Finite Group Databasebeta

← #34317339

Subgroups — #34317339

Order 20224.

874 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579094C211yesmaximal, characteristic
2C2×#19098591C211yesmaximal, characteristic
3#23766024C211yesmaximal, characteristic
4#23766023C211yesmaximal, characteristic
5#23764897C214yesmaximal, complemented
6#23764892C214yesmaximal, complemented
7#23763120C211yesmaximal, characteristic
8#23763116C211yesmaximal, characteristic
9C79×#1736C211yesmaximal, characteristic
10C23×C79⋊C8C2211yescharacteristic
11C2×#14579120C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13C22×C4⋊Dic79C2211yescharacteristic
14C2×#14579094—24no—
15C2×#14579094C2212yes—
16C2×#14579094C2212yes—
17#19098591—24no—
18C2×Dic79⋊C8C2212yes—
19C2×Dic79⋊C8C2212yes—
20#19098462C2214yes—
21#19098459C2214yes—
22#19098438—22no—
23#19098436—22no—
24#19098433—21no—
25#19098432—21no—
26#19098429—21no—
27#19098428—21no—
28#19098326C2214yes—
29#19098325C2212yes—
30#19098325C2212yes—
31C22×M16×C79C2211yescharacteristic
32C2×#117×C79C2211yescharacteristic
33C22×C4⋊C4×C79C2211yescharacteristic
34#340×C79C2214yes—
35C23×Dic79—21no—
36C22×C79⋊C8—22no—
37C22×C79⋊C8—22no—
38C22×C79⋊C8D414yes—
39C22×C79⋊C8C2311yescharacteristic
40C22×C79⋊C8C2311yescharacteristic
41#14579120—22no—
42#14579120—22no—
43#14579120—22no—
44#14579120Q814yes—
45C2×C4⋊Dic79—22no—
46C2×C4⋊Dic79—22no—
47C2×C4⋊Dic79—21no—
48C2×C4⋊Dic79—24no—
49C2×C4⋊Dic79D411yescharacteristic
50C2×C4⋊Dic79D411yescharacteristic
51C2×C4⋊Dic79C2312yes—
52C2×C4⋊Dic79C2312yes—
53C2×Dic79⋊C4—24no—
54C2×C22⋊Dic79—21no—
55C2×C22⋊Dic79D412yes—
56C2×C4×Dic79—21no—
57C2×C4×Dic79C2312yes—
58#14579094—24no—
59#14579094—24no—
60Dic79⋊C8—24no—
61Dic79⋊C8—24no—
62C79⋊#95—24no—
63C79⋊#95—24no—
64C23×C4×C79—21no—
65C23×C4×C79C2311yescharacteristic
66C2×M16×C79—22no—
67C2×M16×C79—22no—
68C2×M16×C79—22no—
69C22×C8×C79C2312yes—
70#117×C79—22no—
71#117×C79Q814yes—
72C2×C4⋊C4×C79—22no—
73C2×C4⋊C4×C79—22no—
74C2×C4⋊C4×C79—21no—
75C2×C4⋊C4×C79C2312yes—
76C2×C4⋊C4×C79C2312yes—
77C2×C22⋊C4×C79—21no—
78C2×C42×C79—21no—
79C22×Dic79—22no—
80C22×Dic79—22no—
81C22×Dic79—42no—
82C22×Dic79—82no—
83C22×Dic79C2×D412yes—
84C22×Dic79C2×D412yes—
85C2×C79⋊C8—28no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8C2×D412yes—
91C2×C79⋊C8C2×D412yes—
92C4⋊Dic79—22no—
93C4⋊Dic79—22no—
94C4⋊Dic79—22no—
95C4⋊Dic79—22no—
96C4⋊Dic79—22no—
97C4⋊Dic79—42no—
98C4⋊Dic79—84no—
99C4⋊Dic79C2×Q814yes—
100Dic79⋊C4—84no—
101C22⋊Dic79—24no—
102C22⋊Dic79—24no—
103C4×Dic79—24no—
104C4×Dic79—22no—
105C4×Dic79—22no—
106C24×C79C2×D411yescharacteristic
107C22×C4×C79—22no—
108C22×C4×C79—22no—
109C22×C4×C79—42no—
110C22×C4×C79—82no—
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D412yes—
113C22×C4×C79C2×Q812yes—
114C22×C4×C79C2×Q812yes—
115C22×C4×C79#4112yes—
116C22×C4×C79#4112yes—
117C22×C4×C79#4112yes—
118C22×C4×C79C2411yescharacteristic
119M16×C79—44no—
120C2×C8×C79—22no—
121C2×C8×C79—22no—
122C2×C8×C79—22no—
123C4⋊C4×C79—22no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—22no—
