Finite Group Databasebeta

← #34183490

Subgroups — #34183490

Order 20224.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773975C211yesmaximal, characteristic
2#23764319C211yesmaximal, characteristic, complemented
3#23761895C211yesmaximal, characteristic, complemented
4#23761798C211yesmaximal, characteristic, complemented
5#23761239C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7C79×#2935C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098966C411yescharacteristic, complemented
10#19098939C411yescharacteristic, complemented
11#19098911C411yescharacteristic, complemented
12#19098909C411yescharacteristic, complemented
13#19098797—22no—
14C4×#11929609—22no—
15#19098616C2211yescharacteristic
16C2×#14579087—21no—
17C2×#14579087—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19C4×C4⋊Dic79C2211yescharacteristic
20#19098297—22no—
21#19098290—21no—
22#19098290—21no—
23C2×#131×C79—21no—
24#532×C79C2211yescharacteristic
25#436×C79—21no—
26#435×C79—22no—
27C8×D4×C79—21no—
28C2×C4⋊C8×C79C2211yescharacteristic
29#408×C79—21no—
30C4×M16×C79C2211yescharacteristic
31C2×#11929624—21no—
32C2×#11929609—22no—
33C22×C79⋊C8—21no—
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579149—21no—
37#14579135—21no—
38#14579135—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43#14579094—22no—
44C4⋊(C79⋊C8)—21no—
45#14579087—42no—
46#14579087—42no—
47#14579082—22no—
48#14579081—22no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C4×C79⋊C8C2×C411yescharacteristic
53C4×C79⋊C8C2×C411yescharacteristic
54C79⋊#95—21no—
55C2×#41×C79—21no—
56#131×C79—42no—
57#131×C79—42no—
58C2×M16×C79—22no—
59C2×M16×C79D411yescharacteristic
60C2×M16×C79D411yescharacteristic
61C22×C8×C79—22no—
62C22×C8×C79—21no—
63C4⋊Q8×C79C2×C411yescharacteristic
64#127×C79C2×C411yescharacteristic
65#124×C79—21no—
66#121×C79—21no—
67#121×C79D412yes—
68C4×D4×C79—21no—
69C4×D4×C79D412yes—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C4⋊C8×C79D411yescharacteristic
73C4⋊C8×C79D411yescharacteristic
74C22⋊C8×C79—22no—
75C22⋊C8×C79—22no—
76#97×C79—21no—
77C4×C8×C79—21no—
78#11929624—41no—
79#11929624—41no—
80#11929624—41no—
81#11929624—41no—
82C2×Dic158—21no—
83C2×D316—21no—
84C2×D316—21no—
85C2×D316—21no—
86C2×C22⋊D79—41no—
87C22×Dic79—41no—
88C2×C4×D79—41no—
89#11929609—44no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—42no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—22no—
95C4⋊Dic79—21no—
96C4⋊Dic79—21no—
97C4⋊Dic79—41no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C4×Dic79—41no—
101C4×Dic79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79—41no—
105#41×C79—41no—
106C2×Q8×C79—21no—
107C2×D4×C79—22no—
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79C22⋊C412yes—
111C22×C4×C79—22no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79#4111yescharacteristic
115M16×C79—21no—
116M16×C79—21no—
117M16×C79—21no—
118M16×C79—44no—
119M16×C79D812yescomplemented
120C2×C8×C79—21no—
121C2×C8×C79—21no—
122C2×C8×C79—21no—
123C2×C8×C79—21no—
124C2×C8×C79—44no—
125C2×C8×C79—41no—
126C2×C8×C79—41no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79C22⋊C412yes—
129C22⋊C4×C79—22no—
130C22⋊C4×C79—22no—
131C42×C79C22×C411yescharacteristic
132C42×C79C2×D411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79#4111yescharacteristic
