Finite Group Databasebeta

← #34199312

Subgroups — #34199312

Order 20224.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#23768305C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic, complemented
3#23761407C212yesmaximal, complemented
4#23761404C212yesmaximal
5C79×#1461C211yesmaximal, characteristic, complemented
6C2×#14579137—21no—
7C2×C4×D316C411yescharacteristic
8#19098587—21no—
9C2×#14579080—21no—
10#19098493C411yescharacteristic
11#19098493C2211yescharacteristic
12#19098480C411yescharacteristic
13#19098480C2211yescharacteristic
14#19098454—21no—
15#19098454—21no—
16#19098429C411yescharacteristic
17#19098420—21no—
18#19098332C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20#431×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C4×D79—21no—
25C2×#11929608—22no—
26#14579137—42no—
27#14579137—42no—
28C4×D316—41no—
29C4×D316—41no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×(C2×D79)⋊C4—21no—
39C4⋊(C79⋊C8)—41no—
40C4⋊(C79⋊C8)—41no—
41#14579080—42no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95C2×C411yescharacteristic
46C79⋊#95C2×C411yescharacteristic
47C22×Q8×C79D411yescharacteristic
48C4×Q8×C79—41no—
49C4×Q8×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79C2311yescharacteristic
53C2×C42×C79C2×C411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C2×C42×C79D411yescharacteristic
56C4⋊C8×C79—41no—
57C22⋊C8×C79—22no—
58C4×C8×C79—41no—
59C23×D79—21no—
60C2×D316—41no—
61C2×D316—42no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C2×C4×D79—42no—
66C2×C4×D79—42no—
67C2×C4×D79—42no—
68#11929608—42no—
69#11929608—42no—
70C2×C79⋊C8—42no—
71C4⋊Dic79—41no—
72C4⋊Dic79—42no—
73Dic79⋊C4—42no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76(C2×D79)⋊C4—42no—
77(C2×D79)⋊C4—42no—
78(C2×D79)⋊C4—41no—
79(C2×D79)⋊C4—41no—
80(C2×D79)⋊C4—42no—
81(C2×D79)⋊C4—41no—
82(C2×D79)⋊C4—41no—
83C2×Q8×C79—41no—
84C2×Q8×C79—41no—
85C2×Q8×C79D812yes—
86C2×Q8×C79SD1612yes—
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22×C411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C2×C8×C79—42no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C42×C79—21no—
102C42×C79—21no—
103C42×C79—21no—
104C23×C79C2×C22⋊C411yescharacteristic
105C22×D79—22no—
106C22×D79—41no—
107C22×D79—41no—
108C22×D79—41no—
109C22×D79—41no—
110C22×D79—41no—
111C22×D79—41no—
112Q8×C79—42no—
113Q8×C79—42no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79#9911yescharacteristic
128C2×C4×C79#9911yescharacteristic
129C2×C4×C79#10212yes—
130C2×C4×C79#10212yes—
131C2×C4×C79C2×D811yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142D316—42no—
143D316—42no—
144C4×D79—84no—
145C8×C79—82no—
146C79⋊C8—82no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79C2×#9911yescharacteristic
150C22×C79C2×#10211yescharacteristic
151C22×C79#42111yescharacteristic
152C2×D79—42no—
153C2×D79—42no—
154C2×D79—42no—
155C2×D79—81no—
156C2×D79—81no—
157C2×D79—81no—
158C2×D79—81no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—22no—
162C4×C79—22no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79—41no—
168Dic79—81no—
169Dic79—81no—
170Dic79—81no—
171#11879—791nomaximal, Sylow
172C2×C79—21no—
173C2×C79—21no—
174C2×C79#151911yescharacteristic
175C2×C79#212011yescharacteristic
176C2×C79#213311yescharacteristic
177D79—82no—
178#2536—791no—
179#1801—791no—
180#1461D7911yescharacteristic, complemented
181#1346—792no—
182#1345—792no—
