Finite Group Databasebeta

← #34205322

Subgroups — #34205322

Order 20224.

476 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
1C2×#19098993C211yesmaximal, characteristic
2C2×#19098681C211yesmaximal, characteristic, complemented
3C2×#19098370C211yesmaximal, characteristic, complemented
4C2×#19098369C211yesmaximal, characteristic, complemented
5#23761493C211yesmaximal, characteristic, complemented
6#23761478C211yesmaximal, characteristic, complemented
7C79×#1974C211yesmaximal, characteristic, complemented
8C2×#14579183C2211yescharacteristic
9C2×#14579167—21no—
10#19098993—22no—
11#19098993—22no—
12C2×#14579107C411yescharacteristic, complemented
13C2×#14579105C411yescharacteristic, complemented
14C2×#14579104C411yescharacteristic, complemented
15C2×#14579102C411yescharacteristic, complemented
16C22×C79⋊C16C2211yescharacteristic
17C22×C79⋊C16C2211yescharacteristic
18#19098681—22no—
19#19098681—22no—
20C2×#14579097C2211yescharacteristic
21C2×#14579089—21no—
22#19098370—22no—
23#19098369—22no—
24C2×#135×C79C2211yescharacteristic
25C2×#108×C79C2211yescharacteristic
26C2×#104×C79—21no—
27#408×C79—21no—
28C2×C4×C8×C79C2211yescharacteristic
29C2×#11929624—21no—
30#14579183—21no—
31#14579183—21no—
32#14579183D412yes—
33#14579183D412yes—
34C2×Dic316C2×C411yescharacteristic
35C2×#11929613—21no—
36C2×D632C2×C411yescharacteristic
37#14579167—24no—
38#14579167—42no—
39C2×#11929608—21no—
40C2×#11929607—41no—
41C2×C8×D79—41no—
42#14579107—22no—
43#14579105—22no—
44#14579104—22no—
45#14579102—22no—
46C2×C79⋊C16—21no—
47C2×C79⋊C16—21no—
48C2×C79⋊C16—21no—
49C2×C79⋊C16—21no—
50C2×C79⋊C16—21no—
51C2×C79⋊C16C2×C411yescharacteristic
52C2×C79⋊C16C2×C411yescharacteristic
53#14579097—21no—
54#14579097D412yes—
55#14579089—24no—
56C2×#41×C79—21no—
57#135×C79—21no—
58#135×C79—21no—
59#135×C79—21no—
60#135×C79—21no—
61C2×Q16×C79C2×C411yescharacteristic
62C2×SD16×C79—21no—
63C2×D8×C79C2×C411yescharacteristic
64C2×M16×C79—21no—
65C22×C8×C79D411yescharacteristic
66C22×C8×C79C2311yescharacteristic
67C2×C42×C79D411yescharacteristic
68#108×C79—21no—
69#108×C79—21no—
70#104×C79—42no—
71C22⋊C8×C79—41no—
72C22⋊C8×C79—41no—
73C4×C8×C79—21no—
74C4×C8×C79—21no—
75#11929624—22no—
76#11929624—22no—
77#11929624—41no—
78#11929624—41no—
79C2×Dic158—21no—
80C2×D316—21no—
81C2×C22⋊D79—41no—
82C2×C4×D79—41no—
83Dic316—21no—
84Dic316C22⋊C412yes—
85#11929613—22no—
86#11929613—22no—
87D632—21no—
88D632C22⋊C412yes—
89#11929608—22no—
90#11929608—41no—
91#11929607—44no—
92C2×C79⋊C8—41no—
93C8×D79—44no—
94C79⋊C16—21no—
95C79⋊C16—21no—
96C79⋊C16—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101C2×Q8×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—41no—
104C22×C4×C79—21no—
105C22×C4×C79—41no—
106C22×C4×C79C2×D411yescharacteristic
107Q16×C79—21no—
108Q16×C79—21no—
109SD16×C79—41no—
110SD16×C79—41no—
111D8×C79—21no—
112D8×C79—21no—
113M16×C79—41no—
114M16×C79—41no—
115C2×C8×C79—21no—
116C2×C8×C79—21no—
117C2×C8×C79—41no—
118C2×C8×C79D811yescharacteristic
119C2×C8×C79D811yescharacteristic
120C2×C8×C79C22×C411yescharacteristic
