—
161#389—1581no—
162#385—1582no—
163#382—791no—
164#343—1581no—
165Q8⋊C8—1584no—
166C22×Q8—791no—
167C22×D4D31611yescharacteristic
168C23×C4C22⋊D7912yes—
169C2×M16—24no—
170C2×M16—1582no—
171C4×Q8—1582no—
172C4×Q8—1582no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C4⋊C4—791no—
176C2×C4⋊C4—1581no—
177C2×C4⋊C4—1581no—
178C2×C4⋊C4—1582no—
179C2×C4⋊C4D31611yescharacteristic
180C2×C22⋊C4—1582no—
181C2×C22⋊C4—1582no—
182C2×C22⋊C4C4×D7912yes—
183C2×C42—1581no—
184C2×C42C22×D7911yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—1582no—
187C4⋊C8—1582no—
188#100—3162no—
189C22⋊C8—24no—
190C22⋊C8—22no—
191C4×C8C22⋊D7914yes—
192#95—792no—
193#95—1582no—
194#95—1581no—
195C24(C2×D79)⋊C412yes—
196C2×Q8—3162no—
197C2×Q8—1582no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—82no—
205C22×C4—1582no—
206C22×C4—1581no—
207C22×C4—1581no—
208C22×C4(C2×D79)⋊C412yes—
209C22×C4C2×C4×D7911yescharacteristic
210C22×C4C2×C22⋊D7911yescharacteristic
211C22×C4C2×D31611yescharacteristic
212M16—84no—
213M16—3162no—
214M16—3162no—
215C2×C8—24no—
216C2×C8—3162no—
217C4⋊C4—22no—
218C4⋊C4—3161no—
219C4⋊C4—3161no—
220C4⋊C4—3161no—
221C4⋊C4—3161no—
222C4⋊C4—6322no—
223C4⋊C4—1582no—
224C22⋊C4—24no—
225C22⋊C4—3164no—
226C22⋊C4—6322no—
227C42—3162no—
228C42C2×C22⋊D7912yes—
229C42#1192962412yes—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23C2×(C2×D79)⋊C411yescharacteristic
236Q8—3162no—
237Q8—1584no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—3161no—
247C2×C4—3161no—
248C2×C4—3162no—
249C2×C4—3161no—
250C2×C4—3162no—
251C2×C4—3161no—
252C2×C4—3161no—
253C2×C4—3162no—
254C2×C4—1582no—
255C2×C4C4×C22⋊D7912yes—
256C2×C4#1457913011yescharacteristic
257C2×C4#1457913512yes—
258C2×C4#1457914411yescharacteristic
259C2×C4#1457918911yescharacteristic
260C2×C4#1457919412yes—
261C2×C4#1457919411yescharacteristic
262C8—44no—
263C8—6322no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#1909844211yescharacteristic
272C22#1909856511yescharacteristic
273C22#1909857311yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—3162no—
279C4—6322no—
280C4—6321no—
281C4#1909880912yes—
282C4#1909882112yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#2376381811yescharacteristic
287C2#2376383311yescharacteristic
288C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2975C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#2975D79
hypocenterC79#9487
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #23761830C2semidirect
4. #23761259C2semidirect
150. C79#9487semidirect

Maximal subgroups

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

subgroupindex
#237687542
#237633922
#237618302
#237612592
C79×#29752
#948779