Finite Group Databasebeta

← #34196462

Subgroups — #34196462

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579093C211yesmaximal, characteristic
2C2×#19098325C212yesmaximal
3C2×#19098317C211yesmaximal, characteristic
4#23761629C212yesmaximal
5C2×#340×C79C211yesmaximal, characteristic
6C23×C79⋊C8C2211yescharacteristic
7C22×C4⋊Dic79C2212yes—
8C22×C4×Dic79C2211yescharacteristic
9C2×#14579093—24no—
10C2×#14579093—22no—
11C2×C79⋊#95—22no—
12C2×C79⋊#95—22no—
13#19098325—24no—
14#19098317—22no—
15C23×C8×C79C2211yescharacteristic
16C22×C4⋊C4×C79C2212yes—
17#340×C79C414yescomplemented
18C23×Dic79—22no—
19C23×Dic79D412yes—
20C22×C79⋊C8—21no—
21C22×C79⋊C8—22no—
22C22×C79⋊C8—21no—
23C22×C79⋊C8—22no—
24C22×C79⋊C8Q812yes—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79—22no—
27C2×C4⋊Dic79—22no—
28C2×C4⋊Dic79—22no—
29C2×C4⋊Dic79—22no—
30C2×C4⋊Dic79D414yes—
31C2×C4×Dic79—21no—
32C2×C4×Dic79—22no—
33C2×C4×Dic79—22no—
34C2×C4×Dic79—21no—
35#14579093—28no—
36C79⋊#95—44no—
37C79⋊#95—44no—
38C23×C4×C79—22no—
39C23×C4×C79C2311yescharacteristic
40C22×C8×C79—22no—
41C22×C8×C79—22no—
42C22×C8×C79C2×C412yes—
43C22×C8×C79D412yes—
44C22×C8×C79Q812yes—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79—22no—
48C2×C4⋊C4×C79—22no—
49C2×C4⋊C4×C79—22no—
50C2×C4⋊C4×C79C2×C414yes—
51C22×Dic79—22no—
52C22×Dic79—24no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C22×Dic79—22no—
56C22×Dic79—22no—
57C22×Dic79—22no—
58C22×Dic79—42no—
59C22×Dic79—42no—
60C22×Dic79—42no—
61C22×Dic79—42no—
62C22×Dic79—42no—
63C22×Dic79—42no—
64C22×Dic79SD1614yes—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—22no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—42no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—24no—
71C4⋊Dic79—24no—
72C4⋊Dic79—42no—
73C4⋊Dic79—42no—
74C4×Dic79—41no—
75C4×Dic79—41no—
76C4×Dic79—42no—
77C24×C79C2×D411yescharacteristic
78C22×C4×C79—24no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79—42no—
83C22×C4×C79—42no—
84C22×C4×C79—42no—
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D412yes—
88C22×C4×C79C2×Q811yescharacteristic
89C22×C4×C79C2×Q811yescharacteristic
90C22×C4×C79#4112yes—
91C22×C4×C79#4112yes—
92C22×C4×C79#4112yes—
93C22×C4×C79#4111yescharacteristic
94C22×C4×C79#4111yescharacteristic
95C2×C8×C79—24no—
96C2×C8×C79—22no—
97C2×C8×C79—24no—
98C2×C8×C79—22no—
99C2×C8×C79C4⋊C414yes—
100C4⋊C4×C79—24no—
101C4⋊C4×C79—24no—
102C4⋊C4×C79—42no—
103C4⋊C4×C79—42no—
104C23×C79C4×D411yescharacteristic
105C23×C79#12112yes—
106C23×C79#12111yescharacteristic
107C23×C79#12312yes—
108C23×C79C4⋊Q811yescharacteristic
109C23×C79C2×SD1612yes—
110C23×C79#13512yes—
111C23×C79AGL(1,ℤ/8)12yes—
112C23×C79#13712yes—
113C2×C4×C79—44no—
114C2×C4×C79—44no—
115C2×C4×C79—44no—
116C2×C4×C79—82no—
117C2×C4×C79—82no—
118C2×C4×C79—82no—
119C2×C4×C79—82no—
120C2×C4×C79C2×C4⋊C411yescharacteristic
121C2×C4×C79#11712yes—
122C2×C4×C79C4×D412yes—
123C2×C4×C79C4×Q811yescharacteristic
124C2×C4×C79#12011yescharacteristic
125C2×C4×C79C22⋊Q812yes—
126C2×C4×C79C22⋊Q812yes—
127C2×C4×C79C22⋊Q811yescharacteristic
