Finite Group Databasebeta

← #34000468

Subgroups — #34000468

Order 18688.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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#22626159C211yesmaximal, characteristic
2#22625807C211yesmaximal, characteristic
3#22609955C211yesmaximal, characteristic
4C2×#14556666C412yescomplemented
5C2×#14556659—21no—
6#17989529—21no—
7#17989482—21no—
8C22⋊#11924031—21no—
9#17989461C412yescomplemented
10C22×C8×D73C2211yescharacteristic
11C2×#14556522—21no—
12#17988444—21no—
13#17988350C411yescharacteristic, complemented
14#340×C73C411yescharacteristic, complemented
15C2×C4⋊#64408—22no—
16C2×C22⋊#64408—41no—
17#14556666—42no—
18C2×#11924031—22no—
19#14556659—42no—
20(C2×#64408)⋊C4—41no—
21C22×C4×D73D411yescharacteristic
22C22×#10596831D411yescharacteristic
23C2×C8×D73—42no—
24C2×C8×D73C2×C411yescharacteristic
25C2×C8×D73C2×C411yescharacteristic
26C2×C8×D73D411yescharacteristic
27C2×C8×D73Q811yescharacteristic
28C2×C4⋊Dic73—21no—
29C2×#11924005—41no—
30#14556522—42no—
31#14556498—41no—
32C22×C8×C73C2×C411yescharacteristic
33C2×C4⋊C4×C73—21no—
34C22×#64408—42no—
35C4⋊#64408—42no—
36C4⋊#64408—42no—
37C22⋊#64408—81no—
38C22⋊#64408—81no—
39#11924031—42no—
40C23×D73D811yescharacteristic
41C22×Dic73—41no—
42C22×Dic73SD1611yescharacteristic
43C2×C4×D73—41no—
44C2×C4×D73—41no—
45C2×C4×D73C22⋊C411yescharacteristic
46C2×C4×D73C4⋊C411yescharacteristic
47C2×C4×D73SD1611yescharacteristic
48C2×C4×D73Q1611yescharacteristic
49C2×#10596831—22no—
50C2×#10596831C22⋊C411yescharacteristic
51C2×#10596831C4⋊C411yescharacteristic
52C8×D73—22no—
53C8×D73—22no—
54C8×D73—41no—
55C8×D73—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924005—82no—
59C22×C4×C73—41no—
60C22×C4×C73C22⋊C411yescharacteristic
61C2×C8×C73—41no—
62C2×C8×C73C4211yescharacteristic
63C2×C8×C73C4⋊C411yescharacteristic
64C4⋊C4×C73—41no—
65C4⋊C4×C73—41no—
66C23×C73#10211yescharacteristic
67C22×D73—21no—
68C22×D73—21no—
69C22×D73—22no—
70C22×D73—41no—
71C22×D73#10211yescharacteristic
72C22×D73#10711yescharacteristic
73C2×#64408—82no—
74C2×#64408—82no—
75C2×#64408—82no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—81no—
79C2×C4×C73—81no—
80C2×C4×C73—81no—
81C2×C4×C73#9511yescharacteristic
82C2×C4×C73#10311yescharacteristic
83C2×Dic73—41no—
84C2×Dic73—81no—
85C2×Dic73—81no—
86C2×Dic73—81no—
87C2×Dic73#10311yescharacteristic
88C2×Dic73#10611yescharacteristic
89C4×D73—21no—
90C4×D73—21no—
91C4×D73—21no—
92C4×D73—21no—
93C4×D73—41no—
94C4×D73—41no—
95C8×C73—21no—
96C8×C73—21no—
97#10596831—21no—
98#10596831#9912yes—
99C22×C73—21no—
100C22×C73—21no—
101C22×C73#34011yescharacteristic
102C22×C73#36011yescharacteristic
103C22×C73#36011yescharacteristic
104C2×D73—21no—
105C2×D73—21no—
106C2×D73—21no—
107C2×D73—21no—
108C2×D73—21no—
109C2×D73—42no—
110C2×D73—42no—
111C2×D73#36012yes—
112#64408—162no—
113C4×C73—21no—
114C4×C73—161no—
115C4×C73H(3,ℤ/4)11yescharacteristic
116C4×C73#34211yescharacteristic
117Dic73—21no—
118Dic73—21no—
119Dic73—161no—
120#7233—731nomaximal, Sylow
