Finite Group Databasebeta

← #34206961

Subgroups — #34206961

Order 20224.

396 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 103 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#23768587C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic
3#23761526C212yesmaximal
4#23761526C212yesmaximal
5C4×#101×C79C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic
7#19098587—21no—
8C2×#14579080C2212yes—
9#19098496—21no—
10#19098496—21no—
11#19098470C2211yescharacteristic
12#19098470C2211yescharacteristic
13#19098447—21no—
14#19098447—21no—
15#19098432C411yescharacteristic
16#19098428C411yescharacteristic
17C4×C4⋊Dic79C411yescharacteristic
18#19098332—21no—
19C2×C4×Q8×C79C2211yescharacteristic
20C2×#101×C79C2212yes—
21C4×M16×C79—21no—
22#343×C79—21no—
23C2×#11929608—22no—
24#14579153—41no—
25#14579153—41no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35C4⋊(C79⋊C8)—41no—
36C4⋊(C79⋊C8)—41no—
37#14579080—24no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C22×Q8×C79C2311yescharacteristic
46C2×M16×C79—22no—
47C4×Q8×C79—41no—
48C4×Q8×C79—41no—
49C2×C4⋊C4×C79C2×C411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C42×C79C2×C411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55#101×C79—22no—
56#101×C79Q814yes—
57#97×C79—41no—
58C4×C8×C79—41no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63#11929608—42no—
64#11929608—42no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—41no—
67C4⋊Dic79—41no—
68C4⋊Dic79—42no—
69Dic79⋊C4—42no—
70Dic79⋊C4—42no—
71Dic79⋊C4—42no—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74C4×Dic79—41no—
75C4×Dic79—41no—
76C2×Q8×C79—41no—
77C2×Q8×C79—41no—
78C2×Q8×C79C2×Q812yes—
79C2×Q8×C79#4112yes—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87M16×C79—24no—
88M16×C79—42no—
89C2×C8×C79—42no—
90C4⋊C4×C79—41no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—21no—
99C23×C79C2×C22⋊C411yescharacteristic
100Q8×C79—42no—
101Q8×C79—42no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79#9911yescharacteristic
116C2×C4×C79#9911yescharacteristic
117C2×C4×C79C4×Q812yes—
118C2×C4×C79C22⋊Q812yes—
119C2×C4×C79#12411yescharacteristic
120C2×C4×C79C4⋊Q811yescharacteristic
121C2×Dic79—22no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—42no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C8×C79—44no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79#39111yescharacteristic
137C22×C79C2×#9911yescharacteristic
138C22×C79#41011yescharacteristic
139C4×C79—21no—
140C4×C79—21no—
141C4×C79—22no—
142C4×C79—22no—
143C4×C79—21no—
144C4×C79—21no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148Dic79—42no—
149Dic79—81no—
150Dic79—81no—
151Dic79—81no—
152#13303—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79#199611yescharacteristic
156C2×C79#213211yescharacteristic
157C2×C79#213311yescharacteristic
158#2565—791no—
159#1801—791no—
160C4×#101D7911yescharacteristic
161#1394—792no—
162#1394—792no—
163C79#1330311yescharacteristic, complemented, Sylow
164C2×#125—791no—
165C2×C4×Q8C2×D7911yescharacteristic
166#423—1581no—
167C2×#101—792no—
168C2×#101C2×D7912yes—
169C4×M16—21no—
170#398—1581no—
171#398—1581no—
172#391—791no—
173#391—791no—
174#387—1581no—
175#387—1581no—
176#384—791no—
177#382—791no—
178C4×C4⋊C4—791no—
179#343—21no—
180#343—1581no—
181C22×Q8C22×D7911yescharacteristic
182C2×M16—22no—
183C2×M16—1582no—
184#125—3161no—
185#125—3161no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—791no—
189C2×C4⋊C4—791no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4C4×D7911yescharacteristic
196C2×C4⋊C4D31611yescharacteristic
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C42—791no—
199C2×C42—791no—
200C2×C42C4×D7911yescharacteristic
201C2×C42D31611yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—3161no—
204C4⋊C8—3161no—
205#101—22no—
206#101—1584no—
207#101Dic15814yes—
208#97—41no—
209C4×C8—41no—
210#95—1581no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215#95—1581no—
216#95—1581no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8C2×Dic15812yes—
220C2×Q8#1192962412yes—
221C22×C4—1581no—
222C22×C4—1581no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4(C2×D79)⋊C411yescharacteristic
226C22×C4(C2×D79)⋊C411yescharacteristic
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4C2×C4×D7911yescharacteristic
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×D31611yescharacteristic
232M16—24no—
233M16—42no—
234M16—3162no—
235M16—3162no—
236C2×C8—42no—
237C2×C8—3162no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—3161no—
242C4⋊C4—3161no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C4⋊C4—3162no—
246C4⋊C4—3162no—
247C4⋊C4—3161no—
248C4⋊C4—3161no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—3161no—
256C42—3161no—
257C23C2×(C2×D79)⋊C411yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—3161no—
274C2×C4—3161no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3162no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—1582no—
284C2×C4#1457907511yescharacteristic
285C2×C4#1457907511yescharacteristic
286C2×C4C4×Dic15812yes—
287C2×C4#1457914012yes—
288C2×C4#1457914911yescharacteristic
289C2×C4#1457916111yescharacteristic
290C8—44no—
291C8—6322no—
292C22—21no—
293C22—21no—
294C22#1909846911yescharacteristic
295C22C2×#1457907511yescharacteristic
296C22#1909853611yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3162no—
307C4—6321no—
308C4—6321no—
309C4—6321no—
310C2—21no—
311C2—21no—
312C2#2376438511yescharacteristic
313C2#2376508311yescharacteristic
314C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC4×#101×C79C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC22×Q8×C79C23
hypercenterC4×#101D79
hypocenterC79#13303
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
163. C79#13303semidirect

Maximal subgroups

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

subgroupindex
#237685872
#237633962
#237615262
C4×#101×C792
#1330379