Finite Group Databasebeta

← #34176377

Subgroups — #34176377

Order 20224.

504 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579140C211yesmaximal, characteristic
2#23763380C211yesmaximal, characteristic
3#23761225C212yesmaximal, complemented
4#23761222C212yesmaximal, complemented
5C79×#1817C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7C2×C4×Dic158C411yescharacteristic
8C2×C4⋊(C79⋊C8)—21no—
9C2×C22⋊(C79⋊C8)—21no—
10#19098469C2211yescharacteristic
11#19098461C2211yescharacteristic
12#19098447—21no—
13#19098436—21no—
14#19098423C411yescharacteristic
15C4×Dic79⋊C4—21no—
16C4×C4⋊Dic79C411yescharacteristic
17C4×C22⋊Dic79—21no—
18#19098318C2212yes—
19C22×C42×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#336×C79—21no—
22#336×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C22×C79⋊C8—22no—
25#14579140—22no—
26#14579140C814yescomplemented
27C4×Dic158—42no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—42no—
39C22⋊(C79⋊C8)—42no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C23×C4×C79C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C22×C8×C79—22no—
49C2×C42×C79—42no—
50C2×C42×C79C2×C411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79C2311yescharacteristic
54C22⋊C8×C79—42no—
55C2×Dic158—21no—
56C2×Dic158—41no—
57C2×Dic158—41no—
58C2×Dic158C2×C812yes—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C79⋊C8—42no—
64C2×C79⋊C8—42no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—21no—
67C4⋊Dic79—41no—
68C4⋊Dic79—41no—
69C4⋊Dic79C2×C812yes—
70Dic79⋊C4—24no—
71Dic79⋊C4—42no—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74C22⋊Dic79—24no—
75C22⋊Dic79—42no—
76C4×Dic79—42no—
77C4×Dic79—42no—
78C24×C79C22⋊C411yescharacteristic
79C22×C4×C79—22no—
80C22×C4×C79—21no—
81C22×C4×C79—22no—
82C22×C4×C79—21no—
83C22×C4×C79—41no—
84C22×C4×C79—41no—
85C22×C4×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79—41no—
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C2×C812yes—
92C22×C4×C79M1612yes—
93C22×C4×C79C22×C411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C2×C8×C79—42no—
97C2×C8×C79—42no—
98C2×C8×C79—42no—
99C42×C79—22no—
100C42×C79—22no—
101C42×C79—41no—
102C42×C79—41no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79C22⋊C812yes—
110C23×C79C2×C22⋊C411yescharacteristic
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—44no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79C22⋊C812yes—
130C2×C4×C79#9912yes—
131C2×C4×C79C22×C811yescharacteristic
132C2×C4×C79C2×M1611yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79#13712yes—
135C2×C4×C79#13711yescharacteristic
136C2×Dic79—22no—
137C2×Dic79—22no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148Dic158—42no—
149Dic158—42no—
150C8×C79—82no—
151C79⋊C8—82no—
152C22×C79—22no—
153C22×C79—22no—
154C22×C79—22no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79—41no—
159C22×C79C2×C22⋊C811yescharacteristic
160C22×C79C2×#9911yescharacteristic
161C22×C79#41311yescharacteristic
162C22×C79#41711yescharacteristic
163C22×C79#41911yescharacteristic
164C22×C79#42111yescharacteristic
165C22×C79#42111yescharacteristic
166C4×C79—22no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—22no—
170C4×C79—22no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—41no—
174Dic79—42no—
175Dic79—42no—
176Dic79—81no—
177Dic79—81no—
178#7600—791nomaximal, Sylow
179C2×C79—42no—
180C2×C79#145911yescharacteristic
181C2×C79#147711yescharacteristic
182C2×C79#147911yescharacteristic
183C2×C79#151111yescharacteristic
184C2×C79#151311yescharacteristic
185C2×C79#153511yescharacteristic
186C2×C79#155911yescharacteristic
