Finite Group Databasebeta

← #22622708

Subgroups — #22622708

Order 9344.

340 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 123 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#17988955C212yesmaximal, complemented
3#17988937C211yesmaximal, characteristic, complemented
4#17988894C212yesmaximal, complemented
5#17988882C211yesmaximal, characteristic, complemented
6#17988789C211yesmaximal, characteristic
7#17988789C212yesmaximal
8#17988718C211yesmaximal, characteristic, complemented
9#17988685C212yesmaximal
10C4×#11924015C211yesmaximal, characteristic
11#557×C73C211yesmaximal, characteristic, complemented
12C2×#11924015C2211yescharacteristic
13C2×#11924015C2212yes—
14#14556591C2211yescharacteristic
15#14556583C2212yes—
16#14556569—22no—
17#14556567—21no—
18C4×D292C2211yescharacteristic
19C4×D292C2212yes—
20C4⋊C4×D73—21no—
21#14556524C2212yes—
22#14556522C2211yescharacteristic
23C4⋊#10596831C2211yescharacteristic
24C4⋊#10596831C2212yes—
25#14556515C2212yes—
26#14556515C2212yes—
27#14556515C2212yes—
28#14556515C2211yescharacteristic
29#14556515C2212yes—
30#14556511C2211yescharacteristic
31#14556502C2212yes—
32C4×#10596831C2211yescharacteristic
33C4⋊Q8×C73C2211yescharacteristic
34C4⋊Q8×C73C2212yes—
35#125×C73—21no—
36#125×C73—22no—
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#11924015—22no—
46#11924015—24no—
47C2×#10596831C2311yescharacteristic
48C2×#10596831C2311yescharacteristic
49C2×#10596831C2312yes—
50C4⋊Dic73—21no—
51C4⋊Dic73C2311yescharacteristic
52C4⋊Dic73C2312yes—
53#11924007—22no—
54#11924007—22no—
55#11924005—22no—
56#11924005—21no—
57C2×Q8×C73D411yescharacteristic
58C2×Q8×C73C2311yescharacteristic
59C2×Q8×C73C2312yes—
60C4⋊C4×C73—22no—
61C4⋊C4×C73—21no—
62C4⋊C4×C73—21no—
63C4⋊C4×C73—22no—
64C4⋊C4×C73—21no—
65C4⋊C4×C73—22no—
66C4⋊C4×C73D411yescharacteristic
67C4⋊C4×C73D412yes—
68C4⋊C4×C73C2311yescharacteristic
69C42×C73—22no—
70C42×C73—21no—
71C42×C73C2311yescharacteristic
72C42×C73C2312yes—
73C22×D73—21no—
74Q8×C73—22no—
75Q8×C73—21no—
76Q8×C73—22no—
77Q8×C73—21no—
78Q8×C73—22no—
79C2×C4×C73—21no—
80C2×C4×C73—22no—
81C2×C4×C73—21no—
82C2×C4×C73C2×D412yes—
83C2×C4×C73C2×D411yescharacteristic
84C2×C4×C73C2×D412yes—
85C2×C4×C73C2×D411yescharacteristic
86C2×C4×C73C2411yescharacteristic
87C2×Dic73—21no—
88C2×Dic73—22no—
89D292—21no—
90D292#4112yes—
91C4×D73—42no—
92C4×D73—82no—
93Dic146—81no—
94#10596831—21no—
95#10596831—22no—
96#10596831—21no—
97C22×C73C22×D411yescharacteristic
98C2×D73—22no—
99C2×D73—41no—
100C2×D73—41no—
101C4×C73—22no—
102C4×C73—21no—
103C4×C73—22no—
104C4×C73—42no—
105C4×C73—41no—
106C4×C73—41no—
107C4×C73#13712yes—
108C4×C73C2×#4111yescharacteristic
109C4×C732−1+411yescharacteristic
110Dic73—41no—
111Dic73—42no—
112C2×C73#54811yescharacteristic
113C2×C73C2×#13711yescharacteristic
114C2×C73#57711yescharacteristic
115D73—42no—
116#3372—731nomaximal, Sylow
117C73#337211yescharacteristic, complemented, Sylow
118#557D7311yescharacteristic, complemented
119#554—731no—
120#491—732no—
121#487—731no—
122#479—732no—
123#477—731no—
124#461—732no—
125#461—731no—
126#446—731no—
127#440—732no—
128C4×SD16—731no—
129C2×SD16—731no—
130C2×SD16—732no—
131C4⋊Q8—731no—
132C4⋊Q8C2×D7311yescharacteristic
133C4⋊Q8C2×D7312yes—
134#125—21no—
135#125—22no—
136#125—732no—
137C22⋊Q8—1461no—
138C22⋊Q8—1462no—
139C4×Q8—21no—
140C4×Q8C2×D7311yescharacteristic
141C4×Q8C2×D7312yes—
142C4×Q8C2×D7311yescharacteristic
143C4×D4—731no—
144C4×D4—732no—
145C2×C4⋊C4—1461no—
146#107—732no—
147#106—731no—
148C4⋊C8—731no—
149C4⋊C8—732no—
150#103—731no—
151#103—732no—
152#103—732no—
153#103—732no—
154#103—732no—
155#102—731no—
156#97—732no—
157C4×C8—731no—
158C2×Q8—1461no—
159C2×Q8C22⋊D7311yescharacteristic
160C2×Q8C22×D7311yescharacteristic
161C2×Q8C22×D7312yes—
162C2×D4—731no—
163C22×C4—1461no—
164C22×C4—1462no—
165SD16—1462no—
166SD16—1464no—
167C2×C8—731no—
168C2×C8—731no—
169C2×C8—732no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—731no—
177C4⋊C4—732no—
178C4⋊C4—1462no—
179C4⋊C4—1462no—
180C4⋊C4—1461no—
181C4⋊C4C22⋊D7311yescharacteristic
182C4⋊C4C22⋊D7312yes—
183C4⋊C4C22×D7311yescharacteristic
184C22⋊C4—1461no—
185C22⋊C4—1462no—
186C42—22no—
187C42—21no—
188C42C22×D7311yescharacteristic
189C42C22×D7312yes—
190C23—1461no—
191Q8—22no—
192Q8—21no—
193Q8—22no—
194Q8—21no—
195Q8—22no—
196Q8—5841no—
197D4—732no—
198D4—1461no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—2922no—
203C2×C4—5842no—
204C2×C4—1461no—
205C2×C4—1462no—
206C2×C4C2×C22⋊D7312yes—
207C2×C4C2×C22⋊D7311yescharacteristic
208C2×C4C2×C22⋊D7312yes—
209C2×C4C2×C22⋊D7311yescharacteristic
210C2×C4C23×D7311yescharacteristic
211C8—1461no—
212C8—1462no—
213C8—1461no—
214C22—2921no—
215C22—2921no—
216C22—1462no—
217C22C22×C22⋊D7311yescharacteristic
218C4—22no—
219C4—22no—
220C4—21no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—2921no—
225C4—2922no—
226C4#1455662212yes—
227C4#41×D7311yescharacteristic
228C4#1455663811yescharacteristic
229C2—2922no—
230C2#1798924411yescharacteristic
231C2C2×#1455662211yescharacteristic
232C2#1798939511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#557×C73C2
socleC22×C73C22×D4
hypercenter#557D73
hypocenterC73#3372
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. #17988955C2semidirect
3. #17988937C2semidirect
4. #17988894C2semidirect
5. #17988882C2semidirect
8. #17988718C2semidirect
11. #557×C73C2semidirect
117. C73#3372semidirect
118. #557D73semidirect