128C2×C4×C79#12311yescharacteristic
129C2×C4×C79#12312yes—
130C2×C4×C79#12412yes—
131C2×C4×C79#12411yescharacteristic
132C2×C4×C79#12411yescharacteristic
133C2×C4×C79#12512yes—
134C2×C4×C79#12511yescharacteristic
135C2×C4×C79#12512yes—
136C2×C4×C79#12612yes—
137C2×C4×C79#12612yes—
138C2×Dic79—24no—
139C2×Dic79—24no—
140C2×Dic79—24no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—42no—
144C2×Dic79—42no—
145C2×Dic79—44no—
146C2×Dic79—44no—
147C2×Dic79—44no—
148C2×Dic79—82no—
149C2×Dic79—82no—
150C2×Dic79—82no—
151C2×Dic79—82no—
152C8×C79—24no—
153C79⋊C8—42no—
154C22×C79#38812yes—
155C22×C79#38911yescharacteristic
156C22×C79#39411yescharacteristic
157C22×C79#39711yescharacteristic
158C22×C79#39812yes—
159C22×C79C4×SD1612yes—
160C22×C79#44111yescharacteristic
161C22×C79#44211yescharacteristic
162C22×C79#46012yes—
163C22×C79#46112yes—
164C22×C79#46312yes—
165C22×C79#46412yes—
166C22×C79#46512yes—
167C22×C79#46812yes—
168C22×C79#48112yes—
169C22×C79#48212yes—
170C22×C79#48312yes—
171C22×C79#48512yes—
172C22×C79#49811yescharacteristic
173C22×C79#49911yescharacteristic
174C22×C79#50112yes—
175C4×C79—84no—
176C4×C79#38211yescharacteristic
177C4×C79#38511yescharacteristic
178C4×C79#38712yes—
179C4×C79#39311yescharacteristic
180C4×C79#40011yescharacteristic
181C4×C79#40012yes—
182Dic79—44no—
183Dic79—84no—
184#11257—791nomaximal, Sylow
185C2×C79#144211yescharacteristic
186C2×C79#192612yes—
187C2×C79#192712yes—
188C2×C79#194111yescharacteristic
189C2×C79#194311yescharacteristic
190C2×C79#205512yes—
191C2×C79#207712yes—
192C2×C79#208412yes—
193C2×C79#208512yes—
194C22×#106—791no—
195C2×#340—792no—
196C2×#340D7911yescharacteristic
197C2×#336—791no—
198C4⋊#95—792no—
199C79#1125711yescharacteristic, complemented, Sylow
200C23×C8—791no—
201C23×C8C2×D7911yescharacteristic
202C22×C4⋊C4—792no—
203C22×C4⋊C4C2×D7912yes—
204C22×C42—791no—
205C2×#106—1584no—
206C2×#106—1582no—
207C2×#95—1582no—
208C2×#95—1582no—
209#340—1584no—
210#340Dic7914yescomplemented
211#336—1582no—
212C23×C4—22no—
213C23×C4—792no—
214C23×C4—1582no—
215C23×C4C22×D7911yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—792no—
219C22×C8—1582no—
220C22×C8—1581no—
221C22×C8—1581no—
222C22×C8—1582no—
223C22×C8Dic15812yes—
224C22×C8D31612yes—
225C22×C8C2×Dic7912yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—794no—
232C2×C4⋊C4—1582no—
233C2×C4⋊C4—1582no—
234C2×C4⋊C4—1582no—
235C2×C4⋊C4—1582no—
236C2×C4⋊C4—1582no—
237C2×C4⋊C4C2×Dic7914yes—
238C2×C42—1581no—
239C2×C42—1582no—
240C2×C42—1582no—
241C2×C42—1581no—
242#106—1588no—
243#95—3164no—
244#95—3164no—
245C24D4×D7911yescharacteristic
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—3162no—
254C22×C4—3162no—
255C22×C4—3162no—
256C22×C4—3162no—
257C22×C4—3162no—
258C22×C4—3162no—
259C22×C4—794no—
260C22×C4—1584no—
261C22×C4—1582no—
262C22×C4—1582no—
263C22×C4—1582no—
264C22×C4—1582no—
265C22×C4—1582no—
266C22×C4—1582no—
267C22×C4C22×Dic7911yescharacteristic
268C22×C4D4×D7912yes—
269C22×C4C2×D31611yescharacteristic
270C22×C4Q8×D7911yescharacteristic
