Finite Group Databasebeta

← #34183673

Subgroups — #34183673

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098620C211yesmaximal, characteristic
2C2×#19098447C211yesmaximal, characteristic
3C2×#19098288C212yesmaximal
4#23761242C212yesmaximal
5C79×#1817C211yesmaximal, characteristic
6C22×Dic79⋊C4C411yescharacteristic
7C22×Dic79⋊C4C2211yescharacteristic
8#19098620—24no—
9C2×C4⋊(C79⋊C8)—21no—
10C2×C22⋊(C79⋊C8)C2212yes—
11C2×#14579072—21no—
12#19098447—21no—
13#19098447—21no—
14#19098447—21no—
15#19098447—21no—
16C2×C79⋊#95C411yescharacteristic
17C2×C79⋊#95C411yescharacteristic
18C2×C79⋊#95C411yescharacteristic
19C2×C79⋊#95C2211yescharacteristic
20#19098288—24no—
21C22×C42×C79C2211yescharacteristic
22C2×C22⋊C8×C79C2212yes—
23#336×C79—21no—
24#336×C79—21no—
25C23×Dic79C2×C411yescharacteristic
26C23×Dic79C2×C411yescharacteristic
27C23×Dic79C2×C411yescharacteristic
28C23×Dic79C2×C411yescharacteristic
29C22×C79⋊C8—22no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—22no—
33C2×Dic79⋊C4—22no—
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—21no—
36C2×Dic79⋊C4—22no—
37C2×Dic79⋊C4—22no—
38C4⋊(C79⋊C8)—24no—
39C22⋊(C79⋊C8)—22no—
40C22⋊(C79⋊C8)D414yes—
41#14579072—24no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C23×C4×C79C2×C411yescharacteristic
51C23×C4×C79C2×C411yescharacteristic
52C23×C4×C79C2311yescharacteristic
53C22×C8×C79—22no—
54C2×C42×C79—22no—
55C2×C42×C79—22no—
56C2×C42×C79D411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C2×C42×C79D411yescharacteristic
60C22⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—22no—
67C22×Dic79—22no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—22no—
71C22×Dic79—22no—
72C22×Dic79—41no—
73C22×Dic79—41no—
74C22×Dic79—41no—
75C22×Dic79—41no—
76C22×Dic79—41no—
77C22×Dic79—41no—
78C22×Dic79M1612yes—
79C22×Dic79M1612yes—
80C2×C79⋊C8—24no—
81C2×C79⋊C8—42no—
82C2×C79⋊C8—42no—
83Dic79⋊C4—24no—
84Dic79⋊C4—24no—
85Dic79⋊C4—41no—
86Dic79⋊C4—42no—
87Dic79⋊C4—41no—
88C24×C79C22×C411yescharacteristic
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—22no—
94C22×C4×C79—22no—
95C22×C4×C79—41no—
96C22×C4×C79—41no—
97C22×C4×C79—41no—
98C22×C4×C79C22⋊C411yescharacteristic
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C22⋊C411yescharacteristic
101C22×C4×C79C22⋊C411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D412yes—
105C22×C4×C79#4112yes—
106C2×C8×C79—42no—
107C2×C8×C79—42no—
108C2×C8×C79—42no—
109C42×C79—21no—
110C42×C79—21no—
111C42×C79—22no—
112C42×C79—22no—
113C42×C79—21no—
114C42×C79—21no—
115C23×C79—21no—
116C23×C79—21no—
117C23×C79—21no—
118C23×C79—21no—
119C23×C79C2×C22⋊C411yescharacteristic
120C23×C79C4×D412yes—
121C23×C79C2×M1611yescharacteristic
122C23×C79C2×M1611yescharacteristic
123C23×C79#13112yes—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—44no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79#10112yes—
145C2×C4×C79#12011yescharacteristic
146C2×C4×C79#12112yes—
147C2×C4×C79#12311yescharacteristic
148C2×Dic79—22no—
149C2×Dic79—22no—
150C2×Dic79—24no—
151C2×Dic79—42no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—42no—
157C2×Dic79—42no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—41no—
162C2×Dic79—41no—
163C2×Dic79—41no—
164C2×Dic79—82no—
165C2×Dic79—82no—
166C8×C79—82no—
167C79⋊C8—44no—
168C22×C79—22no—
169C22×C79—22no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—22no—
175C22×C79—22no—
176C22×C79#38611yescharacteristic
177C22×C79#40711yescharacteristic
178C22×C79#40811yescharacteristic
179C22×C79#41011yescharacteristic
180C22×C79#41011yescharacteristic
181C22×C79C2×#10111yescharacteristic
182C22×C79#41311yescharacteristic
183C22×C79#43512yes—
184C22×C79#43612yes—
185C4×C79—21no—
186C4×C79—22no—
187C4×C79—21no—
188C4×C79—22no—
189C4×C79—41no—
190C4×C79—41no—
191C4×C79—41no—
192Dic79—42no—
193Dic79—42no—
194Dic79—81no—
195Dic79—81no—
196#9086—791nomaximal, Sylow
197C2×C79—22no—
198C2×C79—22no—
199C2×C79#146411yescharacteristic
200C2×C79#146711yescharacteristic
201C2×C79#147011yescharacteristic
202C2×C79#147311yescharacteristic
