Finite Group Databasebeta

← #34215529

Subgroups — #34215529

Order 20224.

2070 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098423C211yesmaximal, characteristic
2C2×#19098421C214yesmaximal
3C22×C79⋊#95C211yesmaximal, characteristic
4#23761664C214yesmaximal, complemented
5#23761663C214yesmaximal, complemented
6C2×#380×C79C211yesmaximal, characteristic
7C24×Dic79C2212yes—
8C22×C4⋊Dic79C2212yes—
9C22×Dic79⋊C4C2218yes—
10C22×C22⋊Dic79C2212yes—
11#19098423—24no—
12#19098421—216no—
13C2×C79⋊#95C418yescomplemented
14C2×C79⋊#95C2218yes—
15C2×C79⋊#95C2212yes—
16C2×C79⋊#95C2212yes—
17C2×C79⋊#95C2212yes—
18C24×C4×C79C2211yescharacteristic
19C22×C4⋊C4×C79C2212yes—
20C22×C22⋊C4×C79C2212yes—
21#380×C79—24no—
22C2×#95×C79C2212yes—
23C23×Dic79—44no—
24C23×Dic79—44no—
25C23×Dic79C2×C418yes—
26C23×Dic79D418yes—
27C23×Dic79Q818yes—
28C23×Dic79C2314yes—
29C23×Dic79C2314yes—
30C2×C4⋊Dic79—24no—
31C2×C4⋊Dic79—44no—
32C2×Dic79⋊C4—216no—
33C2×Dic79⋊C4—48no—
34C2×Dic79⋊C4—48no—
35C2×C22⋊Dic79—28no—
36C2×C22⋊Dic79—24no—
37C79⋊#95—24no—
38C79⋊#95—216no—
39C79⋊#95C2×C418yes—
40C79⋊#95C2×C418yes—
41C25×C79C2311yescharacteristic
42C23×C4×C79—44no—
43C23×C4×C79C2×C414yes—
44C23×C4×C79D414yes—
45C23×C4×C79Q814yes—
46C23×C4×C79C2314yes—
47C23×C4×C79C2311yescharacteristic
48C23×C4×C79C2311yescharacteristic
49C2×C4⋊C4×C79—24no—
50C2×C4⋊C4×C79—44no—
51C2×C22⋊C4×C79—28no—
52C2×C22⋊C4×C79—24no—
53#95×C79—24no—
54C22×Dic79—28no—
55C22×Dic79—24no—
56C22×Dic79—28no—
57C22×Dic79—28no—
58C22×Dic79—24no—
59C22×Dic79—28no—
60C22×Dic79—24no—
61C22×Dic79—48no—
62C22×Dic79—48no—
63C22×Dic79—416no—
64C22×Dic79—416no—
65C22×Dic79C4⋊C4132yescomplemented
66C22×Dic79C22×C418yes—
67C22×Dic79C2×D418yes—
68C22×Dic79C2×Q818yes—
69C4⋊Dic79—48no—
70Dic79⋊C4—432no—
71C22⋊Dic79—48no—
72C24×C79—24no—
73C24×C79—22no—
74C24×C79—22no—
75C24×C79C22×C412yes—
76C24×C79C2×D414yes—
77C24×C79C2×D412yes—
78C24×C79C2×Q814yes—
79C24×C79C2×Q812yes—
80C24×C79C2411yescharacteristic
81C22×C4×C79—24no—
82C22×C4×C79—24no—
83C22×C4×C79—24no—
84C22×C4×C79—28no—
85C22×C4×C79—28no—
86C22×C4×C79—24no—
87C22×C4×C79—24no—
88C22×C4×C79—48no—
89C22×C4×C79—48no—
90C22×C4×C79—48no—
91C22×C4×C79C4⋊C4116yescomplemented
92C22×C4×C79C22×C412yes—
93C22×C4×C79C22×C412yes—
94C22×C4×C79C2×D412yes—
95C22×C4×C79C2×D412yes—
96C22×C4×C79C2×Q812yes—
97C22×C4×C79C2×Q812yes—
98C4⋊C4×C79—48no—
99C22⋊C4×C79—48no—
100C23×C79—22no—
101C23×C79—24no—
102C23×C79—22no—
103C23×C79—28no—
104C23×C79—24no—
105C23×C79—28no—
106C23×C79—24no—
107C23×C79—28no—
108C23×C79—24no—
109C23×C79—24no—
110C23×C79—28no—
111C23×C79C2×C4⋊C418yes—
112C23×C79C2×C4⋊C414yes—
113C23×C79#12714yes—
114C23×C79C4⋊Q814yes—
115C23×C79C4⋊Q818yes—
116C23×C79C23×C411yescharacteristic
117C23×C79C22×D412yes—
118C23×C79C22×D411yescharacteristic
119C23×C79C22×Q812yes—
