Finite Group Databasebeta

← #34182267

Subgroups — #34182267

Order 20224.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763592C212yesmaximal
2C2×#19098289C211yesmaximal, characteristic
3#23761653C211yesmaximal, characteristic
4#23761278C212yesmaximal
5C79×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic79C2211yescharacteristic
7C2×#14579092—22no—
8C2×C22⋊(C79⋊C8)C2212yes—
9C2×C79⋊#95—21no—
10C2×C79⋊#95—21no—
11C2×C79⋊#95C2211yescharacteristic
12#19098325—22no—
13#19098289C414yescomplemented
14C2×#117×C79—21no—
15C22×C4⋊C4×C79C2211yescharacteristic
16#389×C79—22no—
17#389×C79—22no—
18#380×C79—21no—
19#380×C79C2212yes—
20C23×Dic79—21no—
21C23×Dic79—21no—
22C22×C79⋊C8—22no—
23C2×C4⋊Dic79—22no—
24C2×C4⋊Dic79—42no—
25#14579092—44no—
26C22⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)C2×C414yes—
28C79⋊#95—22no—
29C79⋊#95—42no—
30C79⋊#95—42no—
31C23×C4×C79D411yescharacteristic
32C23×C4×C79D411yescharacteristic
33C23×C4×C79C2311yescharacteristic
34#117×C79—42no—
35#117×C79—42no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—22no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79C2×C412yes—
43C2×C4⋊C4×C79D412yes—
44C2×C4⋊C4×C79D412yes—
45C2×C4⋊C4×C79Q812yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79—22no—
48#95×C79—22no—
49#95×C79—22no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—41no—
53C22×Dic79—41no—
54C22×Dic79—41no—
55C22×Dic79—41no—
56C22×Dic79—41no—
57C22×Dic79—41no—
58C22×Dic79—41no—
59C22×Dic79—41no—
60C22×Dic79—41no—
61C22×Dic79—41no—
62C22×Dic79—41no—
63C22×Dic79—41no—
64C2×C79⋊C8—24no—
65C2×C79⋊C8—42no—
66C2×C79⋊C8—42no—
67C4⋊Dic79—24no—
68C4⋊Dic79—42no—
69C24×C79C2×D411yescharacteristic
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—22no—
78C22×C4×C79—41no—
79C22×C4×C79—41no—
80C22×C4×C79—41no—
81C22×C4×C79—41no—
82C22×C4×C79C4⋊C412yes—
83C22×C4×C79C4⋊C412yes—
84C22×C4×C79SD1612yes—
85C22×C4×C79SD1612yes—
86C22×C4×C79C22×C411yescharacteristic
87C22×C4×C79C2×D412yes—
88C22×C4×C79C2×Q811yescharacteristic
89C22×C4×C79#4112yes—
90C4⋊C4×C79—24no—
91C4⋊C4×C79—24no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—42no—
94C4⋊C4×C79—42no—
95C4⋊C4×C79—42no—
96C4⋊C4×C79—81no—
97C4⋊C4×C79—81no—
98C22⋊C4×C79—42no—
99C22⋊C4×C79—42no—
100C42×C79—44no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79C2×C4⋊C411yescharacteristic
106C23×C79#12011yescharacteristic
107C23×C79#12311yescharacteristic
108C23×C79C2×SD1611yescharacteristic
109C23×C79C2×SD1611yescharacteristic
110C23×C79AGL(1,ℤ/8)11yescharacteristic
111C23×C79#13711yescharacteristic
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—81no—
134C2×C4×C79#10614yescomplemented
135C2×C4×C79C4×D412yes—
136C2×C4×C79C22⋊Q812yes—
137C2×Dic79—42no—
138C2×Dic79—42no—
139C2×Dic79—42no—
140C2×Dic79—42no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145C2×Dic79—81no—
146C2×Dic79—81no—
147C2×Dic79—81no—
148C2×Dic79—81no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151C79⋊C8—44no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79#38511yescharacteristic
165C22×C79C2×#10611yescharacteristic
166C22×C79#42811yescharacteristic
167C22×C79#44811yescharacteristic
168C22×C79#45011yescharacteristic
169C22×C79#45411yescharacteristic
170C22×C79#45611yescharacteristic
171C22×C79B(4,𝔽2)11yescharacteristic
172C22×C79#45811yescharacteristic
173C22×C79#48111yescharacteristic
174C22×C79#48311yescharacteristic
175C4×C79—24no—
176C4×C79—22no—
177C4×C79—22no—
178C4×C79—42no—
179C4×C79—82no—
180C4×C79—82no—
181Dic79—82no—
182Dic79—82no—
183#8857—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—21no—
188C2×C79#159911yescharacteristic
189C2×C79#160411yescharacteristic
190C2×C79#161111yescharacteristic
