Finite Group Databasebeta

← #34184039

Subgroups — #34184039

Order 20224.

902 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 139 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#23771687C211yesmaximal, characteristic
2#23763310C211yesmaximal, characteristic
3C2×#19098299C212yesmaximal, complemented
4#23761279C212yesmaximal, complemented
5C2×#381×C79C211yesmaximal, characteristic
6C2×#14579140C412yescomplemented
7C2×#14579140C2211yescharacteristic
8C2×C22⋊(C79⋊C8)—22no—
9C2×C22⋊(C79⋊C8)C2212yes—
10#19098436—21no—
11#19098436—22no—
12#19098423C412yescomplemented
13#19098423C2211yescharacteristic
14#19098419—21no—
15#19098299—24no—
16C24×C4×C79C2211yescharacteristic
17C22×C22⋊C4×C79—21no—
18#381×C79—24no—
19C2×#95×C79—21no—
20C2×#95×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C22×C79⋊C8—22no—
23#14579140—22no—
24#14579140—22no—
25#14579140—44no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C412yes—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—44no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C25×C79D411yescharacteristic
40C23×C4×C79—22no—
41C23×C4×C79—22no—
42C23×C4×C79—22no—
43C23×C4×C79—44no—
44C23×C4×C79C2×C412yes—
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D412yes—
47C23×C4×C79C2311yescharacteristic
48C2×C22⋊C4×C79—24no—
49C2×C22⋊C4×C79—44no—
50#95×C79—24no—
51#95×C79—22no—
52#95×C79D414yes—
53C2×Dic158—22no—
54C2×Dic158—42no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C22×Dic79—22no—
58C2×C79⋊C8—44no—
59C2×C79⋊C8—42no—
60C4⋊Dic79—22no—
61C4⋊Dic79—42no—
62Dic79⋊C4—44no—
63Dic79⋊C4—42no—
64C22⋊Dic79—44no—
65C22⋊Dic79—44no—
66C22⋊Dic79—41no—
67C22⋊Dic79—41no—
68C22⋊Dic79—42no—
69C24×C79—24no—
70C24×C79—21no—
71C24×C79—21no—
72C24×C79—22no—
73C24×C79—22no—
74C24×C79—42no—
75C24×C79C22⋊C412yes—
76C24×C79C2×D411yescharacteristic
77C22×C4×C79—24no—
78C22×C4×C79—24no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—24no—
82C22×C4×C79—24no—
83C22×C4×C79—24no—
84C22×C4×C79—42no—
85C22×C4×C79—42no—
86C22×C4×C79—42no—
87C22×C4×C79—42no—
88C22×C4×C79—48no—
89C22×C4×C79—48no—
90C22×C4×C79—44no—
91C22×C4×C79—44no—
92C22×C4×C79—42no—
93C22×C4×C79—42no—
94C22×C4×C79C22⋊C412yes—
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D412yes—
98C22×C4×C79#4112yes—
99C22⋊C4×C79—48no—
100C23×C79—22no—
101C23×C79—22no—
102C23×C79—24no—
103C23×C79—24no—
104C23×C79—22no—
105C23×C79—22no—
106C23×C79—22no—
107C23×C79—22no—
108C23×C79—22no—
109C23×C79—22no—
110C23×C79—48no—
111C23×C79—48no—
112C23×C79—42no—
113C23×C79—42no—
114C23×C79—42no—
115C23×C79—42no—
116C23×C79#9914yes—
117C23×C79C2×C22⋊C411yescharacteristic
118C23×C79#12011yescharacteristic
119C23×C79#12311yescharacteristic
120C23×C79#13512yes—
121C23×C79#13712yes—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—22no—
125C2×C4×C79—22no—
126C2×C4×C79—22no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—48no—
130C2×C4×C79—48no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—44no—
134C2×C4×C79—44no—
135C2×C4×C79—44no—
136C2×C4×C79—84no—
137C2×C4×C79—84no—
138C2×C4×C79C4×D412yes—
139C2×C4×C79#12112yes—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—42no—
143C2×Dic79—42no—
144C2×Dic79—42no—
145C2×Dic79—42no—
146C2×Dic79—42no—
147Dic158—82no—
148C79⋊C8—82no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—24no—
154C22×C79—24no—
155C22×C79—24no—
156C22×C79—24no—
157C22×C79—24no—
158C22×C79—48no—
159C22×C79—48no—
160C22×C79—42no—
161C22×C79—42no—
162C22×C79—42no—
163C22×C79—42no—
164C22×C79#38611yescharacteristic
165C22×C79C2×#9912yes—
166C22×C79#41611yescharacteristic
