Finite Group Databasebeta

← #34203809

Subgroups — #34203809

Order 20224.

986 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 135 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#23765089C214yesmaximal, complemented
2#23763328C212yesmaximal, complemented
3C79×#3485C211yesmaximal, characteristic
4C2×#14579080C2214yescomplemented
5C2×#14579078—22no—
6#19098527—22no—
7#19098362—24no—
8#19098360—24no—
9C2×H(5,𝔽2)×C79C411yescharacteristic
10C22×#41×C79C2211yescharacteristic
11#547×C79—24no—
12#546×C79—24no—
13#536×C79C2212yes—
14#534×C79—22no—
15#525×C79C412yes—
16C2×#123×C79—22no—
17#518×C79C411yescharacteristic
18C2×#11929608—24no—
19#14579080—28no—
20#14579078—44no—
21C22⋊(C79⋊C8)—44no—
22H(5,𝔽2)×C79—44no—
23C2×#41×C79—22no—
24C2×#41×C79—24no—
25C2×#41×C79—44no—
26C2×#41×C79C2×C412yes—
27C22×Q8×C79C2311yescharacteristic
28C22×D4×C79—22no—
29C22×D4×C79—22no—
30C22×D4×C79C2×C411yescharacteristic
31C22×D4×C79D412yes—
32C23×C4×C79C2×C411yescharacteristic
33C23×C4×C79D412yes—
34C4⋊Q8×C79D414yes—
35#124×C79D414yes—
36#123×C79—28no—
37#123×C79—24no—
38#123×C79—22no—
39#123×C79—24no—
40C22⋊Q8×C79—22no—
41C22⋊Q8×C79—24no—
42#121×C79—28no—
43#121×C79—22no—
44#120×C79—24no—
45#120×C79—22no—
46C4×D4×C79—24no—
47C4×D4×C79—24no—
48#117×C79C2×C412yes—
49C2×C4⋊C4×C79—24no—
50C2×C22⋊C4×C79—24no—
51C2×C22⋊C4×C79—22no—
52#11929608—44no—
53#11929608—44no—
54C2×C79⋊C8—44no—
55C24×C79—22no—
56C24×C79C22⋊C411yescharacteristic
57C24×C79C22⋊C412yes—
58#41×C79—44no—
59#41×C79—416no—
60#41×C79—48no—
61C2×Q8×C79—42no—
62C2×Q8×C79C22⋊C412yes—
63C2×Q8×C79C2×D414yes—
64C2×D4×C79—24no—
65C2×D4×C79—24no—
66C2×D4×C79—24no—
67C2×D4×C79—22no—
68C2×D4×C79—24no—
69C2×D4×C79—22no—
70C2×D4×C79—24no—
71C2×D4×C79—42no—
72C2×D4×C79—48no—
73C2×D4×C79—44no—
74C2×D4×C79—44no—
75C2×D4×C79C22⋊C412yes—
76C22×C4×C79—22no—
77C22×C4×C79—24no—
78C22×C4×C79—24no—
79C22×C4×C79—22no—
80C22×C4×C79—22no—
81C22×C4×C79—24no—
82C22×C4×C79—42no—
83C22×C4×C79—42no—
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D412yes—
87C4⋊C4×C79—24no—
88C4⋊C4×C79—22no—
89C4⋊C4×C79—28no—
90C22⋊C4×C79—24no—
91C22⋊C4×C79—24no—
92C22⋊C4×C79—24no—
93C22⋊C4×C79—22no—
94C22⋊C4×C79—28no—
95C42×C79C22⋊C414yes—
96C23×C79—22no—
97C23×C79—22no—
98C23×C79—22no—
99C23×C79—21no—
100C23×C79—22no—
101C23×C79—44no—
102C23×C79—82no—
103C23×C79—81no—
104C23×C79—84no—
105C23×C79#9914yes—
106C23×C79C2×C22⋊C411yescharacteristic
107Q8×C79—44no—
108Q8×C79—42no—
109D4×C79—28no—
110D4×C79—48no—
111D4×C79—42no—
112D4×C79—48no—
113D4×C79—44no—
114D4×C79—84no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—24no—
119C2×C4×C79—21no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—48no—
123C2×C4×C79—44no—
124C2×C4×C79—44no—
125C2×C4×C79—44no—
126C2×C4×C79—84no—
127C2×C4×C79C2×C22⋊C412yes—
128C2×C4×C79#12014yes—
129C79⋊C8—84no—
130C22×C79—24no—
131C22×C79—41no—
132C22×C79—44no—
133C22×C79—44no—
134C22×C79—42no—
135C22×C79—44no—
136C22×C79—41no—
137C22×C79—42no—
138C22×C79—82no—
