Finite Group Databasebeta

← #34184934

Subgroups — #34184934

Order 20224.

422 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 107 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#23774350C211yesmaximal, characteristic
2#23772337C211yesmaximal, characteristic
3#23764372C211yesmaximal, characteristic
4#23762086C211yesmaximal, characteristic
5#23761963C211yesmaximal, characteristic
6#23761230C211yesmaximal, characteristic
7C79×#1286C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9C2×#14579161C2211yescharacteristic
10#19098943C412yescomplemented
11#19098914C412yescomplemented
12#19098774—22no—
13#19098721—22no—
14#19098692—22no—
15#19098692—22no—
16#19098616C2211yescharacteristic
17#19098591—21no—
18#19098521—21no—
19C2×#14579072C2211yescharacteristic
20#19098325—21no—
21#19098325—21no—
22#19098308—22no—
23#19098299—22no—
24#431×C79C2211yescharacteristic
25#423×C79C2211yescharacteristic
26#382×C79C2211yescharacteristic
27C22×Dic158—21no—
28C2×Dic316—24no—
29C22×C79⋊C8D411yescharacteristic
30C22×C79⋊C8D411yescharacteristic
31#14579161—21no—
32#14579161—21no—
33#14579161—21no—
34#14579161C2×C412yes—
35#14579161C2×C411yescharacteristic
36#14579161C2×C411yescharacteristic
37#14579153—21no—
38#14579140—21no—
39#14579140D412yes—
40C4×Dic158—21no—
41C4×Dic158D412yes—
42#14579120—21no—
43C2×C4⋊Dic79—21no—
44C2×C4⋊Dic79—21no—
45C4⋊(C79⋊C8)—21no—
46Dic79⋊C8—24no—
47#14579086—24no—
48C22⋊(C79⋊C8)—21no—
49#14579072—21no—
50#14579072—21no—
51#14579071—22no—
52C4×C79⋊C8—21no—
53C8×Dic79—22no—
54C2×M16×C79—21no—
55C2×C4⋊C4×C79—21no—
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—21no—
58C4⋊C8×C79—21no—
59C4⋊C8×C79D412yes—
60C22⋊C8×C79—21no—
61C4×C8×C79C2×C412yes—
62#95×C79—21no—
63#95×C79—21no—
64C2×Dic158—22no—
65C2×Dic158—21no—
66C2×Dic158—41no—
67C2×Dic158—82no—
68C2×Dic158C22⋊C412yes—
69C22×Dic79—41no—
70Dic316—84no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—21no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8Q1612yescomplemented
78C4⋊Dic79—21no—
79C4⋊Dic79—21no—
80C4⋊Dic79—21no—
81C4⋊Dic79—22no—
82C4⋊Dic79—22no—
83C4⋊Dic79—41no—
84C4⋊Dic79—41no—
85C4⋊Dic79C22⋊C412yes—
86Dic79⋊C4—22no—
87Dic79⋊C4—22no—
88C22⋊Dic79—22no—
89C4×Dic79—22no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79#4111yescharacteristic
94M16×C79—41no—
95M16×C79—41no—
96C2×C8×C79—22no—
97C2×C8×C79—22no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C42×C79C22×C411yescharacteristic
101C42×C79C2×D411yescharacteristic
102C42×C79C2×D411yescharacteristic
103C42×C79#4111yescharacteristic
104C23×C79#12111yescharacteristic
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—81no—
108C2×C4×C79—81no—
109C2×C4×C79—81no—
110C2×C4×C79C2×C22⋊C411yescharacteristic
111C2×C4×C79C4×D411yescharacteristic
112C2×C4×C79C4×D411yescharacteristic
113C2×C4×C79#12011yescharacteristic
114C2×C4×C79#12111yescharacteristic
115C2×C4×C79#12311yescharacteristic
116C2×C4×C79C2×Q1611yescharacteristic
117C2×C4×C79#13511yescharacteristic
118C2×C4×C79#13711yescharacteristic
119C2×C4×C79#13711yescharacteristic
120C2×Dic79—22no—
121C2×Dic79—22no—
122C2×Dic79—42no—
123C2×Dic79—82no—
124Dic158—42no—
125Dic158—44no—
126Dic158—81no—
127Dic158—82no—
128C8×C79—42no—
129C8×C79—42no—
130C79⋊C8—22no—
131C79⋊C8—41no—
132C79⋊C8—81no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79#38611yescharacteristic
136C22×C79#46211yescharacteristic
137C22×C79#46411yescharacteristic
138C4×C79—21no—
139C4×C79—21no—
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—161no—
143C4×C79C4×Q1611yescharacteristic
144C4×C79#44111yescharacteristic
145C4×C79#45111yescharacteristic
146C4×C79#45211yescharacteristic
147Dic79—42no—
148Dic79—42no—
149Dic79—82no—
150#9290—791nomaximal, Sylow
151C2×C79—21no—
