Finite Group Databasebeta

← #34184467

Subgroups — #34184467

Order 20224.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098850C211yesmaximal, characteristic
2C2×#19098642C211yesmaximal, characteristic
3#23763397C211yesmaximal, characteristic
4C2×#19098312C211yesmaximal, characteristic, complemented
5#23761283C211yesmaximal, characteristic, complemented
6#23761235C211yesmaximal, characteristic, complemented
7C79×#1990C211yesmaximal, characteristic, complemented
8C22×#11929607—21no—
9C2×C4×D316C2211yescharacteristic
10#19098850—22no—
11#19098850C414yescomplemented
12#19098642—22no—
13#19098642—22no—
14C2×#14579092—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×C4⋊(C79⋊C8)C2211yescharacteristic
18C2×#14579081—21no—
19C2×#14579081—21no—
20C2×(C2×D79)⋊C8—21no—
21#19098325—21no—
22#19098312—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4×C8×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#340×C79—21no—
27#340×C79—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C2×#11929607—24no—
31C2×#11929607—42no—
32C2×#11929607—42no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316C2×C412yes—
39C4×D316D412yes—
40C2×C4⋊Dic79D411yescharacteristic
41C2×(C2×D79)⋊C4—21no—
42#14579092—24no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)C2×C412yes—
48C4⋊(C79⋊C8)D412yes—
49#14579081—24no—
50#14579081—42no—
51(C2×D79)⋊C8—24no—
52C22×Q8×C79—21no—
53C22×C8×C79D411yescharacteristic
54C22×C8×C79D411yescharacteristic
55C4⋊Q8×C79—21no—
56C4⋊Q8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79C2×C412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C42×C79C2311yescharacteristic
65C4×C8×C79—21no—
66C4×C8×C79—21no—
67C23×D79—21no—
68C2×D316—21no—
69C2×D316—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316C22⋊C412yes—
74C22×Dic79—21no—
75C2×C4×D79—22no—
76C2×C4×D79—22no—
77C2×C4×D79—41no—
78C2×C4×D79—41no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81#11929607—48no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79C22⋊C412yes—
93(C2×D79)⋊C4—22no—
94(C2×D79)⋊C4—22no—
95C2×Q8×C79—22no—
96C2×Q8×C79—41no—
97C2×Q8×C79—81no—
98C2×Q8×C79—81no—
99C22×C4×C79—41no—
100C22×C4×C79—41no—
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79#4111yescharacteristic
104C2×C8×C79—21no—
105C2×C8×C79—21no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—21no—
109C2×C8×C79SD1612yes—
110C4⋊C4×C79—22no—
111C4⋊C4×C79—22no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79—81no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12111yescharacteristic
124C22×D79—22no—
125C22×D79—41no—
126C22×D79—41no—
127C22×D79—41no—
128C22×D79—41no—
129C22×D79—41no—
130C22×D79—41no—
131Q8×C79—82no—
132Q8×C79—82no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—42no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79C2×C22⋊C411yescharacteristic
142C2×C4×C79C4×D411yescharacteristic
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79#12011yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12311yescharacteristic
147C2×C4×C79C2×SD1611yescharacteristic
148C2×C4×C79#13511yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C79#13711yescharacteristic
152C2×C4×C79#13711yescharacteristic
153C2×C4×C79#13711yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C2×Dic79—22no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158D316—22no—
159D316—22no—
160D316—41no—
161D316—41no—
162C4×D79—42no—
163C4×D79—42no—
164C4×D79—42no—
165C4×D79—42no—
166C8×C79—22no—
167C8×C79—41no—
168C79⋊C8—42no—
169C79⋊C8—42no—
170C79⋊C8—81no—
171C22×C79#38611yescharacteristic
172C22×C79#46111yescharacteristic
173C22×C79#46311yescharacteristic
174C22×C79#46811yescharacteristic
175C22×C79#47011yescharacteristic
176C22×C79#47111yescharacteristic
177C22×C79#47211yescharacteristic
178C2×D79—42no—
179C2×D79—42no—
180C2×D79—42no—
181C2×D79—81no—
182C2×D79—81no—
183C2×D79—81no—
184C2×D79—81no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—82no—
188C4×C79—161no—
189C4×C79#41711yescharacteristic
190C4×C79#41911yescharacteristic
191C4×C79C4×SD1611yescharacteristic
192C4×C79#44011yescharacteristic
193C4×C79#45011yescharacteristic
194C4×C79#45211yescharacteristic
195C4×C79#46811yescharacteristic
196C4×C79#47011yescharacteristic
197Dic79—42no—
198#9223—791nomaximal, Sylow
199C2×C79#169111yescharacteristic
200C2×C79#169511yescharacteristic
201C2×C79#169711yescharacteristic
202C2×C79#170311yescharacteristic
203C2×C79#187711yescharacteristic
204C2×C79#189411yescharacteristic
205C2×C79#190711yescharacteristic
206D79—82no—
207C2×#476—791no—
208C2×#436—791no—
209#1990D7911yescharacteristic, complemented
