Finite Group Databasebeta

← #34199015

Subgroups — #34199015

Order 20224.

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

normal subgroups only

This group has 386 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#23769213C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761426C212yesmaximal, complemented
5C79×#1821C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7#19098527—21no—
8#19098502C411yescharacteristic
9#19098502C2211yescharacteristic
10#19098496—21no—
11#19098496—21no—
12#19098483C411yescharacteristic
13#19098483C2211yescharacteristic
14#19098436—21no—
15#19098426—21no—
16#19098423C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C2×#11929608—41no—
25#14579140—41no—
26#14579140—41no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—41no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48C2×C4⋊C4×C79—41no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C2×Dic158—41no—
57C2×Dic158—81no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62#11929608—82no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—81no—
66Dic79⋊C4—42no—
67Dic79⋊C4—81no—
68Dic79⋊C4—81no—
69C22⋊Dic79—42no—
70C22⋊Dic79—41no—
71C22⋊Dic79—41no—
72C24×C79C22⋊C411yescharacteristic
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—22no—
80C22×C4×C79—41no—
81C22×C4×C79—81no—
82C22×C4×C79—81no—
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22×C411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89M16×C79—42no—
90M16×C79—42no—
91C2×C8×C79—42no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—41no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79#9911yescharacteristic
101C23×C79#9911yescharacteristic
102C23×C79C2×C22⋊C411yescharacteristic
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—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—81no—
120C2×C4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79#13712yes—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—42no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134Dic158—82no—
135C8×C79—82no—
136C79⋊C8—82no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79C2×#9911yescharacteristic
148C22×C79#41911yescharacteristic
149C22×C79#42111yescharacteristic
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156Dic79—81no—
157Dic79—81no—
158Dic79—81no—
159Dic79—81no—
160#11817—791nomaximal, Sylow
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—41no—
164C2×C79—41no—
165C2×C79#151111yescharacteristic
166C2×C79#212111yescharacteristic
167C2×C79#213311yescharacteristic
168#2627—791no—
169#1821D7911yescharacteristic
170#1463—791no—
171#1354—792no—
172#1353—792no—
173C79#1181711yescharacteristic, complemented, Sylow
174C2×C22⋊Q8—791no—
175C22×C4⋊C4C2×D7911yescharacteristic
176#407—21no—
177#407—1581no—
178#400—791no—
179#400—791no—
180#398—1581no—
181#398—1581no—
182#395—791no—
183#395—791no—
184#385—1581no—
185#381—1581no—
186#380—791no—
187#343—21no—
188#343C2×D7912yes—
189#324—792no—
190#324—1581no—
191C22×Q8—791no—
192C23×C4C4×D7911yescharacteristic
193C23×C4D31611yescharacteristic
194C23×C4C22⋊D7911yescharacteristic
195C2×M16—22no—
196C2×M16—3161no—
197C22⋊Q8—3161no—
198C22⋊Q8—3161no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4C4×D7911yescharacteristic
207C2×C4⋊C4D31611yescharacteristic
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C4⋊C4C22×D7911yescharacteristic
210C2×C22⋊C4—1581no—
211C2×C22⋊C4—1581no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—3161no—
215C22⋊C8—1582no—
216#95—791no—
217#95—791no—
218#95—1581no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224C24(C2×D79)⋊C411yescharacteristic
225C2×Q8—3161no—
226C2×Q8—6321no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4—1581no—
240C22×C4—1581no—
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4C2×C4×D7911yescharacteristic
245C22×C4C2×C22⋊D7911yescharacteristic
246C22×C4C2×D31611yescharacteristic
247M16—42no—
248M16—42no—
249M16—6322no—
250C2×C8—42no—
251C2×C8—3162no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3161no—
257C4⋊C4—3162no—
258C4⋊C4—6321no—
259C4⋊C4—6321no—
260C4⋊C4—6321no—
261C22⋊C4—3162no—
262C22⋊C4—3161no—
263C22⋊C4—3161no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1457907511yescharacteristic
269C23#1457907511yescharacteristic
270C23C2×(C2×D79)⋊C411yescharacteristic
271Q8—6322no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3162no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4#1457919112yes—
303C8—82no—
304C8—6322no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1457907511yescharacteristic
316C22#1909857111yescharacteristic
317C22#1909858111yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—6321no—
325C4—6321no—
326C4—6321no—
327C4—6321no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2376193211yescharacteristic
333C2#2376504711yescharacteristic
334C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1821C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1821D79
hypocenterC79#11817
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761426C2semidirect
173. C79#11817semidirect

Maximal subgroups

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

subgroupindex
#237692132
#237617302
#237614292
#237614262
C79×#18212
#1181779