Finite Group Databasebeta

← #34210677

Subgroups — #34210677

Order 20224.

1710 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098429C211yesmaximal, characteristic
2C2×C4×C4⋊Dic79C211yesmaximal, characteristic
3#23761671C212yesmaximal
4#23761668C212yesmaximal
5#23761632C214yesmaximal
6#23761630C214yesmaximal
7C22×C4×Q8×C79C211yesmaximal, characteristic
8C22×C4⋊Dic79C2212yes—
9C22×C4⋊Dic79C2212yes—
10C22×C4×Dic79C2212yes—
11#19098429—24no—
12C4×C4⋊Dic79—24no—
13C2×C79⋊#95C2212yes—
14C2×C79⋊#95C2212yes—
15C2×C79⋊#95C2218yes—
16C2×C79⋊#95C2212yes—
17C2×C79⋊#95C2218yes—
18C23×Q8×C79C2211yescharacteristic
19C2×C4×Q8×C79—24no—
20C2×C4×Q8×C79—24no—
21C2×C4×Q8×C79C418yescomplemented
22C22×C4⋊C4×C79C2211yescharacteristic
23C22×C4⋊C4×C79C2212yes—
24C22×C42×C79C2212yes—
25C22×C42×C79C2211yescharacteristic
26C23×Dic79C2314yes—
27C23×Dic79C2314yes—
28C2×C4⋊Dic79—24no—
29C2×C4⋊Dic79—24no—
30C2×C4⋊Dic79—24no—
31C2×C4⋊Dic79—28no—
32C2×C4⋊Dic79Q818yescomplemented
33C2×C4×Dic79—24no—
34C2×C4×Dic79—44no—
35C79⋊#95—24no—
36C79⋊#95—24no—
37C79⋊#95—24no—
38C79⋊#95—216no—
39C79⋊#95—48no—
40C22×Q8×C79—22no—
41C22×Q8×C79—22no—
42C22×Q8×C79—28no—
43C22×Q8×C79C2×C414yes—
44C23×C4×C79C2312yes—
45C23×C4×C79C2311yescharacteristic
46C23×C4×C79C2311yescharacteristic
47C23×C4×C79C2312yes—
48C23×C4×C79C2311yescharacteristic
49C4×Q8×C79—232no—
50C2×C4⋊C4×C79—24no—
51C2×C4⋊C4×C79—24no—
52C2×C4⋊C4×C79—22no—
53C2×C4⋊C4×C79—22no—
54C2×C4⋊C4×C79C2×C414yes—
55C2×C4⋊C4×C79C2×C418yes—
56C2×C42×C79—24no—
57C2×C42×C79—24no—
58C2×C42×C79C2×C418yes—
59C2×C42×C79C2×C414yes—
60C2×C42×C79D414yes—
61C2×C42×C79D414yes—
62C22×Dic79—24no—
63C22×Dic79—24no—
64C22×Dic79—28no—
65C22×Dic79—24no—
66C22×Dic79—28no—
67C22×Dic79—28no—
68C22×Dic79—28no—
69C22×Dic79—24no—
70C22×Dic79—24no—
71C22×Dic79C2×Q818yes—
72C4⋊Dic79—216no—
73C4⋊Dic79—48no—
74C4×Dic79—48no—
75C24×C79C2411yescharacteristic
76C2×Q8×C79—216no—
77C2×Q8×C79—216no—
78C2×Q8×C79—28no—
79C2×Q8×C79—216no—
80C22×C4×C79—22no—
81C22×C4×C79—22no—
82C22×C4×C79—24no—
83C22×C4×C79—24no—
84C22×C4×C79—22no—
85C22×C4×C79—22no—
86C22×C4×C79—22no—
87C22×C4×C79—22no—
88C22×C4×C79—24no—
89C22×C4×C79—24no—
90C22×C4×C79C22×C414yes—
91C22×C4×C79C22×C412yes—
92C22×C4×C79C22×C412yes—
93C22×C4×C79C22×C414yes—
94C22×C4×C79C22×C412yes—
95C22×C4×C79C2×D412yes—
96C22×C4×C79C2×D412yes—
97C22×C4×C79C2×D412yes—
98C22×C4×C79C2×D412yes—
99C22×C4×C79C2×D412yes—
100C22×C4×C79C2×D412yes—
101C22×C4×C79C2×Q814yes—
102C22×C4×C79#4118yes—
103C22×C4×C79#4114yes—
104C4⋊C4×C79—216no—
105C4⋊C4×C79—28no—
106C42×C79—216no—
107C42×C79C22⋊C4116yes—
108C23×C79C23×C411yescharacteristic
109C23×C79C22×D411yescharacteristic
110C23×C79C22×D411yescharacteristic
111C23×C79C22×Q812yes—
112C23×C79C2×#4114yes—
113C23×C79C2×#4112yes—
