Finite Group Databasebeta

← #22610784

Subgroups — #22610784

Order 9344.

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

normal subgroups only

This group has 348 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
1C22×#11924016C211yesmaximal, characteristic
2C2×#14556522C211yesmaximal, characteristic, complemented
3C2×#14556514C211yesmaximal, characteristic, complemented
4C2×#14556512C211yesmaximal, characteristic
5C2×C4×#10596831C211yesmaximal, characteristic, complemented
6#17988493C211yesmaximal, characteristic, complemented
7#390×C73C211yesmaximal, characteristic
8C22×Dic146C2211yescharacteristic
9C2×#11924016—22no—
10C2×#11924016—22no—
11C2×#11924016C414yescomplemented
12C22×#10596831C2211yescharacteristic
13C22×#10596831C2211yescharacteristic
14C2×C4⋊Dic73C2211yescharacteristic
15#14556522—22no—
16#14556514—22no—
17#14556512—22no—
18C4×#10596831—22no—
19#14556498—22no—
20C22×D4×C73C2211yescharacteristic
21C2×C4⋊C4×C73C2211yescharacteristic
22C2×C22⋊C4×C73—22no—
23C2×C42×C73C2211yescharacteristic
24C2×Dic146—21no—
25C2×Dic146—21no—
26C2×Dic146—21no—
27C2×Dic146—21no—
28C2×Dic146—21no—
29C2×Dic146C2×C412yes—
30C22×Dic73—21no—
31C22×Dic73—21no—
32#11924016—28no—
33C2×#10596831—21no—
34C2×#10596831—21no—
35C2×#10596831C2×C412yes—
36C2×#10596831D412yes—
37C2×#10596831D412yes—
38C2×#10596831D412yes—
39C4⋊Dic73—21no—
40C4⋊Dic73—21no—
41C24×C73—21no—
42C2×D4×C73—21no—
43C2×D4×C73—21no—
44C2×D4×C73—21no—
45C2×D4×C73—21no—
46C2×D4×C73—21no—
47C2×D4×C73C2×C412yes—
48C22×C4×C73—21no—
49C22×C4×C73D412yes—
50C22×C4×C73C2311yescharacteristic
51C4⋊C4×C73—21no—
52C4⋊C4×C73—21no—
53C22⋊C4×C73—44no—
54C42×C73—21no—
55C42×C73—21no—
56C23×C73—22no—
57C23×C73—41no—
58C23×C73—41no—
59C23×C73—42no—
60C23×C73—42no—
61C23×C73C2×D411yescharacteristic
62D4×C73—22no—
63D4×C73—22no—
64D4×C73—41no—
65D4×C73—41no—
66C2×C4×C73—22no—
67C2×C4×C73—22no—
68C2×C4×C73—41no—
69C2×C4×C73—42no—
70C2×C4×C73SD1614yescomplemented
71C2×C4×C73C22×C411yescharacteristic
72C2×C4×C73C2×D411yescharacteristic
73C2×C4×C73C2×D411yescharacteristic
74C2×C4×C73C2×D411yescharacteristic
75C2×C4×C73#4111yescharacteristic
76C2×C4×C73#4111yescharacteristic
77C2×Dic73—22no—
78C2×Dic73—41no—
79C2×Dic73—41no—
80C2×Dic73—41no—
81C2×Dic73—42no—
82Dic146—22no—
83Dic146—22no—
84Dic146—41no—
85Dic146—41no—
86#10596831—22no—
87#10596831C22⋊C414yescomplemented
88C22×C73—42no—
89C22×C73—44no—
90C22×C73—82no—
91C22×C73—81no—
92C22×C73—81no—
93C22×C73C4×D411yescharacteristic
94C22×C73#12111yescharacteristic
95C22×C73#12711yescharacteristic
96C22×C73C2×SD1612yes—
97C22×C73#13512yes—
98C4×C73—24no—
99C4×C73—81no—
100C4×C73C2×C22⋊C411yescharacteristic
101C4×C73C4×D411yescharacteristic
102C4×C73#12111yescharacteristic
103C4×C73#12411yescharacteristic
104Dic73—42no—
105Dic73—81no—
106C2×C73—82no—
