Finite Group Databasebeta

← #34183313

Subgroups — #34183313

Order 20224.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#19098499C211yesmaximal, characteristic
2#23763488C212yesmaximal, complemented
3C2×#19098287C211yesmaximal, characteristic
4#23761278C212yesmaximal, complemented
5C79×#2312C211yesmaximal, characteristic
6C22×C4⋊Dic79C2211yescharacteristic
7C22×Dic79⋊C4—21no—
8C22×C22⋊Dic79—21no—
9C2×C22⋊(C79⋊C8)C2212yes—
10#19098499—22no—
11#19098499C414yescomplemented
12C2×C79⋊#95C2211yescharacteristic
13#19098325—24no—
14#19098287—22no—
15C22×C22⋊C4×C79C2211yescharacteristic
16#388×C79—24no—
17#380×C79C2212yes—
18#379×C79—21no—
19C4×C22⋊C4×C79—21no—
20C23×Dic79—21no—
21C23×Dic79—21no—
22C22×C79⋊C8—22no—
23C2×C4⋊Dic79—22no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—24no—
28C2×Dic79⋊C4—44no—
29C2×C22⋊Dic79—24no—
30C2×C22⋊Dic79—44no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—21no—
34C79⋊#95C2×C412yes—
35C25×C79D411yescharacteristic
36C23×C4×C79D411yescharacteristic
37C23×C4×C79C2311yescharacteristic
38C2×C4⋊C4×C79D414yes—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79C2×C412yes—
45C2×C22⋊C4×C79D412yes—
46C2×C42×C79—22no—
47#95×C79—22no—
48#95×C79—22no—
49C22×Dic79—24no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54C22×Dic79—42no—
55C22×Dic79—41no—
56C22×Dic79—41no—
57C22×Dic79—42no—
58C2×C79⋊C8—44no—
59C2×C79⋊C8—42no—
60C4⋊Dic79—24no—
61C4⋊Dic79—42no—
62Dic79⋊C4—48no—
63C22⋊Dic79—48no—
64C24×C79—21no—
65C24×C79—22no—
66C24×C79—21no—
67C24×C79—22no—
68C24×C79—42no—
69C24×C79—42no—
70C24×C79C22⋊C412yes—
71C24×C79C2×D411yescharacteristic
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—24no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D412yes—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79#4112yes—
82C4⋊C4×C79—44no—
83C22⋊C4×C79—24no—
84C22⋊C4×C79—24no—
85C22⋊C4×C79—42no—
86C22⋊C4×C79—42no—
87C22⋊C4×C79—82no—
88C22⋊C4×C79—82no—
89C42×C79—82no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—22no—
95C23×C79—42no—
96C23×C79—42no—
97C23×C79—48no—
98C23×C79—42no—
99C23×C79—42no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—81no—
103C23×C79—81no—
104C23×C79—84no—
105C23×C79C2×C22⋊C411yescharacteristic
106C23×C79#12011yescharacteristic
107C23×C79#12311yescharacteristic
108C23×C79#13512yes—
109C23×C79#13712yes—
110C2×C4×C79—24no—
111C2×C4×C79—22no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—44no—
118C2×C4×C79—44no—
119C2×C4×C79—44no—
120C2×C4×C79—81no—
121C2×C4×C79C4×D412yes—
122C2×C4×C79#12112yes—
123C2×Dic79—24no—
124C2×Dic79—48no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—82no—
131C2×Dic79—82no—
132C79⋊C8—82no—
133C22×C79—24no—
134C22×C79—24no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—44no—
140C22×C79—42no—
141C22×C79—42no—
142C22×C79—42no—
143C22×C79—82no—
144C22×C79—84no—
145C22×C79—84no—
146C22×C79#38611yescharacteristic
147C22×C79#41611yescharacteristic
148C22×C79#41911yescharacteristic
149C22×C79#45212yes—
150C22×C79#45312yes—
151C22×C79#45812yes—
152C22×C79#48512yes—
153C4×C79—22no—
154C4×C79—22no—
155C4×C79—42no—
156C4×C79—84no—
157Dic79—44no—
158Dic79—82no—
159#9024—791nomaximal, Sylow
