Finite Group Databasebeta

← #22621008

Subgroups — #22621008

Order 9344.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#17989163C211yesmaximal, characteristic
2#17988979C211yesmaximal, characteristic, complemented
3#17988966C211yesmaximal, characteristic, complemented
4#17988957C211yesmaximal, characteristic, complemented
5#17988944C211yesmaximal, characteristic, complemented
6#17988809C212yesmaximal, complemented
7#17988797C212yesmaximal, complemented
8#17988747C212yesmaximal
9#17988742C212yesmaximal, complemented
10#17988648C211yesmaximal, characteristic, complemented
11#541×C73C211yesmaximal, characteristic, complemented
12C2×#11924028C2211yescharacteristic
13C22×Dic146C2211yescharacteristic
14C2×#11924018C2212yes—
15C2×#11924018C2212yes—
16C2×#11924018C2212yes—
17C2×#11924015C2212yes—
18#14556591C2211yescharacteristic
19#14556591C2211yescharacteristic
20#14556579C2211yescharacteristic
21#14556579C2211yescharacteristic
22#14556575—22no—
23#14556571—22no—
24C4⋊#10596831C2212yes—
25#14556514C2212yes—
26#14556514C2212yes—
27#14556514C2212yes—
28#14556511C2212yes—
29C22⋊#10596831C2212yes—
30#14556502C2211yescharacteristic
31C4×#10596831C2211yescharacteristic
32C4⋊Q8×C73C2211yescharacteristic
33#126×C73—22no—
34#125×C73—22no—
35#125×C73C2211yescharacteristic
36C22⋊Q8×C73C2212yes—
37C4×Q8×C73C2212yes—
38#117×C73C2211yescharacteristic
39#11924028—42no—
40C2×Dic146—21no—
41C2×Dic146—42no—
42C2×Dic146C2311yescharacteristic
43C2×Dic146C2311yescharacteristic
44C2×Dic146C2311yescharacteristic
45C2×D292C2311yescharacteristic
46C2×C22⋊D73—21no—
47C22×Dic73—21no—
48C2×C4×D73—21no—
49#11924018—42no—
50#11924018—42no—
51#11924018—42no—
52#11924015—42no—
53C2×#10596831C2312yes—
54C2×#10596831C2312yes—
55C4⋊Dic73—22no—
56C4⋊Dic73—22no—
57#11924005—22no—
58#11924005—22no—
59C2×Q8×C73C2312yes—
60C22×C4×C73C2311yescharacteristic
61C4⋊C4×C73—22no—
62C4⋊C4×C73—22no—
63C4⋊C4×C73—22no—
64C4⋊C4×C73—22no—
65C4⋊C4×C73D412yes—
66C4⋊C4×C73C2312yes—
67C22⋊C4×C73—22no—
68C22⋊C4×C73D412yes—
69C42×C73—22no—
70C42×C73C2311yescharacteristic
71C42×C73C2311yescharacteristic
72C23×C73C2×D411yescharacteristic
73C22×D73—21no—
74Q8×C73—22no—
75Q8×C73—22no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73C2×D412yes—
80C2×C4×C73C2×D412yes—
81C2×C4×C73C2×D411yescharacteristic
82C2×C4×C73C2411yescharacteristic
83C22⋊D73—42no—
84C2×Dic73—21no—
85C2×Dic73—21no—
86C2×Dic73—21no—
87C2×Dic73—42no—
88D292—22no—
89C4×D73—42no—
90Dic146—22no—
91Dic146—22no—
92Dic146—22no—
93Dic146—42no—
94#10596831—22no—
95#10596831—22no—
96C22×C73—21no—
97C22×C73—22no—
98C22×C73C22×D411yescharacteristic
99C2×D73—41no—
100C2×D73—42no—
101C4×C73—21no—
102C4×C73—22no—
103C4×C73—22no—
104C4×C73—42no—
105C4×C73—42no—
106C4×C73H(5,𝔽2)12yes—
107Dic73—41no—
108Dic73—41no—
109Dic73—41no—
110C2×C73—41no—
111C2×C73#53511yescharacteristic
112C2×C73#57812yes—
113D73—81no—
114#3223—731nomaximal, Sylow
115C73#322311yescharacteristic, complemented, Sylow
116#541D7311yescharacteristic, complemented
117#536—731no—
118#497—731no—
119#494—731no—
120#491—731no—
121#488—731no—
122#464—732no—
123#462—732no—
124#452—732no—
125#451—732no—
126#433—731no—
127C2×#41—731no—
128C22×Q8—731no—
129C2×Q16—732no—
130C2×Q16—732no—
131C2×Q16—732no—
132C2×SD16—732no—
133C4⋊Q8—731no—
134C4⋊Q8—731no—
135C4⋊Q8C2×D7311yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D7311yescharacteristic
139#124—731no—
140#124—731no—
141#123—1462no—
142C22⋊Q8—1462no—
143C22⋊Q8C2×D7312yes—
144C4×Q8C2×D7312yes—
145#117C2×D7311yescharacteristic
146C4⋊C8—732no—
147#103—732no—
148#103—732no—
149#103—732no—
150#102—732no—
151C22⋊C8—732no—
152#97—731no—
153C4×C8—731no—
154#41—2922no—
155C2×Q8—2922no—
156C2×Q8—731no—
157C2×Q8—731no—
158C2×Q8—731no—
159C2×Q8—1461no—
160C2×Q8C22×D7312yes—
161C2×D4—731no—
162C2×D4—1461no—
163C22×C4—1461no—
164C22×C4—1461no—
165C22×C4C22×D7311yescharacteristic
166Q16—2922no—
167Q16—2922no—
168Q16—2922no—
169SD16—2922no—
170C2×C8—732no—
171C2×C8—732no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—1462no—
177C4⋊C4—1462no—
178C4⋊C4C22⋊D7312yes—
179C4⋊C4C22×D7312yes—
180C22⋊C4—22no—
181C22⋊C4—1462no—
182C22⋊C4—1462no—
183C22⋊C4C22⋊D7312yes—
184C42—22no—
185C42C22×D7311yescharacteristic
186C42C22×D7311yescharacteristic
187C23—1461no—
188C23C2×C22⋊D7311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—2922no—
192Q8—1462no—
193Q8—1462no—
194Q8—1462no—
195D4—2922no—
196D4—1462no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2922no—
201C2×C4—2922no—
202C2×C4—1461no—
203C2×C4—1461no—
204C2×C4—1461no—
205C2×C4C2×C22⋊D7312yes—
206C2×C4C2×C22⋊D7312yes—
207C2×C4C2×C22⋊D7311yescharacteristic
208C2×C4C23×D7311yescharacteristic
209C8—1462no—
210C8—1462no—
211C22—22no—
212C22—21no—
213C22—2922no—
214C22—2921no—
215C22C22×C22⋊D7311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—2921no—
222C4—2921no—
223C4—2921no—
224C4#1455663712yes—
225C2—41no—
226C2—5841no—
227C2#1798915111yescharacteristic
228C2#1798940212yes—
subgroupG/H
centerC22C22×C22⋊D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#541×C73C2
socleC22×C73C22×D4
hypercenter#541D73
hypocenterC73#3223
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #17988979C2semidirect
3. #17988966C2semidirect
4. #17988957C2semidirect
5. #17988944C2semidirect
6. #17988809C2semidirect
7. #17988797C2semidirect
9. #17988742C2semidirect
10. #17988648C2semidirect
11. #541×C73C2semidirect
115. C73#3223semidirect
116. #541D73semidirect