Finite Group Databasebeta

← #34190842

Subgroups — #34190842

Order 20224.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098850C211yesmaximal, characteristic
2C2×#19098765C211yesmaximal, characteristic
3#23764194C211yesmaximal, characteristic
4C2×#19098312C211yesmaximal, characteristic, complemented
5C2×#19098309C211yesmaximal, characteristic, complemented
6#23761235C211yesmaximal, characteristic, complemented
7C2×#498×C79C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×C4×D316C2211yescharacteristic
10#19098850—22no—
11#19098850—22no—
12#19098765—22no—
13#19098765—22no—
14C2×#14579092—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×C4⋊(C79⋊C8)C2211yescharacteristic
18C2×#14579085—21no—
19C2×#14579081—21no—
20C2×(C2×D79)⋊C8—22no—
21#19098312—22no—
22#19098309—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4⋊Q8×C79C2211yescharacteristic
25#498×C79—22no—
26#498×C79—22no—
27C2×#106×C79—22no—
28C2×C4×C8×C79C2211yescharacteristic
29C22×D316D411yescharacteristic
30C22×C4×D79—21no—
31C2×#11929611—24no—
32C2×#11929611—42no—
33C2×#11929611—42no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C22×C79⋊C8—21no—
37C4×D316—21no—
38C4×D316—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79D411yescharacteristic
42C2×(C2×D79)⋊C4—21no—
43#14579092—24no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)D412yes—
49C4⋊(C79⋊C8)D412yes—
50#14579085—24no—
51#14579081—24no—
52(C2×D79)⋊C8—44no—
53C22×Q8×C79—21no—
54C22×Q8×C79—21no—
55C22×C8×C79D412yes—
56C4⋊Q8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79—21no—
62C4⋊Q8×C79D412yes—
63C4⋊Q8×C79D412yes—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79—21no—
66C2×C4⋊C4×C79—21no—
67C2×C4⋊C4×C79—21no—
68C2×C42×C79C2311yescharacteristic
69#106×C79—28no—
70C4×C8×C79—21no—
71C4×C8×C79—21no—
72C23×D79—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316—21no—
76C2×D316—21no—
77C2×D316—21no—
78C2×D316SD1612yes—
79C22×Dic79—21no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—42no—
83C2×C4×D79—42no—
84#11929611—48no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—42no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95(C2×D79)⋊C4—41no—
96(C2×D79)⋊C4—41no—
97C2×Q8×C79—22no—
98C2×Q8×C79—22no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C2×Q8×C79—41no—
102C2×Q8×C79—41no—
103C2×Q8×C79—41no—
104C2×Q8×C79—41no—
105C2×Q8×C79—82no—
106C22×C4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111C2×C8×C79—22no—
112C2×C8×C79—22no—
113C2×C8×C79SD1614yescomplemented
114C4⋊C4×C79—22no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—22no—
117C4⋊C4×C79—22no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C42×C79C2×D411yescharacteristic
123C42×C79C2×D411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C23×C79#12111yescharacteristic
127C22×D79—22no—
128C22×D79—41no—
129C22×D79—42no—
130C22×D79—41no—
131C22×D79—42no—
132Q8×C79—42no—
133Q8×C79—42no—
134Q8×C79—82no—
135Q8×C79—82no—
136Q8×C79—81no—
137Q8×C79—81no—
138C2×C4×C79—22no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—81no—
142C2×C4×C79—81no—
143C2×C4×C79—82no—
144C2×C4×C79#12011yescharacteristic
145C2×C4×C79#12011yescharacteristic
146C2×C4×C79#12011yescharacteristic
147C2×C4×C79#12111yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79#12711yescharacteristic
150C2×C4×C79C2×SD1611yescharacteristic
151C2×C4×C79C2×SD1612yes—
152C2×C4×C79#13511yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C2×C4×C79#13712yes—
156C2×Dic79—41no—
157C2×Dic79—42no—
158D316—22no—
159D316—22no—
160D316—41no—
161D316—41no—
162C4×D79—82no—
163C4×D79—81no—
164C4×D79—81no—
165C8×C79—24no—
166C79⋊C8—42no—
167C79⋊C8—42no—
168C79⋊C8—81no—
169C22×C79#39211yescharacteristic
170C22×C79#46112yes—
171C22×C79#46511yescharacteristic
172C22×C79#46811yescharacteristic
173C22×C79#47112yes—
174C2×D79—44no—
175C2×D79—42no—
176C2×D79—82no—
177C2×D79—81no—
178C2×D79—81no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—82no—
182C4×C79—82no—
183C4×C79#44811yescharacteristic
184C4×C79#44812yes—
185C4×C79#44911yescharacteristic
186C4×C79#45011yescharacteristic
187C4×C79#45211yescharacteristic
188C4×C79#49211yescharacteristic
189C4×C79#49711yescharacteristic
190Dic79—81no—
191#10271—791nomaximal, Sylow
192C2×C79#167112yes—
193C2×C79#169112yes—
194C2×C79#200011yescharacteristic
195C2×C79#201311yescharacteristic
