Finite Group Databasebeta

← #34204149

Subgroups — #34204149

Order 20224.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19098362C212yesmaximal, complemented
2C2×#19098360C211yesmaximal, characteristic
3#23763804C211yesmaximal, characteristic
4#23763794C212yesmaximal, complemented
5C2×#563×C79C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098572—22no—
8#19098568—21no—
9C2×#14579080C2212yes—
10C2×#14579080C2211yescharacteristic
11#19098362—24no—
12#19098360—22no—
13#19098323—22no—
14#19098323—21no—
15#563×C79—24no—
16#563×C79—22no—
17#563×C79C414yescomplemented
18C2×C4⋊Q8×C79C2212yes—
19C2×#126×C79—22no—
20C2×#125×C79—21no—
21C2×#124×C79C2211yescharacteristic
22C2×#123×C79—21no—
23C2×C22⋊Q8×C79—22no—
24C2×#11929608—21no—
25C2×#11929608—22no—
26Q8×Dic79—42no—
27#14579120—21no—
28#14579120—22no—
29#14579085—44no—
30#14579082—42no—
31#14579080—22no—
32#14579080—22no—
33#14579080—22no—
34C22×Q8×C79D412yes—
35C22×Q8×C79C2311yescharacteristic
36C22×D4×C79D411yescharacteristic
37C23×C4×C79—21no—
38C4⋊Q8×C79—22no—
39C4⋊Q8×C79—22no—
40C4⋊Q8×C79C2×C414yes—
41C4⋊Q8×C79D414yes—
42#126×C79—28no—
43#126×C79—28no—
44#125×C79—24no—
45#125×C79—24no—
46#124×C79—22no—
47#124×C79C2×C412yes—
48#124×C79D412yes—
49#123×C79—24no—
50#123×C79—24no—
51C22⋊Q8×C79—28no—
52C22⋊Q8×C79—28no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79—22no—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79—22no—
57C2×C4⋊C4×C79—22no—
58C2×C22⋊C4×C79—22no—
59C2×C22⋊C4×C79—21no—
60C2×C22⋊C4×C79—22no—
61C2×C42×C79D411yescharacteristic
62C2×C42×C79D412yes—
63#11929608—41no—
64#11929608—42no—
65#11929608—42no—
66#11929608—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—42no—
69C4⋊Dic79—22no—
70C4⋊Dic79—24no—
71C22⋊Dic79—41no—
72C4×Dic79—41no—
73C4×Dic79—42no—
74C24×C79—21no—
75C2×Q8×C79—22no—
76C2×Q8×C79—41no—
77C2×Q8×C79—41no—
78C2×Q8×C79—42no—
79C2×Q8×C79—42no—
80C2×Q8×C79C22⋊C414yes—
81C2×Q8×C79C22×C411yescharacteristic
82C2×Q8×C79C2×D412yes—
83C2×Q8×C79C2×D411yescharacteristic
84C2×D4×C79—21no—
85C2×D4×C79—41no—
86C2×D4×C79—41no—
87C2×D4×C79C22⋊C412yes—
88C22×C4×C79—21no—
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79—24no—
92C22×C4×C79—81no—
93C22×C4×C79—81no—
94C22×C4×C79C2×D412yes—
95C22×C4×C79C2×D411yescharacteristic
96C4⋊C4×C79—24no—
97C4⋊C4×C79—28no—
98C4⋊C4×C79—24no—
99C4⋊C4×C79—24no—
100C4⋊C4×C79—24no—
101C4⋊C4×C79—24no—
102C4⋊C4×C79—24no—
103C22⋊C4×C79—28no—
104C22⋊C4×C79—24no—
105C22⋊C4×C79—24no—
106C22⋊C4×C79—24no—
107C42×C79—21no—
108C42×C79—22no—
109C42×C79C22⋊C412yes—
110C42×C79C22⋊C414yes—
111C23×C79—22no—
112C23×C79—44no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79#12011yescharacteristic
116Q8×C79—41no—
117Q8×C79—41no—
118Q8×C79—42no—
119Q8×C79—44no—
120Q8×C79—82no—
121D4×C79—42no—
122D4×C79—81no—
123C2×C4×C79—21no—
124C2×C4×C79—22no—
125C2×C4×C79—21no—
126C2×C4×C79—22no—
127C2×C4×C79—22no—
128C2×C4×C79—24no—
129C2×C4×C79—24no—
130C2×C4×C79—42no—
131C2×C4×C79—44no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—84no—
135C2×C4×C79C2×C22⋊C412yes—
136C2×C4×C79C2×C22⋊C411yescharacteristic
137C2×C4×C79#12012yes—
138C2×C4×C79#12011yescharacteristic
139C2×Dic79—42no—
140C79⋊C8—81no—
141C79⋊C8—82no—
142C22×C79—22no—
143C22×C79—44no—
144C22×C79—42no—
145C22×C79—82no—
146C22×C79—82no—
147C22×C79#37911yescharacteristic
148C22×C79B(4,𝔽2)11yescharacteristic
