Finite Group Databasebeta

← #34441305

Subgroups — #34441305

Order 20224.

674 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
1C2×M32×D79C211yesmaximal, characteristic
2C2×#19098600C211yesmaximal, characteristic
3#427×D79C211yesmaximal, characteristic
4#23765189C211yesmaximal, characteristic, complemented
5#23765188C211yesmaximal, characteristic, complemented
6#23765187C212yesmaximal, complemented
7#23765186C211yesmaximal, characteristic, complemented
8#23765185C212yesmaximal, complemented
9#23765183C212yesmaximal, complemented
10#23765181C212yesmaximal, complemented
11C79×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D79C2211yescharacteristic
13C2×#14579119—21no—
14#117×D79—21no—
15C2×#14579101C2211yescharacteristic
16C2×#14579100C2212yes—
17M32×D79C418yescomplemented
18C2×C16×D79C2212yes—
19#19098601C2211yescharacteristic
20#19098600—21no—
21#19098600—21no—
22#19098600C2211yescharacteristic
23#19098600C2211yescharacteristic
24#19098600C2212yes—
25#19098600C2212yes—
26#19098599—21no—
27#19098598—21no—
28C2×#14579095C2211yescharacteristic
29#107×D79C2211yescharacteristic
30#107×D79C2211yescharacteristic
31C2×#14579092—21no—
32#106×D79—21no—
33#19098391C2212yescomplemented
34#19098390C2212yescomplemented
35#19098390C2212yescomplemented
36#19098389C2212yescomplemented
37#19098388C2212yescomplemented
38#19098388C2212yescomplemented
39C2×M32×C79C2211yescharacteristic
40#427×C79C2211yescharacteristic
41C2×#107×C79C2211yescharacteristic
42#367×C79C2212yescomplemented
43#366×C79C2212yescomplemented
44C22×C4×D79D411yescharacteristic
45C22×C79⋊C8D411yescharacteristic
46C2×C8×D79C2×C414yes—
47C2×C8×D79D411yescharacteristic
48C2×C8×D79D411yescharacteristic
49C2×C8×D79Q814yes—
50C2×C8×D79C2311yescharacteristic
51C2×C8×D79C2311yescharacteristic
52#14579120—21no—
53#14579119—21no—
54#14579119—21no—
55#14579119—22no—
56#14579119—22no—
57#14579119—41no—
58#14579119—41no—
59#14579118—41no—
60#14579117—41no—
61C2×C4⋊Dic79—21no—
62C4⋊C4×D79—21no—
63C4⋊C4×D79—21no—
64C2×(C2×D79)⋊C4—41no—
65C22⋊C4×D79—41no—
66C2×C4×Dic79—41no—
67C42×D79—41no—
68#14579101C2×C414yescomplemented
69#14579100C2×C418yescomplemented
70C2×C79⋊C16C2312yes—
71C16×D79C2×C418yescomplemented
72#14579095—21no—
73#14579095C2311yescharacteristic
74#14579095C2312yes—
75#14579095C2311yescharacteristic
76#14579094—21no—
77#14579094—21no—
78#14579093—21no—
79#14579092—21no—
80#14579092—21no—
81#14579092—22no—
82#14579092—22no—
83C22×C8×C79C2311yescharacteristic
84#117×C79—21no—
85C2×C4⋊C4×C79—21no—
86M32×C79C2×C414yescomplemented
87C2×C16×C79C2312yes—
88#107×C79—21no—
89#107×C79C2312yes—
90#107×C79C2311yescharacteristic
91#107×C79C2311yescharacteristic
92#106×C79—21no—
93C23×D79D811yescharacteristic
94C22×Dic79—41no—
95C22×Dic79D811yescharacteristic
96C2×C4×D79—41no—
97C2×C4×D79—41no—
98C2×C4×D79C4⋊C414yes—
99C2×C4×D79D811yescharacteristic
100C2×C4×D79D811yescharacteristic
101C2×C4×D79Q1614yes—
102C2×C4×D79C2×D411yescharacteristic
103C2×C4×D79C2×D411yescharacteristic
104C2×C79⋊C8C4⋊C414yes—
105C2×C79⋊C8C2×D411yescharacteristic
106C2×C79⋊C8C2×D411yescharacteristic
107C8×D79C4⋊C414yes—
108C8×D79C4⋊C414yes—
109C8×D79C22×C412yes—
110C8×D79C22×C412yes—
111C8×D79C2×Q812yes—
112C8×D79C2×Q812yes—
113C4⋊Dic79—22no—
114C4⋊Dic79—21no—
115C4⋊Dic79—21no—
116C4⋊Dic79—41no—
117Dic79⋊C4—41no—
118Dic79⋊C4—41no—
119C22⋊Dic79—41no—
120(C2×D79)⋊C4—41no—
121(C2×D79)⋊C4—41no—
122(C2×D79)⋊C4—42no—
