Finite Group Databasebeta

← #34197532

Subgroups — #34197532

Order 20224.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#23766065C211yesmaximal, characteristic
2#23761721C211yesmaximal, characteristic
3#23761347C212yesmaximal, complemented
4#23761329C212yesmaximal, complemented
5C79×#1459C211yesmaximal, characteristic
6C2×#14579142C411yescharacteristic
7C22×Dic79⋊C4C411yescharacteristic
8#19098616—21no—
9#19098458—21no—
10#19098457—21no—
11#19098447—21no—
12#19098443C411yescharacteristic
13#19098436—21no—
14#19098435C2211yescharacteristic
15#19098424C2211yescharacteristic
16C4×C22⋊Dic79C411yescharacteristic
17#19098287—21no—
18#19098287C2212yes—
19C2×C4×D4×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#324×C79—21no—
22#323×C79C2212yes—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25#14579142—22no—
26#14579142C814yescomplemented
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—41no—
32C2×Dic79⋊C4—41no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79C2×C411yescharacteristic
36C2×C22⋊Dic79C2×C411yescharacteristic
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—41no—
39C22⋊(C79⋊C8)—22no—
40C4×C79⋊C8—41no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C22×D4×C79C2×C411yescharacteristic
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C22×C8×C79—41no—
50C4×D4×C79—41no—
51C4×D4×C79—41no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79C2311yescharacteristic
55C2×C42×C79D411yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—41no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—41no—
64C22×Dic79—41no—
65C22×Dic79—41no—
66C22×Dic79—81no—
67C22×Dic79C2×C812yescomplemented
68C2×C79⋊C8—42no—
69Dic79⋊C4—24no—
70Dic79⋊C4—42no—
71Dic79⋊C4—41no—
72Dic79⋊C4—41no—
73Dic79⋊C4—81no—
74C22⋊Dic79—24no—
75C22⋊Dic79—21no—
76C22⋊Dic79—41no—
77C22⋊Dic79—81no—
78C22⋊Dic79C2×C812yescomplemented
79C4×Dic79—81no—
80C24×C79C22⋊C411yescharacteristic
81C24×C79C22×C411yescharacteristic
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C2×D4×C79C2×C812yes—
85C2×D4×C79M1612yes—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—81no—
91C22×C4×C79—81no—
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C2×C8×C79—42no—
98C2×C8×C79—82no—
99C4⋊C4×C79—41no—
100C22⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C42×C79—21no—
103C42×C79—21no—
104C23×C79—21no—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—41no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79C22⋊C812yes—
113C23×C79C2×C22⋊C411yescharacteristic
114C23×C79C22×C811yescharacteristic
115C23×C79C2×M1611yescharacteristic
116D4×C79—42no—
117D4×C79—42no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—81no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79C22⋊C812yes—
132C2×C4×C79#9911yescharacteristic
133C2×C4×C79#9911yescharacteristic
134C2×Dic79—22no—
135C2×Dic79—22no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—42no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—42no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—82no—
149C8×C79—82no—
150C79⋊C8—82no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—22no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—42no—
157C22×C79—41no—
158C22×C79—41no—
159C22×C79—42no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79—41no—
163C22×C79—41no—
164C22×C79—81no—
165C22×C79C2×C22⋊C811yescharacteristic
166C22×C79C2×#9911yescharacteristic
167C22×C79#41311yescharacteristic
168C4×C79—21no—
169C4×C79—21no—
170C4×C79—22no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—41no—
174Dic79—42no—
175Dic79—42no—
176Dic79—81no—
177Dic79—81no—
178#11474—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—41no—
182C2×C79—41no—
183C2×C79—42no—
184C2×C79#145911yescharacteristic