183C79#1187911yescharacteristic, complemented, Sylow
184C2×C22⋊Q8—1581no—
185C2×C4×Q8C2×D7911yescharacteristic
186C2×C4×D4—791no—
187#431—21no—
188#423—1581no—
189C2×#101—1581no—
190#398—791no—
191#398—791no—
192#394—791no—
193#394—791no—
194#388—1581no—
195#388—1581no—
196#382—791no—
197#380—1581no—
198#343—792no—
199#324—21no—
200#324C2×D7912yes—
201C22×Q8D31611yescharacteristic
202C22×D4—791no—
203C23×C4—1581no—
204C2×M16—1582no—
205C22⋊Q8—3162no—
206C22⋊Q8—3162no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—3161no—
210C4×D4—3161no—
211C2×C4⋊C4—791no—
212C2×C4⋊C4—1581no—
213C2×C4⋊C4—1581no—
214C2×C4⋊C4—1581no—
215C2×C4⋊C4C4×D7911yescharacteristic
216C2×C4⋊C4C22⋊D7911yescharacteristic
217C2×C4⋊C4C22×D7911yescharacteristic
218C2×C22⋊C4—1581no—
219C2×C22⋊C4—1581no—
220C2×C22⋊C4—1581no—
221C2×C22⋊C4—1581no—
222C2×C22⋊C4—1581no—
223C2×C42C4×D7911yescharacteristic
224C2×C42D31611yescharacteristic
225C2×C42C22⋊D7911yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—3161no—
228C4⋊C8—3161no—
229#101—3162no—
230C22⋊C8—22no—
231C4×C8—41no—
232#95—791no—
233#95—791no—
234#95—1581no—
235#95—1581no—
236#95—1581no—
237C24—1581no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8D63212yes—
241C2×Q8#1192961312yes—
242C2×D4—3161no—
243C2×D4—3162no—
244C22×C4—3162no—
245C22×C4—3162no—
246C22×C4—3162no—
247C22×C4—1581no—
248C22×C4—1581no—
249C22×C4—1581no—
250C22×C4(C2×D79)⋊C411yescharacteristic
251C22×C4(C2×D79)⋊C411yescharacteristic
252C22×C4(C2×D79)⋊C411yescharacteristic
253C22×C4(C2×D79)⋊C411yescharacteristic
254C22×C4C2×C4×D7911yescharacteristic
255C22×C4C2×C22⋊D7911yescharacteristic
256C22×C4C2×D31611yescharacteristic
257M16—3162no—
258M16—3162no—
259C2×C8—42no—
260C2×C8—3162no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—3161no—
265C4⋊C4—3162no—
266C4⋊C4—3162no—
267C4⋊C4—3161no—
268C4⋊C4—3161no—
269C22⋊C4—3162no—
270C22⋊C4—3162no—
271C22⋊C4—3161no—
272C22⋊C4—3161no—
273C22⋊C4—3162no—
274C22⋊C4—3161no—
275C22⋊C4—3161no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—3161no—
283C23—3161no—
284C23—3161no—
285C23—3161no—
286C23—3161no—
287C23—3161no—
288C23—1582no—
289C23C2×(C2×D79)⋊C411yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—3162no—
293D4—3162no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—6324no—
317C2×C4#1457907511yescharacteristic
318C2×C4#1457907511yescharacteristic
319C2×C4#1457908312yes—
320C2×C4#1457908312yes—
321C2×C4C2×D63211yescharacteristic
322C2×C4C2×#1192961311yescharacteristic
323C8—82no—
324C8—6322no—
325C22—21no—
326C22—21no—
327C22—3162no—
328C22—3162no—
329C22—3162no—
330C22—6321no—
331C22—6321no—
332C22—6321no—
333C22—6321no—
334C22C2×#1457907511yescharacteristic
335C22C2×#1457908311yescharacteristic
336C22#1909857911yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—6321no—
347C4—6321no—
348C4—6321no—
349C2—21no—
350C2—21no—
351C2—6322no—
352C2#2376196911yescharacteristic
353C2#2376503911yescharacteristic
354C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1461C2
socleC22×C79C2×#99
prime radicalC22×D316C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1461D79
hypocenterC79#11879
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #23763396C2semidirect
3. #23761407C2semidirect
5. C79×#1461C2semidirect
180. #1461D79semidirect
183. C79#11879semidirect

Maximal subgroups

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

subgroupindex
#237683052
#237633962
#237614072
#237614042
C79×#14612
#1187979