Finite Group Databasebeta

← #34273554

Subgroups — #34273554

Order 20224.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774327C211yesmaximal, characteristic
2C2×#19098848C211yesmaximal, characteristic, complemented
3#23772872C211yesmaximal, characteristic
4#23762830C214yesmaximal, complemented
5#23762748C214yesmaximal, complemented
6#23762171C211yesmaximal, characteristic, complemented
7#23762084C211yesmaximal, characteristic, complemented
8#23762047C211yesmaximal, characteristic
9C79×#2982C211yesmaximal, characteristic, complemented
10C2×#14579192—21no—
11#19099113C2211yescharacteristic
12C2×#14579161C2211yescharacteristic
13#19098936C2212yes—
14#19098914C2212yes—
15#19098848—22no—
16#19098848C2214yes—
17#19098785—22no—
18#19098773C2212yes—
19#19098769—22no—
20#19098763C2212yes—
21#19098704—22no—
22C2×#14579092—21no—
23#19098587C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25#19098562—21no—
26C2×#14579085—21no—
27#19098314C2214yes—
28#19098310C2214yes—
29#19098308C2212yes—
30#19098306C2212yes—
31C2×C4×Q8×C79C2211yescharacteristic
32C8×Q8×C79—22no—
33C8×Q8×C79C2214yes—
34#431×C79—22no—
35#431×C79C2211yescharacteristic
36#431×C79C2211yescharacteristic
37C2×#11929624—21no—
38C22×Dic158—21no—
39#14579192—44no—
40C2×#11929614—22no—
41C2×#11929613—24no—
42C2×#11929611—22no—
43C2×#11929608—21no—
44C22×C79⋊C8—21no—
45#14579161—21no—
46#14579161—21no—
47#14579161—21no—
48#14579161C2311yescharacteristic
49#14579161C2312yes—
50#14579158C2311yescharacteristic
51#14579149—21no—
52#14579135—21no—
53#14579135—21no—
54C4×D316—21no—
55C4×D316—21no—
56C2×C4⋊Dic79—21no—
57C2×C4⋊Dic79—21no—
58#14579092—24no—
59C4⋊(C79⋊C8)—21no—
60C4⋊(C79⋊C8)—21no—
61C4⋊(C79⋊C8)C2312yes—
62C4⋊(C79⋊C8)C2312yes—
63#14579086—24no—
64#14579085—24no—
65#14579084—22no—
66#14579081—22no—
67C22×Q8×C79D411yescharacteristic
68C4×Q8×C79—21no—
69C4×Q8×C79—21no—
70C4×Q8×C79D412yes—
71C4×Q8×C79C2312yes—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79D411yescharacteristic
74C2×C42×C79—21no—
75C2×C42×C79C2311yescharacteristic
76C4⋊C8×C79—24no—
77C4⋊C8×C79D412yes—
78C4⋊C8×C79C2312yes—
79C22⋊C8×C79—21no—
80C22⋊C8×C79D412yes—
81C4×C8×C79—24no—
82C4×C8×C79—21no—
83C4×C8×C79C2312yes—
84#11929624—42no—
85#11929624—42no—
86C2×Dic158—22no—
87C2×Dic158—21no—
88C2×Dic158—41no—
89C2×Dic158—82no—
90C2×D316—21no—
91C2×D316—21no—
92C2×D316—21no—
93C2×C22⋊D79—41no—
94C22×Dic79—41no—
95C2×C4×D79—41no—
96#11929614—44no—
97#11929613—84no—
98#11929611—44no—
99#11929608—42no—
100C2×C79⋊C8—22no—
101C2×C79⋊C8—22no—
102C2×C79⋊C8—42no—
103C4⋊Dic79—22no—
104C4⋊Dic79—21no—
105C4⋊Dic79—22no—
106C4⋊Dic79—21no—
107C4⋊Dic79—41no—
108C4⋊Dic79—41no—
109(C2×D79)⋊C4—41no—
110(C2×D79)⋊C4—41no—
111C2×Q8×C79—21no—
112C2×Q8×C79—22no—
113C2×Q8×C79SD1614yescomplemented
114C2×Q8×C79C2×D412yes—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120C2×C8×C79—22no—
121C2×C8×C79C2×D414yes—
122C4⋊C4×C79—22no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79C2×D412yes—
