Finite Group Databasebeta

← #34184954

Subgroups — #34184954

Order 20224.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774315C211yesmaximal, characteristic
2#23764224C211yesmaximal, characteristic, complemented
3#23762069C211yesmaximal, characteristic, complemented
4#23761960C211yesmaximal, characteristic, complemented
5#23761260C211yesmaximal, characteristic, complemented
6#23761231C211yesmaximal, characteristic, complemented
7C79×#2932C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9#19098939C412yescomplemented
10#19098911C412yescomplemented
11#19098758—22no—
12#19098696—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15C2×#14579083—21no—
16C2×#14579083—21no—
17#19098429C2211yescharacteristic
18#19098306—22no—
19#19098297—22no—
20C2×#131×C79—21no—
21#532×C79C2211yescharacteristic
22#435×C79—22no—
23#435×C79—22no—
24#423×C79C2211yescharacteristic
25#408×C79—21no—
26C2×#97×C79C2211yescharacteristic
27C22×D316—21no—
28C22×D316—21no—
29C22×D316—21no—
30C2×#11929609—24no—
31#14579158—21no—
32#14579158—21no—
33#14579158—21no—
34#14579158C2×C412yes—
35C2×C4⋊Dic79—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38#14579083—42no—
39#14579083—42no—
40#14579081—24no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)—21no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8C2×C412yes—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C2×#41×C79—21no—
48#131×C79—42no—
49#131×C79—42no—
50C2×M16×C79—22no—
51C2×M16×C79—21no—
52C22×C8×C79—22no—
53C22×C8×C79D411yescharacteristic
54C22×C8×C79D411yescharacteristic
55C4⋊Q8×C79C2×C411yescharacteristic
56#127×C79C2×C411yescharacteristic
57#124×C79—21no—
58#121×C79—21no—
59#121×C79D412yes—
60C4×D4×C79—21no—
61C4×D4×C79D412yes—
62C2×C42×C79C2311yescharacteristic
63C4⋊C8×C79—21no—
64C4⋊C8×C79D412yes—
65C22⋊C8×C79—24no—
66#97×C79—21no—
67#97×C79—21no—
68C23×D79—41no—
69C2×D316—22no—
70C2×D316—22no—
71C2×D316—22no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—82no—
80C2×D316—82no—
81C22×Dic79—41no—
82#11929609—84no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—41no—
85C4⋊Dic79—41no—
86C4⋊Dic79—41no—
87#41×C79—41no—
88#41×C79—41no—
89#41×C79—41no—
90#41×C79—41no—
91C2×Q8×C79—21no—
92C2×D4×C79—22no—
93C2×D4×C79—21no—
94C2×D4×C79—21no—
95C2×D4×C79C22⋊C412yes—
96C22×C4×C79—22no—
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79#4111yescharacteristic
100M16×C79—44no—
101M16×C79—41no—
102M16×C79—41no—
103C2×C8×C79—21no—
104C2×C8×C79—21no—
105C2×C8×C79—22no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—21no—
109C2×C8×C79—44no—
110C2×C8×C79D812yescomplemented
111C4⋊C4×C79—21no—
112C4⋊C4×C79C22⋊C412yes—
113C22⋊C4×C79—22no—
114C22⋊C4×C79—22no—
115C42×C79C22×C411yescharacteristic
116C42×C79C2×D411yescharacteristic
117C42×C79C2×D411yescharacteristic
118C42×C79#4111yescharacteristic
119C23×C79—22no—
120C23×C79#12111yescharacteristic
121C22×D79—42no—
122C22×D79—82no—
123C22×D79—82no—
124C22×D79—81no—
125C22×D79—81no—
126Q8×C79—41no—
127Q8×C79—41no—
128D4×C79—42no—
129D4×C79—42no—
130D4×C79—42no—
131D4×C79—41no—
132D4×C79—41no—
133D4×C79—81no—
134C2×C4×C79—22no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79C2×C22⋊C411yescharacteristic
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79C4×D411yescharacteristic
142C2×C4×C79#12011yescharacteristic
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12311yescharacteristic
145C2×C4×C79C2×D811yescharacteristic
146C2×C4×C79#13511yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×Dic79—81no—
150C2×Dic79—81no—
151C2×Dic79—81no—
152D316—44no—
153D316—42no—
154D316—42no—
155D316—82no—
156D316—82no—
157D316—82no—
158D316—81no—
159D316—81no—
160C8×C79—22no—
161C8×C79—42no—
162C8×C79—41no—
163C79⋊C8—42no—
164C79⋊C8—81no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—42no—
168C22×C79—42no—
169C22×C79—42no—
170C22×C79#38611yescharacteristic
171C22×C79#45911yescharacteristic
172C22×C79#46311yescharacteristic
173C2×D79—82no—
174C2×D79—82no—
175C2×D79—82no—
176C2×D79—162no—
177C2×D79—162no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—42no—
183C4×C79C4×D811yescharacteristic
184C4×C79#44211yescharacteristic
185C4×C79#44711yescharacteristic
186C4×C79#44911yescharacteristic
187Dic79—161no—
188#9293—791nomaximal, Sylow
189C2×C79—21no—
190C2×C79—21no—
191C2×C79—82no—
192C2×C79#187911yescharacteristic
