Finite Group Databasebeta

← #34199628

Subgroups — #34199628

Order 20224.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#23773987C211yesmaximal, characteristic
2#23765202C211yesmaximal, characteristic
3#335×D79C211yesmaximal, characteristic
4#23761455C211yesmaximal, characteristic, complemented
5#23761450C211yesmaximal, characteristic, complemented
6#23761448C211yesmaximal, characteristic, complemented
7C79×#1364C211yesmaximal, characteristic, complemented
8#19099118—21no—
9#19098943C2211yescharacteristic
10#19098915—21no—
11#19098805—21no—
12#19098805C412yes—
13#19098657—21no—
14#19098657C412yes—
15C4⋊C8×D79—21no—
16C4×C8×D79C2211yescharacteristic
17C4⋊(C79⋊C16)C2211yescharacteristic
18(C2×D79)⋊C16—21no—
19#19098338—21no—
20#19098316C2211yescharacteristic
21#19098315—21no—
22#500×C79C2211yescharacteristic
23C4⋊C16×C79C2211yescharacteristic
24#335×C79C2211yescharacteristic
25C2×Dic316—21no—
26C2×Dic316C2×C412yes—
27C2×C8×D79—41no—
28C2×C8×D79C2×C411yescharacteristic
29C2×C8×D79D411yescharacteristic
30#14579161—21no—
31#14579152—41no—
32#14579141—22no—
33#14579137—41no—
34Q8×Dic79—41no—
35#14579126—22no—
36#14579119—41no—
37C42×D79D411yescharacteristic
38C2×C79⋊C16—21no—
39C4⋊(C79⋊C8)—21no—
40Dic79⋊C8—41no—
41#14579084—22no—
42#14579084—22no—
43C4×C79⋊C8D411yescharacteristic
44C8×Dic79C2×C411yescharacteristic
45C8×Dic79D411yescharacteristic
46C4⋊Q8×C79—21no—
47C2×C16×C79—21no—
48#107×C79—21no—
49#107×C79C2×C412yes—
50C4⋊C8×C79—21no—
51C4×C8×C79C2311yescharacteristic
52C2×Dic158—22no—
53C2×C4×D79C22⋊C411yescharacteristic
54C2×C4×D79D811yescharacteristic
55C2×C4×D79SD1611yescharacteristic
56Dic316—42no—
57Dic316—41no—
58Dic316—41no—
59C2×C79⋊C8—41no—
60C2×C79⋊C8C22⋊C411yescharacteristic
61C2×C79⋊C8C22⋊C411yescharacteristic
62C8×D79—21no—
63C8×D79—21no—
64C8×D79—41no—
65C8×D79—82no—
66C4⋊Dic79—42no—
67C4⋊Dic79—41no—
68C4⋊Dic79—41no—
69Dic79⋊C4—42no—
70(C2×D79)⋊C4—42no—
71C4×Dic79—42no—
72C4×Dic79C22⋊C411yescharacteristic
73C4×Dic79D811yescharacteristic
74C4×Dic79SD1611yescharacteristic
75C79⋊C16—41no—
76C2×Q8×C79—41no—
77C2×C8×C79—41no—
78C2×C8×C79C22×C411yescharacteristic
79C2×C8×C79C2×D411yescharacteristic
80C4⋊C4×C79—22no—
81C4⋊C4×C79—41no—
82C42×C79C2×D411yescharacteristic
83C16×C79—41no—
84C22×D79#10211yescharacteristic
85Q8×C79—81no—
86Q8×C79—81no—
87C2×C4×C79—42no—
88C2×C4×C79C2×C22⋊C411yescharacteristic
89C2×C4×C79C2×D811yescharacteristic
90C2×C4×C79C2×SD1611yescharacteristic
91C2×Dic79—42no—
92C2×Dic79#10211yescharacteristic
93C2×Dic79#10211yescharacteristic
94C2×Dic79#10211yescharacteristic
95C4×D79—22no—
96C4×D79—21no—
97C4×D79—21no—
98C4×D79—21no—
99C4×D79—21no—
100Dic158—42no—
101Dic158—42no—
102C8×C79—21no—
103C8×C79—81no—
104C8×C79#13511yescharacteristic
105C8×C79#13511yescharacteristic
106C79⋊C8—21no—
107C79⋊C8—21no—
108C79⋊C8—81no—
109C22×C79C2×#10211yescharacteristic
110C2×D79—21no—
111C2×D79—21no—
112C2×D79#36212yes—
113C4×C79—21no—
114C4×C79—82no—
115C4×C79#41311yescharacteristic
116C4×C79#41611yescharacteristic
117C4×C79#50911yescharacteristic
118C4×C79#51011yescharacteristic
119Dic79—21no—
120Dic79—21no—
121Dic79—21no—
122Dic79—82no—
123#11954—791nomaximal, Sylow