203C2×C79#187011yescharacteristic
204C2×C79#189911yescharacteristic
205C2×C79#190111yescharacteristic
206C2×#432—791no—
207C2×#387—791no—
208#1817D7911yescharacteristic
209C2×#324—792no—
210#1280—792no—
211C79#908611yescharacteristic, complemented, Sylow
212C22×C4⋊C4—791no—
213C22×C4⋊C4—791no—
214C22×C42C2×D7911yescharacteristic
215#432—1584no—
216C2×C4⋊C8—1581no—
217C2×C22⋊C8—792no—
218C2×C22⋊C8C2×D7912yes—
219C2×#97—1581no—
220#387—1581no—
221#387—1581no—
222#387—1581no—
223#387—1581no—
224C2×#95—791no—
225C2×#95—791no—
226C2×#95—791no—
227C2×#95—791no—
228#336—21no—
229#336—21no—
230#324—1584no—
231C23×C4—791no—
232C23×C4—791no—
233C23×C4—791no—
234C23×C4—791no—
235C23×C4C4×D7911yescharacteristic
236C23×C4C4×D7911yescharacteristic
237C23×C4C22×D7911yescharacteristic
238C22×C8—22no—
239C22×C8—1582no—
240C2×C4⋊C4—1582no—
241C2×C4⋊C4—1581no—
242C2×C4⋊C4—1581no—
243C2×C4⋊C4—1582no—
244C2×C4⋊C4—1582no—
245C2×C4⋊C4—1582no—
246C2×C4⋊C4—1581no—
247C2×C4⋊C4—1581no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42D31611yescharacteristic
251C2×C42D31611yescharacteristic
252C2×C42C22⋊D7911yescharacteristic
253C2×C42C22⋊D7911yescharacteristic
254C4⋊C8—1584no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—794no—
258C22⋊C8—1582no—
259#97—1584no—
260#95—1581no—
261#95—1581no—
262#95—1581no—
263#95—1581no—
264#95—1581no—
265#95—1581no—
266#95—1581no—
267#95—1581no—
268C24C2×C4×D7911yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—3161no—
279C22×C4—3161no—
280C22×C4—3161no—
281C22×C4—3161no—
282C22×C4—3161no—
283C22×C4—3161no—
284C22×C4—792no—
285C22×C4—792no—
286C22×C4—1581no—
287C22×C4—1581no—
288C22×C4—1581no—
289C22×C4—1581no—
290C22×C4—1582no—
291C22×C4—1581no—
292C22×C4—1581no—
293C22×C4—1582no—
294C22×C4—1582no—
295C22×C4—1582no—
296C22×C4(C2×D79)⋊C411yescharacteristic
297C22×C4(C2×D79)⋊C411yescharacteristic
298C22×C4(C2×D79)⋊C411yescharacteristic
299C22×C4(C2×D79)⋊C411yescharacteristic
300C22×C4D4×D7912yes—
301C22×C4C2×C22⋊D7911yescharacteristic
302C22×C4C2×D31611yescharacteristic
303C22×C4#1192962312yes—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—3162no—
308C2×C8—3162no—
309C2×C8—1584no—
310C4⋊C4—3161no—
311C4⋊C4—3162no—
312C4⋊C4—3161no—
313C4⋊C4—1584no—
314C4⋊C4—1584no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C2×(C2×D79)⋊C411yescharacteristic
326C23(C22⋊D79)⋊C412yes—
327C23M16×D7911yescharacteristic
328C23M16×D7911yescharacteristic
329C23#1457916712yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—3162no—
351C2×C4—3162no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—3161no—
355C2×C4—3161no—
356C2×C4—3162no—
357C2×C4—3161no—
358C2×C4—3161no—
359C2×C4—3161no—
360C2×C4—3161no—
361C2×C4—3161no—
362C2×C4—3161no—
363C2×C4—6322no—
364C2×C4—6322no—
365C2×C4—1584no—
366C2×C4—1582no—
367C2×C4—1582no—
368C2×C4#1457907912yes—
369C2×C4#1457913111yescharacteristic
370C2×C4#1457913312yes—
371C2×C4#1457914511yescharacteristic
372C8—82no—
373C8—3164no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22#1909844011yescharacteristic
383C22#1909852611yescharacteristic
384C22#1909852911yescharacteristic
385C22#1909853611yescharacteristic
386C22#1909853611yescharacteristic
387C22C2×#1457907911yescharacteristic
388C22#1909854511yescharacteristic
389C22#1909863712yes—
390C22#1909863812yes—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3162no—
399C4—3162no—
400C4—6321no—
401C4—6321no—
402C2—22no—
403C2—22no—
404C2#2376173211yescharacteristic
405C2#2376174311yescharacteristic
406C2#2376175511yescharacteristic
407C2#2376176711yescharacteristic
408C2#2376369511yescharacteristic
409C2#2376385611yescharacteristic
410C2#2376386611yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutator, prime radicalC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
cosocleC23×C4×C79C23
hypercenter#1817D79
hypocenterC79#9086
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
211. C79#9086semidirect

Maximal subgroups

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

subgroupindex
C2×#190986202
C2×#190984472
C2×#190982882
#237612422
C79×#18172
#908679