Finite Group Databasebeta

← #22622802

Subgroups — #22622802

Order 9344.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#17989297C211yesmaximal, characteristic
2#17988986C212yesmaximal, complemented
3#17988983C211yesmaximal, characteristic, complemented
4#17988894C212yesmaximal, complemented
5#17988884C212yesmaximal
6#17988882C211yesmaximal, characteristic, complemented
7#17988864C211yesmaximal, characteristic
8Q8×#10596831C211yesmaximal, characteristic, complemented
9C4×#11924015C211yesmaximal, characteristic
10C4×#11924015C212yesmaximal
11#557×C73C211yesmaximal, characteristic, complemented
12C2×#11924015C2211yescharacteristic
13#14556591C2211yescharacteristic
14#14556583C2212yes—
15#14556569—22no—
16#14556567—21no—
17C4×D292C2211yescharacteristic
18C4×D292C2212yes—
19C4⋊C4×D73—21no—
20#14556524C2212yes—
21#14556524C2212yes—
22#14556522C2212yes—
23#14556522C2212yes—
24#14556522C2211yescharacteristic
25C4⋊#10596831C2211yescharacteristic
26C4⋊#10596831C2212yes—
27#14556515C2212yes—
28#14556515C2211yescharacteristic
29#14556511C2212yes—
30#14556511C2211yescharacteristic
31C4×#10596831C2211yescharacteristic
32C4×#10596831C2212yes—
33C4⋊Q8×C73—21no—
34C4⋊Q8×C73C2211yescharacteristic
35#125×C73—22no—
36#125×C73C2212yes—
37C4×Q8×C73—21no—
38C4×Q8×C73C2211yescharacteristic
39C4×Q8×C73C2212yes—
40C4×Q8×C73C2211yescharacteristic
41C2×Dic146—21no—
42C2×D292C2311yescharacteristic
43C2×C4×D73—21no—
44C2×C4×D73—22no—
45#11924015Q814yescomplemented
46C2×#10596831C2311yescharacteristic
47C2×#10596831C2312yes—
48C2×#10596831C2311yescharacteristic
49C4⋊Dic73—21no—
50C4⋊Dic73C2312yes—
51C4⋊Dic73C2311yescharacteristic
52#11924007—22no—
53#11924007—22no—
54#11924005—22no—
55#11924005—21no—
56C2×Q8×C73—21no—
57C2×Q8×C73D411yescharacteristic
58C2×Q8×C73C2311yescharacteristic
59C4⋊C4×C73—22no—
60C4⋊C4×C73—21no—
61C4⋊C4×C73—22no—
62C4⋊C4×C73—21no—
63C4⋊C4×C73—22no—
64C4⋊C4×C73D411yescharacteristic
65C4⋊C4×C73D412yes—
66C4⋊C4×C73C2311yescharacteristic
67C4⋊C4×C73C2312yes—
68C42×C73—22no—
69C42×C73—21no—
70C42×C73C2312yes—
71C42×C73C2311yescharacteristic
72C22×D73—21no—
73Q8×C73—22no—
74Q8×C73—21no—
75Q8×C73—42no—
76Q8×C73C2×Q812yes—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—22no—
80C2×C4×C73C2×D412yes—
81C2×C4×C73C2×D411yescharacteristic
82C2×C4×C73C2×D412yes—
83C2×C4×C73C2×D411yescharacteristic
84C2×C4×C73C2411yescharacteristic
85C2×Dic73—21no—
86C2×Dic73—22no—
87D292—21no—
88D292C2×Q812yes—
89C4×D73—42no—
90C4×D73—82no—
91Dic146—81no—
92#10596831—21no—
93#10596831—22no—
94#10596831C2×Q812yes—
95C22×C73C22×D411yescharacteristic
96C2×D73—22no—
97C2×D73—41no—
98C2×D73—41no—
99C4×C73—22no—
100C4×C73—22no—
101C4×C73—21no—
102C4×C73—22no—
103C4×C73—42no—
104C4×C73—41no—
105C4×C73#13512yes—
106C4×C73C22×Q811yescharacteristic
