Finite Group Databasebeta

← #34184816

Subgroups — #34184816

Order 20224.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772144C211yesmaximal, characteristic, complemented
2#23764241C211yesmaximal, characteristic, complemented
3#23762077C211yesmaximal, characteristic, complemented
4C2×#19098307C211yesmaximal, characteristic, complemented
5#23761296C211yesmaximal, characteristic, complemented
6#23761249C211yesmaximal, characteristic, complemented
7C79×#3205C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098641—21no—
10#19098635—22no—
11D4×C79⋊C8—21no—
12#19098587C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C2×#14579078—21no—
16C2×#14579078—21no—
17#19098530—21no—
18C4×#11929608C2211yescharacteristic
19#19098309—21no—
20#19098307—21no—
21#19098307—21no—
22#19098307—21no—
23#532×C79C2211yescharacteristic
24C2×#127×C79C2211yescharacteristic
25#493×C79—21no—
26#486×C79—21no—
27#459×C79—22no—
28#447×C79C414yes—
29#431×C79C2211yescharacteristic
30#14579168—42no—
31#14579168—42no—
32C2×#11929608—22no—
33C2×#11929608—21no—
34C2×#11929608D411yescharacteristic
35C2×#11929608D411yescharacteristic
36C22×C79⋊C8—22no—
37C22×C79⋊C8—21no—
38C22×C79⋊C8—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D411yescharacteristic
45C4⋊(C79⋊C8)D411yescharacteristic
46#14579078—42no—
47#14579078—42no—
48C22⋊(C79⋊C8)—22no—
49C22⋊(C79⋊C8)—22no—
50#14579072—21no—
51C4×C79⋊C8—21no—
52C2×#41×C79—21no—
53C22×D4×C79—21no—
54C22×D4×C79—21no—
55C22×D4×C79C2×C412yes—
56C2×D8×C79—24no—
57C4⋊Q8×C79D411yescharacteristic
58#127×C79—21no—
59#127×C79—21no—
60#127×C79—21no—
61#127×C79—21no—
62#127×C79D411yescharacteristic
63#124×C79—21no—
64#121×C79—21no—
65#121×C79C2×C412yes—
66C4×D4×C79—21no—
67C4×D4×C79D412yes—
68C2×C42×C79C2311yescharacteristic
69C4⋊C8×C79—21no—
70#102×C79—24no—
71C22⋊C8×C79C2×C412yes—
72C4×C8×C79—21no—
73#11929608—21no—
74#11929608—21no—
75#11929608—21no—
76#11929608—44no—
77#11929608—41no—
78#11929608—41no—
79#11929608D812yescomplemented
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—44no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—81no—
90C24×C79—22no—
91#41×C79—41no—
92#41×C79—41no—
93#41×C79—41no—
94#41×C79—41no—
95C2×Q8×C79—21no—
96C2×D4×C79—22no—
97C2×D4×C79—22no—
98C2×D4×C79—22no—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—41no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—41no—
109C2×D4×C79—81no—
110C2×D4×C79—81no—
111C2×D4×C79—82no—
112C2×D4×C79C22⋊C412yes—
113C22×C4×C79—22no—
114C22×C4×C79C22×C411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79#4111yescharacteristic
117D8×C79—84no—
118C2×C8×C79—22no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C42×C79C2×D411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79—22no—
128C23×C79—42no—
129C23×C79—42no—
130C23×C79—42no—
131C23×C79—82no—
132C23×C79—82no—
133C23×C79C4×D411yescharacteristic
134Q8×C79—41no—
135Q8×C79—41no—
136D4×C79—42no—
137D4×C79—44no—
138D4×C79—42no—
139D4×C79—42no—
140D4×C79—41no—
141D4×C79—41no—
142D4×C79—41no—
143D4×C79—41no—
144D4×C79—82no—
145D4×C79—82no—
146D4×C79—81no—
147D4×C79—81no—
148D4×C79—81no—
149D4×C79—81no—
150D4×C79—81no—
151D4×C79—81no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—22no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—42no—
158C2×C4×C79C2×C22⋊C411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79#12011yescharacteristic
161C2×C4×C79#12111yescharacteristic
162C2×C4×C79#12111yescharacteristic
163C2×C4×C79#12311yescharacteristic
164C2×C4×C79C2×D811yescharacteristic
165C2×C4×C79#13511yescharacteristic
166C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C8×C79—42no—
169C79⋊C8—22no—
170C79⋊C8—41no—
171C79⋊C8—41no—
172C79⋊C8—41no—
173C79⋊C8—81no—
174C79⋊C8—81no—
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—42no—
178C22×C79—42no—
179C22×C79—42no—
180C22×C79—82no—
181C22×C79—82no—
182C22×C79—82no—
183C22×C79—162no—
184C22×C79—162no—
185C22×C79#38611yescharacteristic
186C22×C79C4×D811yescharacteristic
187C22×C79#44211yescharacteristic
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—21no—
191C4×C79—42no—
192C4×C79#44711yescharacteristic
193C4×C79#44911yescharacteristic
194C4×C79#45911yescharacteristic