193C2×C79#190311yescharacteristic
194C2×C79#191011yescharacteristic
195D79—162no—
196#3205—791no—
197#2932D7911yescharacteristic, complemented
198#1961—791no—
199#1545—791no—
200#1517—791no—
201#1286—791no—
202#1275—791no—
203C79#929311yescharacteristic, complemented, Sylow
204C2×#131—21no—
205#532C2×D7911yescharacteristic
206C2×#127—791no—
207#493—792no—
208#486—792no—
209#459—1582no—
210#447—1582no—
211#435—22no—
212#435—22no—
213#431—791no—
214#431—791no—
215#423C2×D7911yescharacteristic
216C2×#102—1581no—
217C2×#102—1581no—
218#408—21no—
219C2×#97C2×D7911yescharacteristic
220#382—791no—
221#331—1582no—
222#327—1582no—
223C2×#41—21no—
224C22×D4—1581no—
225C22×D4—1581no—
226C22×D4—1581no—
227C2×D8—1584no—
228#131—42no—
229#131—42no—
230C2×M16—22no—
231C2×M16—21no—
232C22×C8—22no—
233C22×C8C22⋊D7911yescharacteristic
234C22×C8C22⋊D7911yescharacteristic
235C4⋊Q8C4×D7911yescharacteristic
236#127—792no—
237#127—1581no—
238#127—1581no—
239#127—1581no—
240#127C4×D7911yescharacteristic
241#124—21no—
242#121—21no—
243#121D31612yes—
244C4×D4—21no—
245C4×D4C22⋊D7912yes—
246C2×C4⋊C4—1581no—
247C2×C42C22×D7911yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—1581no—
250C4⋊C8—1581no—
251C4⋊C8C22⋊D7912yes—
252#102—3162no—
253#102—3162no—
254#102—1584no—
255C22⋊C8—24no—
256C22⋊C8—1581no—
257C22⋊C8—1581no—
258#97—21no—
259#97—21no—
260C4×C8—792no—
261C4×C8—1581no—
262#95—1581no—
263#95—1581no—
264C24—3161no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—3161no—
274C2×D4—3161no—
275C2×D4—3161no—
276C2×D4—3161no—
277C2×D4—3161no—
278C2×D4—3161no—
279C2×D4—3161no—
280C2×D4—6322no—
281C2×D4—6322no—
282C2×D4—1582no—
283C2×D4—1582no—
284C2×D4—1582no—
285C2×D4(C2×D79)⋊C412yes—
286C22×C4—22no—
287C22×C4—3161no—
288C22×C4C2×C22⋊D7911yescharacteristic
289C22×C4C2×D31611yescharacteristic
290C22×C4#1192962411yescharacteristic
291D8—6324no—
292M16—44no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—3161no—
303C2×C8—1582no—
304C2×C8#1192960912yescomplemented
305C4⋊C4—21no—
306C4⋊C4—3161no—
307C4⋊C4—3161no—
308C4⋊C4(C2×D79)⋊C412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C2×C4×D7911yescharacteristic
312C42C2×C22⋊D7911yescharacteristic
313C42C2×C22⋊D7911yescharacteristic
314C42#1192962411yescharacteristic
315C23—22no—
316C23—3162no—
317C23—6322no—
318C23—6322no—
319C23—6321no—
320C23—6321no—
321C23#1457913511yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—3164no—
331D4—3162no—
332D4—3162no—
333D4—6322no—
334D4—6322no—
335D4—6322no—
336D4—6321no—
337D4—6321no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—6321no—
344C2×C4—6321no—
345C2×C4—6321no—
346C2×C4C2×(C2×D79)⋊C411yescharacteristic
347C2×C4C4×C22⋊D7911yescharacteristic
348C2×C4C4×C22⋊D7911yescharacteristic
349C2×C4#1457913011yescharacteristic
350C2×C4#1457913511yescharacteristic
351C2×C4#1457914411yescharacteristic
352C2×C4C2×#1192960911yescharacteristic
353C2×C4#1457918111yescharacteristic
354C2×C4#1457918611yescharacteristic
355C2×C4#1457918711yescharacteristic
356C8—22no—
357C8—42no—
358C8—41no—
359C8—3162no—
360C8—6321no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—6322no—
367C22—6322no—
368C22—6322no—
369C22—12642no—
370C22—12642no—
371C22#1909844211yescharacteristic
372C22#1909875811yescharacteristic
373C22#1909877811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—12641no—
380C4C4×#1192960911yescharacteristic
381C4#1909867311yescharacteristic
382C4#1909869711yescharacteristic
383C4#1909870811yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—12642no—
388C2#2376374811yescharacteristic
389C2#2376388011yescharacteristic
390C2#2376392711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2932C2
socleC22×C79#386
prime radical#23774315C2
cosocleC2×C42×C79C23
hypercenter#2932D79
hypocenterC79#9293
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #23764224C2semidirect
3. #23762069C2semidirect
4. #23761960C2semidirect
5. #23761260C2semidirect
6. #23761231C2semidirect
7. C79×#2932C2semidirect
9. #19098939C4semidirect
10. #19098911C4semidirect
110. C2×C8×C79D8semidirect
197. #2932D79semidirect
203. C79#9293semidirect
304. C2×C8#11929609semidirect

Maximal subgroups

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

subgroupindex
#237743152
#237642242
#237620692
#237619602
#237612602
#237612312
C79×#29322
#929379