Finite Group Databasebeta

← #34205207

Subgroups — #34205207

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098799C211yesmaximal, characteristic
2C2×#19098363C212yesmaximal, complemented
3C2×C22⋊(C79⋊C16)C211yesmaximal, characteristic, complemented
4#23761491C212yesmaximal, complemented
5C22×#107×C79C211yesmaximal, characteristic, complemented
6C22×D632C2211yescharacteristic
7C2×#14579135—21no—
8C22×C79⋊C16C2212yes—
9#19098799—22no—
10#19098799C414yescomplemented
11C2×#14579095C2211yescharacteristic
12C2×#14579083—21no—
13#19098363—24no—
14C22⋊(C79⋊C16)—22no—
15C23×C8×C79C2211yescharacteristic
16C22×C4⋊C4×C79—21no—
17C2×#107×C79—21no—
18C2×#107×C79—24no—
19C2×#107×C79C2212yes—
20C22×D316—21no—
21C22×C22⋊D79—41no—
22C2×D632—22no—
23C2×D632—22no—
24C2×D632—21no—
25C2×D632C2×C412yescomplemented
26#14579135—24no—
27#14579135—42no—
28C2×C4⋊Dic79—21no—
29C2×(C2×D79)⋊C4—41no—
30C2×C79⋊C16—22no—
31C2×C79⋊C16—22no—
32C2×C79⋊C16—22no—
33#14579095—21no—
34#14579095C2×C412yescomplemented
35#14579083—24no—
36C23×C4×C79—41no—
37C23×C4×C79D411yescharacteristic
38C22×C8×C79—22no—
39C22×C8×C79—22no—
40C22×C8×C79C2×C412yes—
41C22×C8×C79D411yescharacteristic
42C22×C8×C79D412yes—
43C22×C8×C79C2311yescharacteristic
44C2×C4⋊C4×C79—24no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—42no—
47C2×C4⋊C4×C79—41no—
48#107×C79—24no—
49#107×C79—22no—
50#107×C79D414yescomplemented
51C23×D79—41no—
52C2×D316—22no—
53C2×D316—41no—
54C2×D316—41no—
55C2×D316—41no—
56C2×D316—41no—
57C2×D316—41no—
58C2×C22⋊D79—44no—
59C2×C22⋊D79—82no—
60C2×C22⋊D79—82no—
61C22×Dic79—41no—
62D632—24no—
63D632—42no—
64C4⋊Dic79—22no—
65C4⋊Dic79—41no—
66(C2×D79)⋊C4—44no—
67C79⋊C16—42no—
68C24×C79D811yescharacteristic
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—44no—
74C22×C4×C79—42no—
75C22×C4×C79—82no—
76C22×C4×C79—82no—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79D811yescharacteristic
79C22×C4×C79SD1612yes—
80C22×C4×C79C2×D411yescharacteristic
81C2×C8×C79—24no—
82C2×C8×C79—24no—
83C2×C8×C79—21no—
84C2×C8×C79—21no—
85C2×C8×C79C22⋊C412yes—
86C2×C8×C79C22×C411yescharacteristic
87C2×C8×C79C2×D411yescharacteristic
88C2×C8×C79C2×D412yes—
89C2×C8×C79#4112yes—
90C4⋊C4×C79—24no—
91C4⋊C4×C79—44no—
92C4⋊C4×C79—42no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79#10212yes—
98C23×C79D1612yes—
99C23×C79SD3212yes—
100C23×C79C2×D811yescharacteristic
101C22×D79—42no—
102C22×D79—81no—
103C22×D79—81no—
104C22×D79—81no—
105C22×D79—81no—
106C22×D79—81no—
107C22×D79—81no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—44no—
115C2×C4×C79—88no—
116C2×C4×C79—82no—
117C2×C4×C79—82no—
118C2×C4×C79#10212yes—
119C2×C4×C79C2×C22⋊C411yescharacteristic
120C2×C4×C79#12011yescharacteristic
121C2×C4×C79#12311yescharacteristic
122C2×C4×C79C2×SD1611yescharacteristic
123C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C79#13711yescharacteristic
125C22⋊D79—88no—
126C2×Dic79—42no—
127C2×Dic79—81no—
128C2×Dic79—81no—
129D316—42no—
130D316—42no—
131D316—81no—
132D316—81no—
133C8×C79—22no—
134C8×C79C4×D412yes—
135C8×C79#12112yes—
136C22×C79—22no—
137C22×C79—22no—
138C22×C79—22no—
139C22×C79—22no—
140C22×C79—22no—
141C22×C79—22no—
142C22×C79#35714yescomplemented
143C22×C79C2×#10211yescharacteristic
144C22×C79#44711yescharacteristic
145C22×C79#48011yescharacteristic
146C22×C79C2×D1611yescharacteristic
147C22×C79C2×SD3211yescharacteristic
148C22×C79#50911yescharacteristic
149C22×C79#51011yescharacteristic
150C2×D79—82no—
151C2×D79—82no—
152C2×D79—82no—
153C2×D79—161no—
154C2×D79—161no—
155C2×D79—161no—
156C2×D79—161no—
157C4×C79—22no—
158C4×C79—84no—
159C4×C79#38611yescharacteristic
160C4×C79#41711yescharacteristic
161C4×C79#44811yescharacteristic
162C4×C79#48111yescharacteristic
163Dic79—82no—
164#12991—791nomaximal, Sylow
165C2×C79—24no—
166C2×C79#187411yescharacteristic
167C2×C79C2×#35711yescharacteristic
168C2×C79#213911yescharacteristic
169C2×C79#218411yescharacteristic
170C2×C79#218511yescharacteristic
171C2×C79#223211yescharacteristic
172C2×C79#223311yescharacteristic
173D79—162no—
174C2×#466—791no—
175C22×#107D7911yescharacteristic, complemented
