Finite Group Databasebeta

← #34188068

Subgroups — #34188068

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098909C211yesmaximal, characteristic
2#23764361C211yesmaximal, characteristic, complemented
3#23764207C211yesmaximal, characteristic
4#23764201C211yesmaximal, characteristic, complemented
5#23763559C211yesmaximal, characteristic, complemented
6C2×#19098285C211yesmaximal, characteristic, complemented
7C79×#1957C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9#19098909—22no—
10#19098909C414yescomplemented
11C2×#14579094—22no—
12C2×#14579084—22no—
13C2×#14579082—22no—
14C2×#14579082—22no—
15C2×C4×C79⋊C8C2211yescharacteristic
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098433C2211yescharacteristic
18#19098326—22no—
19#19098285—22no—
20C2×#127×C79C2211yescharacteristic
21C2×#102×C79—22no—
22C2×C4×C8×C79C2211yescharacteristic
23#384×C79C2211yescharacteristic
24C22×Dic158—21no—
25C22×C79⋊C8D412yes—
26C22×C79⋊C8D412yes—
27#14579161—21no—
28#14579161—21no—
29#14579161—21no—
30#14579161C2×C412yes—
31C2×C4⋊Dic79—21no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79—21no—
36#14579094—44no—
37#14579084—44no—
38#14579082—28no—
39#14579082—44no—
40C4×C79⋊C8—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8C2×C412yes—
43C4×C79⋊C8D412yes—
44C22×D4×C79—21no—
45C22×D4×C79—21no—
46C22×D4×C79—21no—
47C22×C8×C79D412yes—
48#127×C79—21no—
49#127×C79—21no—
50#127×C79—21no—
51#127×C79C2×C412yes—
52C2×C4⋊C4×C79—21no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79—21no—
55C2×C42×C79C2311yescharacteristic
56#102×C79—44no—
57C4×C8×C79—21no—
58C4×C8×C79—21no—
59C2×Dic158—22no—
60C2×Dic158—41no—
61C2×Dic158—42no—
62C2×Dic158—42no—
63C22×Dic79—41no—
64C22×Dic79—41no—
65C2×C79⋊C8—22no—
66C2×C79⋊C8—22no—
67C2×C79⋊C8—22no—
68C2×C79⋊C8—22no—
69C2×C79⋊C8C22⋊C414yes—
70C2×C79⋊C8D814yes—
71C4⋊Dic79—22no—
72C4⋊Dic79—22no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—42no—
76C4⋊Dic79—42no—
77C4⋊Dic79—42no—
78C24×C79—41no—
79C2×D4×C79—22no—
80C2×D4×C79—22no—
81C2×D4×C79—22no—
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79—42no—
86C2×D4×C79—42no—
87C2×D4×C79—42no—
88C2×D4×C79—42no—
89C2×D4×C79—42no—
90C2×D4×C79—42no—
91C22×C4×C79—41no—
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C2×C8×C79—22no—
96C2×C8×C79—22no—
97C2×C8×C79—22no—
98C4⋊C4×C79—42no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—42no—
101C42×C79C22×C411yescharacteristic
102C42×C79C2×D411yescharacteristic
103C42×C79#4111yescharacteristic
104C42×C79#4111yescharacteristic
105C23×C79—42no—
106C23×C79—81no—
107C23×C79—81no—
108C23×C79—81no—
109C23×C79—81no—
110C23×C79—81no—
111C23×C79—81no—
112C23×C79#12711yescharacteristic
113D4×C79—44no—
114D4×C79—44no—
115D4×C79—44no—
116D4×C79—82no—
117D4×C79—82no—
118D4×C79—82no—
119C2×C4×C79—81no—
120C2×C4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79C2×C22⋊C411yescharacteristic
123C2×C4×C79C4×D411yescharacteristic
124C2×C4×C79C4×D411yescharacteristic
125C2×C4×C79#12111yescharacteristic
126C2×C4×C79#12111yescharacteristic
127C2×C4×C79#12411yescharacteristic
128C2×C4×C79C2×D812yes—
129C2×C4×C79#13512yes—
130C2×C4×C79AGL(1,ℤ/8)12yes—
131C2×C4×C79AGL(1,ℤ/8)12yes—
132C2×C4×C79#13712yes—
133C2×C4×C79#13712yes—
134C2×Dic79—42no—
135C2×Dic79—81no—
136C2×Dic79—81no—
137C2×Dic79—81no—
138C2×Dic79—81no—
139C2×Dic79—81no—
140Dic158—44no—
141Dic158—82no—
142C8×C79—42no—
143C79⋊C8—24no—
144C79⋊C8—24no—
145C22×C79—82no—
146C22×C79—82no—
147C22×C79—82no—
148C22×C79—161no—
149C22×C79—161no—
150C22×C79—161no—
151C22×C79—161no—
152C22×C79#39011yescharacteristic
153C22×C79#49311yescharacteristic
154C22×C79#49511yescharacteristic
155C22×C79#49611yescharacteristic
156C22×C79#49611yescharacteristic
157C22×C79#49711yescharacteristic
158C22×C79#49711yescharacteristic
159C4×C79—161no—
160C4×C79#41712yes—
161C4×C79C4×D812yes—
162C4×C79#44012yes—
163C4×C79#46612yes—
164C4×C79#46812yes—
165C4×C79#48912yes—
166Dic79—82no—
167Dic79—161no—
168#9830—791nomaximal, Sylow
169C2×C79—162no—
170C2×C79#171911yescharacteristic
171C2×C79#172011yescharacteristic
