Finite Group Databasebeta

← #34181162

Subgroups — #34181162

Order 20224.

550 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×(C22⋊D79)⋊C8C211yesmaximal, characteristic
2#23763423C212yesmaximal, complemented
3#23763385C212yesmaximal
4#23761268C211yesmaximal, characteristic, complemented
5C79×#1289C211yesmaximal, characteristic, complemented
6C23×C79⋊C8C2211yescharacteristic
7C22×C8×D79C411yescharacteristic, complemented
8C2×C4×C22⋊D79C2211yescharacteristic
9(C22⋊D79)⋊C8—22no—
10(C22⋊D79)⋊C8—22no—
11C2×#14579092—22no—
12C2×Dic79⋊C8C411yescharacteristic, complemented
13C2×#14579075—22no—
14C2×(C2×D79)⋊C8C411yescharacteristic, complemented
15C2×C8×Dic79C411yescharacteristic, complemented
16#19098423C2212yes—
17#19098329—22no—
18#19098325—22no—
19C2×C22⋊C8×C79C2211yescharacteristic
20#380×C79C2212yes—
21C22×C22⋊D79D411yescharacteristic
22C22×C4×D79C2×C411yescharacteristic
23C22×C79⋊C8—42no—
24C22×C79⋊C8C2×C411yescharacteristic
25C22×C79⋊C8C2×C411yescharacteristic
26C22×C79⋊C8D412yes—
27C22×C79⋊C8D412yes—
28C2×C8×D79—21no—
29C2×C8×D79—21no—
30C2×C8×D79—42no—
31C4×C22⋊D79—22no—
32C4×C22⋊D79—22no—
33C2×C4⋊Dic79—22no—
34C2×Dic79⋊C4Q811yescharacteristic
35C2×C22⋊Dic79—22no—
36C2×C22⋊Dic79D411yescharacteristic
37C2×(C2×D79)⋊C4Q811yescharacteristic
38C2×C4×Dic79C2×C411yescharacteristic
39#14579092—44no—
40Dic79⋊C8—22no—
41#14579075—44no—
42(C2×D79)⋊C8—22no—
43C8×Dic79—21no—
44C8×Dic79—21no—
45C79⋊#95—22no—
46C23×C4×C79C2311yescharacteristic
47C22×C8×C79C2×C411yescharacteristic
48C22×C8×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79—22no—
50C2×C22⋊C4×C79—22no—
51C22⋊C8×C79—22no—
52#95×C79—22no—
53C23×D79C4⋊C411yescharacteristic
54C2×C22⋊D79—22no—
55C2×C22⋊D79—21no—
56C2×C22⋊D79—21no—
57C2×C22⋊D79SD1614yes—
58C22×Dic79—42no—
59C22×Dic79C4⋊C411yescharacteristic
60C22×Dic79C4⋊C411yescharacteristic
61C22×Dic79C4⋊C411yescharacteristic
62C2×C4×D79—21no—
63C2×C4×D79—21no—
64C2×C4×D79—41no—
65C2×C4×D79—41no—
66C2×C79⋊C8—24no—
67C2×C79⋊C8—22no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—42no—
70C2×C79⋊C8C22⋊C411yescharacteristic
71C2×C79⋊C8C22⋊C412yes—
72C2×C79⋊C8C22⋊C411yescharacteristic
73C8×D79—42no—
74C8×D79—42no—
75C4⋊Dic79—42no—
76C4⋊Dic79—42no—
77Dic79⋊C4—22no—
78C22⋊Dic79—22no—
79C22⋊Dic79—42no—
80C22⋊Dic79—42no—
81(C2×D79)⋊C4—22no—
82C4×Dic79—21no—
83C4×Dic79—21no—
84C24×C79C2×D411yescharacteristic
85C22×C4×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79—42no—
88C22×C4×C79C22×C411yescharacteristic
89C22×C4×C79C2×D412yes—
90C22×C4×C79C2×Q811yescharacteristic
91C22×C4×C79#4112yes—
92C2×C8×C79—21no—
93C2×C8×C79—21no—
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C4⋊C4×C79—42no—
97C4⋊C4×C79—42no—
98C22⋊C4×C79—42no—
99C22⋊C4×C79—42no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79C2×C4⋊C411yescharacteristic
