Finite Group Databasebeta

← #34297635

Subgroups — #34297635

Order 20224.

1354 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099113C211yesmaximal, characteristic
2#23765798C211yesmaximal, characteristic, complemented
3#23764330C214yesmaximal, complemented
4#23764224C212yesmaximal, complemented
5#23764220C212yesmaximal, complemented
6#23763341C214yesmaximal, complemented
7C2×#432×C79C211yesmaximal, characteristic, complemented
8C22×#11929624C2212yes—
9#19099113—24no—
10#19099113C418yescomplemented
11C2×#14579161C2211yescharacteristic
12C2×#14579158C2211yescharacteristic
13C2×#14579149C2212yes—
14C2×#14579135—21no—
15C2×#14579135—21no—
16C2×C4×D316—21no—
17C2×C4×D316—21no—
18C2×#14579086C2218yes—
19C2×#14579083C2218yes—
20#19098431—21no—
21#19098429C2212yes—
22#19098426—21no—
23#19098423C2212yes—
24C4×C4⋊Dic79C2212yes—
25C4×C22⋊Dic79—21no—
26#19098409—21no—
27C22×C42×C79C2211yescharacteristic
28#432×C79—24no—
29C2×C4⋊C8×C79C2212yes—
30C2×C22⋊C8×C79C2212yes—
31C2×#97×C79C2212yes—
32C2×#11929624—24no—
33C2×#11929624—48no—
34C2×#11929624C2×C418yes—
35C22×Dic158C2312yes—
36C22×D316—21no—
37C22×D316—21no—
38C22×D316C2312yes—
39C22×C22⋊D79—22no—
40C22×C4×D79—22no—
41#14579161—22no—
42#14579161C2×C414yes—
43#14579158—22no—
44#14579158C2×C414yes—
45#14579149—24no—
46#14579149C2×C418yes—
47#14579135—28no—
48#14579135—28no—
49C4×D316—28no—
50C4×D316—28no—
51C2×C4⋊Dic79—21no—
52C2×C4⋊Dic79—21no—
53C2×C4⋊Dic79C2314yes—
54C2×C22⋊Dic79—22no—
55C2×(C2×D79)⋊C4—22no—
56C2×(C2×D79)⋊C4—22no—
57C2×C4×Dic79—22no—
58#14579086—48no—
59#14579083—48no—
60C79⋊#95—22no—
61C79⋊#95—22no—
62C79⋊#95—22no—
63C79⋊#95—22no—
64C23×C4×C79D411yescharacteristic
65C23×C4×C79D411yescharacteristic
66C23×C4×C79C2311yescharacteristic
67C22×C8×C79C2314yes—
68C2×C42×C79C2×C414yes—
69C2×C42×C79D414yes—
70C2×C42×C79D411yescharacteristic
71C2×C42×C79D411yescharacteristic
72C2×C42×C79C2311yescharacteristic
73C2×C42×C79C2311yescharacteristic
74C4⋊C8×C79—24no—
75C22⋊C8×C79—24no—
76#97×C79—24no—
77C23×D79—22no—
78#11929624—416no—
79#11929624—416no—
80C2×Dic158—22no—
81C2×Dic158—28no—
82C2×Dic158C22×C414yes—
83C2×D316—24no—
84C2×D316—24no—
85C2×D316—22no—
86C2×D316—28no—
87C2×D316—44no—
88C2×D316—44no—
89C2×D316C22×C414yes—
90C2×C22⋊D79—28no—
91C2×C22⋊D79—48no—
92C22×Dic79—22no—
93C22×Dic79—24no—
94C2×C4×D79—28no—
95C2×C4×D79—48no—
96C4⋊Dic79—24no—
97C4⋊Dic79—24no—
98C4⋊Dic79—28no—
99C22⋊Dic79—44no—
100(C2×D79)⋊C4—28no—
101(C2×D79)⋊C4—28no—
102C4×Dic79—44no—
103C24×C79C2×D411yescharacteristic
104C22×C4×C79—24no—
105C22×C4×C79—42no—
106C22×C4×C79—42no—
107C22×C4×C79C22⋊C414yes—
108C22×C4×C79C22⋊C414yes—
109C22×C4×C79C22×C412yes—
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D412yes—
116C22×C4×C79C2411yescharacteristic
117C2×C8×C79—24no—
118C2×C8×C79—24no—
119C2×C8×C79—24no—
120C42×C79C22⋊C414yes—
121C42×C79C22⋊C414yes—
122C42×C79C22×C412yes—
123C42×C79C22×C412yes—
