Finite Group Databasebeta

← #34184926

Subgroups — #34184926

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774322C211yesmaximal, characteristic
2#23764373C211yesmaximal, characteristic, complemented
3#23762086C211yesmaximal, characteristic, complemented
4#23761960C211yesmaximal, characteristic
5#23761260C211yesmaximal, characteristic, complemented
6#23761231C211yesmaximal, characteristic, complemented
7C79×#2932C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9#19098935C412yescomplemented
10#19098915C412yescomplemented
11#19098768—22no—
12#19098701—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098429C2211yescharacteristic
16#19098326—21no—
17#19098326—21no—
18#19098308—22no—
19#19098297—22no—
20C2×#131×C79—21no—
21#532×C79C2211yescharacteristic
22#435×C79—22no—
23#435×C79—22no—
24#423×C79C2211yescharacteristic
25#408×C79—21no—
26C2×#97×C79C2211yescharacteristic
27C22×Dic158—21no—
28C2×#11929612—24no—
29#14579161—21no—
30#14579161—21no—
31#14579161—21no—
32#14579161C2×C412yes—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38#14579084—24no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8C2×C412yes—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C2×#41×C79—21no—
46#131×C79—42no—
47#131×C79—42no—
48C2×M16×C79—22no—
49C2×M16×C79—21no—
50C22×C8×C79—22no—
51C22×C8×C79D411yescharacteristic
52C22×C8×C79D411yescharacteristic
53C4⋊Q8×C79C2×C411yescharacteristic
54#127×C79C2×C411yescharacteristic
55#124×C79—21no—
56#121×C79—21no—
57#121×C79D412yes—
58C4×D4×C79—21no—
59C4×D4×C79D412yes—
60C2×C42×C79C2311yescharacteristic
61C4⋊C8×C79—21no—
62C4⋊C8×C79D412yes—
63C22⋊C8×C79—24no—
64#97×C79—21no—
65#97×C79—21no—
66C2×Dic158—22no—
67C2×Dic158—41no—
68C2×Dic158—82no—
69C22×Dic79—41no—
70C22×Dic79—41no—
71#11929612—84no—
72C2×C79⋊C8—22no—
73C2×C79⋊C8—41no—
74C4⋊Dic79—22no—
75C4⋊Dic79—22no—
76C4⋊Dic79—41no—
77C4⋊Dic79—41no—
78C4⋊Dic79—41no—
79C4⋊Dic79—41no—
80#41×C79—41no—
81#41×C79—41no—
82#41×C79—41no—
83#41×C79—41no—
84C2×Q8×C79—21no—
85C2×D4×C79—22no—
86C2×D4×C79—21no—
87C2×D4×C79—21no—
88C2×D4×C79C22⋊C412yes—
89C22×C4×C79—22no—
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79#4111yescharacteristic
93M16×C79—44no—
94M16×C79—41no—
95M16×C79—41no—
96C2×C8×C79—21no—
97C2×C8×C79—21no—
98C2×C8×C79—22no—
99C2×C8×C79—21no—
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—44no—
103C2×C8×C79SD1612yescomplemented
104C4⋊C4×C79—21no—
105C4⋊C4×C79C22⋊C412yes—
106C22⋊C4×C79—22no—
107C22⋊C4×C79—22no—
108C42×C79C22×C411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79#4111yescharacteristic
112C23×C79—22no—
113C23×C79#12111yescharacteristic
114Q8×C79—41no—
115Q8×C79—41no—
116D4×C79—42no—
117D4×C79—42no—
118D4×C79—42no—
119D4×C79—41no—
120D4×C79—41no—
121D4×C79—81no—
122C2×C4×C79—22no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79C2×C22⋊C411yescharacteristic
128C2×C4×C79C4×D411yescharacteristic
129C2×C4×C79C4×D411yescharacteristic
130C2×C4×C79#12011yescharacteristic
131C2×C4×C79#12111yescharacteristic
132C2×C4×C79#12311yescharacteristic
133C2×C4×C79C2×SD1611yescharacteristic
134C2×C4×C79#13511yescharacteristic
135C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C79#13711yescharacteristic
137C2×Dic79—42no—
138C2×Dic79—82no—
139C2×Dic79—81no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142Dic158—44no—
143Dic158—82no—
144C8×C79—22no—
145C8×C79—42no—
146C8×C79—41no—
147C79⋊C8—42no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—42no—
152C22×C79—42no—
153C22×C79—42no—
154C22×C79#38611yescharacteristic
155C22×C79#46111yescharacteristic
156C22×C79#46311yescharacteristic
157C4×C79—21no—
158C4×C79—21no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—42no—
162C4×C79C4×SD1611yescharacteristic
163C4×C79#44011yescharacteristic
164C4×C79#45011yescharacteristic
165C4×C79#45211yescharacteristic
166Dic79—82no—
167Dic79—161no—
168#9289—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—82no—
172C2×C79#187711yescharacteristic