187C4×C22⋊Q8—791no—
188#1817D7911yescharacteristic
189#1797—791no—
190#1272—792no—
191#1271—792no—
192C79#760011yescharacteristic, complemented, Sylow
193C2×C22⋊Q8—791no—
194C2×C4×Q8—791no—
195C22×C42C2×D7911yescharacteristic
196C2×C4⋊C8—1581no—
197C2×C22⋊C8—21no—
198C2×C22⋊C8—1581no—
199#391—791no—
200#389—791no—
201#387—1581no—
202#385—1581no—
203#380—791no—
204C4×C4⋊C4—791no—
205C4×C4⋊C4—1581no—
206C4×C22⋊C4—1581no—
207#336—21no—
208#336—792no—
209#336C2×D7912yes—
210C22×Q8—791no—
211C23×C4C4×D7911yescharacteristic
212C23×C4D31611yescharacteristic
213C23×C4C22⋊D7911yescharacteristic
214C22×C8—22no—
215C22×C8—1582no—
216C22⋊Q8—794no—
217C22⋊Q8—1582no—
218C4×Q8—3162no—
219C2×C4⋊C4—791no—
220C2×C4⋊C4—791no—
221C2×C4⋊C4—791no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4—1581no—
225C2×C22⋊C4—1581no—
226C2×C22⋊C4—1581no—
227C2×C42—42no—
228C2×C42—1581no—
229C2×C42—1581no—
230C2×C42C4×D7911yescharacteristic
231C2×C42D31611yescharacteristic
232C2×C42C22⋊D7911yescharacteristic
233C2×C42C22×D7911yescharacteristic
234C4⋊C8—3162no—
235C22⋊C8—42no—
236C22⋊C8—3162no—
237#95—1581no—
238#95—1581no—
239#95—1581no—
240#95—1581no—
241#95—1581no—
242C24(C2×D79)⋊C411yescharacteristic
243C2×Q8—3161no—
244C2×Q8—3161no—
245C2×Q8—792no—
246C2×Q8—1581no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4—1581no—
259C22×C4—1581no—
260C22×C4(C2×D79)⋊C411yescharacteristic
261C22×C4(C2×D79)⋊C411yescharacteristic
262C22×C4(C2×D79)⋊C411yescharacteristic
263C22×C4C8×D7912yes—
264C22×C4#1192960712yes—
265C22×C4C2×C4×D7911yescharacteristic
266C22×C4C2×C22⋊D7911yescharacteristic
267C22×C4C2×D31611yescharacteristic
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—3162no—
272C2×C8—3162no—
273C2×C8—3162no—
274C4⋊C4—3161no—
275C4⋊C4—3161no—
276C4⋊C4—3162no—
277C4⋊C4—3161no—
278C4⋊C4—3161no—
279C4⋊C4—792no—
280C4⋊C4—1584no—
281C4⋊C4—1581no—
282C22⋊C4—3162no—
283C22⋊C4—1584no—
284C42—22no—
285C42—22no—
286C42—41no—
287C42—41no—
288C42—3162no—
289C42—3162no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23(C2×D79)⋊C812yes—
297C23C2×(C2×D79)⋊C411yescharacteristic
298Q8—3162no—
299Q8—3162no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—3161no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—1582no—
329C2×C4—1582no—
330C2×C4(C2×D79)⋊C812yes—
331C2×C4#1457907512yes—
332C2×C4C2×C8×D7911yescharacteristic
333C2×C4C2×#1192960711yescharacteristic
334C2×C4#1457918911yescharacteristic
335C2×C4#1457919112yes—
336C2×C4#1457919411yescharacteristic
337C8—82no—
338C8—6322no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×(C2×D79)⋊C811yescharacteristic
347C22C2×#1457907511yescharacteristic
348C22#1909854511yescharacteristic
349C22#1909856511yescharacteristic
350C22#1909857111yescharacteristic
351C22#1909857911yescharacteristic
352C22#1909858111yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—3162no—
362C4—3162no—
363C4—6321no—
364C4—6321no—
365C2—42no—
366C2#2376171611yescharacteristic
367C2#2376178811yescharacteristic
368C2#2376179411yescharacteristic
369C2#2376193211yescharacteristic
370C2#2376194411yescharacteristic
371C2#2376202911yescharacteristic
372C2#2376213111yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC2×C4×C79C22×C8
FrattiniC2×C42C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1817D79
hypocenterC79#7600
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #23761225C2semidirect
4. #23761222C2semidirect
26. #14579140C8semidirect
192. C79#7600semidirect

Maximal subgroups

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

subgroupindex
C4×#145791402
#237633802
#237612252
#237612222
C79×#18172
#760079