123(C2×D79)⋊C4—42no—
124C4×Dic79—41no—
125C4×Dic79—41no—
126C4×Dic79—42no—
127C4×Dic79—42no—
128C4×Dic79—41no—
129C79⋊C16C22×C414yes—
130C22×C4×C79—41no—
131C22×C4×C79C2×D411yescharacteristic
132C2×C8×C79C22×C412yes—
133C2×C8×C79C2×D411yescharacteristic
134C2×C8×C79C2×Q812yes—
135C2×C8×C79C2411yescharacteristic
136C4⋊C4×C79—21no—
137C4⋊C4×C79—21no—
138C4⋊C4×C79—22no—
139C4⋊C4×C79—41no—
140C22⋊C4×C79—41no—
141C42×C79—41no—
142C16×C79C22×C414yes—
143C23×C79C2×D811yescharacteristic
144C22×D79—41no—
145C22×D79—41no—
146C22×D79#10714yes—
147C22×D79C2×D811yescharacteristic
148C22×D79C2×D811yescharacteristic
149C2×C4×C79—42no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—82no—
153C2×C4×C79C2×C4⋊C412yes—
154C2×C4×C79C2×D811yescharacteristic
155C2×C4×C79C2×Q1612yes—
156C2×C4×C79C22×D411yescharacteristic
157C2×Dic79—42no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—82no—
161C2×Dic79#10714yes—
162C2×Dic79C2×D811yescharacteristic
163C2×Dic79C2×D811yescharacteristic
164C4×D79—82no—
165C4×D79—82no—
166C4×D79#10714yes—
167C4×D79#10714yes—
168C4×D79C2×C4⋊C412yes—
169C4×D79C2×C4⋊C412yes—
170C4×D79C2×Q1612yes—
171C4×D79C2×Q1612yes—
172C8×C79C2×C4⋊C412yes—
173C8×C79C23×C411yescharacteristic
174C8×C79C22×Q811yescharacteristic
175C79⋊C8C2×C4⋊C412yes—
176C79⋊C8C2×C4⋊C412yes—
177C22×C79—21no—
178C22×C79—21no—
179C22×C79C2×#10712yes—
180C22×C79C22×D811yescharacteristic
181C2×D79—21no—
182C2×D79—21no—
183C2×D79—21no—
184C2×D79—21no—
185C2×D79—44no—
186C2×D79C2×#10712yes—
187C2×D79C2×#10712yes—
188C4×C79—82no—
189C4×C79—81no—
190C4×C79—81no—
191C4×C79C2×#10712yes—
192C4×C79C22×C4⋊C411yescharacteristic
193C4×C79C22×Q1611yescharacteristic
194Dic79—81no—
195Dic79—81no—
196Dic79—82no—
197Dic79C2×#10712yes—
198Dic79C2×#10712yes—
199#33409—791nomaximal, Sylow
200C2×C79—22no—
201C2×C79C22×#10711yescharacteristic
202C2×C79#341411yescharacteristic
203C2×C79#341411yescharacteristic
204D79—22no—
205D79—22no—
206C22×M32—791no—
207C2×#427—791no—
208C2×#427—791no—
209#2159—791no—
210#2159—791no—
211#2159—791no—
212#2159—792no—
213#2159—792no—
214#2159D7911yescharacteristic, complemented
215#2158—792no—
216#2158—792no—
217C79#3340911yescharacteristic, complemented, Sylow
218C23×C8—791no—
219C2×#117—1581no—
220C2×#117—1581no—
221C2×M32—791no—
222C2×M32—798no—
223C2×M32—792no—
224C2×M32C2×D7911yescharacteristic
225C22×C16—792no—
226#427—791no—
227#427—791no—
228#427—792no—
229#427—791no—
230#427—792no—
231#427—1581no—
232#427—1581no—
233#427—1581no—
234#427—1581no—
235#427C2×D7911yescharacteristic
236C2×#107—791no—
237C2×#107—791no—
238C2×#107—791no—
239C2×#107C2×D7911yescharacteristic
240C2×#106—1581no—
241C2×#106—1581no—
242#367—792no—
243#367—792no—
244#367—792no—
245#367C2×D7912yescomplemented
246#366—792no—
247#366—792no—
248#366—792no—
249#366C2×D7912yescomplemented
250C23×C4—791no—
251C22×C8—791no—
252C22×C8—794no—
253C22×C8—791no—
254C22×C8—791no—
255C22×C8—794no—
256C22×C8—791no—
257C22×C8—791no—
258C22×C8C22×D7911yescharacteristic
259#117—21no—
260#117—3161no—
261#117—3161no—
262#117—3161no—
263#117—3161no—
264#117—1581no—
265#117—1581no—
266#117—1582no—
267#117—1582no—
268#117—1581no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—1581no—
271C2×C4⋊C4—1581no—
272C2×C4⋊C4—1581no—
273C2×C22⋊C4—3161no—
274C2×C22⋊C4—3161no—
275C2×C42—3161no—
276C2×C42—3161no—
277M32—794no—
278M32—798no—
279M32C4×D7914yescomplemented
280C2×C16—792no—