173C2×C79#190211yescharacteristic
174C2×C79#191011yescharacteristic
175#3206—791no—
176#2932D7911yescharacteristic
177#1993—791no—
178#1549—791no—
179#1517—791no—
180#1286—791no—
181#1275—791no—
182C79#928911yescharacteristic, complemented, Sylow
183C2×#131—21no—
184#532C2×D7911yescharacteristic
185C2×C4⋊Q8—791no—
186#492—792no—
187#487—792no—
188#461—1582no—
189#448—1582no—
190#435—22no—
191#435—22no—
192#431—791no—
193#431—791no—
194#423C2×D7911yescharacteristic
195#408—21no—
196C2×#97C2×D7911yescharacteristic
197#382—791no—
198#340—1581no—
199#340—1581no—
200#332—1582no—
201#327—1582no—
202C2×#41—21no—
203C22×Q8—1581no—
204C2×SD16—1584no—
205#131—42no—
206#131—42no—
207C2×M16—22no—
208C2×M16—21no—
209C22×C8—22no—
210C22×C8C22⋊D7911yescharacteristic
211C22×C8C22⋊D7911yescharacteristic
212C4⋊Q8—792no—
213C4⋊Q8—1581no—
214C4⋊Q8—1581no—
215C4⋊Q8—1581no—
216C4⋊Q8C4×D7911yescharacteristic
217#127C4×D7911yescharacteristic
218#124—21no—
219#121—21no—
220#121D31612yes—
221C4×D4—21no—
222C4×D4C22⋊D7912yes—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4—1581no—
225C2×C4⋊C4—1581no—
226C2×C42C22×D7911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—1581no—
229C4⋊C8—1581no—
230C4⋊C8C22⋊D7912yes—
231#103—1584no—
232C22⋊C8—24no—
233C22⋊C8—1581no—
234C22⋊C8—1581no—
235#97—21no—
236#97—21no—
237C4×C8—792no—
238C4×C8—1581no—
239#95—1581no—
240#95—1581no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—3161no—
247C2×Q8—6322no—
248C2×Q8—1582no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4(C2×D79)⋊C412yes—
253C22×C4—22no—
254C22×C4—3161no—
255C22×C4—3161no—
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4C2×D31611yescharacteristic
258C22×C4#1192962411yescharacteristic
259SD16—6324no—
260M16—44no—
261M16—41no—
262M16—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—44no—
270C2×C8—3161no—
271C2×C8—1582no—
272C2×C8#1192961212yescomplemented
273C4⋊C4—21no—
274C4⋊C4—3161no—
275C4⋊C4—3161no—
276C4⋊C4—3161no—
277C4⋊C4—3161no—
278C4⋊C4—1582no—
279C4⋊C4—1582no—
280C4⋊C4(C2×D79)⋊C412yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C42C2×C4×D7911yescharacteristic
284C42C2×C22⋊D7911yescharacteristic
285C42C2×C22⋊D7911yescharacteristic
286C42#1192962411yescharacteristic
287C23—22no—
288C23#1457913511yescharacteristic
289Q8—41no—
290Q8—41no—
291Q8—3164no—
292Q8—6322no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298D4—81no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—3162no—
305C2×C4—6322no—
306C2×C4—6321no—
307C2×C4—6321no—
308C2×C4—6321no—
309C2×C4C2×(C2×D79)⋊C411yescharacteristic
310C2×C4C4×C22⋊D7911yescharacteristic
311C2×C4C4×C22⋊D7911yescharacteristic
312C2×C4#1457913011yescharacteristic
313C2×C4#1457913511yescharacteristic
314C2×C4#1457914411yescharacteristic
315C2×C4C2×#1192961211yescharacteristic
316C2×C4#1457918111yescharacteristic
317C2×C4#1457918711yescharacteristic
318C2×C4#1457919111yescharacteristic
319C8—22no—
320C8—42no—
321C8—41no—
322C8—3162no—
323C8—6321no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1909844211yescharacteristic
330C22#1909876811yescharacteristic
331C22#1909877811yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—6322no—
338C4—12641no—
339C4C4×#1192961211yescharacteristic
340C4#1909866311yescharacteristic
341C4#1909871411yescharacteristic
342C4#1909872511yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2#2376373411yescharacteristic
347C2#2376387311yescharacteristic
348C2#2376392711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2932C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#2932D79
hypocenterC79#9289
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #23764373C2semidirect
3. #23762086C2semidirect
5. #23761260C2semidirect
6. #23761231C2semidirect
9. #19098935C4semidirect
10. #19098915C4semidirect
103. C2×C8×C79SD16semidirect
182. C79#9289semidirect
272. C2×C8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237743222
#237643732
#237620862
#237619602
#237612602
#237612312
C79×#29322
#928979