Finite Group Databasebeta

← #34199010

Subgroups — #34199010

Order 20224.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#23766668C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761426C212yesmaximal, complemented
5C79×#1822C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098620—21no—
8#19098502C2211yescharacteristic
9#19098499C411yescharacteristic
10#19098496—21no—
11#19098486—21no—
12#19098483C2211yescharacteristic
13#19098475C411yescharacteristic
14C4×Dic79⋊C4—21no—
15C4×C4⋊Dic79C411yescharacteristic
16C4×C22⋊Dic79—21no—
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79—21no—
35C4⋊(C79⋊C8)—41no—
36C22⋊(C79⋊C8)—22no—
37#14579072—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C2×M16×C79—22no—
46#117×C79—41no—
47#117×C79—41no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79C2311yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C42×C79C2×C411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C2×Dic158—41no—
57C2×Dic158—42no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C2×C79⋊C8—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—42no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68C22⋊Dic79—42no—
69C22⋊Dic79—42no—
70C4×Dic79—42no—
71C4×Dic79—41no—
72C4×Dic79—41no—
73C24×C79C22⋊C411yescharacteristic
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84M16×C79—42no—
85M16×C79—42no—
86C2×C8×C79—42no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C22⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79C2×C22⋊C411yescharacteristic
101C2×C4×C79—21no—
102C2×C4×C79—21no—
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—42no—
110C2×C4×C79—42no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79#9911yescharacteristic
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79#13712yes—
118C2×Dic79—22no—
119C2×Dic79—22no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130Dic158—42no—
131Dic158—42no—
132C8×C79—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—81no—
141C22×C79C2×#9911yescharacteristic
142C22×C79#41911yescharacteristic
143C22×C79#42111yescharacteristic
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—22no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151Dic79—42no—
152Dic79—42no—
153Dic79—81no—
154Dic79—81no—
155#11816—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79—81no—
159C2×C79#151111yescharacteristic
160C2×C79#212011yescharacteristic
161C2×C79#213211yescharacteristic
162#2396—791no—
163#1822D7911yescharacteristic
164#1464—791no—
165#1354—792no—
166#1353—792no—
167C79#1181611yescharacteristic, complemented, Sylow
168C2×C4×Q8—791no—
169C2×#117C2×D7911yescharacteristic
170#432—1581no—
171#407—21no—
172#400—791no—
173#399—791no—
174#398—1581no—
175#396—1581no—
176#395—791no—
177#393—791no—
178C4×C4⋊C4—791no—
179C4×C4⋊C4—1581no—
180C4×C22⋊C4—1581no—
181#343—21no—
182#343C2×D7912yes—
183#324—792no—
184#323—1581no—
185C22×Q8—791no—
186C23×C4C22⋊D7911yescharacteristic
187C2×M16—22no—
188C4×Q8—3161no—
189C4×Q8—3161no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—791no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C4⋊C4C22×D7911yescharacteristic
199C2×C22⋊C4—1581no—
200C2×C22⋊C4—1581no—
201C2×C22⋊C4C4×D7911yescharacteristic
202C2×C22⋊C4D31611yescharacteristic
203C2×C42—1581no—
204C2×C42—1581no—
205C2×C42C4×D7911yescharacteristic
206C2×C42D31611yescharacteristic
207C4⋊C8—3161no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—1582no—
211#97—3161no—
212#95—791no—
213#95—791no—
214#95—1581no—
215#95—1581no—
216#95—1581no—
217#95—1581no—
218C24(C2×D79)⋊C411yescharacteristic
219C2×Q8—3161no—
220C2×Q8—3162no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—1581no—
226C22×C4—1581no—
227C22×C4—1581no—
228C22×C4—1581no—
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4(C2×D79)⋊C411yescharacteristic
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4C2×C4×D7911yescharacteristic
233C22×C4C2×C22⋊D7911yescharacteristic
234C22×C4C2×D31611yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—3162no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—3161no—
242C4⋊C4—3162no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C4⋊C4—3162no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—3162no—
249C22⋊C4—3162no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—3162no—
255C42—3161no—
256C42—3161no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D79)⋊C411yescharacteristic
263Q8—3162no—
264Q8—3162no—
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—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—1582no—
290C2×C4—1582no—
291C2×C4#1457907511yescharacteristic
292C2×C4#1457907511yescharacteristic
293C2×C4#1457919112yes—
294C8—82no—
295C8—6322no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1457907511yescharacteristic
304C22#1909857111yescharacteristic
305C22#1909858111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3162no—
314C4—3162no—
315C4—6321no—
316C4—6321no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2376193211yescharacteristic
321C2#2376504211yescharacteristic
322C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1822C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1822D79
hypocenterC79#11816
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761426C2semidirect
167. C79#11816semidirect

Maximal subgroups

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

subgroupindex
#237666682
#237617342
#237614292
#237614262
C79×#18222
#1181679