Finite Group Databasebeta

← #34442923

Subgroups — #34442923

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773912C211yesmaximal, characteristic, complemented
2C2×#19098645C211yesmaximal, characteristic
3#23765560C212yesmaximal, complemented
4#23765556C212yesmaximal, complemented
5#23765546C212yesmaximal, complemented
6#23765536C212yesmaximal, complemented
7#23765186C211yesmaximal, characteristic, complemented
8#23765155C211yesmaximal, characteristic, complemented
9#23765117C211yesmaximal, characteristic
10#23765115C211yesmaximal, characteristic
11C79×#2159C211yesmaximal, characteristic, complemented
12C22×D632C2211yescharacteristic
13#19099116—21no—
14C2×#14579123—21no—
15C2×#14579101C2211yescharacteristic
16#19098967C2211yescharacteristic, complemented
17#19098967C2211yescharacteristic, complemented
18#19098964C2211yescharacteristic, complemented
19#19098964C2211yescharacteristic, complemented
20#19098963—21no—
21#19098960—21no—
22#19098645—21no—
23#19098645—21no—
24#19098645C2212yes—
25#19098645C2212yes—
26#19098601C2211yescharacteristic
27#19098599—21no—
28C2×#14579081—21no—
29C2×C8×Dic79C2211yescharacteristic
30#19098390C2212yescomplemented
31#19098388C2212yescomplemented
32Dic79⋊C16C2212yescomplemented
33Dic79⋊C16C2212yescomplemented
34#19098365C2212yes—
35#19098365C2212yes—
36#19098363C2212yes—
37#19098363C2212yes—
38C2×M32×C79C2211yescharacteristic
39#427×C79C2211yescharacteristic
40C2×#107×C79C2211yescharacteristic
41#367×C79C2212yescomplemented
42#366×C79C2212yescomplemented
43C22×D316—21no—
44C22×C4×D79—41no—
45C2×D632—21no—
46C2×D632—22no—
47C2×D632Q814yescomplemented
48C2×D632C2312yes—
49C22×C79⋊C8D411yescharacteristic
50#14579160—21no—
51#14579160—21no—
52#14579152—21no—
53#14579152—21no—
54#14579138—41no—
55#14579138—41no—
56C4×Dic158—41no—
57C4×Dic158—41no—
58#14579123—22no—
59#14579123—22no—
60#14579123—41no—
61#14579123—41no—
62#14579120—21no—
63C2×(C2×D79)⋊C4—41no—
64C2×C4×Dic79D411yescharacteristic
65#14579101—22no—
66C2×C79⋊C16C2312yes—
67#14579095C2311yescharacteristic
68#14579095C2311yescharacteristic
69#14579094—21no—
70#14579094—21no—
71#14579093—21no—
72#14579092—21no—
73#14579092—21no—
74#14579081—22no—
75#14579081—22no—
76C8×Dic79D411yescharacteristic
77C8×Dic79D411yescharacteristic
78C8×Dic79C2311yescharacteristic
79C8×Dic79C2311yescharacteristic
80C22×C8×C79C2311yescharacteristic
81#117×C79—21no—
82C2×C4⋊C4×C79—21no—
83M32×C79—22no—
84C2×C16×C79C2312yes—
85#107×C79—21no—
86#107×C79C2312yes—
87#107×C79C2311yescharacteristic
88#107×C79C2311yescharacteristic
89#106×C79—21no—
90C23×D79—41no—
91C2×Dic158—41no—
92C2×Dic158—41no—
93C2×D316—24no—
94C2×D316—22no—
95C2×D316—41no—
96C2×D316—42no—
97C22×Dic79D811yescharacteristic
98C22×Dic79D811yescharacteristic
99C2×C4×D79—42no—
100C2×C4×D79—42no—
101C2×C4×D79—84no—
102D632—24no—
103D632—22no—
104D632C2×Q814yes—
105C2×C79⋊C8—22no—
106C2×C79⋊C8C2×D411yescharacteristic
107C2×C79⋊C8C2×D411yescharacteristic
108C4⋊Dic79—21no—
109C4⋊Dic79—21no—
110C4⋊Dic79—41no—
111C4⋊Dic79—41no—
112Dic79⋊C4—41no—
113Dic79⋊C4—41no—
114Dic79⋊C4—41no—
115Dic79⋊C4—41no—
116C22⋊Dic79—41no—
117(C2×D79)⋊C4—42no—