121C2×C8×C79C2×D411yescharacteristic
122C2×C8×C79C2×D411yescharacteristic
123C2×C8×C79C2×D411yescharacteristic
124C2×C8×C79#4111yescharacteristic
125C2×C8×C79#4111yescharacteristic
126C42×C79—21no—
127C42×C79—21no—
128C23×C79—41no—
129C23×C79#13511yescharacteristic
130C22×D79—41no—
131Q8×C79—41no—
132Q8×C79—41no—
133D4×C79—41no—
134D4×C79—41no—
135D4×C79—81no—
136D4×C79—81no—
137C2×C4×C79—22no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—81no—
142C2×C4×C79—81no—
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79#12011yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79C2×D811yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C22⋊D79—42no—
150C22⋊D79—42no—
151C2×Dic79—41no—
152D316—22no—
153D316—41no—
154C4×D79—42no—
155C4×D79—42no—
156Dic158—22no—
157Dic158—41no—
158C8×C79—21no—
159C8×C79—21no—
160C8×C79—81no—
161C8×C79C2×C22⋊C411yescharacteristic
162C8×C79C4×D411yescharacteristic
163C8×C79#12111yescharacteristic
164C8×C79#12311yescharacteristic
165C79⋊C8—42no—
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—81no—
169C22×C79—81no—
170C22×C79—81no—
171C22×C79C4×D811yescharacteristic
172C22×C79#44911yescharacteristic
173C22×C79#46311yescharacteristic
174C2×D79—42no—
175C2×D79—81no—
176C2×D79—81no—
177C4×C79—42no—
178C4×C79—81no—
179C4×C79#38611yescharacteristic
180C4×C79#44211yescharacteristic
181C4×C79#44711yescharacteristic
182C4×C79#45911yescharacteristic
183Dic79—42no—
184#13011—791nomaximal, Sylow
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—161no—
188C2×C79#187911yescharacteristic
189C2×C79#217811yescharacteristic
190C2×C79#217911yescharacteristic
191D79—82no—
192C2×#508—791no—
193C2×#443—791no—
194C2×#359—791no—
195C2×#359—791no—
196#1974D7911yescharacteristic, complemented
197#1381—791no—
198#1374—791no—
199C79#1301111yescharacteristic, complemented, Sylow
200C2×#135—791no—
201C2×#135C2×D7911yescharacteristic
202C2×#131—1581no—
203#508—1582no—
204#508—1582no—
205C2×Q32—791no—
206C2×SD32—791no—
207C2×SD32—791no—
208C2×D16—791no—
209C22×C16—791no—
210C22×C16—791no—
211#443—1582no—
212#443—1582no—
213C2×#108—791no—
214C2×#108C2×D7911yescharacteristic
215C2×#104—21no—
216C2×#104—1581no—
217#408—21no—
218C2×C4×C8C2×D7911yescharacteristic
219#359—1582no—
220#359—1582no—
221C2×#41—21no—
222C2×#41—1581no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—792no—
228#135—792no—
229#135—1581no—
230#135—1581no—
231C2×Q16—791no—
232C2×Q16C4×D7911yescharacteristic
233C2×SD16—21no—
234C2×SD16—1581no—
235C2×D8—791no—
236C2×D8C4×D7911yescharacteristic
237#131—3162no—
238#131—1584no—
239C2×M16—21no—
240C2×M16—3161no—
241C2×M16—1581no—
242C22×C8—3161no—
243C22×C8C22⋊D7911yescharacteristic
244C22×C8C22×D7911yescharacteristic
245C2×C42C22⋊D7911yescharacteristic
246Q32—1582no—
247SD32—1582no—
248SD32—1582no—
249D16—1582no—
250C2×C16—791no—
251C2×C16—791no—
252C2×C16—1581no—
253C2×C16—1581no—
254C2×C16—1581no—
255C2×C16—1581no—
256C2×C16—1581no—
257#108—21no—
258#108—21no—
259#108—792no—
260#108—1581no—
261#104—42no—
262#104—1584no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C4×C8—21no—