167C22×C79#41911yescharacteristic
168C22×C79#42112yes—
169C22×C79#45212yes—
170C22×C79#48512yes—
171C4×C79—22no—
172C4×C79—22no—
173C4×C79—42no—
174C4×C79—84no—
175Dic79—81no—
176Dic79—81no—
177Dic79—82no—
178#9154—791nomaximal, Sylow
179C2×C79—22no—
180C2×C79—22no—
181C2×C79—44no—
182C2×C79#151112yes—
183C2×C79#151612yes—
184C2×C79#187311yescharacteristic
185C2×C79#189711yescharacteristic
186C2×C79#189811yescharacteristic
187#2848—791no—
188C2×#381D7911yescharacteristic
189#1779—791no—
190C2×#328—792no—
191#1294—792no—
192C79#915411yescharacteristic, complemented, Sylow
193C24×C4C2×D7911yescharacteristic
194C2×C22⋊Q8—792no—
195C2×C22⋊Q8—791no—
196C22×C22⋊C4—21no—
197C2×C22⋊C8—792no—
198C2×C22⋊C8—1582no—
199#385—1581no—
200#385—1582no—
201#381—24no—
202#380—791no—
203#380—792no—
204#379—1581no—
205C2×#95—21no—
206C2×#95C2×D7912yes—
207#328—1584no—
208C25C22⋊D7911yescharacteristic
209C22×Q8—791no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—44no—
214C23×C4C4×D7912yes—
215C23×C4D31612yes—
216C23×C4C22⋊D7911yescharacteristic
217C23×C4C22×D7911yescharacteristic
218C22×C8—1582no—
219C22⋊Q8—3164no—
220C22⋊Q8—1582no—
221C22⋊Q8—1582no—
222C2×C4⋊C4—792no—
223C2×C4⋊C4—791no—
224C2×C4⋊C4—1582no—
225C2×C4⋊C4—1581no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—1582no—
229C2×C22⋊C4—1582no—
230C2×C22⋊C4—1581no—
231C2×C22⋊C4—1581no—
232C22⋊C8—3164no—
233C22⋊C8—1582no—
234C22⋊C8—1582no—
235#95—24no—
236#95—22no—
237#95—1582no—
238#95—1581no—
239#95C22⋊D7914yes—
240C24—24no—
241C24—21no—
242C24—21no—
243C24—22no—
244C24—22no—
245C24—42no—
246C24(C2×D79)⋊C412yes—
247C24C2×C22⋊D7911yescharacteristic
248C2×Q8—3162no—
249C2×Q8—1582no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—1581no—
268C22×C4—1582no—
269C22×C4—1581no—
270C22×C4(C2×D79)⋊C412yes—
271C22×C4C2×C4×D7911yescharacteristic
272C22×C4C2×C22⋊D7912yes—
273C22×C4C2×D31611yescharacteristic
274C22×C4#1192962412yes—
275C2×C8—3164no—
276C2×C8—3162no—
277C4⋊C4—3162no—
278C4⋊C4—3164no—
279C4⋊C4—3162no—
280C4⋊C4—1582no—
281C22⋊C4—48no—
282C22⋊C4—3164no—
283C22⋊C4—3164no—
284C22⋊C4—3161no—
285C22⋊C4—3161no—
286C22⋊C4—3162no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23#1457907514yes—
304C23C2×(C2×D79)⋊C411yescharacteristic
305C23#1457913011yescharacteristic
306C23#1457914411yescharacteristic
307C23#1457918112yes—
308C23#1457919112yes—
309Q8—6322no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—3162no—
329C2×C4—3162no—
330C2×C4—3162no—
331C2×C4—3162no—
332C2×C4—3162no—
333C2×C4C4×C22⋊D7912yes—
334C2×C4#1457913512yes—
335C8—6322no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22#1909844211yescharacteristic
352C22C2×#1457907512yes—
353C22#1909855711yescharacteristic
354C22#1909857111yescharacteristic
355C22#1909858112yes—
356C22#1909872512yes—
357C22#1909890412yes—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—84no—
362C4—6322no—
363C4—6321no—
364C4—6321no—
365C2—22no—
366C2—22no—
367C2—44no—
368C2#2376193212yes—
369C2#2376195412yes—
370C2#2376371211yescharacteristic
371C2#2376384811yescharacteristic
372C2#2376385311yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#381×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenterC2×#381D79
hypocenterC79#9154
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
3. C2×#19098299C2semidirect
4. #23761279C2semidirect
6. C2×#14579140C4semidirect
12. #19098423C4semidirect
192. C79#9154semidirect

Maximal subgroups

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

subgroupindex
#237716872
#237633102
C2×#190982992
#237612792
C2×#381×C792
#915479