120C23×C79C22×Q811yescharacteristic
121C23×C79H(5,𝔽2)14yes—
122C23×C79H(5,𝔽2)12yes—
123C23×C792−1+412yes—
124C2×C4×C79—24no—
125C2×C4×C79—24no—
126C2×C4×C79—24no—
127C2×C4×C79—24no—
128C2×C4×C79—24no—
129C2×C4×C79—24no—
130C2×C4×C79—416no—
131C2×C4×C79—44no—
132C2×C4×C79—48no—
133C2×C4×C79—44no—
134C2×C4×C79—48no—
135C2×C4×C79—44no—
136C2×C4×C79C2×C4⋊C414yes—
137C2×C4×C79C2×C4⋊C414yes—
138C2×Dic79—28no—
139C2×Dic79—28no—
140C2×Dic79—216no—
141C2×Dic79—216no—
142C2×Dic79—432no—
143C2×Dic79—44no—
144C2×Dic79—48no—
145C2×Dic79—44no—
146C2×Dic79—48no—
147C2×Dic79—44no—
148C2×Dic79C2×C4⋊C4116yes—
149C22×C79—24no—
150C22×C79—28no—
151C22×C79—24no—
152C22×C79—28no—
153C22×C79—28no—
154C22×C79—28no—
155C22×C79—216no—
156C22×C79#38418yes—
157C22×C79C22×C4⋊C412yes—
158C22×C79C22×C4⋊C411yescharacteristic
159C22×C79#51812yes—
160C22×C79#51811yescharacteristic
161C22×C79#51911yescharacteristic
162C22×C79C2×#12711yescharacteristic
163C22×C79C2×C4⋊Q811yescharacteristic
164C22×C79C2×C4⋊Q812yes—
165C22×C79#53414yes—
166C22×C79#53412yes—
167C22×C79#53714yes—
168C22×C79#53712yes—
169C22×C79#54314yes—
170C22×C79#54312yes—
171C22×C79#54412yes—
172C22×C79#54414yes—
173C4×C79—24no—
174C4×C79—24no—
175C4×C79—84no—
176Dic79—216no—
177Dic79—84no—
178#14353—791nomaximal, Sylow
179C2×C79—28no—
180C2×C79C2×#38411yescharacteristic
181C2×C79#231511yescharacteristic
182C2×C79#231512yes—
183C2×C79#231611yescharacteristic
184C2×C79#231612yes—
185C2×C79#265612yes—
186C2×C79#265912yes—
187C2×C79#265914yes—
188C2×#380—791no—
189C2×#380—794no—
190C2×#380D7911yescharacteristic
191C22×#95—791no—
192#1444—794no—
193#1444—794no—
194C79#1435311yescharacteristic, complemented, Sylow
195C24×C4—792no—
196C24×C4C2×D7911yescharacteristic
197C22×C4⋊C4—798no—
198C22×C4⋊C4—792no—
199C22×C4⋊C4C2×D7912yes—
200C22×C22⋊C4—792no—
201C22×C22⋊C4C2×D7912yes—
202#380—24no—
203#380—15816no—
204#380—1584no—
205C2×#95—792no—
206C2×#95—792no—
207C2×#95—798no—
208C2×#95—798no—
209C2×#95—792no—
210C2×#95C2×D7912yes—
211C25C22×D7911yescharacteristic
212C23×C4—44no—
213C23×C4—3164no—
214C23×C4—3164no—
215C23×C4—798no—
216C23×C4—798no—
217C23×C4—798no—
218C23×C4—794no—
219C23×C4—794no—
220C23×C4Dic15814yes—
221C23×C4C4×D7914yes—
222C23×C4C22⋊D7914yes—
223C23×C4C22×D7914yes—
224C23×C4C22×D7911yescharacteristic
225C23×C4C22×D7911yescharacteristic
226C2×C4⋊C4—24no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—3168no—
229C2×C4⋊C4—3168no—
230C2×C4⋊C4—3164no—
231C2×C4⋊C4—15816no—
232C2×C4⋊C4—1584no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—1588no—
236C2×C22⋊C4—1584no—
237#95—24no—
238#95—798no—
239#95—798no—
240#95—15816no—
241#95—1584no—
242C24—24no—
243C24—22no—
244C24—22no—
245C24C2×C4×D7912yes—
246C24D4×D7914yes—
247C24C2×C22⋊D7912yes—
248C24Q8×D7914yes—
249C24C2×Dic15812yes—
