Finite Group Databasebeta

← #22625917

Subgroups — #22625917

Order 9344.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×#14556670C211yesmaximal, characteristic
2C2×#14556654C212yesmaximal, complemented
3#17989555C211yesmaximal, characteristic
4#17989460C212yesmaximal, complemented
5C22⋊Q8×D73C211yesmaximal, characteristic
6C2×#11924036C2212yes—
7#14556670—24no—
8C2×C4⋊#64408C2212yes—
9C2×C22⋊#64408—21no—
10C2×C4×#64408—21no—
11#14556654—24no—
12#14556648—21no—
13C22⋊#10596833—21no—
14C2×Q8×D73C2211yescharacteristic
15C22×C4×D73C2211yescharacteristic
16#14556571C411yescharacteristic, complemented
17#14556570C411yescharacteristic, complemented
18#14556569—21no—
19#14556568—21no—
20#14556567C411yescharacteristic, complemented
21C4⋊C4×D73—21no—
22C4⋊C4×D73C2211yescharacteristic
23C22⋊C4×D73—21no—
24C22⋊Q8×C73C411yescharacteristic, complemented
25C22×#64408—22no—
26#11924036—22no—
27#11924036—22no—
28C2×#10596833—22no—
29C4⋊#64408—22no—
30C4⋊#64408D414yes—
31C22⋊#64408—24no—
32C4×#64408—24no—
33C23×D73D411yescharacteristic
34C2×Dic146—21no—
35C2×Dic146C2×C411yescharacteristic
36Q8×D73—22no—
37Q8×D73—22no—
38C22×Dic73D411yescharacteristic
39C2×C4×D73—22no—
40C2×C4×D73—22no—
41C2×C4×D73—21no—
42C2×C4×D73—21no—
43C2×C4×D73C2×C411yescharacteristic
44C2×C4×D73D411yescharacteristic
45C2×C4×D73D411yescharacteristic
46C2×C4×D73C2311yescharacteristic
47C4⋊Dic73—21no—
48C4⋊Dic73C2×C411yescharacteristic
49#11924007—21no—
50#11924007—21no—
51#11924007—21no—
52C22⋊Dic73—21no—
53#11924005—21no—
54#11924005—21no—
55C2×Q8×C73C2×C411yescharacteristic
56C22×C4×C73C2×C411yescharacteristic
57C4⋊C4×C73—21no—
58C4⋊C4×C73C2×C411yescharacteristic
59C22⋊C4×C73—21no—
60C23×C73C22⋊C411yescharacteristic
61C22×D73—21no—
62C22×D73—21no—
63C22×D73—22no—
64C22×D73—22no—
65C22×D73C22⋊C411yescharacteristic
66C22×D73C2×D411yescharacteristic
67C2×#64408—24no—
68C2×#64408—42no—
69C2×#64408—42no—
70Q8×C73—22no—
71C2×C4×C73—22no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73C22⋊C411yescharacteristic
75C2×C4×C73C22×C411yescharacteristic
76C2×Dic73—22no—
77C2×Dic73—21no—
78C2×Dic73—21no—
79C2×Dic73C22⋊C411yescharacteristic
80C2×Dic73C2×D411yescharacteristic
81C4×D73—21no—
82C4×D73—21no—
83C4×D73—22no—
84C4×D73—22no—
85C4×D73—41no—
86C4×D73—41no—
87C4×D73—42no—
88C4×D73C2×D412yes—
89C4×D73#4112yes—
90Dic146—22no—
91Dic146—42no—
92#10596833—42no—
93C22×C73—21no—
94C22×C73—21no—
95C22×C73—21no—
96C22×C73C2×C22⋊C411yescharacteristic
97C2×D73—21no—
98C2×D73—21no—
99C2×D73—21no—
100C2×D73—42no—
101C2×D73—42no—
102C2×D73#12011yescharacteristic
103C2×D73#12311yescharacteristic
104C2×D73#13512yes—
105C2×D73#13712yes—
106#64408—44no—
107C4×C73—21no—
108C4×C73—41no—
109C4×C73—41no—
110C4×C73C4×D412yes—
111Dic73—21no—
112Dic73—41no—
113Dic73—41no—
114Dic73#12112yes—
115C2×C73—41no—
116C2×C73#38611yescharacteristic