103C23×C79#12112yes—
104C23×C79C2×SD1612yes—
105C23×C79#13511yescharacteristic
106C23×C79#13511yescharacteristic
107C22×D79—21no—
108C22×D79—21no—
109C22×D79—41no—
110C22×D79—41no—
111C22×D79#10612yescomplemented
112C2×C4×C79—22no—
113C2×C4×C79—21no—
114C2×C4×C79—22no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79—42no—
118C2×C4×C79—82no—
119C2×C4×C79—82no—
120C2×C4×C79—82no—
121C2×C4×C79C2×C22⋊C411yescharacteristic
122C2×C4×C79#11711yescharacteristic
123C2×C4×C79C22⋊Q812yes—
124C22⋊D79—24no—
125C22⋊D79—24no—
126C2×Dic79—21no—
127C2×Dic79—21no—
128C2×Dic79—21no—
129C2×Dic79—21no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—82no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C2×Dic79#10612yescomplemented
136C4×D79—42no—
137C4×D79—41no—
138C4×D79—41no—
139C8×C79—41no—
140C8×C79—41no—
141C79⋊C8—21no—
142C79⋊C8—22no—
143C79⋊C8#9912yescomplemented
144C22×C79—22no—
145C22×C79—21no—
146C22×C79—22no—
147C22×C79—21no—
148C22×C79—42no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79#38011yescharacteristic
152C22×C79C2×#10611yescharacteristic
153C22×C79#42711yescharacteristic
154C22×C79#46512yes—
155C22×C79#46512yes—
156C2×D79—22no—
157C2×D79—42no—
158C2×D79—41no—
159C2×D79—41no—
160C2×D79—42no—
161C4×C79—21no—
162C4×C79—22no—
163C4×C79—162no—
164C4×C79C2×#9911yescharacteristic
165C4×C79#41011yescharacteristic
166Dic79—22no—
167Dic79—41no—
168Dic79—41no—
169Dic79—162no—
170#8608—791nomaximal, Sylow
171C2×C79—22no—
172C2×C79—22no—
173C2×C79—21no—
174C2×C79—21no—
175C2×C79#179911yescharacteristic
176C2×C79#180711yescharacteristic
177C2×C79#181211yescharacteristic
178D79—42no—
179C2×C8×D4—791no—
180#1807—792no—
181#1799—792no—
182#1289—791no—
183#1289D7911yescharacteristic, complemented
184C79#860811yescharacteristic, complemented, Sylow
185C23×C8—791no—
186C23×C8—791no—
187C2×C4×D4—791no—
188C8×D4—1582no—
189C8×D4—1582no—
190C2×#106—1582no—
191C2×C4⋊C8—791no—
192C2×#99—1582no—
193C2×C22⋊C8—791no—
194C2×C22⋊C8C2×D7911yescharacteristic
195C2×C4×C8—791no—
196#380—792no—
197#380C2×D7912yes—
198#342—1582no—
199#340—1582no—
200C22×D4—791no—
201C23×C4—791no—
202C23×C4C22×D7911yescharacteristic
203C22×C8—3162no—
204C22×C8—3162no—
205C22×C8—791no—
206C22×C8—792no—
207C22×C8—791no—
208C22×C8—792no—
209C22×C8—1581no—
210C22×C8—1581no—
211C22×C8C4×D7911yescharacteristic
212C22×C8C4×D7911yescharacteristic
213C4×D4—1582no—
214C4×D4—1582no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—791no—
217C2×C4⋊C4—1582no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—791no—
220C2×C22⋊C4—791no—
221C2×C22⋊C4—1582no—
222C2×C42—791no—
223#106—3164no—
224C4⋊C8—1582no—
225#99—3164no—
226C22⋊C8—22no—
227C22⋊C8—1582no—
228C4×C8—1581no—
229C4×C8—1581no—
230#95—22no—
231#95—1582no—
232C24—791no—
