Finite Group Databasebeta

← #34201099

Subgroups — #34201099

Order 20224.

346 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764561C211yesmaximal, characteristic
2#23762441C211yesmaximal, characteristic
3#23762373C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic
5#23761427C211yesmaximal, characteristic
6#23761334C211yesmaximal, characteristic
7C79×#2799C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098898—21no—
10#19098719—21no—
11#19098572—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18#400×C79—21no—
19#400×C79C2211yescharacteristic
20#398×C79—21no—
21#395×C79C2211yescharacteristic
22#391×C79—21no—
23#391×C79—21no—
24#391×C79C2211yescharacteristic
25#387×C79—21no—
26#387×C79—21no—
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#14579085—41no—
36#14579085—41no—
37#14579085—41no—
38#14579084—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×Q8×C79D411yescharacteristic
42C22×Q8×C79D411yescharacteristic
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C42×C79—21no—
50C2×C42×C79D411yescharacteristic
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
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—41no—
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—21no—
79C2×Q8×C79—42no—
80C2×Q8×C79—41no—
81C2×Q8×C79—41no—
82C2×Q8×C79—41no—
83C2×Q8×C79Q1612yescomplemented
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C4⋊C4×C79—41no—
94C4⋊C4×C79—42no—
95C4⋊C4×C79—42no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—81no—
98C4⋊C4×C79—81no—
99C42×C79—41no—
100C42×C79—81no—
101C23×C79#12011yescharacteristic
102Q8×C79—42no—
103Q8×C79—42no—
104Q8×C79—81no—
105Q8×C79—81no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
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—42no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79C2×Q1611yescharacteristic
130C2×C4×C79#13711yescharacteristic
131C2×Dic79—41no—
132C2×Dic79—41no—
133Dic158—81no—
134Dic158—81no—
135C79⋊C8—81no—
136C79⋊C8—81no—
137C79⋊C8—81no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79#45111yescharacteristic
141C22×C79#45511yescharacteristic
142C22×C79B(4,𝔽2)11yescharacteristic
143C4×C79—21no—
144C4×C79—21no—
145C4×C79—42no—
146C4×C79—42no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—81no—
150C4×C79—81no—
151C4×C79—81no—
152C4×C79—81no—
153Dic79—81no—
154Dic79—81no—
155#12222—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79#160211yescharacteristic
159C2×C79#219711yescharacteristic
160C2×C79#220311yescharacteristic
161#2799D7911yescharacteristic
162#2028—791no—
163#1617—791no—
164#1605—791no—
165#1354—791no—
166#1353—791no—
167#1319—791no—
168C79#1222211yescharacteristic, complemented, Sylow
169#544—791no—
170#484—1581no—
171#451—1581no—
172#419—1581no—
173#400—21no—
174#400C2×D7911yescharacteristic
175#398—21no—
176#395C2×D7911yescharacteristic
177#391—21no—
178#391—21no—
179#391C2×D7911yescharacteristic
180#387—21no—
181#387—21no—
182#343—791no—
183#339—1581no—
184#328—1581no—
185#328—1581no—
186#324—791no—
187#324—791no—
188C22×Q8C22⋊D7911yescharacteristic
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⋊C4C22⋊D7911yescharacteristic
202C2×C4⋊C4C22×D7911yescharacteristic
203C2×C42—21no—
204C2×C42C22⋊D7911yescharacteristic
205#107—3161no—
206#103—3161no—
207#103—3161no—
208#103—3161no—
209#103—3161no—
210C22⋊C8—1581no—
211C22⋊C8—1581no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C22⋊D7911yescharacteristic
222#95C22⋊D7911yescharacteristic
223C2×Q8—21no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—1581no—
229C2×Q8—1581no—
230C2×Q8#1192961412yescomplemented
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×C22⋊D7911yescharacteristic
238C22×C4C2×C22⋊D7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240Q16—6322no—
241M16—3161no—
242M16—3161no—
243C2×C8—3161no—
244C2×C8—3161no—
245C2×C8—3161no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—3161no—
253C4⋊C4—3161no—
254C4⋊C4—1581no—
255C4⋊C4—1581no—
256C4⋊C4—1581no—
257C4⋊C4—1581no—
258C22⋊C4—3161no—
259C42—41no—
260C42—81no—
261C42—3161no—
262C23#1457913011yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267Q8—6321no—
268Q8—6321no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
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—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4C2×#1192961411yescharacteristic
295C2×C4#1457919211yescharacteristic
296C8—6321no—
297C8—6321no—
298C8—6321no—
299C22—21no—
300C22—21no—
301C22#1909871911yescharacteristic
302C22#1909874111yescharacteristic
303C22#1909875011yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—6321no—
315C4—6321no—
316C2—21no—
317C2—21no—
318C2#2376235711yescharacteristic
319C2#2376539011yescharacteristic
320C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2799C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2799D79
hypocenterC79#12222
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2799

Complements

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

NH ≅ G/Ntype
83. C2×Q8×C79Q16semidirect
168. C79#12222semidirect
230. C2×Q8#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645612
#237624412
#237623732
#237614302
#237614272
#237613342
C79×#27992
#1222279