124C42×C79C2×D412yes—
125C42×C79C2×D412yes—
126C23×C79—22no—
127C23×C79—22no—
128C23×C79—22no—
129C23×C79C2×C22⋊C412yes—
130C23×C79C22×D411yescharacteristic
131C22×D79—24no—
132C22×D79—44no—
133C22×D79—44no—
134C22×D79—44no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—22no—
139C2×C4×C79—22no—
140C2×C4×C79—24no—
141C2×C4×C79—24no—
142C2×C4×C79—44no—
143C2×C4×C79—44no—
144C2×C4×C79C2×C22⋊C412yes—
145C2×C4×C79C2×C22⋊C412yes—
146C2×C4×C79C2×C22⋊C412yes—
147C2×C4×C79C2×C22⋊C412yes—
148C2×C4×C79C2×C22⋊C412yes—
149C2×C4×C79C23×C411yescharacteristic
150C2×C4×C79C22×D411yescharacteristic
151C2×C4×C79H(5,𝔽2)14yes—
152C22⋊D79—416no—
153C2×Dic79—24no—
154C2×Dic79—44no—
155C2×Dic79—44no—
156C2×Dic79—44no—
157C2×Dic79—44no—
158D316—28no—
159D316—48no—
160D316—48no—
161D316—44no—
162C4×D79—416no—
163Dic158—28no—
164Dic158—44no—
165C8×C79—44no—
166C22×C79—22no—
167C22×C79—22no—
168C22×C79—22no—
169C22×C79—44no—
170C22×C79C22×C22⋊C411yescharacteristic
171C22×C79#53512yes—
172C22×C79#57512yes—
173C22×C79#57512yes—
174C2×D79—44no—
175C2×D79—44no—
176C2×D79—44no—
177C2×D79—88no—
178C4×C79—22no—
179C4×C79—22no—
180C4×C79—22no—
181C4×C79—22no—
182C4×C79—22no—
183C4×C79#51812yes—
184C4×C79#53512yes—
185Dic79—44no—
186Dic79—84no—
187#21812—791nomaximal, Sylow
188C2×C79—42no—
189C2×C79#230911yescharacteristic
190C2×C79#289611yescharacteristic
191C2×C79#289611yescharacteristic
192C2×C79#321914yes—
193D79—84no—
194C2×#532—791no—
195C2×#432D7911yescharacteristic, complemented
196#2283—791no—
197#1983—794no—
198#1961—792no—
199#1960—792no—
200#1788—794no—
201C79#2181211yescharacteristic, complemented, Sylow
202C22×#41—792no—
203#532—798no—
204#532—1584no—
205C2×C4⋊Q8—791no—
206C2×#127—791no—
207C2×#124—792no—
208C2×#121—1581no—
209C2×#121—1581no—
210C2×C4×D4—1581no—
211C2×C4×D4—1581no—
212C22×C42C2×D7911yescharacteristic
213#432—24no—
214C2×C4⋊C8C2×D7912yes—
215C2×#103—798no—
216C2×#102—798no—
217C2×C22⋊C8C2×D7912yes—
218C2×#97C2×D7912yes—
219#383—1581no—
220#382—792no—
221#381—1581no—
222#380—792no—
223C4×C4⋊C4—792no—
224C4×C22⋊C4—1581no—
225#376—1581no—
226C2×#41—3168no—
227C2×#41—798no—
228C2×#41—1584no—
229C22×Q8—792no—
230C22×D4—792no—
231C22×D4—1582no—
232C22×D4—1581no—
233C22×D4—1581no—
234C23×C4—1582no—
235C23×C4D31611yescharacteristic
236C23×C4D31611yescharacteristic
237C23×C4C22×D7911yescharacteristic
238C22×C8C22×D7914yes—
239C4⋊Q8—794no—
240C4⋊Q8—1582no—
241#127—794no—
242#127—1582no—
243#124—798no—
244#124—1584no—
245#121—1588no—
246#121—1588no—
247C4×D4—1588no—
248C4×D4—1588no—
249C2×C4⋊C4—794no—
250C2×C4⋊C4—1581no—
251C2×C4⋊C4—1581no—
252C2×C22⋊C4—1582no—
253C2×C22⋊C4—1582no—
254C2×C22⋊C4—1582no—
255C2×C42—1582no—
256C2×C42C4×D7914yes—
257C2×C42D31611yescharacteristic
258C2×C42D31611yescharacteristic
259C2×C42C22⋊D7914yes—
260C2×C42C22×D7911yescharacteristic
261C2×C42C22×D7911yescharacteristic
262C4⋊C8—24no—
263#103—3168no—
264#102—3168no—