185C2×C79#212111yescharacteristic
186C2×C79#212311yescharacteristic
187#2322—791no—
188#1460—791no—
189#1459D7911yescharacteristic
190#1326—792no—
191#1317—792no—
192C79#1147411yescharacteristic, complemented, Sylow
193C2×#123—791no—
194C2×C4×D4C2×D7911yescharacteristic
195C22×C4⋊C4—791no—
196#431—1581no—
197C2×C22⋊C8—21no—
198#389—1581no—
199#388—1581no—
200#387—1581no—
201#386—791no—
202#385—791no—
203#385—1581no—
204#381—791no—
205C4×C22⋊C4—791no—
206#324—21no—
207#323—792no—
208#323—1581no—
209#323C2×D7912yes—
210C22×D4C4×D7911yescharacteristic
211C23×C4—791no—
212C23×C4—791no—
213C23×C4C4×D7911yescharacteristic
214C23×C4C22⋊D7911yescharacteristic
215C22×C8—41no—
216#123—794no—
217#123—1582no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—3161no—
221C2×C4⋊C4—3161no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4—1581no—
225C2×C4⋊C4—1581no—
226C2×C4⋊C4C22⋊D7911yescharacteristic
227C2×C22⋊C4—791no—
228C2×C22⋊C4—791no—
229C2×C22⋊C4—1581no—
230C2×C22⋊C4—1581no—
231C2×C22⋊C4D31611yescharacteristic
232C2×C22⋊C4C22×D7911yescharacteristic
233C2×C42—1581no—
234C2×C42D31611yescharacteristic
235C4⋊C8—3161no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—1582no—
239C4×C8—3161no—
240#95—1581no—
241#95—1581no—
242#95—1581no—
243#95—1581no—
244#95—1581no—
245C24(C2×D79)⋊C411yescharacteristic
246C24C2×C4×D7911yescharacteristic
247C2×D4—41no—
248C2×D4—41no—
249C2×D4C8×D7912yes—
250C2×D4#1192960712yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—3161no—
258C22×C4—3161no—
259C22×C4—3161no—
260C22×C4—792no—
261C22×C4—6321no—
262C22×C4—1581no—
263C22×C4—1581no—
264C22×C4—1581no—
265C22×C4—1581no—
266C22×C4—1581no—
267C22×C4(C2×D79)⋊C411yescharacteristic
268C22×C4(C2×D79)⋊C411yescharacteristic
269C22×C4(C2×D79)⋊C411yescharacteristic
270C22×C4C2×C22⋊D7911yescharacteristic
271C22×C4C2×D31611yescharacteristic
272C2×C8—42no—
273C2×C8—82no—
274C2×C8—3162no—
275C4⋊C4—41no—
276C4⋊C4—3162no—
277C4⋊C4—3161no—
278C4⋊C4—3161no—
279C4⋊C4—6321no—
280C4⋊C4—1584no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—3161no—
284C22⋊C4—792no—
285C22⋊C4—6321no—
286C22⋊C4—1584no—
287C22⋊C4—1581no—
288C42—21no—
289C42—21no—
290C42—6321no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23(C2×D79)⋊C812yes—
300C23C2×(C2×D79)⋊C411yescharacteristic
301C23C2×C8×D7911yescharacteristic
302C23C2×#1192960711yescharacteristic
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3161no—
319C2×C4—3162no—
320C2×C4—3161no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—3162no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—3161no—
329C2×C4—3161no—
330C2×C4—6322no—
331C2×C4—1582no—
332C2×C4—1582no—
333C2×C4(C2×D79)⋊C812yes—
334C2×C4#1457907511yescharacteristic
335C2×C4#1457907511yescharacteristic
336C8—82no—
337C8—6322no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22C2×(C2×D79)⋊C811yescharacteristic
353C22C2×#1457907511yescharacteristic
354C22#1909854511yescharacteristic
355C4—22no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3162no—
362C4—3162no—
363C4—6321no—
364C4—6321no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—42no—
370C2#2376171611yescharacteristic
371C2#2376504411yescharacteristic
372C2#2376505511yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC23×C79C22×C8
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1459C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#1459D79
hypocenterC79#11474
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #23761347C2semidirect
4. #23761329C2semidirect
26. #14579142C8semidirect
67. C22×Dic79C2×C8semidirect
78. C22⋊Dic79C2×C8semidirect
192. C79#11474semidirect

Maximal subgroups

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

subgroupindex
#237660652
#237617212
#237613472
#237613292
C79×#14592
#1147479