126C42×C79—22no—
127C42×C79—22no—
128C42×C79C2×D411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79C2411yescharacteristic
132C23×C79#12111yescharacteristic
133C22×D79—41no—
134Q8×C79—22no—
135Q8×C79—24no—
136Q8×C79C2×SD1614yes—
137C2×C4×C79—22no—
138C2×C4×C79—24no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—41no—
142C2×C4×C79—81no—
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79C2×SD1612yes—
147C2×C4×C79#13512yes—
148C2×C4×C79C22×D411yescharacteristic
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C2×#4111yescharacteristic
151C22⋊D79—82no—
152C2×Dic79—41no—
153C2×Dic79—42no—
154C2×Dic79—82no—
155D316—41no—
156D316—41no—
157D316—42no—
158D316—81no—
159C4×D79—82no—
160Dic158—41no—
161Dic158—41no—
162Dic158—44no—
163Dic158—82no—
164C8×C79—24no—
165C8×C79—42no—
166C79⋊C8—42no—
167C79⋊C8—42no—
168C22×C79—21no—
169C22×C79—21no—
170C22×C79#46512yes—
171C22×C79C2×#12111yescharacteristic
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C4×C79—21no—
176C4×C79—24no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—42no—
180C4×C79#46512yes—
181C4×C79C22×SD1611yescharacteristic
182C4×C79C2×#13511yescharacteristic
183C4×C79#57712yes—
184Dic79—82no—
185Dic79—81no—
186#19397—791nomaximal, Sylow
187C2×C79—22no—
188C2×C79#304311yescharacteristic
189C2×C79C2×#46511yescharacteristic
190C2×C79#306911yescharacteristic
191D79—161no—
192#3206—791no—
193C2×#475—791no—
194#3037—791no—
195#2982D7911yescharacteristic, complemented
196#1680—794no—
197#1666—794no—
198#1570—791no—
199#1549—791no—
200#1540—791no—
201C79#1939711yescharacteristic, complemented, Sylow
202C2×#137—1581no—
203#532—791no—
204C2×C4⋊Q8—791no—
205C2×C4×Q8C2×D7911yescharacteristic
206#492—792no—
207#487—792no—
208#475—794no—
209#475—1582no—
210#464—1582no—
211#462—792no—
212#461—1582no—
213#460—792no—
214#448—1582no—
215C8×Q8—22no—
216C8×Q8C2×D7914yes—
217#431—22no—
218#431C2×D7911yescharacteristic
219#431C2×D7911yescharacteristic
220C2×#106—1581no—
221#423—791no—
222C2×C4⋊C8—791no—
223#417—1581no—
224C2×#103—1581no—
225#334—794no—
226#332—792no—
227#332—794no—
228#331—792no—
229C2×#41—1581no—
230C22×Q8—1581no—
231C22×Q8D31611yescharacteristic
232#137—3164no—
233C2×Q16—1582no—
234C2×SD16—1582no—
235C2×SD16—1584no—
236C2×M16—1581no—
237C22×C8—1581no—
238C4⋊Q8—792no—
239C4⋊Q8—791no—
240C4⋊Q8—1581no—
241C4⋊Q8—1581no—
242C4⋊Q8—1581no—
243#127—791no—
244#124—1581no—
245#121—1581no—
246#121—1581no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D31612yes—
250C4×Q8C22×D7912yes—
251C4×D4—1581no—
252C4×D4—1581no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4—1581no—
256C2×C4⋊C4D31611yescharacteristic
257C2×C42—21no—
258C2×C42C22×D7911yescharacteristic
259#106—1584no—
260C4⋊C8—24no—
261C4⋊C8—792no—
262C4⋊C8—792no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265C4⋊C8C22⋊D7912yes—
266C4⋊C8C22×D7912yes—
267#103—1584no—
268#103—1584no—
269#103—1582no—