121C2×C73—21no—
122C2×C73—21no—
123C2×C73#128911yescharacteristic
124C2×C73#138611yescharacteristic
125C2×C73#138611yescharacteristic
126D73—22no—
127D73—41no—
128#2112—731no—
129#1807—731no—
130#1807—731no—
131C73#723311yescharacteristic, complemented, Sylow
132C23×C8—731no—
133C2×M32—732no—
134C2×#106—1461no—
135C2×#106—1461no—
136#380—1461no—
137#380—1461no—
138C22⋊C16—1461no—
139C22⋊C16—1461no—
140#340—732no—
141#340—731no—
142#340#6440811yescharacteristic, complemented
143C23×C4—731no—
144C22×C8—2922no—
145C22×C8—731no—
146C22×C8—731no—
147C22×C8—731no—
148C22×C8—731no—
149C22×C8—731no—
150C22×C8C2×#6440811yescharacteristic
151C2×C4⋊C4—21no—
152C2×C4⋊C4—1462no—
153C2×C4⋊C4—1461no—
154C2×C22⋊C4—2921no—
155C2×C22⋊C4—2921no—
156M32—2922no—
157C2×C16—1462no—
158#106—2922no—
159#106—2922no—
160#95—2921no—
161#95—2921no—
162C24—731no—
163C22×C4—41no—
164C22×C4—2921no—
165C22×C4—2921no—
166C22×C4—2922no—
167C22×C4—2921no—
168C22×C4—731no—
169C22×C4—731no—
170C22×C4—731no—
171C22×C4—731no—
172C22×C4—731no—
173C22×C4C22⋊#6440811yescharacteristic
174C2×C8—41no—
175C2×C8—2921no—
176C2×C8—2921no—
177C2×C8—731no—
178C2×C8—731no—
179C2×C8—1462no—
180C2×C8—1462no—
181C2×C8—1462no—
182C2×C8C4×#6440811yescharacteristic
183C2×C8C4⋊#6440811yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—2921no—
187C4⋊C4—2921no—
188C4⋊C4—2922no—
189C4⋊C4—2922no—
190C22⋊C4—5842no—
191C22⋊C4—5841no—
192C22⋊C4—5841no—
193C16—2922no—
194C23—2921no—
195C23—731no—
196C23—731no—
197C23—1462no—
198C23—1461no—
199C23—1461no—
200C23#1455665311yescharacteristic
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—2921no—
207C2×C4—2921no—
208C2×C4—2921no—
209C2×C4—5841no—
210C2×C4—5841no—
211C2×C4—5842no—
212C2×C4—5842no—
213C2×C4—5842no—
214C2×C4—5841no—
215C2×C4—731no—
216C2×C4—731no—
217C2×C4—1461no—
218C2×C4—1461no—
219C2×C4—1461no—
220C2×C4—1461no—
221C2×C4(C2×#64408)⋊C411yescharacteristic
222C2×C4#1455665411yescharacteristic
223C8—21no—
224C8—21no—
225C8—732no—
226C8—1461no—
227C22—21no—
228C22—21no—
229C22—2922no—
230C22—2922no—
231C22—732no—
232C22—1461no—
233C22—1461no—
234C22—1461no—
235C22—1461no—
236C22—1461no—
237C22#1798946011yescharacteristic
238C22#1798950311yescharacteristic
239C22#1798950411yescharacteristic
240C4—21no—
241C4—161no—
242C4—11681no—
243C4—11682no—
244C4—1461no—
245C4—1461no—
246C4#1798945411yescharacteristic
247C4#1798946411yescharacteristic
248C2—21no—
249C2—21no—
250C2—2921no—
251C2—1462no—
252C2#2262479311yescharacteristic
253C2#2262500211yescharacteristic
254C2#2262500411yescharacteristic
subgroupG/H
centerC22#17989460
commutatorC2×C8×C73C42
FrattiniC22×C8C2×#64408
Fitting#340×C73C4
socleC22×C73#340
prime radicalC23×D73D8
cosocleC22×C8×D73C22
hypercenter#340#64408
hypocenterC73#7233
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
4. C2×#14556666C4semidirect
9. #17989461C4semidirect
13. #17988350C4semidirect
14. #340×C73C4semidirect
131. C73#7233semidirect
142. #340#64408semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226258072
#226099552
#723373