135C23×C79—22no—
136C23×C79#12111yescharacteristic
137C22×D79—41no—
138Q8×C79—41no—
139Q8×C79—41no—
140D4×C79—42no—
141D4×C79—42no—
142D4×C79—42no—
143D4×C79—41no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—41no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—22no—
150C2×C4×C79—41no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79C2×C22⋊C411yescharacteristic
154C2×C4×C79C4×D411yescharacteristic
155C2×C4×C79C4×D411yescharacteristic
156C2×C4×C79#12011yescharacteristic
157C2×C4×C79#12111yescharacteristic
158C2×C4×C79#12311yescharacteristic
159C2×C4×C79C2×D811yescharacteristic
160C2×C4×C79#13511yescharacteristic
161C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
163C22⋊D79—81no—
164C22⋊D79—81no—
165C2×Dic79—42no—
166C2×Dic79—41no—
167C2×Dic79—81no—
168C2×Dic79—81no—
169D316—42no—
170D316—41no—
171D316—41no—
172D316—41no—
173D316—41no—
174C4×D79—81no—
175C4×D79—81no—
176Dic158—41no—
177Dic158—41no—
178C8×C79—22no—
179C8×C79—41no—
180C8×C79—41no—
181C8×C79—41no—
182C79⋊C8—22no—
183C79⋊C8—41no—
184C79⋊C8—81no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—42no—
188C22×C79—42no—
189C22×C79—42no—
190C22×C79#38611yescharacteristic
191C22×C79#45911yescharacteristic
192C22×C79#46311yescharacteristic
193C2×D79—81no—
194C2×D79—81no—
195C2×D79—81no—
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—42no—
200C4×C79C4×D811yescharacteristic
201C4×C79#44211yescharacteristic
202C4×C79#44311yescharacteristic
203C4×C79#44411yescharacteristic
204C4×C79#44711yescharacteristic
205C4×C79#44911yescharacteristic
206Dic79—82no—
207Dic79—81no—
208#9053—791nomaximal, Sylow
209C2×C79—21no—
210C2×C79—21no—
211C2×C79—82no—
212C2×C79#187611yescharacteristic
213C2×C79#187911yescharacteristic
214C2×C79#188411yescharacteristic
215D79—161no—
216#3169—791no—
217#2935D7911yescharacteristic, complemented
218#1980—791no—
219#1502—791no—
220#1480—791no—
221#1279—791no—
222#1278—791no—
223C79#905311yescharacteristic, complemented, Sylow
224C2×#131—21no—
225#532—791no—
226#532C2×D7911yescharacteristic
227#500—791no—
228#493—791no—
229#486—791no—
230#486—791no—
231#466—1582no—
232C4×D8—1582no—
233#436—21no—
234#435—22no—
235C8×D4—21no—
236#431—791no—
237C2×C4⋊C8C2×D7911yescharacteristic
238C2×#104—1581no—
239C2×#104—1581no—
240#408—21no—
241C4×M16C2×D7911yescharacteristic
242C2×C4×C8—791no—
243C4×C4⋊C4—791no—
244#327—1582no—
245D4⋊C8—1581no—
246D4⋊C8—1581no—
247C2×#41—21no—
248C2×#41—1581no—
249C2×D8—1582no—
250#131—42no—
251#131—42no—
252C2×M16—22no—
253C2×M16C22⋊D7911yescharacteristic
254C2×M16C22⋊D7911yescharacteristic
255C22×C8—22no—
256C22×C8—21no—
257C22×C8—1581no—
258C4⋊Q8—791no—
259C4⋊Q8C4×D7911yescharacteristic
260#127—791no—
261#127C4×D7911yescharacteristic
262#124—21no—
263#124—1581no—
264#121—21no—
265#121—1581no—
266#121—1581no—
267#121D31612yes—
268C4×D4—21no—
269C4×D4—1581no—
270C4×D4—1581no—
271C4×D4C22⋊D7912yes—
272C2×C4⋊C4—1581no—
273C2×C42—1581no—
274C2×C42C22×D7911yescharacteristic
275#107—1582no—
276C4⋊C8—21no—
277C4⋊C8—1581no—
278C4⋊C8C22⋊D7911yescharacteristic
279C4⋊C8C22⋊D7911yescharacteristic
280#104—3162no—
281#104—3162no—
282#102—1582no—
283#102—1582no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—1581no—
287#97—21no—
288C4×C8—21no—
289C4×C8—791no—
290C4×C8—791no—
291C4×C8—1581no—
292C4×C8—1581no—