191C2×C79#161611yescharacteristic
192C2×C79#185311yescharacteristic
193C2×C79#185711yescharacteristic
194C2×C79#186711yescharacteristic
195#2316D7911yescharacteristic
196#1853—792no—
197C2×#324—791no—
198#1441—791no—
199#1294—792no—
200C79#885711yescharacteristic, complemented, Sylow
201C2×#117—21no—
202C22×C4⋊C4—791no—
203C22×C4⋊C4C2×D7911yescharacteristic
204C2×#106—1582no—
205C2×C22⋊C8—792no—
206#389—22no—
207#389—22no—
208#380—21no—
209#380C2×D7912yes—
210C2×#95—791no—
211C2×#95—1581no—
212C2×#95—1581no—
213#340—1582no—
214#324—794no—
215C23×C4—1581no—
216C23×C4—1581no—
217C23×C4C22⋊D7911yescharacteristic
218C23×C4C22⋊D7911yescharacteristic
219C23×C4C22×D7911yescharacteristic
220C22×C8—1582no—
221#117—42no—
222#117—42no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—3162no—
230C2×C4⋊C4—1582no—
231C2×C4⋊C4Dic15812yes—
232C2×C4⋊C4C4×D7912yes—
233C2×C4⋊C4C22⋊D7912yes—
234C2×C4⋊C4C22⋊D7912yes—
235C2×C22⋊C4—22no—
236C2×C42—22no—
237#106—3164no—
238C22⋊C8—794no—
239C22⋊C8—1582no—
240#95—22no—
241#95—22no—
242#95—3162no—
243#95—3162no—
244#95—1582no—
245C24C2×C22⋊D7911yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—3161no—
259C22×C4—3161no—
260C22×C4—3161no—
261C22×C4—3161no—
262C22×C4—3161no—
263C22×C4—3161no—
264C22×C4—3161no—
265C22×C4—3161no—
266C22×C4—3161no—
267C22×C4—3161no—
268C22×C4—3161no—
269C22×C4—3161no—
270C22×C4—1582no—
271C22×C4—1582no—
272C22×C4Dic79⋊C412yes—
273C22×C4Dic79⋊C412yes—
274C22×C4#1192961112yes—
275C22×C4#1192961212yes—
276C22×C4C2×C4×D7911yescharacteristic
277C22×C4C2×C22⋊D7912yes—
278C22×C4C2×Dic15811yescharacteristic
279C22×C4#1192962412yes—
280C2×C8—3162no—
281C2×C8—3162no—
282C2×C8—1584no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—3162no—
292C4⋊C4—1584no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—44no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×Dic79⋊C411yescharacteristic
301C23#1457913011yescharacteristic
302C23#1457914411yescharacteristic
303C23C2×#1192961111yescharacteristic
304C23C2×#1192961211yescharacteristic
305C23#1457918711yescharacteristic
306C23#1457919211yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—3162no—
330C2×C4—3162no—
331C2×C4—3162no—
332C2×C4—3162no—
333C2×C4—3162no—
334C2×C4—3162no—
335C2×C4—6321no—
336C2×C4—6321no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4—6321no—
341C2×C4—6321no—
342C2×C4—6321no—
343C2×C4#1457909214yescomplemented
344C2×C4C4×C22⋊D7912yes—
345C2×C4#1457914012yes—
346C8—3164no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1909843611yescharacteristic
360C22C2×#1457909211yescharacteristic
361C22#1909860311yescharacteristic
362C22#1909870211yescharacteristic
363C22#1909871411yescharacteristic
364C22#1909873611yescharacteristic
365C22#1909874611yescharacteristic
366C22#1909875011yescharacteristic
367C22#1909875311yescharacteristic
368C22#1909888111yescharacteristic
369C22#1909889311yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—6322no—
377C4—6322no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2376234011yescharacteristic
383C2#2376236911yescharacteristic
384C2#2376240811yescharacteristic
385C2#2376243711yescharacteristic
386C2#2376359311yescharacteristic
387C2#2376361511yescharacteristic
388C2#2376368111yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2316C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2316D79
hypocenterC79#8857
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
13. #19098289C4semidirect
134. C2×C4×C79#106semidirect
200. C79#8857semidirect
343. C2×C4#14579092semidirect

Maximal subgroups

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

subgroupindex
#237635922
C2×#190982892
#237616532
#237612782
C79×#23162
#885779