Finite Group Databasebeta

← #34199193

Subgroups — #34199193

Order 20224.

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

normal subgroups only

This group has 438 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#23768880C211yesmaximal, characteristic
2#23763480C211yesmaximal, characteristic, complemented
3#23761428C212yesmaximal, complemented
4#23761404C212yesmaximal, complemented
5C2×#324×C79C211yesmaximal, characteristic, complemented
6C2×#14579135C411yescharacteristic
7#19098527—21no—
8#19098502C411yescharacteristic
9#19098502C2211yescharacteristic
10#19098480C411yescharacteristic
11#19098480C2211yescharacteristic
12#19098453—21no—
13#19098453—21no—
14#19098442—21no—
15#19098442—21no—
16C4×C22⋊Dic79C411yescharacteristic
17#19098332—21no—
18#19098332C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C2×#11929608—22no—
26#14579135—41no—
27#14579135—41no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C22⋊Dic79—21no—
30C2×(C2×D79)⋊C4—21no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×C4×Dic79—21no—
36C22⋊(C79⋊C8)—41no—
37C22⋊(C79⋊C8)—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C22×C8×C79—41no—
48C2×C4⋊C4×C79—41no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C22⋊C8×C79—22no—
55C22⋊C8×C79—41no—
56C23×D79—21no—
57C2×D316—41no—
58C2×D316—81no—
59C2×C22⋊D79—42no—
60C2×C22⋊D79—42no—
61C2×C22⋊D79—41no—
62C2×C22⋊D79—41no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66#11929608—42no—
67#11929608—42no—
68C2×C79⋊C8—42no—
69C4⋊Dic79—41no—
70C22⋊Dic79—42no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—42no—
74(C2×D79)⋊C4—81no—
75(C2×D79)⋊C4—81no—
76C4×Dic79—81no—
77C24×C79C22⋊C411yescharacteristic
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—22no—
85C22×C4×C79—41no—
86C22×C4×C79—81no—
87C22×C4×C79—81no—
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22×C411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C2×C8×C79—42no—
95C2×C8×C79—82no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C23×C79—22no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79#9911yescharacteristic
105C23×C79#9911yescharacteristic
106C23×C79C2×C22⋊C411yescharacteristic
107C22×D79—41no—
108C22×D79—41no—
109C22×D79—41no—
110C22×D79—41no—
111C22×D79—41no—
112C22×D79—41no—
113C22×D79—41no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—42no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79#13512yes—
134C22⋊D79—82no—
135C22⋊D79—82no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145D316—82no—
146C8×C79—82no—
147C79⋊C8—82no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—42no—
156C22×C79—42no—
157C22×C79#35512yes—
158C22×C79C2×#9911yescharacteristic
159C22×C79#41611yescharacteristic
160C22×C79#42111yescharacteristic
161C2×D79—81no—
162C2×D79—81no—
163C2×D79—81no—
164C2×D79—81no—
165C2×D79—81no—
166C2×D79—81no—
167C2×D79—81no—
168C4×C79—22no—
169C4×C79—41no—
170C4×C79—41no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—41no—
174Dic79—81no—
175Dic79—81no—
176Dic79—81no—
177#11852—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—22no—
181C2×C79—41no—
182C2×C79#151611yescharacteristic
183C2×C79#212111yescharacteristic
184C2×C79C2×#35511yescharacteristic
185D79—161no—
186#2591—791no—
187#1821—791no—
188C2×#324D7911yescharacteristic, complemented
189#1353—792no—
190#1345—792no—
191C79#1185211yescharacteristic, complemented, Sylow
192C2×#121—791no—
193C22×C4⋊C4C2×D7911yescharacteristic
194#407—1581no—
195C2×C22⋊C8—21no—
196#400—791no—
197#400—791no—
198#394—791no—
199#394—791no—
200#388—1581no—
201#388—1581no—
202#386—1581no—
203#386—1581no—
204C4×C22⋊C4—791no—
205#343—792no—
206#343—1581no—
207#324—21no—
208#324C2×D7912yes—
209C22×D4—791no—
210C22×D4—1581no—
211C23×C4C4×D7911yescharacteristic
212C23×C4D31611yescharacteristic
213C23×C4C22⋊D7911yescharacteristic
214C2×M16—1582no—
215C22×C8—41no—
216#121—3161no—
217#121—3161no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—791no—
221C2×C4⋊C4C4×D7911yescharacteristic
222C2×C4⋊C4D31611yescharacteristic
223C2×C4⋊C4C22⋊D7911yescharacteristic
224C2×C4⋊C4C22×D7911yescharacteristic
225C2×C22⋊C4—1581no—
226C2×C22⋊C4—1581no—
227C2×C22⋊C4—1581no—
228C2×C22⋊C4—1581no—
229C2×C22⋊C4—1581no—
230C2×C22⋊C4—1581no—
231C2×C42—1581no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—3161no—
235C22⋊C8—3161no—
236#95—791no—
237#95—791no—
238#95—1581no—
239#95—1581no—
240#95—1581no—
241#95—1581no—
242C24—1581no—
243C24(C2×D79)⋊C411yescharacteristic
244C2×D4—3161no—
245C2×D4—3162no—
246C2×D4—3162no—
247C2×D4—3161no—
248C2×D4—3161no—
249C2×D4—6321no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—1581no—
261C22×C4—1581no—
262C22×C4—1581no—
263C22×C4(C2×D79)⋊C411yescharacteristic
264C22×C4(C2×D79)⋊C411yescharacteristic
265C22×C4(C2×D79)⋊C411yescharacteristic
266C22×C4C2×C4×D7911yescharacteristic
267C22×C4C2×C22⋊D7911yescharacteristic
268C22×C4C2×D31611yescharacteristic
269M16—3162no—
270M16—3162no—
271C2×C8—42no—
272C2×C8—82no—
273C2×C8—3162no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—3161no—
279C22⋊C4—3162no—
280C22⋊C4—3162no—
281C22⋊C4—3162no—
282C22⋊C4—3162no—
283C22⋊C4—6321no—
284C22⋊C4—6321no—
285C42—6321no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—3161no—
291C23—3161no—
292C23—3161no—
293C23—3161no—
294C23—3161no—
295C23—3161no—
296C23—3161no—
297C23#1457907511yescharacteristic
298C23#1457907511yescharacteristic
299C23C2×(C2×D79)⋊C411yescharacteristic
300D4—6322no—
301D4—6322no—
302D4—6322no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—3161no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4#1457918312yes—
332C8—82no—
333C8—6322no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—6321no—
344C22—6321no—
345C22—6321no—
346C22—6321no—
347C22—6321no—
348C22—6321no—
349C22—6321no—
350C22#1909835912yes—
351C22C2×#1457907511yescharacteristic
352C22#1909856011yescharacteristic
353C22#1909857911yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6321no—
361C4—6321no—
362C4—6321no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—12641no—
368C2#2376195711yescharacteristic
369C2#2376504411yescharacteristic
370C2C2×#1909835911yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC2×#324×C79C2
socleC22×C79C2×#99
prime radicalC2×#14579135C4
cosocleC2×C4⋊C4×C79C23
hypercenterC2×#324D79
hypocenterC79#11852
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #23763480C2semidirect
3. #23761428C2semidirect
4. #23761404C2semidirect
5. C2×#324×C79C2semidirect
188. C2×#324D79semidirect
191. C79#11852semidirect

Maximal subgroups

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

subgroupindex
#237688802
#237634802
#237614282
#237614042
C2×#324×C792
#1185279