126C4⋊C4×C79—42no—
127C4⋊C4×C79C2×Q814yes—
128C22⋊C4×C79—24no—
129C42×C79—24no—
130C23×C79—22no—
131C23×C79—22no—
132C23×C79#12112yes—
133C23×C79C22⋊Q812yes—
134C23×C79C22⋊Q812yes—
135C23×C79C22×D411yescharacteristic
136C2×C4×C79—24no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79—42no—
140C2×C4×C79—44no—
141C2×C4×C79—84no—
142C2×C4×C79#12112yes—
143C2×C4×C79C22⋊Q812yes—
144C2×C4×C79C22⋊Q812yes—
145C2×C4×C79#12314yes—
146C2×C4×C79#12514yes—
147C2×C4×C79#12514yes—
148C2×C4×C79C4⋊Q814yes—
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C22×Q811yescharacteristic
151C2×C4×C79C22×Q811yescharacteristic
152C2×C4×C79C2×#4111yescharacteristic
153C2×C4×C79C2×#4111yescharacteristic
154C2×C4×C79C2×#4111yescharacteristic
155C2×Dic79—22no—
156C2×Dic79—22no—
157C2×Dic79—24no—
158C2×Dic79—24no—
159C2×Dic79—24no—
160C2×Dic79—42no—
161C2×Dic79—42no—
162C2×Dic79—42no—
163C2×Dic79—44no—
164C2×Dic79—84no—
165C8×C79—42no—
166C79⋊C8—22no—
167C79⋊C8—22no—
168C22×C79—24no—
169C22×C79—24no—
170C22×C79—22no—
171C22×C79—22no—
172C22×C79#39814yes—
173C22×C79C2×#12111yescharacteristic
174C22×C79C2×C22⋊Q811yescharacteristic
175C22×C79C2×C22⋊Q811yescharacteristic
176C22×C79#57612yes—
177C22×C79#57812yes—
178C4×C79—44no—
179C4×C79—82no—
180C4×C79#39814yes—
181C4×C79C2×#12311yescharacteristic
182C4×C79C2×#12511yescharacteristic
183C4×C79C2×#12511yescharacteristic
184C4×C79C2×C4⋊Q811yescharacteristic
185Dic79—44no—
186Dic79—42no—
187Dic79—42no—
188Dic79—82no—
189#23667—791nomaximal, Sylow
190C2×C79—24no—
191C2×C79C2×#39811yescharacteristic
192C2×C79#305811yescharacteristic
193C2×C79#305911yescharacteristic
194C2×C79#308312yes—
195C2×C79#308312yes—
196C22×#107—791no—
197C2×#424—791no—
198#2317—791no—
199#2317—791no—
200#2087—794no—
201#2086—794no—
202#1736—791no—
203#1736D7911yescharacteristic
204#1735—791no—
205C79#2366711yescharacteristic, complemented, Sylow
206C22×M16C2×D7911yescharacteristic
207C23×C8—791no—
208C2×#117—791no—
209C2×#117—791no—
210C2×#117C2×D7911yescharacteristic
211C22×C4⋊C4—791no—
212C22×C4⋊C4C2×D7911yescharacteristic
213C2×#107—792no—
214C2×#107—792no—
215C2×#107—1584no—
216#424—1584no—
217C2×C4⋊C8—792no—
218C2×C4⋊C8—792no—
219#389—794no—
220#389—794no—
221#385—1582no—
222#385—1582no—
223#384—1581no—
224#384—1581no—
225#382—1581no—
226#382—1581no—
227#340—794no—
228#340—792no—
229#340—792no—
230#340C2×D7914yes—
231C23×C4—21no—
232C23×C4—1581no—
233C23×C4C22×D7911yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C22×C8—794no—
238C22×C8—791no—
239C22×C8—791no—
240C22×C8—1582no—
241C22×C8—1582no—
242C22×C8C22×D7912yes—
243#117—22no—
244#117—794no—
245#117—1582no—
246#117—1582no—
247#117—1582no—
248#117Dic15814yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—792no—
253C2×C4⋊C4—791no—
254C2×C4⋊C4—791no—
255C2×C4⋊C4—792no—
256C2×C4⋊C4—1581no—
257C2×C4⋊C4—1582no—
258C2×C4⋊C4—1582no—
259C2×C4⋊C4—1584no—
260C2×C4⋊C4—1584no—
261C2×C4⋊C4C22×D7912yes—