172C2×C79#172111yescharacteristic
173C2×C79#172211yescharacteristic
174C2×C79#195911yescharacteristic
175C2×C79#197011yescharacteristic
176C2×C79#197311yescharacteristic
177C2×#486—791no—
178#1990—791no—
179#1957—791no—
180#1957D7911yescharacteristic
181#1956—791no—
182#1845—791no—
183C2×#322—791no—
184C79#983011yescharacteristic, complemented, Sylow
185C2×C4⋊Q8—791no—
186C2×#127C2×D7911yescharacteristic
187#486—794no—
188#486—1582no—
189C2×#107—1582no—
190C2×#103—1582no—
191C2×#102—22no—
192C2×#102—1582no—
193C2×#102—1582no—
194C2×C4×C8—791no—
195C2×C4×C8—791no—
196C2×C4×C8C2×D7911yescharacteristic
197#384—791no—
198#384C2×D7911yescharacteristic
199#340—1582no—
200#322—1582no—
201C22×Q8—1581no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×C8—792no—
206C22×C8—792no—
207C22×C8C22⋊D7912yes—
208C4⋊Q8—792no—
209C4⋊Q8—1581no—
210C4⋊Q8—1581no—
211C4⋊Q8—1581no—
212#127—21no—
213#127—21no—
214#127—21no—
215#127C4×D7912yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1581no—
220C2×C4⋊C4—1581no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C42C22×D7911yescharacteristic
225#107—3164no—
226#103—3164no—
227#102—44no—
228#102—3164no—
229#102—1588no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—792no—
233C4×C8—792no—
234C4×C8—1581no—
235C4×C8—1581no—
236C24—41no—
237C2×Q8—3161no—
238C2×Q8—3162no—
239C2×Q8—3162no—
240C2×Q8—1582no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C22×C4—41no—
254C22×C4—3161no—
255C22×C4—3161no—
256C22×C4D4×D7911yescharacteristic
257C22×C4D4×D7911yescharacteristic
258C22×C4C2×C22⋊D7911yescharacteristic
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—794no—
263C2×C8—794no—
264C2×C8—1582no—
265C2×C8—1582no—
266C2×C8—1582no—
267C2×C8—1582no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—3161no—
272C4⋊C4—3161no—
273C4⋊C4—3162no—
274C4⋊C4—3162no—
275C4⋊C4—3162no—
276C4⋊C4—1582no—
277C4⋊C4—1582no—
278C42C2×C4×D7911yescharacteristic
279C42D4×D7911yescharacteristic
280C42#1192962311yescharacteristic
281C42#1192962411yescharacteristic
282C23—42no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23#1457915711yescharacteristic
290Q8—3164no—
291Q8—6322no—
292D4—44no—
293D4—44no—
294D4—44no—
295D4—82no—
296D4—82no—
297D4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—3162no—
302C2×C4—6321no—
303C2×C4—6321no—
304C2×C4—6321no—
305C2×C4—6321no—
306C2×C4—6321no—
307C2×C4C22⋊C4×D7911yescharacteristic
308C2×C4(C22⋊D79)⋊C411yescharacteristic
309C2×C4C4×C22⋊D7911yescharacteristic
310C2×C4#1457913311yescharacteristic
311C2×C4#1457913411yescharacteristic
312C2×C4#1457914711yescharacteristic
313C2×C4D8×D7912yes—
314C2×C4#1457918212yes—
315C2×C4#1457918512yes—
316C2×C4#1457918712yes—
317C2×C4#1457919012yes—
318C2×C4#1457919112yes—
319C8—42no—
320C8—1584no—
321C8—1584no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22—161no—
329C22#1909846311yescharacteristic
330C22#1909893811yescharacteristic
331C22#1909894411yescharacteristic
332C22#1909895011yescharacteristic
333C22#1909895111yescharacteristic
334C22#1909895411yescharacteristic
335C22#1909895811yescharacteristic
336C4—161no—
337C4—6322no—
338C4—12641no—
339C4#1909856112yes—
340C4#1909864312yes—
341C4#1909866312yes—
342C4#1909879812yes—
343C4#1909880712yes—
344C4#1909892112yes—
345C2—162no—
346C2#2376305111yescharacteristic
347C2#2376306211yescharacteristic
348C2#2376306511yescharacteristic
349C2#2376307311yescharacteristic
350C2#2376421511yescharacteristic
351C2#2376426311yescharacteristic
352C2#2376428011yescharacteristic
subgroupG/H
centerC23#14579157
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1957C2
socleC23×C79#127
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenter#1957D79
hypocenterC79#9830
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. #23764361C2semidirect
4. #23764201C2semidirect
5. #23763559C2semidirect
6. C2×#19098285C2semidirect
10. #19098909C4semidirect
184. C79#9830semidirect

Maximal subgroups

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

subgroupindex
C2×#190989092
#237643612
#237642072
#237642012
#237635592
C2×#190982852
C79×#19572
#983079