160C2×C79—21no—
161C2×C79—21no—
162C2×C79—22no—
163C2×C79—82no—
164C2×C79#161814yes—
165C2×C79#187311yescharacteristic
166C2×C79#188711yescharacteristic
167C2×C79#189011yescharacteristic
168C2×#399—791no—
169#2312D7911yescharacteristic
170#1824—792no—
171C2×#323—791no—
172#1294—792no—
173C79#902411yescharacteristic, complemented, Sylow
174C22×C4⋊C4—791no—
175C22×C4⋊C4—1581no—
176C22×C22⋊C4—1581no—
177C22×C22⋊C4C2×D7911yescharacteristic
178C2×C22⋊C8—792no—
179#399—794no—
180#399—1582no—
181#388—24no—
182#380C2×D7912yes—
183#379—21no—
184C4×C22⋊C4—21no—
185C2×#95—791no—
186#340—1584no—
187#323—1582no—
188C25C22⋊D7911yescharacteristic
189C23×C4—1581no—
190C23×C4—1581no—
191C23×C4C22⋊D7911yescharacteristic
192C23×C4C22×D7911yescharacteristic
193C22×C8—1582no—
194C2×C4⋊C4—3164no—
195C2×C4⋊C4—792no—
196C2×C4⋊C4—1584no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4—1582no—
199C2×C4⋊C4—1582no—
200C2×C4⋊C4C22⋊D7914yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—3164no—
207C2×C22⋊C4—1584no—
208C2×C22⋊C4C4×D7912yes—
209C2×C22⋊C4D31612yes—
210C2×C42—22no—
211C22⋊C8—1582no—
212C22⋊C8—1582no—
213#95—22no—
214#95—22no—
215#95—792no—
216#95—1581no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24(C2×D79)⋊C412yes—
224C24C2×C22⋊D7911yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—3162no—
231C22×C4—3162no—
232C22×C4—3162no—
233C22×C4—3162no—
234C22×C4—3161no—
235C22×C4—3161no—
236C22×C4—1582no—
237C22×C4—1584no—
238C22×C4—1582no—
239C22×C4(C2×D79)⋊C412yes—
240C22×C4C2×C4×D7911yescharacteristic
241C22×C4C2×C22⋊D7912yes—
242C22×C4C2×D31611yescharacteristic
243C22×C4#1192962412yes—
244C2×C8—3164no—
245C2×C8—3162no—
246C4⋊C4—44no—
247C4⋊C4—3168no—
248C4⋊C4—3162no—
249C4⋊C4—1584no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—3168no—
257C42—82no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×(C2×D79)⋊C411yescharacteristic
274C23#1457913011yescharacteristic
275C23#1457914411yescharacteristic
276C23#1457918112yes—
277C23#1457919112yes—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—3168no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3162no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4—6322no—
296C2×C4—6322no—
297C2×C4—1584no—
298C2×C4C4×C22⋊D7912yes—
299C2×C4#1457913512yes—
300C8—6322no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1909844211yescharacteristic
315C22#1909855711yescharacteristic
316C22#1909857111yescharacteristic
317C22#1909872512yes—
318C22#1909873112yes—
319C22#1909875312yes—
320C22#1909890412yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—84no—
325C4—3164no—
326C4—6322no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#2376244814yes—
332C2#2376371211yescharacteristic
333C2#2376379811yescharacteristic
334C2#2376381311yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2312C2
socleC23×C79#120
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenter#2312D79
hypocenterC79#9024
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #23763488C2semidirect
4. #23761278C2semidirect
11. #19098499C4semidirect
173. C79#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#190984992
#237634882
C2×#190982872
#237612782
C79×#23122
#902479