118(C2×D79)⋊C4—42no—
119C4×Dic79—41no—
120C4×Dic79D811yescharacteristic
121C4×Dic79D811yescharacteristic
122C4×Dic79C2×D411yescharacteristic
123C4×Dic79C2×D411yescharacteristic
124C79⋊C16—22no—
125C22×C4×C79—41no—
126C22×C4×C79C2×D411yescharacteristic
127C2×C8×C79C2×D411yescharacteristic
128C2×C8×C79C2×Q812yes—
129C2×C8×C79#4112yes—
130C2×C8×C79C2411yescharacteristic
131C4⋊C4×C79—21no—
132C4⋊C4×C79—21no—
133C4⋊C4×C79—22no—
134C4⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79—41no—
137C16×C79—22no—
138C23×C79C2×D811yescharacteristic
139C22×D79—42no—
140C22×D79—44no—
141C22×D79—82no—
142C22×D79—82no—
143C2×C4×C79—42no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—82no—
147C2×C4×C79C22⋊Q812yes—
148C2×C4×C79C2×D811yescharacteristic
149C2×C4×C79#13712yes—
150C2×C4×C79C22×D411yescharacteristic
151C2×Dic79—22no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79C2×D811yescharacteristic
156C2×Dic79C2×D811yescharacteristic
157C2×Dic79C2×D811yescharacteristic
158C2×Dic79C2×D811yescharacteristic
159D316—24no—
160D316—42no—
161D316—44no—
162C4×D79—84no—
163C4×D79—84no—
164Dic158—82no—
165Dic158—82no—
166C8×C79C22⋊Q812yes—
167C8×C79C22×Q811yescharacteristic
168C8×C79C2×#4111yescharacteristic
169C79⋊C8—21no—
170C79⋊C8—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79#47412yes—
174C22×C79C22×D811yescharacteristic
175C2×D79—44no—
176C2×D79—88no—
177C2×D79—82no—
178C2×D79—82no—
179C4×C79—82no—
180C4×C79—81no—
181C4×C79—81no—
182C4×C79#47412yes—
183C4×C79C2×C22⋊Q811yescharacteristic
184C4×C79C2×#13711yescharacteristic
185Dic79—21no—
186Dic79—21no—
187Dic79—21no—
188Dic79—81no—
189Dic79—81no—
190Dic79#50912yes—
191#33627—791nomaximal, Sylow
192C2×C79—22no—
193C2×C79C2×#47411yescharacteristic
194C2×C79C2×#50911yescharacteristic
195C2×C79#341411yescharacteristic
196D79—84no—
197#3160—791no—
198C2×C4×D8—791no—
199#2228—792no—
200#2228—792no—
201#2226—792no—
202#2224—792no—
203#2159—791no—
204#2159D7911yescharacteristic, complemented
205#2150—791no—
206#2140—791no—
207#2140—791no—
208C79#3362711yescharacteristic, complemented, Sylow
209C22×D8—791no—
210#533—1581no—
211C2×C4×D4—1581no—
212C2×M32—791no—
213C2×M32C2×D7911yescharacteristic
214#500—791no—
215#500—791no—
216#499—791no—
217#499—791no—
218#499—1581no—
219#498—1581no—
220C4×D8—792no—
221C4×D8—792no—
222C4×D8—1581no—
223C4×D8—1581no—
224#427—791no—
225#427—1581no—
226#427C2×D7911yescharacteristic
227C2×#107C2×D7911yescharacteristic
228C2×#102—1581no—
229C2×C4×C8—791no—
230#367—792no—
231#367C2×D7912yescomplemented
232#366—792no—
233#366C2×D7912yescomplemented
234C4⋊C16—792no—
235C4⋊C16—792no—
236#357—792no—
237#357—792no—
238#357—792no—
239#357—792no—
240C22×D4—1581no—
241C23×C4—3161no—
242C2×D8—794no—
243C2×D8—792no—
244C2×D8—1581no—
245C2×D8—1582no—
246C22×C8—791no—
247C22×C8C22×D7911yescharacteristic
248C4⋊Q8—1581no—
249C4⋊Q8—1581no—
250#125—1581no—
251#125—1581no—
252C22⋊Q8—3161no—
253C22⋊Q8—3161no—
254C4×Q8—3161no—
255C4×Q8—3161no—
256C4×D4—3161no—
257C4×D4—3161no—
258C4×D4—1582no—
259C4×D4—1582no—
260#117—21no—
261#117—1581no—
262C2×C4⋊C4—21no—
263C2×C22⋊C4—3161no—
264C2×C42—791no—
265M32—22no—
