Finite Group Databasebeta

← #34000467

Subgroups — #34000467

Order 18688.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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#22626156C211yesmaximal, characteristic
2#22625807C211yesmaximal, characteristic
3#22609968C211yesmaximal, characteristic
4C2×#14556664C412yescomplemented
5C2×#14556659—21no—
6#17989529—21no—
7#17989482—21no—
8C22⋊#11924031—21no—
9#17989461C412yes—
10C22×C8×D73C2211yescharacteristic
11C2×#14556526—21no—
12#17988550—21no—
13#17988352C411yescharacteristic, complemented
14#341×C73C411yescharacteristic, complemented
15C2×C4⋊#64408—22no—
16C2×C22⋊#64408—41no—
17#14556664—42no—
18C2×#11924031—22no—
19#14556659—42no—
20(C2×#64408)⋊C4—41no—
21C22×C4×D73D411yescharacteristic
22C2×#11924012—21no—
23C22×#10596831D411yescharacteristic
24C2×C8×D73—42no—
25C2×C8×D73C2×C411yescharacteristic
26C2×C8×D73C2×C411yescharacteristic
27C2×C8×D73D411yescharacteristic
28C2×C8×D73Q811yescharacteristic
29#14556526—42no—
30#14556503—41no—
31#14556503—41no—
32C2×M16×C73—21no—
33C22×C8×C73C2×C411yescharacteristic
34C22×#64408—42no—
35C4⋊#64408—42no—
36C4⋊#64408—42no—
37C22⋊#64408—82no—
38#11924031—42no—
39C23×D73SD1611yescharacteristic
40C22×Dic73Q1611yescharacteristic
41C2×C4×D73—41no—
42C2×C4×D73—41no—
43C2×C4×D73C22⋊C411yescharacteristic
44C2×C4×D73C4⋊C411yescharacteristic
45C2×C4×D73D811yescharacteristic
46C2×C4×D73SD1611yescharacteristic
47#11924012—41no—
48#11924012—41no—
49C2×#10596831—22no—
50C2×#10596831—41no—
51C2×#10596831C22⋊C411yescharacteristic
52C2×#10596831C4⋊C411yescharacteristic
53C8×D73—22no—
54C8×D73—22no—
55C8×D73—41no—
56C8×D73—41no—
57C22×C4×C73C22⋊C411yescharacteristic
58M16×C73—41no—
59M16×C73—41no—
60C2×C8×C73—41no—
61C2×C8×C73—41no—
62C2×C8×C73C4211yescharacteristic
63C2×C8×C73C4⋊C411yescharacteristic
64C23×C73#10311yescharacteristic
65C22×D73—22no—
66C22×D73—22no—
67C22×D73—41no—
68C22×D73#10211yescharacteristic
69C22×D73#10611yescharacteristic
70C2×#64408—82no—
71C2×#64408—82no—
72C2×#64408—82no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73#9511yescharacteristic
76C2×C4×C73#10211yescharacteristic
77C2×Dic73—41no—
78C2×Dic73#10311yescharacteristic
79C2×Dic73#10711yescharacteristic
80C4×D73—21no—
81C4×D73—21no—
82C4×D73—21no—
83C4×D73—21no—
84C4×D73—41no—
85C4×D73—41no—
86C8×C73—21no—
87C8×C73—21no—
88C8×C73—81no—
89#10596831—21no—
90#10596831—81no—
91#10596831#9912yes—
92C22×C73—21no—
93C22×C73—21no—
94C22×C73—21no—
95C22×C73#34011yescharacteristic
96C2×D73—21no—
97C2×D73—21no—
98C2×D73—21no—
99C2×D73—21no—
100C2×D73—44no—
101C2×D73#35912yes—
102C2×D73#36512yes—
103#64408—162no—
104C4×C73—21no—
105C4×C73H(3,ℤ/4)11yescharacteristic
106C4×C73#34211yescharacteristic
107Dic73—21no—
108Dic73—21no—
109#7232—731nomaximal, Sylow
110C2×C73—41no—
111C2×C73#128911yescharacteristic
112C2×C73#138211yescharacteristic