270#102—1582no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D7912yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D7912yes—
276#41—3162no—
277#41—3162no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—3161no—
281C2×Q8—6322no—
282C2×Q8—1582no—
283C2×Q8—1581no—
284C2×Q8#1192961314yescomplemented
285C2×Q8C2×D31612yes—
286C2×D4—3161no—
287C2×D4—1581no—
288C2×D4—1581no—
289C2×D4—1581no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—3161no—
293C22×C4—3161no—
294C22×C4C2×C22⋊D7911yescharacteristic
295C22×C4C2×D31611yescharacteristic
296C22×C4#1192962411yescharacteristic
297Q16—3164no—
298SD16—3164no—
299SD16—6324no—
300M16—3162no—
301C2×C8—22no—
302C2×C8—3162no—
303C2×C8—1582no—
304C2×C8—1582no—
305C2×C8C2×C22⋊D7914yes—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—3161no—
310C4⋊C4—3161no—
311C4⋊C4—1582no—
312C4⋊C4—1581no—
313C4⋊C4—1581no—
314C4⋊C4—1582no—
315C4⋊C4C2×D31612yes—
316C22⋊C4—3161no—
317C22⋊C4—3161no—
318C42—22no—
319C42—22no—
320C42C2×C22⋊D7911yescharacteristic
321C42C2×D31611yescharacteristic
322C42#1192962411yescharacteristic
323C42C23×D7911yescharacteristic
324C23—3161no—
325C23#1457913511yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—3161no—
329Q8—3161no—
330Q8—3164no—
331Q8—6322no—
332Q8C2×#1192961314yes—
333D4—3161no—
334D4—3161no—
335D4—3162no—
336D4—6321no—
337D4—6322no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—3161no—
345C2×C4—3162no—
346C2×C4—6322no—
347C2×C4—6322no—
348C2×C4#1457913511yescharacteristic
349C2×C4#1457913511yescharacteristic
350C2×C4#1457913511yescharacteristic
351C2×C4C2×#1192961312yes—
352C2×C4#1457918312yes—
353C2×C4C22×C22⋊D7911yescharacteristic
354C2×C4C22×D31611yescharacteristic
355C2×C4C2×#1192962411yescharacteristic
356C8—24no—
357C8—42no—
358C8—3162no—
359C8—3162no—
360C22—21no—
361C22—21no—
362C22—6321no—
363C22—6321no—
364C22—6321no—
365C22#1909879212yes—
366C22C2×#1457913511yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—6321no—
373C4—6322no—
374C4#1909879212yes—
375C4C22×#1192961311yescharacteristic
376C4C2×#1457918311yescharacteristic
377C4#1909937112yes—
378C2—22no—
379C2—12641no—
380C2#2377292111yescharacteristic
381C2C2×#1909879211yescharacteristic
382C2#2377313711yescharacteristic
subgroupG/H
centerC22C2×#14579135
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#2982C2
socleC22×C79C2×#121
prime radical#19099113C22
hypercenter#2982D79
hypocenterC79#19397
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#19098848C2semidirect
4. #23762830C2semidirect
5. #23762748C2semidirect
6. #23762171C2semidirect
7. #23762084C2semidirect
9. C79×#2982C2semidirect
113. C2×Q8×C79SD16semidirect
195. #2982D79semidirect
201. C79#19397semidirect
284. C2×Q8#11929613semidirect

Maximal subgroups

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

subgroupindex
#237743272
C2×#190988482
#237728722
#237628302
#237627482
#237621712
#237620842
#237620472
C79×#29822
#1939779