Finite Group Databasebeta

← #34201221

Subgroups — #34201221

Order 20224.

350 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 59 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#23764745C211yesmaximal, characteristic
2#23762407C211yesmaximal, characteristic
3#23762373C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic
5#23761406C211yesmaximal, characteristic
6#23761346C211yesmaximal, characteristic
7C79×#2792C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098893—21no—
10#19098719—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#398×C79—21no—
19#398×C79C2211yescharacteristic
20#395×C79—21no—
21#395×C79C2211yescharacteristic
22#389×C79—21no—
23#389×C79—21no—
24#387×C79—21no—
25#387×C79—21no—
26#384×C79C2211yescharacteristic
27C2×#11929614—41no—
28C2×#11929608—21no—
29#14579160—41no—
30#14579152—41no—
31#14579140—21no—
32#14579140—21no—
33#14579120—21no—
34#14579094—41no—
35#14579092—41no—
36#14579092—41no—
37#14579085—41no—
38#14579084—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×Q8×C79D411yescharacteristic
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79C2311yescharacteristic
52C2×C42×C79—21no—
53C2×C42×C79D411yescharacteristic
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79D411yescharacteristic
61#95×C79D411yescharacteristic
62C2×Dic158—21no—
63C2×Dic158—21no—
64#11929614—82no—
65#11929608—22no—
66#11929608—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74Dic79⋊C4—41no—
75Dic79⋊C4—41no—
76C22⋊Dic79—41no—
77C4×Dic79—41no—
78C2×Q8×C79—41no—
79C2×Q8×C79—81no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C4⋊C4×C79—24no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C4⋊C4×C79—41no—
95C4⋊C4×C79—81no—
96C4⋊C4×C79—81no—
97C4⋊C4×C79—81no—
98C4⋊C4×C79—81no—
99C4⋊C4×C79—81no—
100C42×C79—21no—
101C42×C79—21no—
102C42×C79—81no—
103C23×C79#12011yescharacteristic
104Q8×C79—42no—
105Q8×C79—81no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79#13511yescharacteristic
131C2×C4×C79#13711yescharacteristic
132C2×Dic79—41no—
133C2×Dic79—41no—
134Dic158—81no—
135Dic158—81no—
136C79⋊C8—42no—
137C79⋊C8—81no—
138C79⋊C8—81no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79#45211yescharacteristic
142C22×C79#45611yescharacteristic
143C22×C79B(4,𝔽2)11yescharacteristic
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—42no—
147C4×C79—42no—
148C4×C79—42no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—81no—
152C4×C79—81no—
153C4×C79—81no—
154Dic79—81no—
155Dic79—81no—
156#12240—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79#160311yescharacteristic
160C2×C79#220011yescharacteristic
161C2×C79#220511yescharacteristic
162#2792D7911yescharacteristic
163#2060—791no—
164#1611—791no—
165#1605—791no—
166#1354—791no—
167#1346—791no—
168#1325—791no—
169C79#1224011yescharacteristic, complemented, Sylow
170#544—791no—
171#483—1581no—
172#451—1581no—
173#428—1581no—
174#398—21no—
175#398C2×D7911yescharacteristic
176#395—21no—
177#395C2×D7911yescharacteristic
178#389—21no—
179#389—21no—
180#387—21no—
181#387—21no—
182#384C2×D7911yescharacteristic
183#343—791no—
184#339—1581no—
185#328—1581no—
186#328—1581no—
187#324—791no—
188#324—791no—
189C22×Q8C22⋊D7911yescharacteristic
190C2×Q16—3161no—
191C2×M16—1581no—
192C4⋊Q8—3161no—
193#125—3161no—
194C22⋊Q8—1581no—
195C22⋊Q8—1581no—
196#117—1581no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D7911yescharacteristic
205C2×C4⋊C4C22⋊D7911yescharacteristic
206C2×C4⋊C4C22×D7911yescharacteristic
207C2×C42—21no—
208C2×C42C22⋊D7911yescharacteristic
209#107—3161no—
210#106—3161no—
211#106—3161no—
212#103—3161no—
213#103—3161no—
214C22⋊C8—1581no—
215C22⋊C8—1581no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C22⋊D7911yescharacteristic
223#95C22⋊D7911yescharacteristic
224C2×Q8—41no—
225C2×Q8—81no—
226C2×Q8—1581no—
227C2×Q8—1581no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4C2×C22⋊D7911yescharacteristic
237Q16—6322no—
238M16—3161no—
239M16—1582no—
240C2×C8—3161no—
241C2×C8—3161no—
242C2×C8—3161no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—3161no—
255C4⋊C4—3161no—
256C4⋊C4—1581no—
257C4⋊C4—1581no—
258C4⋊C4—1581no—
259C4⋊C4—1581no—
260C22⋊C4—3161no—
261C42—21no—
262C42—21no—
263C42—81no—
264C42—3161no—
265C23#1457913011yescharacteristic
266Q8—42no—
267Q8—81no—
268Q8—6321no—
269Q8—6321no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4#1457918111yescharacteristic
297C2×C4#1457919111yescharacteristic
298C8—3162no—
299C8—6321no—
300C8—6321no—
301C22—21no—
302C22—21no—
303C22#1909872511yescharacteristic
304C22#1909874611yescharacteristic
305C22#1909875011yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—6321no—
317C4—6321no—
318C2—21no—
319C2—21no—
320C2#2376236311yescharacteristic
321C2#2376540711yescharacteristic
322C2#2376543011yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2792C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2792D79
hypocenterC79#12240
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2792

Complements

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

NH ≅ G/Ntype
169. C79#12240semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237624072
#237623732
#237614302
#237614062
#237613462
C79×#27922
#1224079