149C22×C79#45811yescharacteristic
150C4×C79—21no—
151C4×C79—22no—
152C4×C79—21no—
153C4×C79—22no—
154C4×C79—42no—
155C4×C79—44no—
156C4×C79—44no—
157Dic79—82no—
158#12775—791nomaximal, Sylow
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—82no—
162C2×C79#179011yescharacteristic
163C2×C79#220312yes—
164C2×#563D7911yescharacteristic
165C2×#356—792no—
166C2×#355—791no—
167#1888—791no—
168#1886—792no—
169C79#1277511yescharacteristic, complemented, Sylow
170#563—24no—
171#563—22no—
172#563Dic7914yescomplemented
173C2×C4⋊Q8C2×D7912yes—
174C2×#126—22no—
175C2×#125—21no—
176C2×#124C2×D7911yescharacteristic
177C2×#123—21no—
178C2×C22⋊Q8—22no—
179#519—791no—
180#419—1582no—
181#418—1581no—
182C2×#101—792no—
183C2×#101—791no—
184#356—1584no—
185#355—1582no—
186#339—1582no—
187#339—1581no—
188C22×Q8C22⋊D7912yes—
189C22×Q8C22×D7911yescharacteristic
190C22×D4C22⋊D7911yescharacteristic
191C23×C4—21no—
192C2×M16—1581no—
193C2×M16—1582no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C2×Dic7914yes—
197C4⋊Q8C22⋊D7914yes—
198#126—28no—
199#126—28no—
200#125—24no—
201#125—24no—
202#124—22no—
203#124C2×Dic7912yes—
204#124C22⋊D7912yes—
205#123—24no—
206#123—24no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C4×Q8—3162no—
210#117—1581no—
211#117—1582no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42C22⋊D7911yescharacteristic
221C2×C42C22⋊D7912yes—
222#103—3164no—
223#102—3162no—
224#101—1582no—
225#101—1582no—
226#101—1582no—
227C24—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C22⋊Dic7914yes—
234C2×Q8C22×Dic7911yescharacteristic
235C2×Q8C2×C22⋊D7912yes—
236C2×Q8C2×C22⋊D7911yescharacteristic
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊Dic7912yes—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×C22⋊D7912yes—
248C22×C4C2×C22⋊D7911yescharacteristic
249M16—3161no—
250M16—3162no—
251M16—3162no—
252M16—3161no—
253C2×C8—3161no—
254C2×C8—3162no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—1582no—
263C4⋊C4—1584no—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—3161no—
269C42—21no—
270C42—22no—
271C42—3161no—
272C42—3162no—
273C42C22⋊Dic7912yes—
274C42C22⋊Dic7914yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23#1457913011yescharacteristic
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—44no—
284Q8—82no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4—3162no—
300C2×C4C2×C22⋊Dic7912yes—
301C2×C4C2×C22⋊Dic7911yescharacteristic
302C2×C4#1457913012yes—
303C2×C4#1457913011yescharacteristic
304C8—6321no—
305C8—6322no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22#1909841911yescharacteristic
312C22#1909875011yescharacteristic
313C22#1909875311yescharacteristic
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—6322no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2#2376334811yescharacteristic
326C2#2376542112yes—
subgroupG/H
centerC22#19098750
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#563×C79C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
cosocleC22×Q8×C79C23
hypercenterC2×#563D79
hypocenterC79#12775
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. C2×#19098362C2semidirect
4. #23763794C2semidirect
17. #563×C79C4semidirect
169. C79#12775semidirect
172. #563Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983622
C2×#190983602
#237638042
#237637942
C2×#563×C792
#1277579