265C22⋊C8—24no—
266#97—24no—
267#95—1582no—
268#95—1582no—
269#95—1582no—
270#95—1582no—
271C24—1582no—
272C24C2×D31611yescharacteristic
273#41—31616no—
274#41—31616no—
275C2×Q8—794no—
276C2×Q8—1588no—
277C2×Q8—1582no—
278C2×D4—3164no—
279C2×D4—3164no—
280C2×D4—3168no—
281C2×D4—794no—
282C2×D4—1588no—
283C2×D4—1588no—
284C2×D4—1584no—
285C2×D4—1584no—
286C2×D4—1582no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—3168no—
291C22×C4—1584no—
292C22×C4—1582no—
293C22×C4—1588no—
294C22×C4(C2×D79)⋊C414yes—
295C22×C4(C2×D79)⋊C414yes—
296C22×C4C2×C4×D7912yes—
297C22×C4C2×C22⋊D7912yes—
298C22×C4C2×D31611yescharacteristic
299C22×C4C2×D31611yescharacteristic
300C22×C4C2×D31611yescharacteristic
301C22×C4C2×D31611yescharacteristic
302C22×C4C2×D31611yescharacteristic
303C22×C4C23×D7911yescharacteristic
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C4⋊C4—1588no—
308C4⋊C4—1584no—
309C4⋊C4—1584no—
310C22⋊C4—3164no—
311C22⋊C4—1588no—
312C22⋊C4—1588no—
313C42—3164no—
314C42(C2×D79)⋊C414yes—
315C42(C2×D79)⋊C414yes—
316C42C2×C4×D7912yes—
317C42C2×C4×D7912yes—
318C42C2×C22⋊D7912yes—
319C42C2×C22⋊D7912yes—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—3164no—
324C23—3164no—
325C23—3164no—
326C23—1584no—
327C23C2×(C2×D79)⋊C412yes—
328C23C22×D31611yescharacteristic
329Q8—3164no—
330Q8—1588no—
331D4—3168no—
332D4—3168no—
333D4—31616no—
334D4—3164no—
335D4—1588no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—31616no—
346C2×C4—3164no—
347C2×C4—3164no—
348C2×C4—3164no—
349C2×C4—3164no—
350C2×C4—1584no—
351C2×C4C2×(C2×D79)⋊C412yes—
352C2×C4C2×(C2×D79)⋊C412yes—
353C2×C4C2×(C2×D79)⋊C412yes—
354C2×C4C2×(C2×D79)⋊C412yes—
355C2×C4C2×(C2×D79)⋊C412yes—
356C2×C4C22×C4×D7911yescharacteristic
357C2×C4C22×C22⋊D7911yescharacteristic
358C2×C4#1457920714yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3164no—
365C22—3164no—
366C22—3164no—
367C22—6328no—
368C22C22×(C2×D79)⋊C411yescharacteristic
369C22#1909913212yes—
370C22#1909936312yes—
371C22#1909936312yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—3164no—
378C4—6324no—
379C4#1909903712yes—
380C4#1909912712yes—
381C2—42no—
382C2—6324no—
383C2#2376598011yescharacteristic
384C2#2377192011yescharacteristic
385C2#2377192011yescharacteristic
386C2#2377443914yes—
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#432×C79C2
socleC23×C79C22×D4
prime radicalC2×#19099113C2
cosocleC22×C4×C79C24
hypercenterC2×#432D79
hypocenterC79#21812
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
2. #23765798C2semidirect
3. #23764330C2semidirect
4. #23764224C2semidirect
5. #23764220C2semidirect
6. #23763341C2semidirect
7. C2×#432×C79C2semidirect
10. #19099113C4semidirect
195. C2×#432D79semidirect
201. C79#21812semidirect

Maximal subgroups

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

subgroupindex
C2×#190991132
#237657982
#237643302
#237642242
#237642202
#237633412
C2×#432×C792
#2181279