114C23×C79H(5,𝔽2)12yes—
115C23×C792−1+412yes—
116Q8×C79—232no—
117C2×C4×C79—24no—
118C2×C4×C79—24no—
119C2×C4×C79—24no—
120C2×C4×C79—24no—
121C2×C4×C79—24no—
122C2×C4×C79—24no—
123C2×C4×C79—28no—
124C2×C4×C79—28no—
125C2×C4×C79—22no—
126C2×C4×C79—22no—
127C2×C4×C79—24no—
128C2×C4×C79—24no—
129C2×C4×C79—28no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—44no—
133C2×C4×C79C2×C22⋊C414yes—
134C2×C4×C79C2×C22⋊C414yes—
135C2×C4×C79C2×C22⋊C414yes—
136C2×C4×C79#11718yes—
137C2×C4×C79C4×Q818yes—
138C2×C4×C79C22⋊Q818yes—
139C2×C4×C79#12414yes—
140C2×C4×C79C4⋊Q814yes—
141C2×Dic79—28no—
142C2×Dic79—216no—
143C2×Dic79—44no—
144C2×Dic79—44no—
145C2×Dic79—48no—
146C2×Dic79—44no—
147C2×Dic79—44no—
148C2×Dic79—44no—
149C2×Dic79—48no—
150C2×Dic79—48no—
151C22×C79C22×C22⋊C411yescharacteristic
152C22×C79C2×#11712yes—
153C22×C79C2×C4×Q812yes—
154C22×C79#52012yes—
155C22×C79C2×C22⋊Q812yes—
156C22×C79#52512yes—
157C22×C79#52512yes—
158C22×C79C2×#12411yescharacteristic
159C22×C79C2×C4⋊Q811yescharacteristic
160C22×C79#53511yescharacteristic
161C22×C79#53611yescharacteristic
162C22×C79#53712yes—
163C22×C79#55414yes—
164C22×C79#55614yes—
165C22×C79#55714yes—
166C22×C79#55914yes—
167C4×C79—24no—
168C4×C79—24no—
169C4×C79—28no—
170C4×C79—44no—
171C4×C79#39118yes—
172Dic79—48no—
173Dic79—84no—
174#13925—791nomaximal, Sylow
175C2×C79#228011yescharacteristic
176C2×C79C2×#39111yescharacteristic
177C2×C79#231111yescharacteristic
178C2×C79#235514yes—
179C2×C79#258012yes—
180C2×C79#258112yes—
181C2×C79#258412yes—
182C2×C79#259212yes—
183C22×C4×Q8D7911yescharacteristic
184C2×#382—791no—
185C2×C4×C4⋊C4—791no—
186#1446—792no—
187#1445—792no—
188#1436—794no—
189C4⋊#95—794no—
190C79#1392511yescharacteristic, complemented, Sylow
191C23×Q8C2×D7911yescharacteristic
192C2×C4×Q8—24no—
193C2×C4×Q8—24no—
194C2×C4×Q8Dic7918yescomplemented
195C22×C4⋊C4—792no—
196C22×C4⋊C4—792no—
197C22×C4⋊C4C2×D7912yes—
198C22×C4⋊C4C2×D7911yescharacteristic
199C22×C42—792no—
200C22×C42C2×D7912yes—
201C22×C42C2×D7911yescharacteristic
202#382—1584no—
203C4×C4⋊C4—1584no—
204C2×#95—798no—
205C2×#95—798no—
206C2×#95—792no—
207C2×#95—792no—
208C2×#95—792no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—28no—
212C22×Q8C2×Dic7914yes—
213C23×C4—794no—
214C23×C4—794no—
215C23×C4C22×D7912yes—
216C23×C4C22×D7911yescharacteristic
217C23×C4C22×D7911yescharacteristic
218C23×C4C22×D7912yes—
219C23×C4C22×D7911yescharacteristic
220C4×Q8—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—798no—
226C2×C4⋊C4—1584no—
227C2×C4⋊C4—1588no—
228C2×C4⋊C4—1584no—
229C2×C4⋊C4—1584no—
230C2×C4⋊C4C2×Dic7914yes—
231C2×C4⋊C4C2×Dic7918yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—3164no—
235C2×C42—1584no—
236C2×C42C2×Dic7918yes—
237C2×C42C2×Dic7914yes—
238C2×C42C22⋊D7914yes—
