Finite Group Databasebeta

← #34199083

Subgroups — #34199083

Order 20224.

384 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 83 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#23766691C211yesmaximal, characteristic
2#23761744C211yesmaximal, characteristic
3#23761429C212yesmaximal
4#23761405C212yesmaximal
5C79×#1801C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098620—21no—
8#19098494—21no—
9#19098493C411yescharacteristic
10#19098493C2211yescharacteristic
11#19098483C411yescharacteristic
12#19098483C2211yescharacteristic
13#19098482—21no—
14C4×Dic79⋊C4—21no—
15C4×Dic79⋊C4—21no—
16#19098409C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20#423×C79—21no—
21C2×#101×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79—21no—
34C4⋊(C79⋊C8)—41no—
35C22⋊(C79⋊C8)—22no—
36#14579072—41no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×Q8×C79D411yescharacteristic
45C2×M16×C79—22no—
46C4×Q8×C79—41no—
47C4×Q8×C79—41no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C4⋊C8×C79—41no—
55C4⋊C8×C79—41no—
56#101×C79—42no—
57C2×Dic158—41no—
58C2×Dic158—42no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71C4×Dic79—24no—
72C4×Dic79—41no—
73C4×Dic79—41no—
74C2×Q8×C79—21no—
75C2×Q8×C79—21no—
76C2×Q8×C79—41no—
77C2×Q8×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85M16×C79—42no—
86M16×C79—42no—
87C2×C8×C79—42no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C23×C79C2×C22⋊C411yescharacteristic
98Q8×C79—42no—
99Q8×C79—42no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79#9911yescharacteristic
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C79#13711yescharacteristic
119C2×Dic79—22no—
120C2×Dic79—22no—
121C2×Dic79—22no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131Dic158—42no—
132Dic158—42no—
133C8×C79—82no—
134C79⋊C8—82no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79C2×#9911yescharacteristic
138C22×C79#41711yescharacteristic
139C22×C79#42111yescharacteristic
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—41no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79#35612yes—
149Dic79—42no—
150Dic79—42no—
151Dic79—42no—
152Dic79—81no—
153#11831—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79#151311yescharacteristic
157C2×C79#212011yescharacteristic
158C2×C79C2×#35611yescharacteristic
159#2399—791no—
160#1801D7911yescharacteristic
161#1467—791no—
162#1354—792no—
163#1346—792no—
164C79#1183111yescharacteristic, complemented, Sylow
165C2×C4×Q8—791no—
166C2×C4×Q8C2×D7911yescharacteristic
167#432—1581no—
168#423—21no—
169C2×#101—21no—
170#398—791no—
171#398—791no—
172#398—1581no—
173#395—791no—
174#395—791no—
175#395—1581no—
176C4×C4⋊C4—1581no—
177C4×C4⋊C4—1581no—
178#376—791no—
179#343C2×D7912yes—
180#324—792no—
181#324—1581no—
182C22×Q8—791no—
183C22×Q8C22⋊D7911yescharacteristic
184C2×M16—22no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—3161no—
188C4×Q8—3161no—
189C2×C4⋊C4—791no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4C4×D7911yescharacteristic
196C2×C4⋊C4D31611yescharacteristic
197C2×C4⋊C4C22×D7911yescharacteristic
198C2×C42—1581no—
199C2×C42—1581no—
200C2×C42C4×D7911yescharacteristic
201C2×C42D31611yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—3161no—
206#101—42no—
207C22⋊C8—1582no—
208#97—3161no—
209#95—791no—
210#95—791no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215#95—1581no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—3161no—
221C2×Q8—3162no—
222C22×C4—1581no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4—1581no—
226C22×C4(C2×D79)⋊C411yescharacteristic
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4C2×C4×D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×D31611yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—3162no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—3161no—
241C4⋊C4—3162no—
242C4⋊C4—3162no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C4⋊C4—3161no—
246C4⋊C4—3161no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—3161no—
254C42—3161no—
255C42—1584no—
256C23C2×(C2×D79)⋊C411yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—3162no—
260Q8—3162no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—1582no—
286C2×C4—1582no—
287C2×C4—1582no—
288C2×C4#1457907511yescharacteristic
289C2×C4#1457907511yescharacteristic
290C2×C4#1457918711yescharacteristic
291C2×C4#1457919211yescharacteristic
292C8—82no—
293C8—6322no—
294C22—21no—
295C22—21no—
296C22C2×#1457907511yescharacteristic
297C22#1909856211yescharacteristic
298C22#1909858111yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3162no—
308C4—3162no—
309C4—3162no—
310C4—6321no—
311C4#1909836112yes—
312C2—21no—
313C2—21no—
314C2#2376194111yescharacteristic
315C2#2376504211yescharacteristic
316C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1801C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1801D79
hypocenterC79#11831
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
164. C79#11831semidirect

Maximal subgroups

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

subgroupindex
#237666912
#237617442
#237614292
#237614052
C79×#18012
#1183179