Finite Group Databasebeta

← #34177346

Subgroups — #34177346

Order 20224.

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

normal subgroups only

This group has 394 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#23770431C211yesmaximal, characteristic
2#23764872C211yesmaximal, characteristic
3#23764809C211yesmaximal, characteristic
4#23761225C211yesmaximal, characteristic
5#23761223C211yesmaximal, characteristic
6#23761222C211yesmaximal, characteristic
7C79×#2099C211yesmaximal, characteristic
8C2×#14579160—21no—
9C2×#14579094—21no—
10C2×#14579084—21no—
11#19098496—21no—
12#19098484—21no—
13#19098483—21no—
14#19098469C2211yescharacteristic
15#19098461C2211yescharacteristic
16#19098449—21no—
17#19098446—21no—
18#19098325—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22#389×C79C2211yescharacteristic
23#384×C79C2211yescharacteristic
24#340×C79—21no—
25#340×C79—21no—
26#336×C79C2211yescharacteristic
27C22×Dic158—21no—
28C22×Dic158D411yescharacteristic
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31#14579160—42no—
32#14579160—42no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79D411yescharacteristic
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4⋊Dic79D411yescharacteristic
37C2×Dic79⋊C4—21no—
38C2×Dic79⋊C4—21no—
39C2×Dic79⋊C4—21no—
40C2×C4×Dic79—21no—
41#14579094—42no—
42#14579084—42no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C22×C8×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79—21no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C42×C79C2311yescharacteristic
57#95×C79—21no—
58C2×Dic158—21no—
59C2×Dic158—41no—
60C2×Dic158—41no—
61C2×Dic158—42no—
62C2×Dic158—81no—
63C2×Dic158—81no—
64C2×Dic158D812yes—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C2×C79⋊C8—22no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—21no—
76C4⋊Dic79—41no—
77C4⋊Dic79—41no—
78C4⋊Dic79—42no—
79C4⋊Dic79SD1612yes—
80Dic79⋊C4—42no—
81Dic79⋊C4—41no—
82Dic79⋊C4—41no—
83Dic79⋊C4—81no—
84C4×Dic79—42no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—41no—
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C4⋊C4×C79—21no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—42no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79Q1612yes—
104C42×C79—41no—
105C23×C79#12011yescharacteristic
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—22no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—81no—
119C2×C4×C79—81no—
120C2×C4×C79—81no—
121C2×C4×C79C2×D811yescharacteristic
122C2×C4×C79C2×SD1611yescharacteristic
123C2×C4×C79C2×Q1611yescharacteristic
124C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C79#13711yescharacteristic
126C2×C4×C79#13711yescharacteristic
127C2×Dic79—22no—
128C2×Dic79—22no—
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—
138C2×Dic79—41no—
139Dic158—42no—
140Dic158—81no—
141Dic158—84no—
142C8×C79—81no—
143C79⋊C8—42no—
144C79⋊C8—81no—
145C22×C79#44711yescharacteristic
146C22×C79#44811yescharacteristic
147C22×C79#45111yescharacteristic
148C22×C79#45411yescharacteristic
149C22×C79#45511yescharacteristic
150C22×C79#45611yescharacteristic
151C22×C79#45811yescharacteristic
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—21no—
157C4×C79—21no—
158C4×C79—42no—
159C4×C79—81no—
160C4×C79—161no—
161Dic79—42no—
162Dic79—42no—
163Dic79—81no—
164Dic79—81no—
165#7794—791nomaximal, Sylow
166C2×C79#160711yescharacteristic
167C2×C79#161111yescharacteristic
168C2×C79#161711yescharacteristic
169C2×C79#162711yescharacteristic
170C2×C79#164511yescharacteristic
171C2×C79#164911yescharacteristic
172C2×C79#166411yescharacteristic
173#2734—791no—
174#2099D7911yescharacteristic
175#2083—791no—
176#2072—791no—
177#1272—791no—
178#1271—791no—
179#1271—791no—
180C79#779411yescharacteristic, complemented, Sylow
181C2×C4⋊Q8—1581no—
182C2×#107—1581no—
183C2×#103—1581no—
184#398—1581no—
185#395—1581no—
186#395—1581no—
187#391—791no—
188#389—791no—
189#389C2×D7911yescharacteristic
190#387—1581no—
191#387—1581no—
192#384C2×D7911yescharacteristic
193#340—21no—
194#340—21no—
195#340—1581no—
196#340—1581no—
197#336—791no—
198#336—791no—
199#336C2×D7911yescharacteristic
200C22×Q8—791no—
201C22×Q8—1581no—
202C22×C8—21no—
203C22×C8—1581no—
204C22×C8—1581no—
205C4⋊Q8—3162no—
206C4⋊Q8—3162no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—791no—
212C2×C4⋊C4—791no—
213C2×C4⋊C4—791no—
214C2×C4⋊C4—1581no—
215C2×C4⋊C4—1581no—
216C2×C4⋊C4—1581no—
217C2×C4⋊C4—1581no—
218C2×C4⋊C4C22⋊D7911yescharacteristic
219C2×C4⋊C4C22⋊D7911yescharacteristic
220C2×C42—1581no—
221C2×C42C22×D7911yescharacteristic
222#107—3162no—
223#103—3162no—
224#95—21no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230#95—1581no—
231C2×Q8—3161no—
232C2×Q8—3161no—
233C2×Q8—3162no—
234C2×Q8—792no—
235C2×Q8—6321no—
236C2×Q8—6321no—
237C2×Q8—1581no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—1581no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4—1581no—
245C22×C4D4×D7911yescharacteristic
246C22×C4D4×D7911yescharacteristic
247C22×C4C2×C22⋊D7911yescharacteristic
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—3161no—
252C2×C8—3161no—
253C2×C8—3161no—
254C2×C8—3161no—
255C2×C8—3161no—
256C2×C8—1582no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—3161no—
267C4⋊C4—3161no—
268C4⋊C4—3162no—
269C4⋊C4—3162no—
270C4⋊C4—3161no—
271C4⋊C4—3161no—
272C4⋊C4—792no—
273C4⋊C4—6321no—
274C4⋊C4—1581no—
275C4⋊C4#1192961412yes—
276C42—41no—
277C42—3162no—
278C23#1457912911yescharacteristic
279Q8—3162no—
280Q8—6321no—
281Q8—6324no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
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—1582no—
309C2×C4D8×D7911yescharacteristic
310C2×C4SD16×D7911yescharacteristic
311C2×C4C2×#1192961411yescharacteristic
312C2×C4#1457918511yescharacteristic
313C2×C4#1457919011yescharacteristic
314C2×C4#1457919111yescharacteristic
315C8—81no—
316C8—3162no—
317C8—6321no—
318C22#1909869511yescharacteristic
319C22#1909870011yescharacteristic
320C22#1909871811yescharacteristic
321C22#1909873511yescharacteristic
322C22#1909873911yescharacteristic
323C22#1909874411yescharacteristic
324C22#1909875211yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—81no—
333C4—161no—
334C4—3162no—
335C4—3162no—
336C4—6321no—
337C4—6321no—
338C2#2376238411yescharacteristic
339C2#2376240611yescharacteristic
340C2#2376244111yescharacteristic
341C2#2376250011yescharacteristic
342C2#2376262011yescharacteristic
343C2#2376264811yescharacteristic
344C2#2376273111yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2099C2
socle, prime radicalC23×C79#120
hypercenter#2099D79
hypocenterC79#7794
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
180. C79#7794semidirect

Maximal subgroups

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

subgroupindex
#237704312
#237648722
#237648092
#237612252
#237612232
#237612222
C79×#20992
#779479