Finite Group Databasebeta

← #34296248

Subgroups — #34296248

Order 20224.

878 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098913C211yesmaximal, characteristic
2C4⋊Q8×Dic79C211yesmaximal, characteristic
3#23765967C211yesmaximal, characteristic
4#23764361C211yesmaximal, characteristic
5#23764073C214yesmaximal
6#23763527C211yesmaximal, characteristic
7#103×Dic79C214yesmaximal
8C4×C8×Dic79C211yesmaximal, characteristic
9C79×#1956C211yesmaximal, characteristic
10C2×#14579161C2211yescharacteristic
11C2×Q8×Dic79C2212yes—
12C2×#14579126—21no—
13#19098913C418yescomplemented
14C2×#14579085C2214yes—
15C2×C4×C79⋊C8C2211yescharacteristic
16C2×C8×Dic79C2212yes—
17C2×C8×Dic79C2212yes—
18#19098470C2212yes—
19#19098468—21no—
20#19098462—21no—
21#19098462—21no—
22#19098459—21no—
23#19098459C2212yes—
24#19098433C2211yescharacteristic
25C4⋊C4×Dic79—21no—
26C4⋊C4×Dic79—21no—
27C4⋊C4×Dic79—21no—
28C4⋊C4×Dic79C2212yes—
29C42×Dic79C2211yescharacteristic
30#19098326C2214yes—
31#19098325C2214yes—
32C2×C4⋊Q8×C79C2211yescharacteristic
33C2×#103×C79C2214yes—
34C2×C4×C8×C79C2211yescharacteristic
35#384×C79C2211yescharacteristic
36C22×Dic158—21no—
37C22×C79⋊C8C2312yes—
38#14579161—22no—
39#14579161C2×C414yes—
40Q8×Dic79—28no—
41#14579126—44no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79—21no—
44C2×C4⋊Dic79—21no—
45C2×C4⋊Dic79—22no—
46C2×C4⋊Dic79C2312yes—
47C2×C4⋊Dic79C2312yes—
48C2×Dic79⋊C4—22no—
49C2×C4×Dic79—22no—
50C2×C4×Dic79—22no—
51C2×C4×Dic79D411yescharacteristic
52C2×C4×Dic79D411yescharacteristic
53C2×C4×Dic79D411yescharacteristic
54C2×C4×Dic79D411yescharacteristic
55C2×C4×Dic79C2311yescharacteristic
56C2×C4×Dic79C2311yescharacteristic
57#14579085C2×C4116yes—
58C4×C79⋊C8C2×C414yes—
59C8×Dic79—24no—
60C8×Dic79—24no—
61C79⋊#95—22no—
62C79⋊#95—22no—
63C79⋊#95—22no—
64C79⋊#95—22no—
65C22×Q8×C79C2312yes—
66C22×C8×C79C2312yes—
67C4⋊Q8×C79—22no—
68C4⋊Q8×C79C2×C414yes—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79—21no—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79C2312yes—
74C2×C42×C79C2311yescharacteristic
75#103×C79—28no—
76C4×C8×C79—22no—
77C2×Dic158—24no—
78C2×Dic158—42no—
79C2×Dic158—42no—
80C22×Dic79—22no—
81C22×Dic79—22no—
82C22×Dic79C2×D411yescharacteristic
83C22×Dic79C2×D411yescharacteristic
84C22×Dic79C2×D411yescharacteristic
85C22×Dic79C2×D411yescharacteristic
86C2×C79⋊C8—22no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8C22×C414yes—
89C4⋊Dic79—24no—
90C4⋊Dic79—22no—
91C4⋊Dic79—24no—
92C4⋊Dic79—44no—
93C4⋊Dic79—42no—
94C4⋊Dic79—42no—
95C4⋊Dic79C22×C414yes—
96Dic79⋊C4—44no—
97C4×Dic79—22no—
98C4×Dic79—22no—
99C4×Dic79—22no—
100C4×Dic79—44no—
101C4×Dic79—44no—
102C4×Dic79SD1614yes—
103C4×Dic79Q1614yes—
104C4×Dic79#4114yes—
105C2×Q8×C79—22no—
106C2×Q8×C79—24no—
107C2×Q8×C79—24no—
108C2×Q8×C79C22×C414yes—
109C22×C4×C79—22no—
110C22×C4×C79—22no—
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2411yescharacteristic
114C2×C8×C79—22no—
115C2×C8×C79—22no—