233C24D4×D7911yescharacteristic
234C2×D4—794no—
235C2×D4—1582no—
236C2×D4—1581no—
237C2×D4—1581no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—3161no—
242C22×C4—3161no—
243C22×C4—3162no—
244C22×C4—791no—
245C22×C4—791no—
246C22×C4—791no—
247C22×C4—1581no—
248C22×C4—1581no—
249C22×C4C2×C4×D7911yescharacteristic
250C22×C4D4×D7912yes—
251C22×C4Q8×D7911yescharacteristic
252C22×C4#1192962412yes—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—3162no—
258C2×C8—3162no—
259C2×C8—3162no—
260C2×C8—3161no—
261C2×C8—791no—
262C2×C8—791no—
263C2×C8—792no—
264C2×C8—1584no—
265C2×C8—1582no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—3162no—
269C4⋊C4—3162no—
270C4⋊C4—1582no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—3162no—
274C22⋊C4—3162no—
275C22⋊C4—1582no—
276C22⋊C4—1582no—
277C42—1581no—
278C42—1581no—
279C23—41no—
280C23—41no—
281C23—3161no—
282C23—3161no—
283C23—792no—
284C23—1581no—
285C23—1581no—
286C23C4⋊C4×D7911yescharacteristic
287C23#1457913412yes—
288C23SD16×D7912yes—
289C23#1457918011yescharacteristic
290C23#1457918411yescharacteristic
291D4—1584no—
292D4—1584no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—3162no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—792no—
308C2×C4—6322no—
309C2×C4—6322no—
310C2×C4—6322no—
311C2×C4—1581no—
312C2×C4—1581no—
313C2×C4—1581no—
314C2×C4—1581no—
315C2×C4C22⋊C4×D7911yescharacteristic
316C2×C4#1457911811yescharacteristic
317C2×C4#1457913912yes—
318C8—41no—
319C8—41no—
320C8—792no—
321C8—1581no—
322C8—1582no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—3162no—
331C22—3162no—
332C22—3161no—
333C22—3161no—
334C22—1582no—
335C22#1909842211yescharacteristic
336C22#106×D7911yescharacteristic
337C22#1909860011yescharacteristic
338C22#1909879112yes—
339C22#1909879512yes—
340C4—21no—
341C4—22no—
342C4—162no—
343C4—3161no—
344C4—3161no—
345C4—1582no—
346C4—12642no—
347C4#99×D7911yescharacteristic
348C4#1909853511yescharacteristic
349C2—22no—
350C2—22no—
351C2—21no—
352C2—21no—
353C2—3162no—
354C2#2376338711yescharacteristic
355C2#2376342411yescharacteristic
356C2#2376344511yescharacteristic
subgroupG/H
centerC22#19098422
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1289C2
socleC22×C79#380
prime radicalC22×C22⋊D79D4
cosocleC23×C4×C79C23
hypercenter#1289D79
hypocenterC79#8608
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #23763423C2semidirect
4. #23761268C2semidirect
5. C79×#1289C2semidirect
7. C22×C8×D79C4semidirect
12. C2×Dic79⋊C8C4semidirect
14. C2×(C2×D79)⋊C8C4semidirect
15. C2×C8×Dic79C4semidirect
111. C22×D79#106semidirect
135. C2×Dic79#106semidirect
143. C79⋊C8#99semidirect
183. #1289D79semidirect
184. C79#8608semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D79)⋊C82
#237634232
#237633852
#237612682
C79×#12892
#860879