Finite Group Databasebeta

← #34206594

Subgroups — #34206594

Order 20224.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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#23768385C211yesmaximal, characteristic
2C2×#19098361C214yesmaximal
3C4×#14579080C211yesmaximal, characteristic
4C79×#1801C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic
6C2×#14579159C2212yes—
7C2×#14579080C2212yes—
8C4×#11929608—21no—
9#19098493—21no—
10#19098493—21no—
11#19098468C412yes—
12#19098461—21no—
13#19098461—21no—
14C4×C4⋊Dic79C411yescharacteristic
15#19098361—28no—
16#19098332—21no—
17C2×C4×Q8×C79C2211yescharacteristic
18#423×C79—21no—
19C2×#101×C79C2212yes—
20#343×C79—21no—
21C22×Dic158C2×C412yes—
22C2×#11929608—22no—
23#14579161—41no—
24#14579161—41no—
25#14579159—24no—
26#14579159—24no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×Dic79⋊C4—22no—
31C2×Dic79⋊C4—22no—
32C2×C4×Dic79C2×C412yes—
33#14579080—22no—
34#14579080D414yes—
35#14579072—41no—
36C4×C79⋊C8—41no—
37C79⋊#95—21no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×Q8×C79C2311yescharacteristic
41C2×M16×C79—22no—
42C4×Q8×C79—41no—
43C4×Q8×C79—41no—
44C2×C4⋊C4×C79C2×C411yescharacteristic
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C42×C79C2×C411yescharacteristic
48C2×C42×C79D411yescharacteristic
49C2×C42×C79D411yescharacteristic
50C4⋊C8×C79—41no—
51C4⋊C8×C79—41no—
52#101×C79—24no—
53C2×Dic158—24no—
54C2×Dic158—42no—
55C2×Dic158—42no—
56C22×Dic79—22no—
57C22×Dic79—22no—
58#11929608—24no—
59#11929608—42no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—24no—
62C4⋊Dic79—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65Dic79⋊C4—44no—
66Dic79⋊C4—42no—
67Dic79⋊C4—42no—
68C4×Dic79—24no—
69C2×Q8×C79—41no—
70C2×Q8×C79—41no—
71C2×Q8×C79C2×D412yes—
72C2×Q8×C79#4112yes—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80M16×C79—42no—
81M16×C79—42no—
82C2×C8×C79—42no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C42×C79—21no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C23×C79C2×C22⋊C411yescharacteristic
93Q8×C79—42no—
94Q8×C79—42no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79#9911yescharacteristic
109C2×C4×C79#9911yescharacteristic
110C2×C4×C79C4×D412yes—
111C2×C4×C79#12112yes—
112C2×C4×C79#12411yescharacteristic
113C2×C4×C79#12711yescharacteristic
114C2×Dic79—24no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120Dic158—84no—
121C8×C79—82no—
122C79⋊C8—44no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79#39011yescharacteristic
126C22×C79C2×#9911yescharacteristic
127C22×C79#41011yescharacteristic
128C4×C79—21no—
129C4×C79—22no—
130C4×C79—22no—
131C4×C79—21no—
132C4×C79—21no—
133C4×C79—41no—
134C4×C79—41no—
135C4×C79—41no—
136C4×C79#35612yescomplemented
137Dic79—44no—
138Dic79—82no—
139#13227—791nomaximal, Sylow
140C2×C79—21no—
141C2×C79—21no—
142C2×C79#196411yescharacteristic
143C2×C79C2×#35611yescharacteristic
144C2×C79#213511yescharacteristic
145#2546—791no—
146C2×#356—794no—
147#1801D7911yescharacteristic
148C4×#101—791no—
149C79#1322711yescharacteristic, complemented, Sylow
150C2×C4⋊Q8—792no—
151C2×C4⋊Q8—791no—
152C2×C4×Q8C2×D7911yescharacteristic
153#423—21no—
154C2×#101—792no—
155C2×#101C2×D7912yes—
156C4×M16—1581no—
157#398—1581no—
158#398—1581no—
159#391—792no—
160#389—1581no—
161#389—1581no—
162C4×C4⋊C4—791no—
163#356—1588no—
164#343—21no—
165#343—1581no—
166C22×Q8—792no—
167C22×Q8C22×D7911yescharacteristic
168C2×M16—22no—
169C2×M16—1582no—
170C4⋊Q8—3161no—
171C4⋊Q8—3161no—
172C4⋊Q8—1584no—
173C4⋊Q8—1584no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1581no—
177C2×C4⋊C4—1581no—
178C2×C4⋊C4—1582no—
179C2×C4⋊C4—1582no—
180C2×C4⋊C4—1581no—
181C2×C4⋊C4C4×D7911yescharacteristic
182C2×C4⋊C4D31611yescharacteristic
183C2×C4⋊C4C22⋊D7911yescharacteristic
184C2×C42—792no—
185C2×C42C4×D7911yescharacteristic
186C2×C42D31611yescharacteristic
187C2×C42C22⋊D7911yescharacteristic
188C4⋊C8—41no—
189C4⋊C8—41no—
190#101—24no—
191#101—794no—
192#101—1582no—
193#97—3161no—
194C4×C8—3161no—
195#95—1581no—
196#95—1582no—
197#95—1582no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—3162no—
201C2×Q8—3162no—
202C2×Q8—1584no—
203C2×Q8D4×D7912yes—
204C2×Q8#1192962512yes—
205C22×C4—1582no—
206C22×C4—1582no—
207C22×C4(C2×D79)⋊C411yescharacteristic
208C22×C4(C2×D79)⋊C411yescharacteristic
209C22×C4(C2×D79)⋊C411yescharacteristic
210C22×C4(C2×D79)⋊C411yescharacteristic
211C22×C4C2×C4×D7911yescharacteristic
212C22×C4C2×C22⋊D7911yescharacteristic
213C22×C4C2×D31611yescharacteristic
214M16—42no—
215M16—42no—
216M16—3162no—
217M16—1584no—
218C2×C8—42no—
219C2×C8—3162no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—3162no—
224C4⋊C4—3161no—
225C4⋊C4—3161no—
226C4⋊C4—3164no—
227C4⋊C4—3162no—
228C4⋊C4—3162no—
229C4⋊C4—1584no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1584no—
237C23C2×(C2×D79)⋊C411yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—6324no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—3162no—
255C2×C4—3162no—
256C2×C4—3162no—
257C2×C4—3162no—
258C2×C4—3162no—
259C2×C4—1584no—
260C2×C4#1457907511yescharacteristic
261C2×C4#1457907511yescharacteristic
262C2×C4#1457912312yes—
263C2×C4#1457913612yes—
264C2×C4#1457915011yescharacteristic
265C2×C4#1457915711yescharacteristic
266C8—82no—
267C8—3164no—
268C22—21no—
269C22—21no—
270C22#1909846411yescharacteristic
271C22C2×#1457907511yescharacteristic
272C22#1909853611yescharacteristic
273C4—21no—
274C4—22no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—3164no—
282C4—6322no—
283C4#1909836112yescomplemented
284C2—21no—
285C2—21no—
286C2#2376423511yescharacteristic
287C2C2×#1909836111yescharacteristic
288C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#1801C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC22×Q8×C79C23
hypercenter#1801D79
hypocenterC79#13227
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
136. C4×C79#356semidirect
149. C79#13227semidirect
283. C4#19098361semidirect

Maximal subgroups

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

subgroupindex
#237683852
C2×#190983612
C4×#145790802
C79×#18012
#1322779