293#95—1581no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—3161no—
299#41—3161no—
300#41—3161no—
301#41—3161no—
302C2×Q8—21no—
303C2×Q8—1581no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—3161no—
308C2×D4—1581no—
309C2×D4—1581no—
310C2×D4—1581no—
311C2×D4(C2×D79)⋊C412yes—
312C22×C4—22no—
313C22×C4—3161no—
314C22×C4—3161no—
315C22×C4C2×C22⋊D7911yescharacteristic
316C22×C4C2×D31611yescharacteristic
317C22×C4#1192962411yescharacteristic
318D8—3164no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—44no—
323M16#1192960912yescomplemented
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—44no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—3162no—
332C2×C8—3161no—
333C2×C8—1581no—
334C2×C8—1581no—
335C4⋊C4—21no—
336C4⋊C4—3161no—
337C4⋊C4—1582no—
338C4⋊C4—1581no—
339C4⋊C4—1581no—
340C4⋊C4(C2×D79)⋊C412yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—3161no—
344C22⋊C4—3161no—
345C42—3161no—
346C42—3161no—
347C42C2×C4×D7911yescharacteristic
348C42C2×C22⋊D7911yescharacteristic
349C42C2×C22⋊D7911yescharacteristic
350C42#1192962411yescharacteristic
351C23—22no—
352C23—3161no—
353C23#1457913511yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—3161no—
357Q8—3161no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—3162no—
366D4—3161no—
367D4—3161no—
368D4—3161no—
369D4—3161no—
370D4—6321no—
371D4—6321no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—3162no—
379C2×C4—3161no—
380C2×C4—6321no—
381C2×C4—6321no—
382C2×C4—6321no—
383C2×C4—6321no—
384C2×C4C2×(C2×D79)⋊C411yescharacteristic
385C2×C4C4×C22⋊D7911yescharacteristic
386C2×C4C4×C22⋊D7911yescharacteristic
387C2×C4#1457913011yescharacteristic
388C2×C4#1457913511yescharacteristic
389C2×C4#1457914411yescharacteristic
390C2×C4C2×#1192960911yescharacteristic
391C2×C4#1457918111yescharacteristic
392C2×C4#1457918611yescharacteristic
393C2×C4#1457918711yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—3161no—
399C8—6321no—
400C8—1582no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—6321no—
407C22—6321no—
408C22—6321no—
409C22#1909844211yescharacteristic
410C22#1909875811yescharacteristic
411C22#1909877811yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—6322no—
417C4—6321no—
418C4C4×#1192960911yescharacteristic
419C4#1909867311yescharacteristic
420C4#1909867811yescharacteristic
421C4#1909868311yescharacteristic
422C4#1909869711yescharacteristic
423C4#1909870811yescharacteristic
424C2—21no—
425C2—21no—
426C2—82no—
427C2—12641no—
428C2#2376372711yescharacteristic
429C2#2376374811yescharacteristic
430C2#2376378311yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2935C2
socleC22×C79#386
prime radical#23773975C2
cosocleC2×C42×C79C23
hypercenter#2935D79
hypocenterC79#9053
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #23764319C2semidirect
3. #23761895C2semidirect
4. #23761798C2semidirect
5. #23761239C2semidirect
6. #23761238C2semidirect
7. C79×#2935C2semidirect
9. #19098966C4semidirect
10. #19098939C4semidirect
11. #19098911C4semidirect
12. #19098909C4semidirect
119. M16×C79D8semidirect
217. #2935D79semidirect
223. C79#9053semidirect
323. M16#11929609semidirect

Maximal subgroups

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

subgroupindex
#237739752
#237643192
#237618952
#237617982
#237612392
#237612382
C79×#29352
#905379