195C4×C79#46311yescharacteristic
196C4×C79#47111yescharacteristic
197C4×C79#47211yescharacteristic
198#9273—791nomaximal, Sylow
199C2×C79—21no—
200C2×C79—21no—
201C2×C79—82no—
202C2×C79—162no—
203C2×C79#187911yescharacteristic
204C2×C79#190711yescharacteristic
205C2×C79#191411yescharacteristic
206#3205D7911yescharacteristic
207#2935—791no—
208#1965—791no—
209#1547—791no—
210C2×#331—791no—
211#1301—791no—
212#1283—791no—
213C79#927311yescharacteristic, complemented, Sylow
214C2×#131—1581no—
215#532C2×D7911yescharacteristic
216C2×#127C2×D7911yescharacteristic
217#493—21no—
218#486—21no—
219#459—22no—
220#447Dic7914yes—
221#436—1581no—
222#435—1582no—
223C8×D4—1581no—
224#431C2×D7911yescharacteristic
225#423—791no—
226C2×C4⋊C8—791no—
227C2×C4⋊C8—791no—
228C2×#100—1581no—
229C2×#100—1581no—
230#408—1581no—
231C4×M16—791no—
232#332—1581no—
233#331—1581no—
234#331—1581no—
235#331—1581no—
236C2×#41—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×Dic7912yes—
240C2×D8—24no—
241#131—3162no—
242#131—3162no—
243C2×M16—791no—
244C2×M16—791no—
245C2×M16—1581no—
246C2×M16—1582no—
247C22×C8—1582no—
248C22×C8—1581no—
249C22×C8—1581no—
250C4⋊Q8C22⋊D7911yescharacteristic
251#127—21no—
252#127—21no—
253#127—21no—
254#127—21no—
255#127C22⋊D7911yescharacteristic
256#124—21no—
257#121—21no—
258#121C2×Dic7912yes—
259C4×D4—21no—
260C4×D4C22⋊D7912yes—
261C2×C42C22×D7911yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—791no—
264C4⋊C8—791no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269C4⋊C8—1581no—
270#102—24no—
271#100—3162no—
272#100—3162no—
273C22⋊C8—1582no—
274C22⋊C8—1582no—
275C22⋊C8C2×Dic7912yes—
276#97—1581no—
277C4×C8—21no—
278C4×C8—1581no—
279C24—22no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—82no—
301C2×D4C22⋊Dic7912yes—
302C22×C4—22no—
303C22×C4C22×Dic7911yescharacteristic
304C22×C4D4×D7911yescharacteristic
305C22×C4#1192962311yescharacteristic
306D8—84no—
307M16—3164no—
308M16—3161no—
309M16—3161no—
310M16—792no—
311M16—1581no—
312M16—1581no—
313M16—1581no—
314C2×C8—22no—
315C2×C8—3161no—
316C2×C8—3161no—
317C2×C8—3164no—
318C2×C8—3161no—
319C2×C8—3161no—
320C2×C8—6321no—
321C2×C8—1581no—
322C2×C8—1581no—
323C2×C8—1581no—
324C2×C8—1581no—
325C4⋊C4—21no—
326C4⋊C4C22⋊Dic7912yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42D4×D7911yescharacteristic
330C42C2×C22⋊D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C42#1192962311yescharacteristic
333C23—22no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—82no—
338C23—82no—
339C23D4×Dic7911yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—44no—
344D4—42no—
345D4—42no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×C22⋊Dic7911yescharacteristic
365C2×C4D4×Dic7911yescharacteristic
366C2×C4#1457912911yescharacteristic
367C2×C4#1457913311yescharacteristic
368C2×C4#1457913311yescharacteristic
369C2×C4#1457914211yescharacteristic
370C2×C4D8×D7911yescharacteristic
371C2×C4#1457918011yescharacteristic
372C2×C4#1457918511yescharacteristic
373C2×C4#1457918511yescharacteristic
374C8—42no—
375C8—3161no—
376C8—3161no—
377C8—3161no—
378C8—6321no—
379C8—6321no—
380C8—1582no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22#1909844311yescharacteristic
392C22D8×Dic7911yescharacteristic
393C22#1909867511yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1909869511yescharacteristic
399C4#1909870511yescharacteristic
400C4#1909875611yescharacteristic
401C4#1909877611yescharacteristic
402C4#1909882411yescharacteristic
403C4#1909883111yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—162no—
408C2#2376375011yescharacteristic
409C2#2376391011yescharacteristic
410C2#2376394811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3205C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3205D79
hypocenterC79#9273
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #23772144C2semidirect
2. #23764241C2semidirect
3. #23762077C2semidirect
4. C2×#19098307C2semidirect
5. #23761296C2semidirect
6. #23761249C2semidirect
79. #11929608D8semidirect
213. C79#9273semidirect

Maximal subgroups

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

subgroupindex
#237721442
#237642412
#237620772
C2×#190983072
#237612962
#237612492
C79×#32052
#927379