Finite Group Databasebeta

← #34185938

Subgroups — #34185938

Order 20224.

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

normal subgroups only

This group has 490 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#23768596C211yesmaximal, characteristic
2#23762026C212yesmaximal, complemented
3#23761741C211yesmaximal, characteristic
4#23761255C212yesmaximal, complemented
5C79×#2973C211yesmaximal, characteristic
6C2×#14579153C2211yescharacteristic
7#19098620C2212yes—
8#19098486—21no—
9#19098486—21no—
10#19098466C411yescharacteristic
11#19098466C411yescharacteristic
12#19098436—21no—
13#19098436—21no—
14#19098432C411yescharacteristic
15#19098428C411yescharacteristic
16C4×C4⋊Dic79C2211yescharacteristic
17#19098304—24no—
18#19098287—21no—
19#19098287—21no—
20C2×C4×D4×C79C2211yescharacteristic
21#435×C79—24no—
22#432×C79C2212yes—
23#407×C79—21no—
24C2×C22⋊C8×C79—21no—
25#14579153—22no—
26#14579153—22no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C4×Dic79C2×C411yescharacteristic
39C2×C4×Dic79C2×C411yescharacteristic
40C4⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—22no—
42#14579072—22no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C22×D4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—22no—
50C22×C8×C79—22no—
51C4×D4×C79—22no—
52C4×D4×C79—22no—
53C2×C4⋊C4×C79C2×C411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—22no—
58C22⋊C8×C79—22no—
59C22⋊C8×C79—22no—
60C22⋊C8×C79—22no—
61#97×C79D414yes—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C2×C79⋊C8—42no—
67C4⋊Dic79—22no—
68C4⋊Dic79—22no—
69C4⋊Dic79—42no—
70Dic79⋊C4—41no—
71Dic79⋊C4—42no—
72Dic79⋊C4—41no—
73Dic79⋊C4—81no—
74Dic79⋊C4—81no—
75C22⋊Dic79—42no—
76C22⋊Dic79—42no—
77C22⋊Dic79—81no—
78C22⋊Dic79—81no—
79C4×Dic79—41no—
80C4×Dic79—41no—
81C24×C79C22⋊C411yescharacteristic
82C24×C79C22⋊C411yescharacteristic
83C2×D4×C79—22no—
84C2×D4×C79—22no—
85C2×D4×C79—41no—
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
97M16×C79—82no—
98C2×C8×C79—24no—
99C2×C8×C79—82no—
100C4⋊C4×C79—22no—
101C22⋊C4×C79—22no—
102C22⋊C4×C79—22no—
103C42×C79C2×D412yes—
104C42×C79#4112yes—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79#10112yes—
115C23×C79C2×C22⋊C411yescharacteristic
116D4×C79—44no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—42no—
124C2×C4×C79—42no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—81no—
128C2×C4×C79—81no—
129C2×C4×C79C4×D412yes—
130C2×C4×C79#12011yescharacteristic
131C2×C4×C79#12112yes—
132C2×C4×C79#12311yescharacteristic
133C2×Dic79—22no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C8×C79—44no—
146C79⋊C8—82no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—22no—
150C22×C79—42no—
151C22×C79—42no—
152C22×C79—41no—
153C22×C79—41no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79#38611yescharacteristic
160C22×C79C2×#10111yescharacteristic
161C22×C79#41311yescharacteristic
162C4×C79—22no—
163C4×C79—22no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—41no—
167C4×C79—41no—
168Dic79—42no—
169Dic79—81no—
170Dic79—81no—
171Dic79—81no—
172#9449—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—42no—
176C2×C79—81no—
177C2×C79#188411yescharacteristic
178C2×C79#190111yescharacteristic
179C2×C79#191111yescharacteristic
180#2973D7911yescharacteristic
181#2566—791no—
182#1534—792no—
183#1466—791no—
184#1285—792no—
185C79#944911yescharacteristic, complemented, Sylow
186C2×#125—791no—
187C2×C4×D4C2×D7911yescharacteristic
188#435—24no—
189#432—792no—
190#432C2×D7912yes—
191#407—21no—
192C2×C22⋊C8—21no—
193#396—1581no—
194#396—1581no—
195#390—791no—
196#390—791no—
197#385—1581no—
198#385—1581no—
199#384—791no—
200#382—791no—
201C4×C4⋊C4—791no—
202#330—1584no—
203#323—1581no—
204#323—1581no—
205C22×D4C4×D7911yescharacteristic
206C23×C4C22⋊D7911yescharacteristic
207C23×C4C22⋊D7911yescharacteristic
208C2×M16—22no—
209C22×C8—22no—
210#125—1582no—
211#125—1582no—
212C4×D4—22no—
213C4×D4—22no—
214C2×C4⋊C4—791no—
215C2×C4⋊C4—791no—
216C2×C4⋊C4—1581no—
217C2×C4⋊C4—1581no—
218C2×C4⋊C4—1581no—
219C2×C4⋊C4—1581no—
220C2×C4⋊C4—1581no—
221C2×C4⋊C4C4×D7911yescharacteristic
222C2×C22⋊C4—1581no—
223C2×C22⋊C4—1581no—
224C2×C22⋊C4—1581no—
225C2×C22⋊C4—1581no—
226C2×C22⋊C4D31611yescharacteristic
227C2×C22⋊C4D31611yescharacteristic
228C2×C42—791no—
229C2×C42—791no—
230C2×C42C22×D7911yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—1582no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—1582no—
237#97—1582no—
238#97C22⋊D7914yes—
239#95—1581no—
240#95—1581no—
241#95—1581no—
242C24(C2×D79)⋊C411yescharacteristic
243C24(C2×D79)⋊C411yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1581no—
254C22×C4—1581no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4(C2×D79)⋊C411yescharacteristic
258C22×C4(C2×D79)⋊C411yescharacteristic
259C22×C4C2×C4×D7911yescharacteristic
260C22×C4C2×C22⋊D7911yescharacteristic
261C22×C4C2×D31611yescharacteristic
262M16—82no—
263C2×C8—24no—
264C2×C8—82no—
265C2×C8—3162no—
266C4⋊C4—22no—
267C4⋊C4—3162no—
268C4⋊C4—3161no—
269C4⋊C4—3162no—
270C4⋊C4—3161no—
271C4⋊C4—6321no—
272C4⋊C4—6321no—
273C4⋊C4—1582no—
274C4⋊C4—1582no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—3162no—
278C22⋊C4—3162no—
279C22⋊C4—6321no—
280C22⋊C4—6321no—
281C42—3161no—
282C42—3161no—
283C42C2×C22⋊D7912yes—
284C42#1192962412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1457907912yes—
295C23C2×(C2×D79)⋊C411yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—3161no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—1582no—
321C2×C4C4×C22⋊D7912yes—
322C2×C4#1457913011yescharacteristic
323C2×C4#1457913512yes—
324C2×C4#1457914411yescharacteristic
325C8—44no—
326C8—6322no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1909844211yescharacteristic
340C22C2×#1457907911yescharacteristic
341C22#1909854511yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—3162no—
349C4—6321no—
350C4—6321no—
351C4—6321no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2376378311yescharacteristic
357C2#2376386811yescharacteristic
358C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2973C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#2973D79
hypocenterC79#9449
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #23762026C2semidirect
4. #23761255C2semidirect
185. C79#9449semidirect

Maximal subgroups

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

subgroupindex
#237685962
#237620262
#237617412
#237612552
C79×#29732
#944979