196C2×C79#201911yescharacteristic
197D79—82no—
198C2×#498D7911yescharacteristic, complemented
199C2×#476—791no—
200C2×#461—791no—
201#1954—791no—
202C2×#333—791no—
203C2×#332—791no—
204#1277—791no—
205C79#1027111yescharacteristic, complemented, Sylow
206C22×SD16—1581no—
207C2×C4⋊Q8C2×D7911yescharacteristic
208C2×C4⋊Q8C2×D7911yescharacteristic
209C2×C4×D4—791no—
210#498—22no—
211#498—22no—
212#476—1582no—
213#476—1582no—
214#461—1582no—
215#461—1582no—
216C2×#106—22no—
217C2×#106—1581no—
218C2×C4⋊C8—791no—
219C2×C4⋊C8—791no—
220C2×C4⋊C8—791no—
221C2×#103—1581no—
222C2×#102—1581no—
223C2×C22⋊C8—1582no—
224C2×C4×C8C2×D7911yescharacteristic
225#333—1582no—
226#332—1582no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—791no—
230C23×C4—1581no—
231C2×SD16—3162no—
232C2×SD16—3162no—
233C2×SD16—1584no—
234C22×C8—1581no—
235C22×C8—1581no—
236C22×C8—1581no—
237C22×C8C22⋊D7912yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D7912yes—
245C4⋊Q8C22⋊D7912yes—
246C4×D4—1581no—
247C4×D4—1581no—
248C4×D4—1581no—
249C4×D4—1581no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—791no—
255C2×C22⋊C4—1581no—
256C2×C42C22×D7911yescharacteristic
257#106—28no—
258#106—1584no—
259C4⋊C8—792no—
260C4⋊C8—792no—
261C4⋊C8—1581no—
262C4⋊C8—1581no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265#103—1584no—
266#102—1584no—
267C22⋊C8—3164no—
268C4×C8—21no—
269C4×C8—21no—
270C24—1581no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—792no—
281C2×D4—1581no—
282C2×D4—1581no—
283C2×D4—1581no—
284C2×D4—1581no—
285C2×D4—1581no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—3161no—
289C22×C4—3161no—
290C22×C4—3162no—
291C22×C4—3162no—
292C22×C4—1581no—
293C22×C4D4×D7911yescharacteristic
294C22×C4C2×C22⋊D7911yescharacteristic
295C22×C4#1192962311yescharacteristic
296SD16—3168no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—3161no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—3161no—
303C2×C8—3161no—
304C2×C8—3162no—
305C2×C8—1582no—
306C2×C8—1582no—
307C2×C8#1192961114yescomplemented
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1581no—
317C4⋊C4—1581no—
318C22⋊C4—3161no—
319C22⋊C4—3161no—
320C42D4×D7911yescharacteristic
321C42D4×D7911yescharacteristic
322C42C2×C22⋊D7911yescharacteristic
323C42C2×C22⋊D7911yescharacteristic
324C23—3161no—
325C23—3162no—
326C23—3161no—
327C23—3162no—
328C23—1582no—
329C23#1457913211yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—3161no—
337D4—3161no—
338D4—1582no—
339D4—1582no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—3161no—
347C2×C4—3162no—
348C2×C4—6321no—
349C2×C4—6321no—
350C2×C4—6322no—
351C2×C4#1457912911yescharacteristic
352C2×C4#1457912911yescharacteristic
353C2×C4#1457913011yescharacteristic
354C2×C4#1457913311yescharacteristic
355C2×C4#1457913311yescharacteristic
356C2×C4#1457915711yescharacteristic
357C2×C4SD16×D7911yescharacteristic
358C2×C4C2×#1192961112yes—
359C2×C4#1457918211yescharacteristic
360C2×C4#1457918811yescharacteristic
361C2×C4#1457919011yescharacteristic
362C2×C4#1457919212yes—
363C8—24no—
364C8—3162no—
365C8—3162no—
366C8—6321no—
367C22—3164no—
368C22—3162no—
369C22—6321no—
370C22—6321no—
371C22—6322no—
372C22#1909847111yescharacteristic
373C22#1909876512yes—
374C22#1909878911yescharacteristic
375C22#1909880611yescharacteristic
376C22#1909882312yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—6321no—
382C4#1909870011yescharacteristic
383C4#1909870212yes—
384C4#1909870711yescharacteristic
385C4#1909871211yescharacteristic
386C4#1909872211yescharacteristic
387C4#1909893711yescharacteristic
388C4#1909895511yescharacteristic
389C2—6322no—
390C2#2376277212yes—
391C2#2376289612yes—
392C2#2376440011yescharacteristic
393C2#2376448311yescharacteristic
394C2#2376451211yescharacteristic
subgroupG/H
centerC23#14579132
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#498×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#498D79
hypocenterC79#10271
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. C2×#19098312C2semidirect
5. C2×#19098309C2semidirect
6. #23761235C2semidirect
7. C2×#498×C79C2semidirect
113. C2×C8×C79SD16semidirect
198. C2×#498D79semidirect
205. C79#10271semidirect
307. C2×C8#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190988502
C2×#190987652
#237641942
C2×#190983122
C2×#190983092
#237612352
C2×#498×C792
#1027179