Finite Group Databasebeta

← #34183306

Subgroups — #34183306

Order 20224.

616 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#19098502C211yesmaximal, characteristic
2#23766018C211yesmaximal, characteristic
3#23763491C212yesmaximal
4#23761280C212yesmaximal
5C2×#324×C79C211yesmaximal, characteristic
6C2×#14579120—21no—
7C22×C4⋊Dic79C2211yescharacteristic
8#19098502—22no—
9#19098502C414yescomplemented
10#19098462—24no—
11#19098423—21no—
12#19098423C2212yes—
13C2×C79⋊#95—21no—
14C2×C79⋊#95—21no—
15C2×C79⋊#95C2211yescharacteristic
16#19098326—24no—
17C22×C4⋊C4×C79C2211yescharacteristic
18C2×C22⋊C8×C79C2212yes—
19#324×C79—22no—
20C23×Dic79—21no—
21C23×Dic79—21no—
22#14579120—42no—
23#14579120—42no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79—22no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—22no—
29C2×C4⋊Dic79C2×C412yes—
30C2×C4⋊Dic79D414yes—
31C2×C22⋊Dic79—22no—
32C2×C4×Dic79—22no—
33C79⋊#95—22no—
34C79⋊#95—24no—
35C79⋊#95—24no—
36C79⋊#95—21no—
37C79⋊#95—22no—
38C79⋊#95C2×C412yes—
39C23×C4×C79D411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C23×C4×C79C2311yescharacteristic
42C22×C8×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79C2×C412yes—
46C2×C4⋊C4×C79D412yes—
47C22⋊C8×C79—22no—
48C22⋊C8×C79—22no—
49C22×Dic79—22no—
50C22×Dic79—22no—
51C22×Dic79—24no—
52C22×Dic79—42no—
53C22×Dic79—44no—
54C22×Dic79—42no—
55C22×Dic79—41no—
56C22×Dic79—41no—
57C22×Dic79—41no—
58C22×Dic79—41no—
59C4⋊Dic79—24no—
60C4⋊Dic79—44no—
61C4⋊Dic79—44no—
62C4⋊Dic79—42no—
63C4⋊Dic79—82no—
64C22⋊Dic79—42no—
65C22⋊Dic79—42no—
66C4×Dic79—44no—
67C24×C79C2×D411yescharacteristic
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79SD1612yes—
79C22×C4×C79Q1612yes—
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D412yes—
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79#4112yes—
84C2×C8×C79—44no—
85C2×C8×C79—42no—
86C4⋊C4×C79—24no—
87C4⋊C4×C79—24no—
88C23×C79—22no—
89C23×C79—22no—
90C23×C79C2×C22⋊C411yescharacteristic
91C23×C79#12011yescharacteristic
92C23×C79#12311yescharacteristic
93C23×C79C2×SD1611yescharacteristic
94C23×C79C2×Q1611yescharacteristic
95C23×C79AGL(1,ℤ/8)11yescharacteristic
96C23×C79#13711yescharacteristic
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—24no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—81no—
111C2×C4×C79#10314yes—
112C2×C4×C79C4×D412yes—
113C2×C4×C79#12112yes—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—44no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—44no—
120C2×Dic79—44no—
121C2×Dic79—44no—
122C2×Dic79—82no—
123C2×Dic79—82no—
124C2×Dic79—82no—
125C2×Dic79—82no—
126C8×C79—82no—
127C22×C79—22no—
128C22×C79—22no—
129C22×C79—22no—
130C22×C79—22no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79#38611yescharacteristic
136C22×C79C2×#10311yescharacteristic
137C22×C79#41711yescharacteristic
138C22×C79#45011yescharacteristic
139C22×C79#45111yescharacteristic
140C22×C79#45412yes—
141C22×C79#45812yes—
142C22×C79#48311yescharacteristic
143C22×C79#48411yescharacteristic
144C4×C79—24no—
145C4×C79—22no—
146C4×C79—22no—
147C4×C79—42no—
148Dic79—82no—
149Dic79—84no—
150Dic79—82no—
151#9023—791nomaximal, Sylow
152C2×C79—21no—
153C2×C79—21no—
154C2×C79—21no—
155C2×C79—21no—
156C2×C79#161612yes—
157C2×C79#161712yes—
158C2×C79#187211yescharacteristic
159C2×C79#188811yescharacteristic
