Finite Group Databasebeta

← #34201228

Subgroups — #34201228

Order 20224.

324 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 59 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#23764750C211yesmaximal, characteristic
2#23762441C211yesmaximal, characteristic
3#23762407C211yesmaximal, characteristic
4#23761427C211yesmaximal, characteristic
5#23761406C211yesmaximal, characteristic
6#23761344C211yesmaximal, characteristic
7C79×#2769C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098898—21no—
10#19098893—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#400×C79—21no—
19#400×C79C2211yescharacteristic
20#398×C79—21no—
21#398×C79C2211yescharacteristic
22#389×C79—21no—
23#387×C79—21no—
24#387×C79—21no—
25#383×C79—21no—
26#382×C79C2211yescharacteristic
27C2×#11929608—21no—
28#14579160—41no—
29#14579152—41no—
30#14579140—21no—
31#14579140—21no—
32#14579120—21no—
33#14579094—41no—
34#14579094—41no—
35#14579092—41no—
36#14579092—41no—
37#14579084—41no—
38#14579084—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C42×C79—21no—
50C2×C42×C79D411yescharacteristic
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79D411yescharacteristic
61#95×C79D411yescharacteristic
62C2×Dic158—21no—
63C2×Dic158—21no—
64#11929608—22no—
65#11929608—41no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C4⋊Dic79—21no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73Dic79⋊C4—41no—
74Dic79⋊C4—41no—
75C22⋊Dic79—41no—
76C4×Dic79—41no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C4⋊C4×C79—41no—
87C4⋊C4×C79—42no—
88C4⋊C4×C79—42no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—81no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—42no—
98C23×C79#12011yescharacteristic
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79#13511yescharacteristic
124C2×C4×C79#13711yescharacteristic
125C2×Dic79—41no—
126C2×Dic79—41no—
127Dic158—81no—
128Dic158—81no—
129C79⋊C8—42no—
130C79⋊C8—81no—
131C79⋊C8—81no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79#45211yescharacteristic
135C22×C79#45611yescharacteristic
136C22×C79B(4,𝔽2)11yescharacteristic
137C4×C79—21no—
138C4×C79—21no—
139C4×C79—42no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—81no—
143C4×C79—81no—
144C4×C79—81no—
145C4×C79—81no—
146C4×C79—81no—
147Dic79—81no—
148Dic79—81no—
149#12241—791nomaximal, Sylow
150C2×C79—21no—
151C2×C79—21no—
152C2×C79#160311yescharacteristic
153C2×C79#220111yescharacteristic
154C2×C79#220411yescharacteristic
155#2769D7911yescharacteristic
156#2061—791no—
157#1617—791no—
158#1611—791no—
159#1353—791no—
160#1346—791no—
161#1324—791no—
162C79#1224111yescharacteristic, complemented, Sylow
163#544—791no—
164#484—1581no—
165#483—1581no—
166#428—1581no—
167#400—21no—
168#400C2×D7911yescharacteristic
169#398—21no—
170#398C2×D7911yescharacteristic
171#389—21no—
172#387—21no—
173#387—21no—
174#383—21no—
175#382C2×D7911yescharacteristic
176#343—791no—
177#339—1581no—
178#328—1581no—
179#328—1581no—
180#324—791no—
181#324—791no—
182C2×M16—1581no—
183C4⋊Q8—3161no—
184#125—3161no—
185C22⋊Q8—1581no—
186C22⋊Q8—1581no—
187#117—1581no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22⋊D7911yescharacteristic
193C2×C4⋊C4C22⋊D7911yescharacteristic
194C2×C4⋊C4C22⋊D7911yescharacteristic
195C2×C4⋊C4C22×D7911yescharacteristic
196C2×C42—21no—
197C2×C42C22⋊D7911yescharacteristic
198#107—3161no—
199#107—3161no—
200#106—3161no—
201#106—3161no—
202#103—3161no—
203#103—3161no—
204C22⋊C8—1581no—
205C22⋊C8—1581no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95C22⋊D7911yescharacteristic
216#95C22⋊D7911yescharacteristic
217C2×Q8—1581no—
218C2×Q8—1581no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C2×C22⋊D7911yescharacteristic
226C22×C4C2×C22⋊D7911yescharacteristic
227C22×C4C2×C22⋊D7911yescharacteristic
228M16—3161no—
229M16—1582no—
230C2×C8—3161no—
231C2×C8—3161no—
232C2×C8—3161no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—81no—
242C4⋊C4—3161no—
243C4⋊C4—3161no—
244C4⋊C4—1581no—
245C4⋊C4—1581no—
246C4⋊C4—1581no—
247C4⋊C4—1581no—
248C22⋊C4—3161no—
249C42—21no—
250C42—21no—
251C42—42no—
252C42—3161no—
253C23#1457913011yescharacteristic
254Q8—6321no—
255Q8—6321no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4#1457918111yescharacteristic
283C2×C4#1457919111yescharacteristic
284C8—3162no—
285C8—6321no—
286C8—6321no—
287C22—21no—
288C22—21no—
289C22#1909872511yescharacteristic
290C22#1909874611yescharacteristic
291C22#1909875011yescharacteristic
292C4—21no—
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—6321no—
303C4—6321no—
304C2—21no—
305C2—21no—
306C2#2376236311yescharacteristic
307C2#2376541311yescharacteristic
308C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2769C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2769D79
hypocenterC79#12241
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2769

Complements

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

NH ≅ G/Ntype
162. C79#12241semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237624412
#237624072
#237614272
#237614062
#237613442
C79×#27692
#1224179