116C2×C8×C79#4114yes—
117C4⋊C4×C79—24no—
118C4⋊C4×C79—24no—
119C4⋊C4×C79—24no—
120C4⋊C4×C79—42no—
121C4⋊C4×C79—42no—
122C42×C79C22×C412yes—
123C42×C79#4112yes—
124C23×C79C22×D411yescharacteristic
125Q8×C79—28no—
126Q8×C79—44no—
127C2×C4×C79—24no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79C4×D412yes—
134C2×C4×C79C4×D412yes—
135C2×C4×C79C2×SD1612yes—
136C2×C4×C79C2×Q1612yes—
137C2×C4×C79#13512yes—
138C2×C4×C79#13512yes—
139C2×C4×C79C23×C411yescharacteristic
140C2×C4×C79C2×#4112yes—
141C2×C4×C79C2×#4112yes—
142C2×C4×C79C2×#4111yescharacteristic
143C2×Dic79—21no—
144C2×Dic79—21no—
145C2×Dic79—21no—
146C2×Dic79—21no—
147C2×Dic79—21no—
148C2×Dic79—21no—
149C2×Dic79—24no—
150C2×Dic79—42no—
151C2×Dic79—42no—
152C2×Dic79—42no—
153C2×Dic79—42no—
154C2×Dic79—42no—
155C2×Dic79#12412yes—
156C2×Dic79#12412yes—
157C2×Dic79C2×SD1612yes—
158C2×Dic79C2×SD1612yes—
159C2×Dic79C2×Q1612yes—
160C2×Dic79C2×Q1612yes—
161Dic158—48no—
162C8×C79—24no—
163C79⋊C8—24no—
164C22×C79C2×C4×D411yescharacteristic
165C22×C79C2×#12411yescharacteristic
166C22×C79#53211yescharacteristic
167C22×C79C22×SD1611yescharacteristic
168C22×C79C22×Q1611yescharacteristic
169C22×C79C2×#13511yescharacteristic
170C22×C79C2×#13511yescharacteristic
171C4×C79—44no—
172C4×C79—82no—
173C4×C79C4×SD1614yes—
174C4×C79C4×Q1614yes—
175C4×C79C4×#4112yes—
176C4×C79#52912yes—
177Dic79—22no—
178Dic79—22no—
179Dic79—22no—
180Dic79—44no—
181Dic79—82no—
182Dic79#48714yescomplemented
183C4×#487—791nomaximal, Sylow
184C2×C79C4×#12411yescharacteristic
185C2×C79C2×C4×SD1611yescharacteristic
186C2×C79C2×C4×Q1611yescharacteristic
187C2×C79C2×#48711yescharacteristic
188C2×C79#313111yescharacteristic
189C2×C79#316711yescharacteristic
190C2×C79#317011yescharacteristic
191C2×#487—791no—
192C4×C4⋊Q8—791no—
193C4×C4⋊Q8—791no—
194#1990—791no—
195#1956D7911yescharacteristic
196#1933—794no—
197#1834—791no—
198C4×#103—794no—
199C42×C8—791no—
200C79C4×#48711yescharacteristic, complemented, Sylow
201C2×C4⋊Q8—791no—
202C2×C4⋊Q8C2×D7911yescharacteristic
203C2×C4×Q8—792no—
204C2×C4×Q8—1581no—
205#487—798no—
206C2×#103—794no—
207C2×#103C2×D7914yes—
208C2×C4×C8—791no—
209C2×C4×C8—792no—
210C2×C4×C8—792no—
211C2×C4×C8C2×D7911yescharacteristic
212#391—792no—
213#391—1581no—
214#389—792no—
215#389—1581no—
216#389—1581no—
217#389—1581no—
218#384—791no—
219#384C2×D7911yescharacteristic
220C4×C4⋊C4—792no—
221C4×C4⋊C4—1581no—
222C4×C4⋊C4—1581no—
223C4×C4⋊C4—1581no—
224C43—791no—
225#340—794no—
226#340—794no—
227C22×Q8—1581no—
228C22×Q8C22×D7912yes—
229C22×C8—792no—
230C22×C8C22×D7912yes—
231C4⋊Q8—22no—
232C4⋊Q8—794no—
233C4⋊Q8—1582no—
234C4⋊Q8C4×D7914yes—
235C4×Q8—3164no—
236C4×Q8—1588no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—792no—
242C2×C4⋊C4—792no—
243C2×C4⋊C4—1582no—
244C2×C4⋊C4—1582no—
245C2×C4⋊C4—1581no—
246C2×C4⋊C4—1581no—
247C2×C4⋊C4—1581no—
248C2×C4⋊C4C22×D7912yes—
249C2×C42—791no—
250C2×C42—791no—
251C2×C42—791no—
252C2×C42—791no—
253C2×C42—791no—