124C2×C79#154011yescharacteristic
125C2×C79#213911yescharacteristic
126C2×C79C2×#36211yescharacteristic
127D79—22no—
128#3170—791no—
129#2163—791no—
130C2×#335—791no—
131#1364—791no—
132#1364D7911yescharacteristic, complemented
133#1362—791no—
134#1362—791no—
135C79#1195411yescharacteristic, complemented, Sylow
136#533—1581no—
137#500C2×D7911yescharacteristic
138#494—791no—
139#487—1581no—
140#467—792no—
141#467—1581no—
142C4×Q16—792no—
143C4×Q16—1581no—
144C2×C4⋊C8—1581no—
145C2×C4×C8—791no—
146C4⋊C16—791no—
147C4⋊C16C2×D7911yescharacteristic
148C22⋊C16—1581no—
149#346—1581no—
150#335—791no—
151#335—1581no—
152#335C2×D7911yescharacteristic
153C2×Q16—792no—
154C2×Q16—1581no—
155C22×C8—3161no—
156C22×C8—791no—
157C22×C8—791no—
158C4⋊Q8—21no—
159C4⋊Q8—1581no—
160#125—3161no—
161C22⋊Q8—3161no—
162C22⋊Q8—1582no—
163C4×Q8—3161no—
164C4×Q8—1582no—
165#117—3161no—
166C2×C42—791no—
167C2×C16—21no—
168C2×C16—1581no—
169#107—21no—
170#107C4×D7912yes—
171C4⋊C8—21no—
172C4⋊C8—3161no—
173C4⋊C8—1581no—
174#103—1582no—
175#103—1582no—
176C4×C8—791no—
177C4×C8—791no—
178C4×C8—791no—
179C4×C8C22×D7911yescharacteristic
180C2×Q8—41no—
181C2×Q8—1582no—
182C22×C4—791no—
183C22×C4—791no—
184C22×C4—791no—
185Q16—3162no—
186Q16—3161no—
187Q16—3161no—
188C2×C8—41no—
189C2×C8—3161no—
190C2×C8—3161no—
191C2×C8—791no—
192C2×C8—791no—
193C2×C8—6322no—
194C2×C8—1581no—
195C2×C8—1581no—
196C2×C8C2×C4×D7911yescharacteristic
197C2×C8D4×D7911yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—3162no—
201C4⋊C4—3162no—
202C4⋊C4—3161no—
203C4⋊C4—3161no—
204C22⋊C4—3162no—
205C42—3162no—
206C42—791no—
207C42—791no—
208C42—791no—
209C42D4×D7911yescharacteristic
210C16—41no—
211C16—3161no—
212C23—791no—
213Q8—81no—
214Q8—81no—
215Q8—3162no—
216Q8—3162no—
217C2×C4—42no—
218C2×C4—3162no—
219C2×C4—791no—
220C2×C4—791no—
221C2×C4—791no—
222C2×C4—1582no—
223C2×C4—1581no—
224C2×C4—1581no—
225C2×C4—1581no—
226C2×C4—1581no—
227C2×C4C22⋊C4×D7911yescharacteristic
228C2×C4D8×D7911yescharacteristic
229C2×C4SD16×D7911yescharacteristic
230C8—21no—
231C8—81no—
232C8—6321no—
233C8—1581no—
234C8—1581no—
235C8#1457918211yescharacteristic
236C8#1457918411yescharacteristic
237C22—792no—
238C22—1581no—
239C22—1581no—
240C22#102×D7911yescharacteristic
241C4—21no—
242C4—82no—
243C4—6322no—
244C4—1581no—
245C4—1581no—
246C4—1581no—
247C4#1909854611yescharacteristic
248C4#1909855811yescharacteristic
249C4#1909899711yescharacteristic
250C4#1909900211yescharacteristic
251C2—1582no—
252C2#2376204811yescharacteristic
253C2#2376510911yescharacteristic
254C2#362×D7911yescharacteristic
subgroupG/H
centerC22#102×D79
commutatorC2×C8×C79C22×C4
FrattiniC4×C8C22×D79
FittingC79×#1364C2
socleC22×C79C2×#102
prime radicalC22×D79#102
cosocleC4×C8×C79C23
hypercenter#1364D79
hypocenterC79#11954
LCSG ▷ C2×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
4. #23761455C2semidirect
5. #23761450C2semidirect
6. #23761448C2semidirect
7. C79×#1364C2semidirect
132. #1364D79semidirect
135. C79#11954semidirect

Maximal subgroups

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

subgroupindex
#237739872
#237652022
#335×D792
#237614552
#237614502
#237614482
C79×#13642
#1195479