107C2×C73#39011yescharacteristic
108C2×C73C4×SD1612yes—
109C2×C73#46512yes—
110C2×C73#49211yescharacteristic
111C2×C73#49511yescharacteristic
112#1968—731nomaximal, Sylow
113C73#196811yescharacteristic, complemented, Sylow
114C22×SD16—731no—
115C2×#106—731no—
116C2×#103—731no—
117C2×#102—731no—
118C2×C4×C8—731no—
119#391—731no—
120#390D7311yescharacteristic
121C22×Q8—731no—
122C22×D4C2×D7311yescharacteristic
123C2×SD16—734no—
124C2×SD16—1462no—
125C2×SD16—1462no—
126C22×C8—731no—
127C22×C8—731no—
128C2×C4⋊C4—731no—
129C2×C4⋊C4C2×D7311yescharacteristic
130C2×C22⋊C4—22no—
131C2×C42C2×D7311yescharacteristic
132#106—1462no—
133#103—1462no—
134#102—1462no—
135C4×C8—1462no—
136#95—1462no—
137C24—21no—
138C2×Q8—732no—
139C2×Q8—1461no—
140C2×Q8—1461no—
141C2×Q8—1461no—
142C2×Q8—1461no—
143C2×Q8—1461no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D7312yes—
150C22×C4—21no—
151C22×C4—1461no—
152C22×C4—1461no—
153C22×C4C22⋊D7312yes—
154C22×C4C22×D7311yescharacteristic
155SD16—1468no—
156C2×C8—732no—
157C2×C8—732no—
158C2×C8—732no—
159C2×C8—732no—
160C2×C8—1461no—
161C2×C8—1461no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—1461no—
165C4⋊C4—1461no—
166C22⋊C4—44no—
167C42—21no—
168C42—21no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23C2×C22⋊D7311yescharacteristic
175Q8—2921no—
176Q8—2921no—
177Q8—1462no—
178Q8—1462no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—2921no—
188C2×C4—2921no—
189C2×C4—2921no—
190C2×C4—2922no—
191C2×C4—1462no—
192C2×C4#1192401614yescomplemented
193C2×C4C2×C4×D7311yescharacteristic
194C2×C4D4×D7311yescharacteristic
195C2×C4D4×D7311yescharacteristic
196C2×C4D4×D7311yescharacteristic
197C2×C4#1192402711yescharacteristic
198C2×C4#1192402811yescharacteristic
199C8—734no—
200C8—1462no—
201C22—44no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—82no—
206C22C4×C22⋊D7311yescharacteristic
207C22#1455656311yescharacteristic
208C22#1455658711yescharacteristic
209C22C2×#1192401612yes—
210C22#1455661112yes—
211C4—24no—
212C4—81no—
213C4—2922no—
214C4—5841no—
215C4C22⋊C4×D7311yescharacteristic
216C4#1455655111yescharacteristic
217C4#1455656411yescharacteristic
218C4#1455657711yescharacteristic
219C2—82no—
220C2#1798848711yescharacteristic
221C2C4×#1192401612yes—
222C2#1798881412yes—
223C2#1798895911yescharacteristic
224C2#1798896911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D73
commutatorC2×C4×C73C22×C4
FrattiniC22×C4C22×D73
Fitting#390×C73C2
socleC23×C73C2×D4
hypercenter#390D73
hypocenterC73#1968
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#14556522C2semidirect
3. C2×#14556514C2semidirect
5. C2×C4×#10596831C2semidirect
6. #17988493C2semidirect
11. C2×#11924016C4semidirect
70. C2×C4×C73SD16semidirect
87. #10596831C22⋊C4semidirect
113. C73#1968semidirect
192. C2×C4#11924016semidirect