152C2×C79—21no—
153C2×C79#187811yescharacteristic
154C2×C79#190211yescharacteristic
155C2×C79#191011yescharacteristic
156#3209—791no—
157#2961—791no—
158#1993—791no—
159#1549—791no—
160#1518—791no—
161#1286D7911yescharacteristic
162#1275—791no—
163C79#929011yescharacteristic, complemented, Sylow
164#533—791no—
165C2×C4⋊Q8—791no—
166#494—792no—
167#487—792no—
168#462—1582no—
169#451—1582no—
170#446—1582no—
171#446—1582no—
172#431—791no—
173#431C2×D7911yescharacteristic
174#424—1581no—
175#423C2×D7911yescharacteristic
176#405—1581no—
177C2×#97—791no—
178#382C2×D7911yescharacteristic
179#340—1581no—
180#340—1581no—
181#332—1582no—
182#328—1582no—
183C22×Q8—1581no—
184C2×Q16—1584no—
185C2×M16—21no—
186C22×C8—791no—
187C22×C8—791no—
188C4⋊Q8—791no—
189C4⋊Q8—791no—
190C4⋊Q8—792no—
191C4⋊Q8—1581no—
192C4⋊Q8—1581no—
193C4⋊Q8—1581no—
194#125—1581no—
195C22⋊Q8—792no—
196C22⋊Q8—1581no—
197C4×Q8—792no—
198C4×Q8—1581no—
199#117—1581no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1581no—
203C2×C42C22×D7911yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1584no—
207C4⋊C8—1581no—
208C4⋊C8D31612yes—
209#103—1584no—
210C22⋊C8—21no—
211C22⋊C8—1581no—
212#97—1582no—
213#97—1581no—
214#97—1581no—
215C4×C8—1582no—
216C4×C8—1581no—
217C4×C8C4×D7912yes—
218#95—21no—
219#95—21no—
220C2×Q8—3161no—
221C2×Q8—792no—
222C2×Q8—6322no—
223C2×Q8—1582no—
224C2×Q8—1581no—
225C22×C4—41no—
226C22×C4—3161no—
227C22×C4D4×D7911yescharacteristic
228C22×C4D4×D7911yescharacteristic
229C22×C4#1192962411yescharacteristic
230Q16—6324no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—3161no—
236C2×C8—792no—
237C2×C8—1581no—
238C2×C8—1581no—
239C2×C8—1581no—
240C2×C8—1581no—
241C2×C8—1581no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—3161no—
245C4⋊C4—3161no—
246C4⋊C4—792no—
247C4⋊C4—1582no—
248C4⋊C4—1582no—
249C4⋊C4—1581no—
250C4⋊C4—1581no—
251C4⋊C4—1581no—
252C4⋊C4—1582no—
253C4⋊C4—1582no—
254C22⋊C4—1582no—
255C42—1582no—
256C42C2×C4×D7911yescharacteristic
257C42D4×D7911yescharacteristic
258C42C2×D31611yescharacteristic
259C42#1192962511yescharacteristic
260C23#1457913411yescharacteristic
261Q8—3162no—
262Q8—3164no—
263Q8—6321no—
264Q8—6322no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—3162no—
271C2×C4—6322no—
272C2×C4—1582no—
273C2×C4—1582no—
274C2×C4C22⋊C4×D7911yescharacteristic
275C2×C4#1457912311yescharacteristic
276C2×C4C4×D31611yescharacteristic
277C2×C4#1457913111yescharacteristic
278C2×C4#1457913611yescharacteristic
279C2×C4#1457914611yescharacteristic
280C2×C4Q16×D7911yescharacteristic
281C2×C4#1457918011yescharacteristic
282C2×C4#1457919011yescharacteristic
283C2×C4#1457919011yescharacteristic
284C8—42no—
285C8—42no—
286C8—3161no—
287C8—6321no—
288C8—1582no—
289C22—21no—
290C22—21no—
291C22#1909844111yescharacteristic
292C22#1909877211yescharacteristic
293C22#1909878411yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—161no—
299C4—3162no—
300C4—3162no—
301C4—6322no—
302C4#1909865711yescharacteristic
303C4#1909866911yescharacteristic
304C4#1909872011yescharacteristic
305C4#1909872611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2376374211yescharacteristic
309C2#2376387211yescharacteristic
310C2#2376392611yescharacteristic
subgroupG/H
centerC22#19098441
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1286C2
socleC22×C79#386
prime radicalC23×C79#121
cosocleC2×C42×C79C23
hypercenter#1286D79
hypocenterC79#9290
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
10. #19098943C4semidirect
11. #19098914C4semidirect
77. C2×C79⋊C8Q16semidirect
163. C79#9290semidirect

Maximal subgroups

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

subgroupindex
#237743502
#237723372
#237643722
#237620862
#237619632
#237612302
C79×#12862
#929079