266C4×C8—21no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—3161no—
272#41—3161no—
273#41—1582no—
274#41—1582no—
275C2×Q8—21no—
276C2×Q8—1581no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—3161no—
280C2×D4—1581no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—3161no—
284C22×C4C2×C22⋊D7911yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—792no—
288Q16—1581no—
289SD16—41no—
290SD16—41no—
291SD16—1582no—
292SD16—1582no—
293D8—21no—
294D8—21no—
295D8—792no—
296D8—1581no—
297M16—41no—
298M16—41no—
299M16—3164no—
300M16—3161no—
301M16—1582no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—3164no—
306C2×C8—3161no—
307C2×C8#1192960911yescharacteristic
308C2×C8#1192960911yescharacteristic
309C2×C8C2×C4×D7911yescharacteristic
310C2×C8D4×D7911yescharacteristic
311C2×C8D4×D7911yescharacteristic
312C2×C8D4×D7911yescharacteristic
313C2×C8#1192962311yescharacteristic
314C2×C8#1192962411yescharacteristic
315C42—21no—
316C42—21no—
317C16—3161no—
318C16—1581no—
319C16—1581no—
320C23—41no—
321C23—3161no—
322C23#1457918111yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—3161no—
326Q8—1582no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—3162no—
332D4—3162no—
333D4—3161no—
334D4—1582no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—3161no—
342C2×C4—3162no—
343C2×C4—3162no—
344C2×C4C4×C22⋊D7911yescharacteristic
345C2×C4#1457912911yescharacteristic
346C2×C4#1457913211yescharacteristic
347C2×C4C2×#1192960911yescharacteristic
348C2×C4#1457918611yescharacteristic
349C2×C4#1457918711yescharacteristic
350C8—21no—
351C8—21no—
352C8—81no—
353C8—3162no—
354C8C22⋊C4×D7911yescharacteristic
355C8(C22⋊D79)⋊C411yescharacteristic
356C8#1457913411yescharacteristic
357C8#1457914311yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—3162no—
364C22—6321no—
365C22—6321no—
366C22C4×#1192960911yescharacteristic
367C22#1909870611yescharacteristic
368C22#1909877511yescharacteristic
369C4—42no—
370C4—81no—
371C4—3162no—
372C4#1909843911yescharacteristic
373C4#1909867311yescharacteristic
374C4#1909869611yescharacteristic
375C4#1909875511yescharacteristic
376C2—21no—
377C2—21no—
378C2—161no—
379C2—6322no—
380C2#2376375111yescharacteristic
381C2#2376527911yescharacteristic
382C2#2376528411yescharacteristic
subgroupG/H
centerC2×C4C2×#11929609
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1974C2
socleC22×C79C4×D8
prime radicalC2×#19098993C2
cosocleC22×C8×C79C23
hypercenter#1974D79
hypocenterC79#13011
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1974

Complements

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

NH ≅ G/Ntype
2. C2×#19098681C2semidirect
3. C2×#19098370C2semidirect
4. C2×#19098369C2semidirect
5. #23761493C2semidirect
6. #23761478C2semidirect
7. C79×#1974C2semidirect
12. C2×#14579107C4semidirect
13. C2×#14579105C4semidirect
14. C2×#14579104C4semidirect
15. C2×#14579102C4semidirect
196. #1974D79semidirect
199. C79#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#190989932
C2×#190986812
C2×#190983702
C2×#190983692
#237614932
#237614782
C79×#19742
#1301179