250C24C23×D7911yescharacteristic
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—31616no—
262C22×C4—3168no—
263C22×C4—3168no—
264C22×C4—31616no—
265C22×C4—7932no—
266C22×C4—798no—
267C22×C4—798no—
268C22×C4—798no—
269C22×C4—1588no—
270C22×C4—1584no—
271C22×C4—1584no—
272C22×C4—1588no—
273C22×C4—1584no—
274C22×C4—1588no—
275C22×C4—1588no—
276C22×C4Dic79⋊C4116yescomplemented
277C22×C4C2×C4×D7912yes—
278C22×C4C2×C4×D7912yes—
279C22×C4C2×C22⋊D7912yes—
280C22×C4C2×C22⋊D7912yes—
281C22×C4C2×Dic15812yes—
282C22×C4C2×Dic15812yes—
283C4⋊C4—48no—
284C4⋊C4—3168no—
285C4⋊C4—31632no—
286C22⋊C4—48no—
287C22⋊C4—3168no—
288C23—28no—
289C23—24no—
290C23—28no—
291C23—24no—
292C23—24no—
293C23—28no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—28no—
298C23—24no—
299C23C4⋊C4×D7918yes—
300C23C2×Dic79⋊C414yes—
301C23#1457915714yes—
302C23#1457915914yes—
303C23#1457916018yes—
304C23C22×C4×D7911yescharacteristic
305C23C2×D4×D7912yes—
306C23C22×C22⋊D7911yescharacteristic
307C23C2×Q8×D7912yes—
308C23C22×Dic15811yescharacteristic
309C23#1457920614yes—
310C23#1457920712yes—
311C23#1457920812yes—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—416no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—3168no—
325C2×C4—3164no—
326C2×C4—3164no—
327C2×C4—3168no—
328C2×C4—3164no—
329C2×C4—31632no—
330C2×C4—7916no—
331C2×C4—15816no—
332C2×C4—1588no—
333C2×C4—1588no—
334C2×C4—15816no—
335C2×C4C2×Dic79⋊C414yes—
336C2×C4C2×Dic79⋊C414yes—
337C22—24no—
338C22—28no—
339C22—28no—
340C22—28no—
341C22—216no—
342C22—24no—
343C22—28no—
344C22#1909843218yes—
345C22C2×C4⋊C4×D7912yes—
346C22C22×Dic79⋊C411yescharacteristic
347C22#1909903612yes—
348C22#1909903711yescharacteristic
349C22#1909904011yescharacteristic
350C22C2×#1457915711yescharacteristic
351C22C2×#1457915911yescharacteristic
352C22C2×#1457916012yes—
353C22#1909912114yes—
354C22#1909912212yes—
355C22#1909914014yes—
356C22#1909914112yes—
357C22#1909918814yes—
358C22#1909918912yes—
359C22#1909919112yes—
360C22#1909919214yes—
361C4—24no—
362C4—24no—
363C4—84no—
364C4—6324no—
365C4—15816no—
366C2—28no—
367C2C2×#1909843211yescharacteristic
368C2#2376601211yescharacteristic
369C2#2376601312yes—
370C2#2376601611yescharacteristic
371C2#2376601712yes—
372C2#2376950412yes—
373C2#2376951612yes—
374C2#2376951714yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#380×C79C2
socle, cosocleC24×C79C24
prime radicalC25×C79C23
hypercenterC2×#380D79
hypocenterC79#14353
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #23761664C2semidirect
5. #23761663C2semidirect
13. C2×C79⋊#95C4semidirect
65. C22×Dic79C4⋊C4semidirect
91. C22×C4×C79C4⋊C4semidirect
194. C79#14353semidirect
276. C22×C4Dic79⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#190984232
C2×#190984212
C22×C79⋊#952
#237616642
#237616632
C2×#380×C792
#1435379