Finite Group Databasebeta

← #34198644

Subgroups — #34198644

Order 20224.

440 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 87 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#23766659C211yesmaximal, characteristic
2C4×#14579078C211yesmaximal, characteristic
3#23761428C212yesmaximal, complemented
4#23761407C212yesmaximal, complemented
5C79×#1459C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C2×#14579078—21no—
8C4×#11929608—21no—
9#19098502C2211yescharacteristic
10#19098499C411yescharacteristic
11#19098493C2211yescharacteristic
12#19098491—21no—
13#19098488C411yescharacteristic
14#19098477—21no—
15C4×Dic79⋊C4—21no—
16C4×C22⋊Dic79—21no—
17C4×C22⋊Dic79C411yescharacteristic
18#19098332C2212yes—
19C2×C4×D4×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C2×#11929608—22no—
24#14579120—41no—
25#14579120—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36#14579078—24no—
37#14579072—41no—
38C4×C79⋊C8—41no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95C2×C411yescharacteristic
44C79⋊#95C2×C411yescharacteristic
45C22×D4×C79D411yescharacteristic
46C23×C4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C22×C8×C79—41no—
49C4×D4×C79—41no—
50C4×D4×C79—41no—
51C2×C4⋊C4×C79C2311yescharacteristic
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C22⋊C8×C79—22no—
56C22⋊C8×C79—41no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61#11929608—24no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—41no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68C22⋊Dic79—42no—
69C22⋊Dic79—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C4×Dic79—42no—
73C4×Dic79—41no—
74C4×Dic79—41no—
75C24×C79C22⋊C411yescharacteristic
76C24×C79C22⋊C411yescharacteristic
77C2×D4×C79—22no—
78C2×D4×C79—22no—
79C2×D4×C79—41no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—42no—
85C22×C4×C79—42no—
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22×C411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C2×C8×C79—42no—
92C2×C8×C79—82no—
93C4⋊C4×C79—41no—
94C22⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C42×C79—21no—
97C42×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79#10412yes—
108C23×C79C2×C22⋊C411yescharacteristic
109D4×C79—44no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—22no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—81no—
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#13512yes—
126C2×Dic79—22no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C8×C79—82no—
139C79⋊C8—44no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—22no—
143C22×C79—42no—
144C22×C79—41no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—81no—
150C22×C79—81no—
151C22×C79—81no—
152C22×C79C2×#9911yescharacteristic
153C22×C79#41611yescharacteristic
154C22×C79C2×#10411yescharacteristic
155C4×C79—21no—
156C4×C79—22no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79#35212yes—
161Dic79—42no—
162Dic79—81no—
163Dic79—81no—
164Dic79—81no—
165#11735—791nomaximal, Sylow
166C2×C79—21no—
167C2×C79—21no—
168C2×C79—42no—
169C2×C79—81no—
170C2×C79#150111yescharacteristic
171C2×C79C2×#35211yescharacteristic
172C2×C79#213211yescharacteristic
173#2395—791no—
174C4×#100—791no—
175#1459D7911yescharacteristic
176#1353—792no—
177#1346—792no—
178C79#1173511yescharacteristic, complemented, Sylow
179C2×C4×D4C2×D7911yescharacteristic
180C2×#117—791no—
181C2×#100—1581no—
182C2×C22⋊C8—21no—
183C4×M16—1581no—
184#400—791no—
185#399—791no—
186#398—791no—
187#397—791no—
188#397—1581no—
189#394—1581no—
190C4×C4⋊C4—1581no—
191C4×C22⋊C4—791no—
192C4×C22⋊C4—1581no—
193#343—792no—
194#324C2×D7912yes—
195#323—21no—
196C22×D4D31611yescharacteristic
197C23×C4C4×D7911yescharacteristic
198C23×C4C22⋊D7911yescharacteristic
199C2×M16—1582no—
200C22×C8—41no—
201C4×D4—41no—
202C4×D4—41no—
203#117—3161no—
204#117—3161no—
205C2×C4⋊C4—791no—
206C2×C4⋊C4—791no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4C22×D7911yescharacteristic
210C2×C22⋊C4—1581no—
211C2×C22⋊C4—1581no—
212C2×C22⋊C4—1581no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4C4×D7911yescharacteristic
215C2×C22⋊C4C22⋊D7911yescharacteristic
216C2×C42—1581no—
217C2×C42—1581no—
218C2×C42D31611yescharacteristic
219#100—1584no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222#97—3161no—
223C4×C8—3161no—
224#95—791no—
225#95—791no—
226#95—1581no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230C24(C2×D79)⋊C411yescharacteristic
231C24(C2×D79)⋊C411yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—1581no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4—1581no—
245C22×C4(C2×D79)⋊C411yescharacteristic
246C22×C4(C2×D79)⋊C411yescharacteristic
247C22×C4C2×C4×D7911yescharacteristic
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×D31611yescharacteristic
250M16—3162no—
251M16—1584no—
252C2×C8—42no—
253C2×C8—82no—
254C2×C8—3162no—
255C4⋊C4—41no—
256C4⋊C4—3161no—
257C4⋊C4—3161no—
258C4⋊C4—3162no—
259C4⋊C4—3162no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—3162no—
263C22⋊C4—3162no—
264C22⋊C4—3162no—
265C22⋊C4—3162no—
266C42—21no—
267C42—21no—
268C42—3162no—
269C42—3161no—
270C42—3161no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1457908712yes—
281C23C2×(C2×D79)⋊C411yescharacteristic
282D4—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—1582no—
308C2×C4#1457907511yescharacteristic
309C2×C4#1457907511yescharacteristic
310C2×C4#1457918312yes—
311C8—82no—
312C8—3164no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#1457907511yescharacteristic
326C22#1909856011yescharacteristic
327C22C2×#1457908711yescharacteristic
328C4—21no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3162no—
334C4—6321no—
335C4—6321no—
336C4—6321no—
337C4#1909835212yes—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#2376189211yescharacteristic
343C2C2×#1909835211yescharacteristic
344C2#2376508311yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1459C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenter#1459D79
hypocenterC79#11735
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #23761428C2semidirect
4. #23761407C2semidirect
178. C79#11735semidirect

Maximal subgroups

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

subgroupindex
#237666592
C4×#145790782
#237614282
#237614072
C79×#14592
#1173579