271C22×C4C2×Dic15811yescharacteristic
272C22×C4#1192962312yes—
273C22×C4#1192962311yescharacteristic
274C22×C4#1192962412yes—
275C22×C4#1192962412yes—
276C22×C4#1192962511yescharacteristic
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—3161no—
282C2×C8—3162no—
283C2×C8—3161no—
284C2×C8—1584no—
285C2×C8—1582no—
286C2×C8C4⋊Dic7914yes—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—3162no—
292C4⋊C4—3162no—
293C4⋊C4—1584no—
294C4⋊C4—1584no—
295C42—3161no—
296C42—3161no—
297C42—3162no—
298C23D4×Dic7911yescharacteristic
299C23#1457913412yes—
300C23#1457913611yescharacteristic
301C23#1457914312yes—
302C23#1457916011yescharacteristic
303C23SD16×D7912yes—
304C23#1457918212yes—
305C23#1457918512yes—
306C23#1457919312yes—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—3162no—
315C2×C4—3162no—
316C2×C4—3162no—
317C2×C4—3162no—
318C2×C4—3164no—
319C2×C4—3164no—
320C2×C4—3164no—
321C2×C4—6322no—
322C2×C4—6322no—
323C2×C4—6322no—
324C2×C4—6322no—
325C2×C4—1584no—
326C2×C4—1584no—
327C2×C4—1584no—
328C2×C4C2×C4⋊Dic7911yescharacteristic
329C2×C4#1457912012yes—
330C2×C4D4×Dic7912yes—
331C2×C4Q8×Dic7911yescharacteristic
332C2×C4#1457913111yescharacteristic
333C2×C4#1457913812yes—
334C2×C4#1457913912yes—
335C2×C4#1457914111yescharacteristic
336C2×C4#1457914511yescharacteristic
337C2×C4#1457914612yes—
338C2×C4#1457914712yes—
339C2×C4#1457914911yescharacteristic
340C2×C4#1457914911yescharacteristic
341C2×C4#1457915112yes—
342C2×C4#1457915211yescharacteristic
343C2×C4#1457915312yes—
344C2×C4#1457915412yes—
345C2×C4#1457915612yes—
346C8—24no—
347C8—3162no—
348C22#1909845612yes—
349C22#1909846211yescharacteristic
350C22#1909848111yescharacteristic
351C22#1909849011yescharacteristic
352C22#1909849712yes—
353C22SD16×Dic7912yes—
354C22#1909867011yescharacteristic
355C22#1909867511yescharacteristic
356C22#1909876212yes—
357C22#1909876712yes—
358C22#1909877712yes—
359C22#1909878812yes—
360C22#1909879312yes—
361C22#1909881012yes—
362C22#1909888012yes—
363C22#1909888612yes—
364C22#1909889212yes—
365C22#1909890212yes—
366C22#1909896011yescharacteristic
367C22#1909896311yescharacteristic
368C22#1909897112yes—
369C4—84no—
370C4—3164no—
371C4—6324no—
372C4#1909842911yescharacteristic
373C4#1909843811yescharacteristic
374C4#1909844912yes—
375C4#1909847611yescharacteristic
376C4#1909850311yescharacteristic
377C4#1909850512yes—
378C2#2376165711yescharacteristic
379C2#2376402912yes—
380C2#2376403612yes—
381C2#2376412911yescharacteristic
382C2#2376413911yescharacteristic
383C2#2376471912yes—
384C2#2376483512yes—
385C2#2376488012yes—
386C2#2376488612yes—
subgroupG/H
centerC24D4×D79
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#340×C79C2
socle, prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#340D79
hypocenterC79#11257
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
17. #340×C79C4semidirect
199. C79#11257semidirect
210. #340Dic79semidirect

Maximal subgroups

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

subgroupindex
C22×#145790932
C2×#190983252
C2×#190983172
#237616292
C2×#340×C792
#1125779