239C2×C42C22⋊D7914yes—
240#95—3168no—
241#95—1584no—
242#95—1584no—
243#95—1584no—
244#95—15816no—
245C24C23×D7911yescharacteristic
246C2×Q8—216no—
247C2×Q8—216no—
248C2×Q8—28no—
249C2×Q8—216no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—798no—
261C22×C4—1588no—
262C22×C4—1588no—
263C22×C4—1584no—
264C22×C4—1584no—
265C22×C4—1584no—
266C22×C4—1588no—
267C22×C4—1584no—
268C22×C4—1588no—
269C22×C4—1584no—
270C22×C4C22×Dic7914yes—
271C22×C4C22×Dic7912yes—
272C22×C4C22×Dic7912yes—
273C22×C4C22×Dic7914yes—
274C22×C4C22×Dic7912yes—
275C22×C4C2×C22⋊D7912yes—
276C22×C4C2×C22⋊D7912yes—
277C22×C4C2×C22⋊D7912yes—
278C22×C4C2×C22⋊D7912yes—
279C22×C4C2×C22⋊D7912yes—
280C22×C4C2×C22⋊D7912yes—
281C22×C4Q8×D7914yes—
282C22×C4#1192962418yes—
283C22×C4#1192962514yes—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—3168no—
287C4⋊C4—15816no—
288C42—216no—
289C42—3168no—
290C42C22⋊Dic79116yes—
291C23C23×Dic7911yescharacteristic
292C23C22×C22⋊D7911yescharacteristic
293C23C22×C22⋊D7911yescharacteristic
294C23C2×Q8×D7912yes—
295C23C2×#1192962414yes—
296C23C2×#1192962512yes—
297C23#1457920712yes—
298C23#1457920812yes—
299Q8—232no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—3168no—
317C2×C4—3164no—
318C2×C4—3164no—
319C2×C4—3164no—
320C2×C4—3168no—
321C2×C4—3168no—
322C2×C4—3164no—
323C2×C4—3164no—
324C2×C4—15816no—
325C2×C4—1588no—
326C2×C4C2×C22⋊Dic7914yes—
327C2×C4C2×C22⋊Dic7914yes—
328C2×C4C2×C22⋊Dic7914yes—
329C2×C4#1457912018yes—
330C2×C4Q8×Dic7918yes—
331C2×C4#1457913718yes—
332C2×C4#1457915014yes—
333C2×C4#1457915914yes—
334C22C22×C22⋊Dic7911yescharacteristic
335C22C2×#1457912012yes—
336C22C2×Q8×Dic7912yes—
337C22#1909904612yes—
338C22C2×#1457913712yes—
339C22#1909907412yes—
340C22#1909907412yes—
341C22C2×#1457915011yescharacteristic
342C22C2×#1457915911yescharacteristic
343C22#1909912711yescharacteristic
344C22#1909913411yescharacteristic
345C22#1909914112yes—
346C22#1909927314yes—
347C22#1909928614yes—
348C22#1909929114yes—
349C22#1909929814yes—
350C4—24no—
351C4—28no—
352C4—24no—
353C4—44no—
354C4—3168no—
355C4—6324no—
356C4#1909847018yes—
357C2#2376578311yescharacteristic
358C2C2×#1909847011yescharacteristic
359C2#2376599211yescharacteristic
360C2#2376635714yes—
361C2#2376875812yes—
362C2#2376876612yes—
363C2#2376878812yes—
364C2#2376888812yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC22×C4×Q8×C79C2
socle, prime radical, cosocleC24×C79C24
hypercenterC22×C4×Q8D79
hypocenterC79#13925
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
21. C2×C4×Q8×C79C4semidirect
32. C2×C4⋊Dic79Q8semidirect
190. C79#13925semidirect
194. C2×C4×Q8Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190984292
C2×C4×C4⋊Dic792
#237616712
#237616682
#237616322
#237616302
C22×C4×Q8×C792
#1392579