160C2×C79#189011yescharacteristic
161C2×#400—791no—
162#2316—791no—
163#1824—792no—
164C2×#324D7911yescharacteristic
165#1294—792no—
166C79#902311yescharacteristic, complemented, Sylow
167C2×#117—1581no—
168C22×C4⋊C4—791no—
169C22×C4⋊C4C2×D7911yescharacteristic
170C2×C22⋊C8C2×D7912yes—
171#400—794no—
172#400—1582no—
173#389—1584no—
174#380—792no—
175#380—1581no—
176C2×#95—791no—
177C2×#95—1581no—
178C2×#95—1581no—
179#340—1584no—
180#324—22no—
181C23×C4—1581no—
182C23×C4—1581no—
183C23×C4D31611yescharacteristic
184C23×C4D31611yescharacteristic
185C23×C4C22×D7911yescharacteristic
186C22×C8—22no—
187#117—3162no—
188#117—3162no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—794no—
192C2×C4⋊C4—792no—
193C2×C4⋊C4—1582no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1582no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4—1582no—
198C2×C4⋊C4C4×D7912yes—
199C2×C4⋊C4C22⋊D7912yes—
200C2×C22⋊C4—1582no—
201C2×C42—1582no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#95—792no—
205#95—1582no—
206#95—1584no—
207#95—1584no—
208#95—1582no—
209#95—1581no—
210C24C2×D31611yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—3162no—
220C22×C4—3162no—
221C22×C4—3164no—
222C22×C4—3161no—
223C22×C4—3161no—
224C22×C4—3161no—
225C22×C4—3161no—
226C22×C4—1582no—
227C22×C4—1582no—
228C22×C4—1584no—
229C22×C4(C2×D79)⋊C412yes—
230C22×C4(C2×D79)⋊C412yes—
231C22×C4#1192961312yes—
232C22×C4Dic31612yes—
233C22×C4C2×C4×D7911yescharacteristic
234C22×C4D4×D7912yes—
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4#1192962312yes—
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—3164no—
242C4⋊C4—3164no—
243C4⋊C4—3162no—
244C4⋊C4—6322no—
245C4⋊C4—1584no—
246C22⋊C4—3162no—
247C22⋊C4—3162no—
248C42—3164no—
249C23—22no—
250C23—22no—
251C23C2×(C2×D79)⋊C411yescharacteristic
252C23#1457913111yescharacteristic
253C23#1457914511yescharacteristic
254C23C2×#1192961311yescharacteristic
255C23C2×Dic31611yescharacteristic
256C23#1457918911yescharacteristic
257C23#1457919411yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—3164no—
275C2×C4—3162no—
276C2×C4—3164no—
277C2×C4—3164no—
278C2×C4—3164no—
279C2×C4—3162no—
280C2×C4—6322no—
281C2×C4—6322no—
282C2×C4—6322no—
283C2×C4—6322no—
284C2×C4#1457908614yes—
285C2×C4(C22⋊D79)⋊C412yes—
286C2×C4#1457913312yes—
287C8—82no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#1909844011yescharacteristic
297C22C2×#1457908611yescharacteristic
298C22#1909856511yescharacteristic
299C22#1909871611yescharacteristic
300C22#1909872111yescharacteristic
301C22#1909873812yes—
302C22#1909875412yes—
303C22#1909889411yescharacteristic
304C22#1909889911yescharacteristic
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—6322no—
310C4—6324no—
311C4—6322no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2376243912yes—
317C2#2376244412yes—
318C2#2376370511yescharacteristic
319C2#2376380111yescharacteristic
320C2#2376381111yescharacteristic
subgroupG/H
centerC23#14579131
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#324×C79C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#324D79
hypocenterC79#9023
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
9. #19098502C4semidirect
166. C79#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#190985022
#237660182
#237634912
#237612802
C2×#324×C792
#902379