254C2×C42—791no—
255C2×C42—1582no—
256C2×C42—1582no—
257C2×C42C22×D7911yescharacteristic
258#103—28no—
259#103—7916no—
260C4×C8—22no—
261C4×C8—794no—
262C4×C8—1584no—
263C4×C8—1584no—
264#95—1582no—
265#95—1582no—
266#95—1582no—
267#95—1582no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—3162no—
272C2×Q8—3162no—
273C2×Q8—1584no—
274C2×Q8C2×C4×D7914yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—791no—
278C22×C4—791no—
279C22×C4—791no—
280C22×C4—791no—
281C22×C4—1582no—
282C22×C4—1582no—
283C22×C4D4×D7911yescharacteristic
284C22×C4D4×D7911yescharacteristic
285C22×C4C23×D7911yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—794no—
289C2×C8—1582no—
290C2×C8—1582no—
291C2×C8#1192962314yes—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—3164no—
298C4⋊C4—3164no—
299C4⋊C4—3162no—
300C4⋊C4—3162no—
301C4⋊C4—794no—
302C4⋊C4—1584no—
303C4⋊C4—1582no—
304C4⋊C4—1584no—
305C42—3164no—
306C42—3164no—
307C42—794no—
308C42—794no—
309C42—794no—
310C42—1582no—
311C42—1582no—
312C42—1582no—
313C42C2×C4×D7912yes—
314C42#1192962312yes—
315C23C2×D4×D7911yescharacteristic
316Q8—28no—
317Q8—44no—
318Q8—3168no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—3162no—
326C2×C4—3162no—
327C2×C4—3162no—
328C2×C4—3162no—
329C2×C4—3162no—
330C2×C4—792no—
331C2×C4—792no—
332C2×C4—792no—
333C2×C4—792no—
334C2×C4—792no—
335C2×C4—792no—
336C2×C4—1584no—
337C2×C4—1581no—
338C2×C4—1581no—
339C2×C4—1581no—
340C2×C4—1581no—
341C2×C4—1581no—
342C2×C4—1581no—
343C2×C4(C22⋊D79)⋊C412yes—
344C2×C4(C22⋊D79)⋊C412yes—
345C2×C4SD16×D7912yes—
346C2×C4Q16×D7912yes—
347C2×C4#1457918212yes—
348C2×C4#1457918412yes—
349C2×C4C22×C4×D7911yescharacteristic
350C2×C4#41×D7912yes—
351C2×C4C2×#1192962312yes—
352C2×C4C2×#1192962311yescharacteristic
353C8—24no—
354C8—1584no—
355C22C2×(C22⋊D79)⋊C411yescharacteristic
356C22#124×D7911yescharacteristic
357C22#1909911111yescharacteristic
358C22C2×SD16×D7911yescharacteristic
359C22C2×Q16×D7911yescharacteristic
360C22C2×#1457918211yescharacteristic
361C22C2×#1457918411yescharacteristic
362C4—44no—
363C4—82no—
364C4—3164no—
365C4—794no—
366C4—6322no—
367C4—1582no—
368C4—1582no—
369C4—1582no—
370C4#1909864814yes—
371C4#1909865514yes—
372C4C4×#1192962312yes—
373C4#1909909412yes—
374C2#2376594711yescharacteristic
375C2C2×#1909864811yescharacteristic
376C2C2×#1909865511yescharacteristic
377C2#487×D7911yescharacteristic
378C2#2377372411yescharacteristic
379C2#2377395711yescharacteristic
380C2#2377398411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1956C2
socle, prime radicalC23×C79C22×D4
cosocleC22×C4×C79C24
hypercenter#1956D79
hypocenterC79C4×#487
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
13. #19098913C4semidirect
182. Dic79#487semidirect
200. C79C4×#487semidirect

Maximal subgroups

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

subgroupindex
C2×#190989132
C4⋊Q8×Dic792
#237659672
#237643612
#237640732
#237635272
#103×Dic792
C4×C8×Dic792
C79×#19562
C4×#48779