266M32—1582no—
267C2×C16—792no—
268C2×C16C22×D7912yes—
269#107—21no—
270#107—791no—
271#107—791no—
272#107—1581no—
273#107—1581no—
274#107C22×D7911yescharacteristic
275#107C22×D7911yescharacteristic
276#107C22×D7912yes—
277#106—21no—
278#106—1581no—
279#106—1581no—
280#106—1581no—
281#102—1582no—
282#102—1582no—
283C4×C8—791no—
284C4×C8—791no—
285C4×C8—791no—
286C4×C8—791no—
287C24—3161no—
288C2×Q8—3161no—
289C2×Q8—3161no—
290C2×D4—3161no—
291C2×D4—3162no—
292C2×D4—1584no—
293C2×D4—1582no—
294C22×C4—41no—
295C22×C4—3162no—
296C22×C4—3162no—
297C22×C4—791no—
298C22×C4—791no—
299C22×C4—6324no—
300C22×C4D4×D7911yescharacteristic
301D8—794no—
302D8—1582no—
303D8—1584no—
304C2×C8—791no—
305C2×C8—791no—
306C2×C8—1582no—
307C2×C8D4×D7911yescharacteristic
308C2×C8Q8×D7912yes—
309C2×C8#1192962412yes—
310C2×C8C23×D7911yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—3161no—
319C4⋊C4—3161no—
320C4⋊C4—3161no—
321C4⋊C4—1581no—
322C4⋊C4—1581no—
323C22⋊C4—41no—
324C22⋊C4—3162no—
325C22⋊C4—3162no—
326C22⋊C4—3161no—
327C42—41no—
328C42—3161no—
329C42—791no—
330C42—791no—
331C42—791no—
332C42—791no—
333C16—22no—
334C16—1582no—
335C23—3162no—
336C23—3164no—
337C23—6322no—
338C23—6322no—
339C23D8×D7911yescharacteristic
340Q8—6322no—
341Q8—6322no—
342D4—3164no—
343D4—3162no—
344D4—1584no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—791no—
353C2×C4—791no—
354C2×C4—791no—
355C2×C4—791no—
356C2×C4—6324no—
357C2×C4—6324no—
358C2×C4—1582no—
359C2×C4#1457913912yes—
360C2×C4D8×D7911yescharacteristic
361C2×C4#1457919312yes—
362C2×C4C2×D4×D7911yescharacteristic
363C8—1581no—
364C8—1581no—
365C8#1457913912yes—
366C8C2×Q8×D7911yescharacteristic
367C8C2×#1192962411yescharacteristic
368C22—21no—
369C22—21no—
370C22—3164no—
371C22—6328no—
372C22—6322no—
373C22—6322no—
374C22#1909884212yes—
375C22C2×D8×D7911yescharacteristic
376C4—82no—
377C4—81no—
378C4—81no—
379C4—792no—
380C4—6321no—
381C4—6321no—
382C4—1581no—
383C4—1581no—
384C4—1581no—
385C4#1909884212yes—
386C4C2×#1457913911yescharacteristic
387C4C2×#1457919311yescharacteristic
388C2—22no—
389C2—6324no—
390C2C2×#1909884211yescharacteristic
391C2#509×D7911yescharacteristic
392C2#2377653511yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2159C2
socleC22×C79C22×D8
prime radicalC22×D632C22
hypercenter#2159D79
hypocenterC79#33627
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. #23773912C2semidirect
3. #23765560C2semidirect
4. #23765556C2semidirect
5. #23765546C2semidirect
6. #23765536C2semidirect
7. #23765186C2semidirect
8. #23765155C2semidirect
11. C79×#2159C2semidirect
16. #19098967C22semidirect
17. #19098967C22semidirect
18. #19098964C22semidirect
19. #19098964C22semidirect
30. #19098390C22semidirect
31. #19098388C22semidirect
32. Dic79⋊C16C22semidirect
33. Dic79⋊C16C22semidirect
41. #367×C79C22semidirect
42. #366×C79C22semidirect
47. C2×D632Q8semidirect
204. #2159D79semidirect
208. C79#33627semidirect
231. #367C2×D79semidirect
233. #366C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237739122
C2×#190986452
#237655602
#237655562
#237655462
#237655362
#237651862
#237651552
#237651172
#237651152
C79×#21592
#3362779