Finite Group Databasebeta

← #34176803

Subgroups — #34176803

Order 20224.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771731C211yesmaximal, characteristic
2#23763162C211yesmaximal, characteristic, complemented
3#23761222C212yesmaximal, complemented
4#23761220C212yesmaximal, complemented
5C79×#1797C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C411yescharacteristic
8C2×C22⋊(C79⋊C8)—21no—
9C2×#14579072—21no—
10#19098465C411yescharacteristic
11#19098465C2211yescharacteristic
12#19098461C411yescharacteristic
13#19098461C2211yescharacteristic
14#19098454—21no—
15#19098454—21no—
16#19098442—21no—
17#19098423C411yescharacteristic
18#19098318C2212yes—
19C22×C42×C79C2211yescharacteristic
20C2×C4⋊C8×C79—21no—
21C2×C22⋊C8×C79—21no—
22#336×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C22×C79⋊C8—22no—
26#14579144—44no—
27#14579135—42no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×(C2×D79)⋊C4—21no—
39C22⋊(C79⋊C8)—24no—
40#14579072—42no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C22×C8×C79—22no—
48C2×C42×C79—42no—
49C2×C42×C79C2×C411yescharacteristic
50C2×C42×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79C2311yescharacteristic
53C4⋊C8×C79—42no—
54C22⋊C8×C79—42no—
55C23×D79—21no—
56C2×D316—22no—
57C2×D316—41no—
58C2×D316—41no—
59C2×C22⋊D79—42no—
60C2×C22⋊D79—42no—
61C2×C22⋊D79—42no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—42no—
67C2×C79⋊C8—42no—
68C4⋊Dic79—22no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71Dic79⋊C4—42no—
72Dic79⋊C4—42no—
73C22⋊Dic79—42no—
74(C2×D79)⋊C4—42no—
75(C2×D79)⋊C4—42no—
76(C2×D79)⋊C4—42no—
77(C2×D79)⋊C4—41no—
78(C2×D79)⋊C4—42no—
79(C2×D79)⋊C4—41no—
80C24×C79C22⋊C411yescharacteristic
81C22×C4×C79—22no—
82C22×C4×C79—22no—
83C22×C4×C79—22no—
84C22×C4×C79—21no—
85C22×C4×C79—22no—
86C22×C4×C79—21no—
87C22×C4×C79—41no—
88C22×C4×C79—41no—
89C22×C4×C79—41no—
90C22×C4×C79—41no—
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22×C411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C2×C8×C79—42no—
98C2×C8×C79—42no—
99C2×C8×C79—42no—
100C42×C79—22no—
101C42×C79—22no—
102C42×C79—41no—
103C42×C79—41no—
104C23×C79—21no—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79#10412yes—
111C23×C79C2×C22⋊C411yescharacteristic
112C22×D79—41no—
113C22×D79—41no—
114C22×D79—41no—
115C22×D79—41no—
116C22×D79—41no—
117C22×D79—41no—
118C22×D79—41no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
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—44no—
134C2×C4×C79—44no—
135C2×C4×C79—42no—
136C2×C4×C79—42no—
137C2×C4×C79#9912yes—
138C2×C4×C79#13112yes—
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79AGL(1,ℤ/8)12yes—
141C2×C4×C79#13711yescharacteristic
142C22⋊D79—84no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152D316—82no—
153C8×C79—82no—
154C79⋊C8—44no—
155C22×C79—22no—
156C22×C79—22no—
157C22×C79—22no—
158C22×C79—41no—
159C22×C79—41no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79#40811yescharacteristic
163C22×C79C2×#9911yescharacteristic
164C22×C79#41311yescharacteristic
165C22×C79#41711yescharacteristic
166C22×C79#41811yescharacteristic
167C22×C79C2×#10411yescharacteristic
168C22×C79#42111yescharacteristic
169C2×D79—81no—
170C2×D79—81no—
171C2×D79—81no—
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C2×D79—81no—
176C4×C79—22no—
177C4×C79—22no—
178C4×C79—22no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—41no—
182C4×C79—41no—
183Dic79—81no—
184Dic79—81no—
185Dic79—81no—
186#7698—791nomaximal, Sylow
187C2×C79—42no—
188C2×C79#146811yescharacteristic
189C2×C79#148311yescharacteristic
190C2×C79#149611yescharacteristic
191C2×C79#150711yescharacteristic
192C2×C79#151311yescharacteristic
193C2×C79#153211yescharacteristic
194C2×C79#155711yescharacteristic
195D79—161no—
196#2854—791no—