117C2×C73#41611yescharacteristic
118C2×C73#41911yescharacteristic
119D73—41no—
120D73#45212yes—
121D73#48512yes—
122#1873—731nomaximal, Sylow
123C73#187311yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—731no—
125C2×#117—731no—
126C2×#103—732no—
127C2×C22⋊C8—731no—
128#340—732no—
129C22×Q8—731no—
130C23×C4—731no—
131C22×C8—732no—
132C22⋊Q8—731no—
133C22⋊Q8—731no—
134C22⋊Q8—731no—
135C22⋊Q8—1461no—
136C22⋊Q8—1461no—
137C22⋊Q8#6440811yescharacteristic, complemented
138#117—1464no—
139C2×C4⋊C4—732no—
140C2×C4⋊C4—731no—
141C2×C4⋊C4—1461no—
142C2×C22⋊C4—1461no—
143C2×C22⋊C4—1461no—
144C2×C42—1461no—
145#103—1464no—
146C22⋊C8—1461no—
147C22⋊C8—1461no—
148C24—731no—
149C2×Q8—731no—
150C2×Q8—1461no—
151C2×Q8—1462no—
152C2×Q8—1462no—
153C2×Q8C2×#6440811yescharacteristic
154C22×C4—731no—
155C22×C4—731no—
156C22×C4—731no—
157C22×C4—731no—
158C22×C4—731no—
159C22×C4—1461no—
160C22×C4—1461no—
161C22×C4—1462no—
162C22×C4—1462no—
163C22×C4—1462no—
164C22×C4C2×#6440811yescharacteristic
165C2×C8—1462no—
166C2×C8—1462no—
167C2×C8—1462no—
168C4⋊C4—21no—
169C4⋊C4—734no—
170C4⋊C4—731no—
171C4⋊C4—1461no—
172C4⋊C4—1461no—
173C4⋊C4—1461no—
174C4⋊C4—1462no—
175C4⋊C4—1461no—
176C4⋊C4C2×#6440811yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—1461no—
179C22⋊C4—1461no—
180C22⋊C4—1464no—
181C22⋊C4—1461no—
182C42—1464no—
183C23—731no—
184C23—731no—
185C23—1461no—
186C23—1461no—
187C23—1462no—
188C23—1462no—
189C23C22⋊#6440811yescharacteristic
190Q8—22no—
191Q8—2922no—
192Q8—1462no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—2921no—
197C2×C4—2921no—
198C2×C4—2922no—
199C2×C4—2922no—
200C2×C4—2922no—
201C2×C4—731no—
202C2×C4—732no—
203C2×C4—732no—
204C2×C4—731no—
205C2×C4—1462no—
206C2×C4—1461no—
207C2×C4—1461no—
208C2×C4—1464no—
209C2×C4—1461no—
210C2×C4—1461no—
211C2×C4—1462no—
212C2×C4—1462no—
213C2×C4C22⋊#6440811yescharacteristic
214C2×C4C22×#6440811yescharacteristic
215C8—2922no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—2922no—
220C22—2922no—
221C22—731no—
222C22—731no—
223C22—732no—
224C22—732no—
225C22—1461no—
226C22—1461no—
227C22—1461no—
228C22C2×C22⋊#6440811yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—2924no—
233C4—2921no—
234C4—2921no—
235C4—732no—
236C4—1461no—
237C4D4×#6440812yes—
238C2—41no—
239C2—2921no—
240C2—732no—
241C2—732no—
242C2#1798953511yescharacteristic
243C2#1798957711yescharacteristic
244C2#1798958011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#64408
commutatorC2×C4×C73C22×C4
FrattiniC2×C4C22×#64408
FittingC22⋊Q8×C73C4
socleC22×C73C2×C22⋊C4
hypercenterC22⋊Q8#64408
hypocenterC73#1873
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14556654C2semidirect
4. #17989460C2semidirect
16. #14556571C4semidirect
17. #14556570C4semidirect
20. #14556567C4semidirect
24. C22⋊Q8×C73C4semidirect
123. C73#1873semidirect
137. C22⋊Q8#64408semidirect