139C22×C79—82no—
140C22×C79—164no—
141C22×C79#37911yescharacteristic
142C22×C79C2×#9912yes—
143C4×C79—22no—
144C4×C79—24no—
145C4×C79—44no—
146C4×C79—41no—
147C4×C79—42no—
148#12709—791nomaximal, Sylow
149C2×C79—21no—
150C2×C79—22no—
151C2×C79—44no—
152C2×C79—81no—
153C2×C79—82no—
154C2×C79#178111yescharacteristic
155#3485D7911yescharacteristic
156#2133—794no—
157#1785—792no—
158C79#1270911yescharacteristic, complemented, Sylow
159C2×H(5,𝔽2)Dic7911yescharacteristic
160C22×#41C2×D7911yescharacteristic
161#547—24no—
162#546—24no—
163#536C2×D7912yes—
164#534—22no—
165#525Dic7912yes—
166C2×#123—22no—
167#518Dic7911yescharacteristic
168C2×#101—794no—
169C2×#100—1582no—
170#407—1582no—
171#356—1584no—
172#355—1584no—
173H(5,𝔽2)—44no—
174C2×#41—22no—
175C2×#41—24no—
176C2×#41—44no—
177C2×#41C2×Dic7912yes—
178C22×Q8C22×D7911yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4C2×Dic7911yescharacteristic
182C22×D4C22⋊D7912yes—
183C23×C4C2×Dic7911yescharacteristic
184C23×C4C22⋊D7912yes—
185C2×M16—1584no—
186C4⋊Q8C22⋊D7914yes—
187#124C22⋊D7914yes—
188#123—28no—
189#123—24no—
190#123—22no—
191#123—24no—
192C22⋊Q8—22no—
193C22⋊Q8—24no—
194#121—28no—
195#121—22no—
196#120—24no—
197#120—22no—
198C4×D4—24no—
199C4×D4—24no—
200#117C2×Dic7912yes—
201C2×C4⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204#101—1588no—
205#100—3164no—
206C22⋊C8—3164no—
207C24—22no—
208C24C22⋊Dic7911yescharacteristic
209C24C22⋊Dic7912yes—
210#41—44no—
211#41—416no—
212#41—48no—
213C2×Q8—42no—
214C2×Q8C22⋊Dic7912yes—
215C2×Q8C2×C22⋊D7914yes—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4C22⋊Dic7912yes—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic7911yescharacteristic
237C22×C4C22×Dic7911yescharacteristic
238C22×C4C2×C22⋊D7912yes—
239M16—3164no—
240M16—3164no—
241C2×C8—3164no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—28no—
250C42C22⋊Dic7914yes—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—44no—
257C23—82no—
258C23—81no—
259C23—84no—
260C23#1457907614yes—
261C23C2×C22⋊Dic7911yescharacteristic
262Q8—44no—
263Q8—42no—
264D4—28no—
265D4—48no—
266D4—42no—
267D4—48no—
268D4—44no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4C2×C22⋊Dic7912yes—
283C2×C4#1457913014yes—
284C8—6324no—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#1909841911yescharacteristic
297C22C2×#1457907612yes—
298C4—24no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—42no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2#2376331511yescharacteristic
subgroupG/H
centerC2#23763315
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
Fitting, prime radicalC79×#3485C2
socleC2×C79#1781
cosocleC22×Q8×C79C23
hypercenter#3485D79
hypocenterC79#12709
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #23765089C2semidirect
2. #23763328C2semidirect
4. C2×#14579080C22semidirect
158. C79#12709semidirect

Maximal subgroups

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

subgroupindex
#237650892
#237633282
C79×#34852
#1270979