262C2×C4⋊C4C22×D7912yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—792no—
265C2×C22⋊C4—1581no—
266C2×C42—21no—
267C2×C42—792no—
268C2×C42—1581no—
269#107—1584no—
270#107—1584no—
271C4⋊C8—1584no—
272C4⋊C8—1584no—
273#95—1584no—
274#95—1584no—
275C24D4×D7911yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—3162no—
281C22×C4—792no—
282C22×C4—792no—
283C22×C4—6322no—
284C22×C4—1582no—
285C22×C4—1582no—
286C22×C4D4×D7911yescharacteristic
287C22×C4D4×D7912yes—
288C22×C4Q8×D7912yes—
289C22×C4C2×Dic15812yes—
290C22×C4#1192962312yes—
291C22×C4#1192962412yes—
292C22×C4#1192962412yes—
293C22×C4C23×D7911yescharacteristic
294M16—44no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—792no—
299C2×C8—792no—
300C2×C8—1581no—
301C2×C8—1581no—
302C2×C8—1581no—
303C2×C8—1581no—
304C2×C8—1588no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—3162no—
310C4⋊C4—794no—
311C4⋊C4—6324no—
312C4⋊C4—6324no—
313C4⋊C4—1582no—
314C4⋊C4—1582no—
315C4⋊C4—1582no—
316C4⋊C4—1582no—
317C4⋊C4—1582no—
318C4⋊C4C2×Dic15814yes—
319C22⋊C4—24no—
320C22⋊C4—1584no—
321C22⋊C4—1584no—
322C42—24no—
323C42—1582no—
324C42—1584no—
325C42—1582no—
326C23—22no—
327C23—22no—
328C23#1457913412yes—
329C23#1457913812yes—
330C23#1457913912yes—
331C23C2×D4×D7911yescharacteristic
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—3162no—
339C2×C4—3162no—
340C2×C4—3162no—
341C2×C4—3164no—
342C2×C4—6324no—
343C2×C4—1582no—
344C2×C4—1582no—
345C2×C4—1584no—
346C2×C4—1584no—
347C2×C4—1584no—
348C2×C4#1457913412yes—
349C2×C4#1457913812yes—
350C2×C4#1457913912yes—
351C2×C4#1457914314yes—
352C2×C4#1457915114yes—
353C2×C4#1457915314yes—
354C2×C4#1457916014yes—
355C2×C4C2×D4×D7911yescharacteristic
356C2×C4C2×Q8×D7911yescharacteristic
357C2×C4C22×Dic15811yescharacteristic
358C2×C4C2×#1192962311yescharacteristic
359C2×C4C2×#1192962411yescharacteristic
360C2×C4C2×#1192962411yescharacteristic
361C8—42no—
362C8—1582no—
363C8—1582no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22#1909849714yes—
369C22C2×#1457913411yescharacteristic
370C22C2×#1457913811yescharacteristic
371C22C2×#1457913911yescharacteristic
372C22#1909936712yes—
373C22#1909937912yes—
374C4—44no—
375C4—82no—
376C4—3164no—
377C4—3162no—
378C4—3162no—
379C4—6322no—
380C4#1909849714yes—
381C4C2×#1457914311yescharacteristic
382C4C2×#1457915111yescharacteristic
383C4C2×#1457915311yescharacteristic
384C4C2×#1457916011yescharacteristic
385C2—24no—
386C2C2×#1909849711yescharacteristic
387C2#2377304711yescharacteristic
388C2#2377305411yescharacteristic
389C2#2377325012yes—
390C2#2377325512yes—
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1736C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1736D79
hypocenterC79#23667
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
5. #23764897C2semidirect
6. #23764892C2semidirect
205. C79#23667semidirect

Maximal subgroups

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

subgroupindex
C22×#145790942
C2×#190985912
#237660242
#237660232
#237648972
#237648922
#237631202
#237631162
C79×#17362
#2366779