176C2×#357—792no—
177C2×C22⋊C16—791no—
178#1380—792no—
179C79#1299111yescharacteristic, complemented, Sylow
180C22×D8—791no—
181C23×C8C2×D7911yescharacteristic
182C2×#121—1581no—
183C22×C4⋊C4—21no—
184C22×C16—792no—
185#466—794no—
186#466—1582no—
187C2×#107—21no—
188C2×#107—24no—
189C2×#107—791no—
190C2×#107C2×D7912yes—
191C2×#102—1581no—
192#357—1584no—
193C22⋊C16—1582no—
194C22×D4—3161no—
195C22×D4—1581no—
196C23×C4—41no—
197C23×C4C22⋊D7911yescharacteristic
198C2×D8—792no—
199C2×D8—1582no—
200C2×D8—1581no—
201C2×D8—1582no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C4×D7912yes—
205C22×C8D31612yes—
206C22×C8C22⋊D7911yescharacteristic
207C22×C8C22×D7911yescharacteristic
208#121—3162no—
209#121—1584no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—1581no—
215C2×C22⋊C4—3161no—
216C2×C16—1582no—
217C2×C16—1582no—
218C2×C16—1582no—
219#107—24no—
220#107—22no—
221#107—792no—
222#107—1581no—
223#107C22⋊D7914yescomplemented
224#102—1584no—
225C24—3161no—
226C24#1192960911yescharacteristic
227C2×D4—3164no—
228C2×D4—3161no—
229C2×D4—3161no—
230C2×D4—3161no—
231C2×D4—3161no—
232C2×D4—3161no—
233C2×D4—6322no—
234C2×D4—6322no—
235C2×D4—1582no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—3161no—
245C22×C4(C2×D79)⋊C412yes—
246C22×C4#1192960911yescharacteristic
247C22×C4#1192961112yes—
248C22×C4C2×C22⋊D7911yescharacteristic
249D8—3162no—
250D8—1584no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8(C2×D79)⋊C412yes—
256C2×C8C2×C4×D7911yescharacteristic
257C2×C8C2×C22⋊D7912yes—
258C2×C8C2×D31611yescharacteristic
259C2×C8#1192962412yes—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—3161no—
264C4⋊C4—1582no—
265C22⋊C4—3164no—
266C16—3162no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—3162no—
272C23—6321no—
273C23—6321no—
274C23—6321no—
275C23—6321no—
276C23—6321no—
277C23—6321no—
278C23#1457908112yes—
279C23#1457910212yes—
280C23#1457910412yes—
281C23C2×#1192960911yescharacteristic
282D4—3162no—
283D4—3162no—
284D4—6328no—
285D4—6321no—
286D4—6321no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—3162no—
298C2×C4—6321no—
299C2×C4—6321no—
300C2×C4#1457908112yes—
301C2×C4C2×(C2×D79)⋊C411yescharacteristic
302C2×C4#1457913011yescharacteristic
303C2×C4#1457914411yescharacteristic
304C2×C4C2×#1192961111yescharacteristic
305C2×C4#1457918711yescharacteristic
306C2×C4#1457919211yescharacteristic
307C8—22no—
308C8C4×C22⋊D7912yes—
309C8#1457913512yes—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—6322no—
317C22—6322no—
318C22—6322no—
319C22—12641no—
320C22—12641no—
321C22—12641no—
322C22—12641no—
323C22#1909836314yescomplemented
324C22C2×#1457908111yescharacteristic
325C22#1909869711yescharacteristic
326C22#1909887611yescharacteristic
327C22C2×#1457910211yescharacteristic
328C22C2×#1457910411yescharacteristic
329C22#1909899911yescharacteristic
330C22#1909900411yescharacteristic
331C4—22no—
332C4—84no—
333C4—6322no—
334C4#1909844211yescharacteristic
335C4#1909856211yescharacteristic
336C4#1909870211yescharacteristic
337C4#1909888111yescharacteristic
338C2—24no—
339C2—12642no—
340C2#2376371711yescharacteristic
341C2C2×#1909836311yescharacteristic
342C2#2376511011yescharacteristic
343C2#2376531211yescharacteristic
344C2#2376531711yescharacteristic
345C2#2376558011yescharacteristic
346C2#2376558511yescharacteristic
subgroupG/H
centerC23C2×#11929609
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC22×#107×C79C2
socleC23×C79C2×D8
prime radicalC2×#19098799C2
cosocleC22×C8×C79C23
hypercenterC22×#107D79
hypocenterC79#12991
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C2×#19098363C2semidirect
3. C2×C22⋊(C79⋊C16)C2semidirect
4. #23761491C2semidirect
5. C22×#107×C79C2semidirect
10. #19098799C4semidirect
25. C2×D632C2×C4semidirect
34. #14579095C2×C4semidirect
50. #107×C79D4semidirect
142. C22×C79#357semidirect
175. C22×#107D79semidirect
179. C79#12991semidirect
223. #107C22⋊D79semidirect
323. C22#19098363semidirect

Maximal subgroups

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

subgroupindex
C2×#190987992
C2×#190983632
C2×C22⋊(C79⋊C16)2
#237614912
C22×#107×C792
#1299179