210#1802—791no—
211C2×#333—791no—
212#1296—791no—
213#1277—791no—
214C79#922311yescharacteristic, complemented, Sylow
215C22×M16—1581no—
216C2×C4⋊Q8C2×D7911yescharacteristic
217C2×C4×D4—791no—
218#476—794no—
219#476—1582no—
220#436—1582no—
221#436—1582no—
222C2×#106—1581no—
223C2×C4⋊C8—791no—
224C2×C4⋊C8—791no—
225C2×C4⋊C8—791no—
226C2×#102—1581no—
227C2×#102—1581no—
228C2×C22⋊C8—1581no—
229C2×C4×C8C2×D7911yescharacteristic
230#384C2×D7911yescharacteristic
231#340—21no—
232#340—21no—
233#340—1581no—
234#333—1582no—
235C22×Q8—21no—
236C22×D4—791no—
237C23×C4—1581no—
238C2×M16—3162no—
239C2×M16—3162no—
240C2×M16—1584no—
241C22×C8—1581no—
242C22×C8—1581no—
243C22×C8—1581no—
244C22×C8C22⋊D7911yescharacteristic
245C22×C8C22⋊D7911yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C4×D7912yes—
250C4×D4—792no—
251C4×D4—792no—
252C4×D4—1581no—
253C4×D4—1581no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—791no—
260C2×C22⋊C4—1581no—
261C2×C42C22×D7911yescharacteristic
262#106—1584no—
263C4⋊C8—792no—
264C4⋊C8—792no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269#102—3162no—
270#102—1584no—
271C22⋊C8—1584no—
272C4×C8—21no—
273C4×C8—21no—
274C24—1581no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—792no—
280C2×D4—1581no—
281C2×D4—1581no—
282C2×D4—1581no—
283C2×D4—1581no—
284C2×D4—1581no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—3161no—
288C22×C4—3161no—
289C22×C4—3161no—
290C22×C4—3161no—
291C22×C4—1581no—
292C22×C4—1582no—
293C22×C4—1582no—
294C22×C4D4×D7911yescharacteristic
295C22×C4C2×C22⋊D7911yescharacteristic
296C22×C4#1192962311yescharacteristic
297M16—3168no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—3161no—
304C2×C8—3161no—
305C2×C8—3161no—
306C2×C8—3161no—
307C2×C8—3161no—
308C2×C8—3161no—
309C2×C8—3161no—
310C2×C8—1582no—
311C2×C8—1582no—
312C2×C8#1192961112yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C4⋊C4—792no—
323C4⋊C4—1581no—
324C22⋊C4—1582no—
325C22⋊C4—1582no—
326C42C2×C4×D7911yescharacteristic
327C42D4×D7911yescharacteristic
328C42D4×D7911yescharacteristic
329C42#1192962411yescharacteristic
330C23—3161no—
331C23—3161no—
332C23—3161no—
333C23—3161no—
334C23—3161no—
335C23—3161no—
336C23—1582no—
337C23#1457913211yescharacteristic
338Q8—82no—
339Q8—82no—
340D4—3161no—
341D4—3161no—
342D4—1582no—
343D4—1582no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—3162no—
355C2×C4—3162no—
356C2×C4—3162no—
357C2×C4—3162no—
358C2×C4—1582no—
359C2×C4C22⋊C4×D7911yescharacteristic
360C2×C4(C22⋊D79)⋊C411yescharacteristic
361C2×C4C4×C22⋊D7911yescharacteristic
362C2×C4#1457912911yescharacteristic
363C2×C4#1457913411yescharacteristic
364C2×C4#1457914311yescharacteristic
365C2×C4C2×#1192961111yescharacteristic
366C2×C4#1457918111yescharacteristic
367C2×C4#1457918611yescharacteristic
368C2×C4#1457918811yescharacteristic
369C2×C4#1457919011yescharacteristic
370C2×C4#1457919211yescharacteristic
371C2×C4#1457919311yescharacteristic
372C2×C4#1457919311yescharacteristic
373C8—22no—
374C8—41no—
375C8—3162no—
376C8—3162no—
377C8—6321no—
378C22—3162no—
379C22—3162no—
380C22—3162no—
381C22—6321no—
382C22—6321no—
383C22—6321no—
384C22—6321no—
385C22#1909843911yescharacteristic
386C22#1909876511yescharacteristic
387C22#1909877511yescharacteristic
388C22#1909880611yescharacteristic
389C22#1909881811yescharacteristic
390C22#1909882311yescharacteristic
391C22#1909883011yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—161no—
396C4—3162no—
397C4#1909856311yescharacteristic
398C4#1909857011yescharacteristic
399C4C4×#1192961111yescharacteristic
400C4#1909866111yescharacteristic
401C4#1909871311yescharacteristic
402C4#1909872311yescharacteristic
403C4#1909881211yescharacteristic
404C4#1909882011yescharacteristic
405C2—6322no—
406C2#2376289611yescharacteristic
407C2#2376292011yescharacteristic
408C2#2376293011yescharacteristic
409C2#2376297211yescharacteristic
410C2#2376373711yescharacteristic
411C2#2376383011yescharacteristic
412C2#2376391111yescharacteristic
subgroupG/H
centerC23#14579132
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1990C2
socleC23×C79#121
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1990D79
hypocenterC79#9223
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#19098312C2semidirect
5. #23761283C2semidirect
6. #23761235C2semidirect
7. C79×#1990C2semidirect
11. #19098850C4semidirect
209. #1990D79semidirect
214. C79#9223semidirect

Maximal subgroups

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

subgroupindex
C2×#190988502
C2×#190986422
#237633972
C2×#190983122
#237612832
#237612352
C79×#19902
#922379