Finite Group Databasebeta

← #34198664

Subgroups — #34198664

Order 20224.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#23769317C211yesmaximal, characteristic
2C4×#14579078C211yesmaximal, characteristic
3#23761431C212yesmaximal, complemented
4#23761407C212yesmaximal, complemented
5C79×#1459C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7C2×#14579078—21no—
8C4×#11929608—21no—
9#19098493C2211yescharacteristic
10#19098491—21no—
11#19098488C411yescharacteristic
12#19098483C2211yescharacteristic
13#19098477—21no—
14#19098475C411yescharacteristic
15#19098447—21no—
16#19098436—21no—
17#19098426C411yescharacteristic
18#19098332C2212yes—
19C2×C4×D4×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×Dic158C2×C411yescharacteristic
24C2×#11929608—22no—
25#14579140—41no—
26#14579140—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35#14579078—24no—
36#14579072—41no—
37C4×C79⋊C8—41no—
38C79⋊#95—21no—
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—
57C2×Dic158—41no—
58C2×Dic158—81no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63#11929608—24no—
64#11929608—42no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—41no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—42no—
70Dic79⋊C4—81no—
71C22⋊Dic79—42no—
72C22⋊Dic79—42no—
73C22⋊Dic79—81no—
74C22⋊Dic79—81no—
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—81no—
85C22×C4×C79—81no—
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—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—41no—
116C2×C4×C79—42no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#9911yescharacteristic
125C2×C4×C79#13512yes—
126C2×Dic79—41no—
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—
138Dic158—82no—
139C8×C79—82no—
140C79⋊C8—44no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—41no—
150C22×C79—81no—
151C22×C79—81no—
152C22×C79—81no—
153C22×C79C2×#9911yescharacteristic
154C22×C79#41611yescharacteristic
155C22×C79C2×#10411yescharacteristic
156C4×C79—21no—
157C4×C79—21no—
158C4×C79—22no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162Dic79—81no—
163Dic79—81no—
164Dic79—81no—
165Dic79—81no—
166#11739—791nomaximal, Sylow
167C2×C79—21no—
168C2×C79—21no—
169C2×C79—42no—
170C2×C79—81no—
171C2×C79#150111yescharacteristic
172C2×C79#212111yescharacteristic
173C2×C79#213511yescharacteristic
174#2638—791no—
175C4×#100—791no—
176#1459D7911yescharacteristic
177#1354—792no—
178#1346—792no—
179C79#1173911yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—791no—
181C2×C4×D4C2×D7911yescharacteristic
182C2×#100—1581no—
183C2×C22⋊C8—21no—
184C4×M16—1581no—
185#398—791no—
186#397—791no—
187#397—1581no—
188#395—791no—
189#394—1581no—
190#393—791no—
191#387—1581no—
192#385—1581no—
193#381—791no—
194#343—792no—
195#324C2×D7912yes—
196#323—21no—
197C22×Q8—791no—
198C22×D4D31611yescharacteristic
199C23×C4C4×D7911yescharacteristic
200C23×C4C22⋊D7911yescharacteristic
201C2×M16—1582no—
202C22×C8—41no—
203C22⋊Q8—3161no—
204C22⋊Q8—3161no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—791no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4C22×D7911yescharacteristic
212C2×C22⋊C4—1581no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C22⋊C4C4×D7911yescharacteristic
217C2×C22⋊C4C22⋊D7911yescharacteristic
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—
230#95—1581no—
231C24(C2×D79)⋊C411yescharacteristic
232C24(C2×D79)⋊C411yescharacteristic
233C2×Q8—3161no—
234C2×Q8—6321no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—1581no—
245C22×C4—1581no—
246C22×C4—1581no—
247C22×C4—1581no—
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4(C2×D79)⋊C411yescharacteristic
250C22×C4C2×C4×D7911yescharacteristic
251C22×C4C2×C22⋊D7911yescharacteristic
252C22×C4C2×D31611yescharacteristic
253M16—3162no—
254M16—1584no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—3162no—
258C4⋊C4—41no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—3161no—
262C4⋊C4—3162no—
263C4⋊C4—6321no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—3162no—
267C22⋊C4—3162no—
268C22⋊C4—6321no—
269C22⋊C4—6321no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1457908712yes—
282C23C2×(C2×D79)⋊C411yescharacteristic
283Q8—6322no—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
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—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4#1457907511yescharacteristic
311C2×C4#1457907511yescharacteristic
312C2×C4#1457918312yes—
313C8—82no—
314C8—3164no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#1457907511yescharacteristic
328C22#1909856011yescharacteristic
329C22C2×#1457908711yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—6321no—
337C4—6321no—
338C4—6321no—
339C4—6321no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2376189211yescharacteristic
345C2#2376504411yescharacteristic
346C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#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#11739
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #23761431C2semidirect
4. #23761407C2semidirect
179. C79#11739semidirect

Maximal subgroups

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

subgroupindex
#237693172
C4×#145790782
#237614312
#237614072
C79×#14592
#1173979