107C4×C732−1+411yescharacteristic
108Dic73—41no—
109Dic73—42no—
110C2×C73D4×Q811yescharacteristic
111C2×C73C2×#13511yescharacteristic
112C2×C73#57711yescharacteristic
113D73—42no—
114#3381—731nomaximal, Sylow
115C73#338111yescharacteristic, complemented, Sylow
116#557D7311yescharacteristic, complemented
117#554—731no—
118#499—732no—
119#498—731no—
120#479—732no—
121#478—732no—
122#477—731no—
123#474—731no—
124C8×Q8—731no—
125C4×SD16—731no—
126C4×SD16—732no—
127C2×SD16—731no—
128C4⋊Q8—21no—
129C4⋊Q8—731no—
130C4⋊Q8C2×D7311yescharacteristic
131#125—22no—
132#125—732no—
133#125C2×D7312yes—
134C22⋊Q8—1461no—
135C22⋊Q8—1462no—
136C4×Q8—21no—
137C4×Q8C2×D7311yescharacteristic
138C4×Q8C2×D7312yes—
139C4×Q8C2×D7311yescharacteristic
140C4×D4—731no—
141C4×D4—732no—
142C2×C4⋊C4—1461no—
143#107—732no—
144#107—732no—
145#106—732no—
146#106—731no—
147#106—732no—
148C4⋊C8—731no—
149C4⋊C8—732no—
150#103—732no—
151#103—731no—
152#102—732no—
153#102—731no—
154C4×C8—731no—
155C4×C8—732no—
156C2×Q8—21no—
157C2×Q8—1461no—
158C2×Q8C22⋊D7311yescharacteristic
159C2×Q8C22×D7311yescharacteristic
160C2×D4—731no—
161C22×C4—1461no—
162C22×C4—1462no—
163SD16—734no—
164C2×C8—731no—
165C2×C8—731no—
166C2×C8—732no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—732no—
173C4⋊C4—731no—
174C4⋊C4—1462no—
175C4⋊C4—1462no—
176C4⋊C4—1461no—
177C4⋊C4C22⋊D7311yescharacteristic
178C4⋊C4C22⋊D7312yes—
179C4⋊C4C22×D7312yes—
180C4⋊C4C22×D7311yescharacteristic
181C22⋊C4—1461no—
182C22⋊C4—1462no—
183C42—22no—
184C42—21no—
185C42C22×D7312yes—
186C42C22×D7311yescharacteristic
187C23—1461no—
188Q8—22no—
189Q8—21no—
190Q8—42no—
191Q8—5841no—
192Q8Q8×D7312yes—
193D4—732no—
194D4—1461no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—2922no—
199C2×C4—5842no—
200C2×C4—1461no—
201C2×C4—1462no—
202C2×C4C2×C22⋊D7312yes—
203C2×C4C2×C22⋊D7312yes—
204C2×C4C2×C22⋊D7311yescharacteristic
205C2×C4C2×C22⋊D7311yescharacteristic
206C2×C4C23×D7311yescharacteristic
207C8—732no—
208C8—1461no—
209C8—1462no—
210C22—2921no—
211C22—2921no—
212C22—1462no—
213C22C22×C22⋊D7311yescharacteristic
214C4—22no—
215C4—21no—
216C4—22no—
217C4—22no—
218C4—41no—
219C4—42no—
220C4—2922no—
221C4—2921no—
222C4#1455661112yes—
223C4C2×Q8×D7311yescharacteristic
224C4#1455663911yescharacteristic
225C2—2922no—
226C2Q8×C22⋊D7311yescharacteristic
227C2C2×#1455661111yescharacteristic
228C2#1798939511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#557×C73C2
socleC22×C73C22×D4
hypercenter#557D73
hypocenterC73#3381
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #17988986C2semidirect
3. #17988983C2semidirect
4. #17988894C2semidirect
6. #17988882C2semidirect
8. Q8×#10596831C2semidirect
11. #557×C73C2semidirect
45. #11924015Q8semidirect
115. C73#3381semidirect
116. #557D73semidirect