197#1797D7911yescharacteristic, complemented
198#1749—791no—
199#1271—792no—
200#1270—792no—
201C79#769811yescharacteristic, complemented, Sylow
202C2×#123—1581no—
203C2×#121—791no—
204C22×C42C2×D7911yescharacteristic
205C2×C4⋊C8—21no—
206C2×C22⋊C8—21no—
207C2×C22⋊C8—1581no—
208C2×#97—1581no—
209#390—791no—
210#390—791no—
211#389—791no—
212#389—791no—
213#388—1581no—
214#388—1581no—
215#386—1581no—
216#380—791no—
217#336—792no—
218#336C2×D7912yes—
219C22×D4—791no—
220C22×D4—1581no—
221C23×C4C4×D7911yescharacteristic
222C23×C4D31611yescharacteristic
223C23×C4C22⋊D7911yescharacteristic
224C22×C8—22no—
225C22×C8—1582no—
226#123—3164no—
227#121—3162no—
228C2×C4⋊C4—791no—
229C2×C4⋊C4—791no—
230C2×C4⋊C4—791no—
231C2×C4⋊C4—1581no—
232C2×C4⋊C4—1581no—
233C2×C22⋊C4—1581no—
234C2×C22⋊C4—1581no—
235C2×C22⋊C4—1581no—
236C2×C22⋊C4—1581no—
237C2×C22⋊C4—1581no—
238C2×C22⋊C4—1581no—
239C2×C42—42no—
240C2×C42C4×D7911yescharacteristic
241C2×C42D31611yescharacteristic
242C2×C42C22⋊D7911yescharacteristic
243C2×C42C22×D7911yescharacteristic
244C4⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—1584no—
247#97—3162no—
248#95—1581no—
249#95—1581no—
250#95—1581no—
251C24—1581no—
252C24(C2×D79)⋊C411yescharacteristic
253C2×D4—3161no—
254C2×D4—3161no—
255C2×D4—3162no—
256C2×D4—3162no—
257C2×D4—3162no—
258C2×D4—1582no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1581no—
270C22×C4—1581no—
271C22×C4—1581no—
272C22×C4(C2×D79)⋊C411yescharacteristic
273C22×C4(C2×D79)⋊C411yescharacteristic
274C22×C4(C2×D79)⋊C411yescharacteristic
275C22×C4C2×C4×D7911yescharacteristic
276C22×C4C2×C22⋊D7911yescharacteristic
277C22×C4C2×D31611yescharacteristic
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—3162no—
282C2×C8—3162no—
283C2×C8—1584no—
284C4⋊C4—3161no—
285C4⋊C4—3161no—
286C4⋊C4—3162no—
287C4⋊C4—3162no—
288C4⋊C4—1582no—
289C22⋊C4—3162no—
290C22⋊C4—3162no—
291C22⋊C4—3162no—
292C22⋊C4—3162no—
293C22⋊C4—3161no—
294C22⋊C4—3162no—
295C22⋊C4—3161no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—3161no—
307C23—3161no—
308C23—3161no—
309C23—3161no—
310C23—3161no—
311C23—3161no—
312C23—3161no—
313C23#1457908712yes—
314C23C2×(C2×D79)⋊C411yescharacteristic
315D4—6322no—
316D4—6324no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—3161no—
336C2×C4—3161no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3161no—
343C2×C4—3161no—
344C2×C4#1457907512yes—
345C2×C4#1457916712yes—
346C2×C4#1457918711yescharacteristic
347C2×C4#1457918912yes—
348C2×C4#1457919211yescharacteristic
349C8—82no—
350C8—3164no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—6321no—
359C22—6321no—
360C22—6321no—
361C22—6321no—
362C22—6321no—
363C22—6321no—
364C22—6321no—
365C22#1909852911yescharacteristic
366C22C2×#1457907511yescharacteristic
367C22#1909854511yescharacteristic
368C22#1909856211yescharacteristic
369C22#1909856911yescharacteristic
370C22C2×#1457908711yescharacteristic
371C22#1909858111yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—6321no—
380C4—6321no—
381C4—6321no—
382C2—42no—
383C2—12641no—
384C2#2376174611yescharacteristic
385C2#2376181111yescharacteristic
386C2#2376187011yescharacteristic
387C2#2376191711yescharacteristic
388C2#2376194111yescharacteristic
389C2#2376202111yescharacteristic
390C2#2376212011yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1797C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C4
cosocleC2×C42×C79C23
hypercenter#1797D79
hypocenterC79#7698
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #23763162C2semidirect
3. #23761222C2semidirect
4. #23761220C2semidirect
5. C79×#1797C2semidirect
197. #1797D79semidirect
201. C79#7698semidirect

Maximal subgroups

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

subgroupindex
#237717312
#237631622
#237612222
#237612202
C79×#17972
#769879