281C2×C16—798no—
282C2×C16C22×D7912yes—
283#107—21no—
284#107—791no—
285#107—792no—
286#107—791no—
287#107—1581no—
288#107—1581no—
289#107—1581no—
290#107C22×D7911yescharacteristic
291#107C22×D7912yes—
292#107C22×D7911yescharacteristic
293#106—21no—
294#106—1581no—
295#106—1581no—
296#106—1582no—
297#106—1582no—
298#106—1581no—
299C24—791no—
300C22×C4—41no—
301C22×C4—3161no—
302C22×C4—3161no—
303C22×C4—3161no—
304C22×C4—791no—
305C22×C4—794no—
306C22×C4—791no—
307C22×C4—791no—
308C22×C4—794no—
309C22×C4—791no—
310C22×C4—791no—
311C22×C4D4×D7911yescharacteristic
312C2×C8—794no—
313C2×C8—791no—
314C2×C8—791no—
315C2×C8—792no—
316C2×C8—792no—
317C2×C8—794no—
318C2×C8—794no—
319C2×C8—792no—
320C2×C8—792no—
321C2×C8C2×C4×D7912yes—
322C2×C8D4×D7911yescharacteristic
323C2×C8Q8×D7912yes—
324C2×C8C23×D7911yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—3161no—
330C4⋊C4—3161no—
331C4⋊C4—3161no—
332C4⋊C4—1582no—
333C4⋊C4—1581no—
334C4⋊C4—1581no—
335C22⋊C4—41no—
336C22⋊C4—3161no—
337C22⋊C4—3161no—
338C22⋊C4—3162no—
339C22⋊C4—3162no—
340C22⋊C4—3161no—
341C42—41no—
342C42—3161no—
343C42—3161no—
344C42—3162no—
345C42—3162no—
346C42—3161no—
347C16—794no—
348C16C2×C4×D7914yes—
349C23—3161no—
350C23—3161no—
351C23—794no—
352C23—791no—
353C23—791no—
354C23D8×D7911yescharacteristic
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—3161no—
360C2×C4—3162no—
361C2×C4—3161no—
362C2×C4—791no—
363C2×C4—791no—
364C2×C4—794no—
365C2×C4—792no—
366C2×C4—792no—
367C2×C4—792no—
368C2×C4—792no—
369C2×C4—794no—
370C2×C4—794no—
371C2×C4—6322no—
372C2×C4—6322no—
373C2×C4—6322no—
374C2×C4C4⋊C4×D7912yes—
375C2×C4D8×D7911yescharacteristic
376C2×C4Q16×D7912yes—
377C2×C4C2×D4×D7911yescharacteristic
378C8—792no—
379C8—792no—
380C8C4⋊C4×D7912yes—
381C8C22×C4×D7911yescharacteristic
382C8C2×Q8×D7911yescharacteristic
383C22—21no—
384C22—21no—
385C22—3164no—
386C22—792no—
387C22—792no—
388C22—1581no—
389C22—1581no—
390C22—1581no—
391C22—1581no—
392C22#107×D7912yes—
393C22C2×D8×D7911yescharacteristic
394C4—82no—
395C4—81no—
396C4—81no—
397C4—792no—
398C4—792no—
399C4—6321no—
400C4—6321no—
401C4—6322no—
402C4#107×D7912yes—
403C4C2×C4⋊C4×D7911yescharacteristic
404C4C2×Q16×D7911yescharacteristic
405C2—22no—
406C2—1582no—
407C2—1582no—
408C2C2×#107×D7911yescharacteristic
409C2#2377653211yescharacteristic
410C2#2377653511yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2159C2
socleC22×C79C22×D8
prime radicalC23×D79D8
cosocleC2×C8×C79C24
hypercenter#2159D79
hypocenterC79#33409
LCSG ▷ 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
4. #23765189C2semidirect
5. #23765188C2semidirect
6. #23765187C2semidirect
7. #23765186C2semidirect
8. #23765185C2semidirect
9. #23765183C2semidirect
10. #23765181C2semidirect
11. C79×#2159C2semidirect
17. M32×D79C4semidirect
33. #19098391C22semidirect
34. #19098390C22semidirect
35. #19098390C22semidirect
36. #19098389C22semidirect
37. #19098388C22semidirect
38. #19098388C22semidirect
42. #367×C79C22semidirect
43. #366×C79C22semidirect
68. #14579101C2×C4semidirect
69. #14579100C2×C4semidirect
71. C16×D79C2×C4semidirect
86. M32×C79C2×C4semidirect
214. #2159D79semidirect
217. C79#33409semidirect
245. #367C2×D79semidirect
249. #366C2×D79semidirect
279. M32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D792
C2×#190986002
#427×D792
#237651892
#237651882
#237651872
#237651862
#237651852
#237651832
#237651812
C79×#21592
#3340979