113C2×C73#138311yescharacteristic
114D73—22no—
115D73—22no—
116C2×C22⋊C16—731no—
117#1810—731no—
118#1807—731no—
119C73#723211yescharacteristic, complemented, Sylow
120C23×C8—731no—
121C22×C16—732no—
122C2×#108—1461no—
123C2×#106—1461no—
124#407—1461no—
125#380—1461no—
126C22⋊C16—1461no—
127C22⋊C16—1461no—
128#341—731no—
129#341#6440811yescharacteristic, complemented
130#340—732no—
131C23×C4—731no—
132C2×M16—21no—
133C2×M16—1461no—
134C22×C8—2922no—
135C22×C8—731no—
136C22×C8—731no—
137C22×C8—731no—
138C22×C8—731no—
139C22×C8—731no—
140C22×C8C2×#6440811yescharacteristic
141C2×C4⋊C4—1462no—
142C2×C22⋊C4—2921no—
143C2×C16—2922no—
144C2×C16—1462no—
145#108—2922no—
146#106—2922no—
147C22⋊C8—2921no—
148C22⋊C8—2921no—
149#95—2921no—
150C24—731no—
151C22×C4—2921no—
152C22×C4—2921no—
153C22×C4—2922no—
154C22×C4—731no—
155C22×C4—731no—
156C22×C4—731no—
157C22×C4—731no—
158C22×C4—731no—
159C22×C4C22⋊#6440811yescharacteristic
160M16—41no—
161M16—41no—
162M16—2921no—
163M16—2921no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—2921no—
167C2×C8—2921no—
168C2×C8—2921no—
169C2×C8—731no—
170C2×C8—731no—
171C2×C8—1462no—
172C2×C8—1462no—
173C2×C8—1462no—
174C2×C8C4×#6440811yescharacteristic
175C2×C8C4⋊#6440811yescharacteristic
176C4⋊C4—2922no—
177C4⋊C4—2922no—
178C22⋊C4—5842no—
179C16—2922no—
180C23—2921no—
181C23—731no—
182C23—731no—
183C23—1462no—
184C23—1462no—
185C23#1455665411yescharacteristic
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—2921no—
189C2×C4—2921no—
190C2×C4—2921no—
191C2×C4—5842no—
192C2×C4—5842no—
193C2×C4—5842no—
194C2×C4—731no—
195C2×C4—731no—
196C2×C4—1461no—
197C2×C4—1461no—
198C2×C4—1461no—
199C2×C4—1461no—
200C2×C4(C2×#64408)⋊C411yescharacteristic
201C2×C4#1455665311yescharacteristic
202C8—21no—
203C8—21no—
204C8—81no—
205C8—5841no—
206C8—732no—
207C8—1461no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—2924no—
212C22—732no—
213C22—732no—
214C22—1461no—
215C22—1461no—
216C22—1461no—
217C22—1461no—
218C22#1798946011yescharacteristic
219C4—21no—
220C4—11682no—
221C4—1461no—
222C4—1461no—
223C4#1798945411yescharacteristic
224C4#1798946411yescharacteristic
225C2—41no—
226C2—1462no—
227C2—1462no—
228C2#2262479311yescharacteristic
229C2#2262499411yescharacteristic
230C2#2262499711yescharacteristic
subgroupG/H
centerC22#17989460
commutatorC2×C8×C73C42
FrattiniC22×C8C2×#64408
Fitting#341×C73C4
socleC22×C73#340
prime radicalC23×D73SD16
cosocleC22×C8×D73C22
hypercenter#341#64408
hypocenterC73#7232
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #341

Complements

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

NH ≅ G/Ntype
4. C2×#14556664C4semidirect
13. #17988352C4semidirect
14. #341×C73C4semidirect
119. C73#7232semidirect
129. #341#64408semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226261562
#226258072
#226099682
#723273