Finite Group Databasebeta

← #34186156

Subgroups — #34186156

Order 20224.

458 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 95 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#23772038C211yesmaximal, characteristic
2#23768977C211yesmaximal, characteristic
3#23761846C212yesmaximal, complemented
4#23761266C212yesmaximal, complemented
5C79×#1464C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7#19098624—24no—
8#19098587—21no—
9#19098527—21no—
10C4×#11929608C2212yes—
11#19098493C411yescharacteristic
12#19098488C411yescharacteristic
13#19098483C411yescharacteristic
14#19098475C411yescharacteristic
15#19098451—21no—
16#19098447—21no—
17#19098444—21no—
18#19098436—21no—
19#19098431C2211yescharacteristic
20#19098293—24no—
21C2×#117×C79C2211yescharacteristic
22#432×C79C2212yes—
23#324×C79—21no—
24#323×C79—21no—
25C22×Dic158C2×C411yescharacteristic
26C2×#11929608—22no—
27C4×Dic158—22no—
28C4×Dic158—22no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—24no—
37C22⋊(C79⋊C8)—24no—
38#14579072—22no—
39C4×C79⋊C8D414yes—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95C2×C411yescharacteristic
47C79⋊#95C2×C411yescharacteristic
48C23×C4×C79D411yescharacteristic
49#117×C79—22no—
50#117×C79—22no—
51C2×C4⋊C4×C79C2×C411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79C2×C411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C42×C79D411yescharacteristic
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—22no—
58C22⋊C8×C79—22no—
59#97×C79—22no—
60C2×Dic158—22no—
61C2×Dic158—42no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66#11929608—42no—
67#11929608—42no—
68C2×C79⋊C8—24no—
69C4⋊Dic79—22no—
70Dic79⋊C4—42no—
71Dic79⋊C4—42no—
72Dic79⋊C4—41no—
73Dic79⋊C4—41no—
74C22⋊Dic79—42no—
75C22⋊Dic79—42no—
76C4×Dic79—42no—
77C24×C79C22⋊C411yescharacteristic
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—41no—
81C22×C4×C79—41no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D411yescharacteristic
88C2×C8×C79—42no—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—22no—
91C22⋊C4×C79—22no—
92C22⋊C4×C79—22no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79C2×D412yes—
96C42×C79#4112yes—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79C2×C22⋊C411yescharacteristic
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—41no—
111C2×C4×C79—41no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79C4×D412yes—
117C2×C4×C79#12011yescharacteristic
118C2×C4×C79#12112yes—
119C2×C4×C79#12311yescharacteristic
120C2×Dic79—22no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132Dic158—24no—
133Dic158—42no—
134C8×C79—82no—
135C79⋊C8—44no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—42no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—81no—
143C22×C79#38611yescharacteristic
144C22×C79#42111yescharacteristic
145C22×C79#42111yescharacteristic
146C4×C79—22no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—22no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153Dic79—42no—
154Dic79—81no—
155Dic79—81no—
156Dic79—81no—
157#9493—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—81no—
161C2×C79#189111yescharacteristic
162C2×C79#189711yescharacteristic
163C2×C79#190911yescharacteristic
164#2921—791no—
165#2599—791no—
166#1491—792no—
167#1464D7911yescharacteristic
168#1288—792no—
169C79#949311yescharacteristic, complemented, Sylow
170C2×C4×Q8—791no—
171C2×#117C2×D7911yescharacteristic
172#433—1584no—
173#432C2×D7912yes—
174#423—1581no—
175#407—1581no—
176C4×M16—792no—
177#398—791no—
178#397—791no—
179#395—791no—
180#393—791no—
181#388—1581no—
182#387—1581no—
183#387—1581no—
184#385—1581no—
185#383—791no—
186Q8⋊C8—1584no—
187#324—21no—
188#323—21no—
189C22×Q8—791no—
190C23×C4D31611yescharacteristic
191C2×M16—1582no—
192C4×Q8—1582no—
193C4×Q8—1582no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—791no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4C4×D7911yescharacteristic
201C2×C4⋊C4C22⋊D7911yescharacteristic
202C2×C22⋊C4—1581no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4C4×D7911yescharacteristic
205C2×C22⋊C4C22⋊D7911yescharacteristic
206C2×C42—1581no—
207C2×C42D31611yescharacteristic
208C2×C42C22×D7911yescharacteristic
209C4⋊C8—22no—
210C4⋊C8—1584no—
211C22⋊C8—22no—
212C22⋊C8—1584no—
213#97—22no—
214#97—1582no—
215C4×C8—794no—
216#95—791no—
217#95—791no—
218#95—1581no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224C24(C2×D79)⋊C411yescharacteristic
225C2×Q8—3162no—
226C2×Q8—1582no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—1581no—
232C22×C4—1581no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4C2×D31611yescharacteristic
241M16—3162no—
242M16—3162no—
243C2×C8—42no—
244C2×C8—1584no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—3162no—
248C4⋊C4—3162no—
249C4⋊C4—3161no—
250C4⋊C4—3161no—
251C4⋊C4—1582no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—3162no—
255C22⋊C4—3162no—
256C42—21no—
257C42—21no—
258C42—3162no—
259C42D4×D7912yes—
260C42#1192962312yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×(C2×D79)⋊C411yescharacteristic
267Q8—3162no—
268Q8—1584no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—1582no—
294C2×C4(C22⋊D79)⋊C412yes—
295C2×C4#1457913111yescharacteristic
296C2×C4#1457913312yes—
297C2×C4#1457914511yescharacteristic
298C8—82no—
299C8—3164no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1909844011yescharacteristic
308C22#1909857911yescharacteristic
309C22#1909857911yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—3162no—
318C4—6321no—
319C4—6321no—
320C4—6321no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2376381611yescharacteristic
325C2#2376384611yescharacteristic
326C2#2376392011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1464C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1464D79
hypocenterC79#9493
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #23761846C2semidirect
4. #23761266C2semidirect
169. C79#9493semidirect

Maximal subgroups

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

subgroupindex
#237720382
#237689772
#237618462
#237612662
C79×#14642
#949379