Finite Group Databasebeta

← #34198898

Subgroups — #34198898

Order 20224.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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#23767717C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761426C214yesmaximal, complemented
4C4×#100×C79C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic
6#19098527—21no—
7#19098502C2212yes—
8#19098499C412yes—
9#19098461—21no—
10#19098443—21no—
11#19098435—21no—
12#19098423—21no—
13#19098423C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#100×C79—21no—
18C4×M16×C79—21no—
19#343×C79C2212yes—
20C23×Dic79—21no—
21C2×#11929608—41no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—41no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—21no—
29C2×C22⋊Dic79—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—41no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95C2×C412yes—
35C22×D4×C79D411yescharacteristic
36C23×C4×C79C2×C411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C2×M16×C79—22no—
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79C2311yescharacteristic
42C2×C22⋊C4×C79C2×C411yescharacteristic
43C2×C22⋊C4×C79D411yescharacteristic
44C2×C42×C79D411yescharacteristic
45#100×C79—24no—
46#97×C79—41no—
47C4×C8×C79—41no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50C22×Dic79—42no—
51C22×Dic79—82no—
52#11929608—82no—
53C2×C79⋊C8—42no—
54C4⋊Dic79—42no—
55C4⋊Dic79—42no—
56C4⋊Dic79—41no—
57C4⋊Dic79—41no—
58Dic79⋊C4—82no—
59C22⋊Dic79—42no—
60C22⋊Dic79—82no—
61C24×C79C22⋊C411yescharacteristic
62C24×C79C22⋊C411yescharacteristic
63C2×D4×C79—22no—
64C2×D4×C79—22no—
65C2×D4×C79—41no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—42no—
71C22×C4×C79—42no—
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22×C411yescharacteristic
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77M16×C79—24no—
78M16×C79—42no—
79C2×C8×C79—42no—
80C4⋊C4×C79—41no—
81C22⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C42×C79—21no—
84C42×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79—41no—
94C23×C79#10412yes—
95C23×C79C2×C22⋊C411yescharacteristic
96D4×C79—44no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—81no—
110C2×C4×C79#9911yescharacteristic
111C2×C4×C79#9911yescharacteristic
112C2×C4×C79#13712yes—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—84no—
120C8×C79—44no—
121C79⋊C8—82no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—22no—
125C22×C79—42no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79—41no—
129C22×C79—41no—
130C22×C79—41no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—81no—
134C22×C79C2×#9911yescharacteristic
135C22×C79#41911yescharacteristic
136C22×C79C2×#10411yescharacteristic
137C4×C79—21no—
138C4×C79—22no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79#35212yes—
143Dic79—82no—
144Dic79—82no—
145#11790—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79—42no—
149C2×C79—81no—
150C2×C79#150811yescharacteristic
151C2×C79C2×#35211yescharacteristic
152C2×C79#213411yescharacteristic
153#2487—791no—
154C4×#100D7911yescharacteristic
155#1465—791no—
156#1353—794no—
157C79#1179011yescharacteristic, complemented, Sylow
158C2×C4×D4C2×D7911yescharacteristic
159C22×C4⋊C4—791no—
160C2×#100—21no—
161#407—1581no—
162C4×M16—21no—
163#400—792no—
164#399—792no—
165#389—1581no—
166#386—1581no—
167#385—1581no—
168#380—791no—
169#380—1581no—
170#343C2×D7912yes—
171#324—792no—
172#323—1581no—
173C22×D4C22⋊D7911yescharacteristic
174C23×C4—1581no—
175C23×C4C4×D7911yescharacteristic
176C23×C4D31611yescharacteristic
177C2×M16—22no—
178C2×M16—3161no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—3161no—
182C2×C4⋊C4—3161no—
183C2×C4⋊C4—792no—
184C2×C4⋊C4—1581no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4—1582no—
187C2×C4⋊C4C22×D7911yescharacteristic
188C2×C22⋊C4—1581no—
189C2×C22⋊C4—1582no—
190C2×C22⋊C4C4×D7911yescharacteristic
191C2×C22⋊C4D31611yescharacteristic
192C2×C42C22⋊D7911yescharacteristic
193#100—24no—
194C22⋊C8—3161no—
195C22⋊C8—1582no—
196#97—41no—
197C4×C8—41no—
198#95—792no—
199#95—1582no—
200#95—1582no—
201C24(C2×D79)⋊C411yescharacteristic
202C24(C2×D79)⋊C411yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—3162no—
213C22×C4—6322no—
214C22×C4—1582no—
215C22×C4—1582no—
216C22×C4(C2×D79)⋊C411yescharacteristic
217C22×C4(C2×D79)⋊C411yescharacteristic
218C22×C4C2×C4×D7911yescharacteristic
219C22×C4C2×C22⋊D7911yescharacteristic
220C22×C4C2×D31611yescharacteristic
221M16—24no—
222M16—42no—
223M16—6322no—
224C2×C8—42no—
225C2×C8—3162no—
226C4⋊C4—41no—
227C4⋊C4—3162no—
228C4⋊C4—3162no—
229C4⋊C4—3161no—
230C4⋊C4—3161no—
231C4⋊C4—6322no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—3162no—
235C22⋊C4—6322no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1457908912yes—
248C23C2×(C2×D79)⋊C411yescharacteristic
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—3162no—
264C2×C4—3162no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—3162no—
269C2×C4—6324no—
270C2×C4#1457907511yescharacteristic
271C2×C4#1457907511yescharacteristic
272C2×C4#1457919112yes—
273C8—44no—
274C8—6322no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22C2×#1457907511yescharacteristic
288C22#1909857111yescharacteristic
289C22C2×#1457908911yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—6322no—
296C4—6322no—
297C4#1909835412yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2376191911yescharacteristic
303C2C2×#1909835411yescharacteristic
304C2#2376509111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#100×C79C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#100D79
hypocenterC79#11790
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #23761426C2semidirect
157. C79#11790semidirect

Maximal subgroups

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

subgroupindex
#237677172
#237617382
#237614262
C4×#100×C792
#1179079