Finite Group Databasebeta

← #32996830

Subgroups — #32996830

Order 18688.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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#22620669C211yesmaximal, characteristic
2#22611743C211yesmaximal, characteristic, complemented
3#22607921C212yesmaximal, complemented
4#22607915C212yesmaximal, complemented
5C73×#1841C211yesmaximal, characteristic, complemented
6#17989137—21no—
7#17988985C411yescharacteristic, complemented
8#17988964C411yescharacteristic, complemented
9#17988934—21no—
10#17988840—21no—
11#17988840C2212yes—
12#17988700—21no—
13#17988700C412yescomplemented
14#17988408—21no—
15#17988369C2212yes—
16#17988365—21no—
17C4×M16×C73C2211yescharacteristic
18#344×C73—21no—
19#339×C73—21no—
20#338×C73C2211yescharacteristic
21H(3,ℤ/4)×C73C2211yescharacteristic
22C2×#11924028—41no—
23C2×D584—21no—
24C2×D584C2×C412yes—
25#14556591—21no—
26#14556588—21no—
27#14556579—41no—
28#14556565—22no—
29#14556565—41no—
30C4×D292—22no—
31C4×D292—41no—
32#14556531—22no—
33#14556523—21no—
34#14556523C2×C412yes—
35#14556513—22no—
36#14556513—22no—
37C2×M16×C73—21no—
38C2×M16×C73D411yescharacteristic
39C2×M16×C73C2311yescharacteristic
40#117×C73—21no—
41C2×C42×C73D411yescharacteristic
42#97×C73C2×C411yescharacteristic
43#97×C73D411yescharacteristic
44C4×C8×C73C2×C411yescharacteristic
45C4×C8×C73D411yescharacteristic
46#11924028—84no—
47C2×Dic146—41no—
48C2×D292—22no—
49C2×D292—41no—
50C2×D292—41no—
51C2×C22⋊D73—42no—
52C2×C4×D73—42no—
53D584—42no—
54D584—42no—
55C4⋊Dic73—22no—
56C4⋊Dic73—41no—
57#11924005—42no—
58#11924005—42no—
59#11924003—42no—
60C22×C4×C73—21no—
61C22×C4×C73C2×D411yescharacteristic
62M16×C73—41no—
63M16×C73—41no—
64M16×C73—41no—
65M16×C73—41no—
66C2×C8×C73—41no—
67C2×C8×C73C22⋊C411yescharacteristic
68C2×C8×C73C22⋊C411yescharacteristic
69C2×C8×C73C22×C411yescharacteristic
70C2×C8×C73C2×D411yescharacteristic
71C4⋊C4×C73—41no—
72C22⋊C4×C73—41no—
73C42×C73—21no—
74C42×C73—41no—
75C42×C73C22⋊C411yescharacteristic
76C42×C73C22⋊C411yescharacteristic
77C23×C73AGL(1,ℤ/8)11yescharacteristic
78C22×D73—42no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—42no—
84C2×C4×C73—81no—
85C2×C4×C73C2×C22⋊C411yescharacteristic
86C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
87C22⋊D73—84no—
88C2×Dic73—42no—
89D292—42no—
90D292—42no—
91D292—82no—
92D292—82no—
93C4×D73—84no—
94Dic146—82no—
95C8×C73—21no—
96C8×C73—21no—
97C8×C73—21no—
98C8×C73—21no—
99C8×C73—81no—
100C22×C73—21no—
101C22×C73—41no—
102C22×C73#41811yescharacteristic
103C2×D73—82no—
104C2×D73—82no—
105C2×D73—82no—
106C4×C73—22no—
107C4×C73—21no—
108C4×C73—41no—
109C4×C73—161no—
110C4×C73#41011yescharacteristic
111C4×C73#42111yescharacteristic
112Dic73—82no—
113#11837—731nomaximal, Sylow
114C2×C73—21no—
115C2×C73—41no—
116C2×C73#151411yescharacteristic
117D73—162no—
118#3184—731no—
119#2165—731no—
120#1841D7311yescharacteristic, complemented
121#1341—732no—
122#1339—732no—
123C73#1183711yescharacteristic, complemented, Sylow
124#532—1461no—
125#498—731no—
126#493—731no—
127#486—1461no—
128#469—732no—
129#469—1461no—
130#442—732no—
131#442—1461no—
132C4×M16C2×D7311yescharacteristic
133#364—1461no—
134#349—732no—
135#347—1461no—
136#344—21no—
137#339—21no—
138#338C2×D7311yescharacteristic
139H(3,ℤ/4)C2×D7311yescharacteristic
140C2×#41—2921no—
141C2×D8—732no—
142C2×D8—1461no—
143C2×M16—21no—
144C2×M16C22⋊D7311yescharacteristic
145C2×M16C22×D7311yescharacteristic
146C4⋊Q8—1461no—
147#127—1461no—
148#124—2921no—
149#121—2921no—
150#121—1462no—
151C4×D4—2921no—
152C4×D4—1462no—
153#117—21no—
154C2×C42C22⋊D7311yescharacteristic
155M32—1462no—
156#106—732no—
157#106—1461no—
158#102—1462no—
159#102—1462no—
160#97C4×D7311yescharacteristic
161#97D29211yescharacteristic
162C4×C8C4×D7311yescharacteristic
163C4×C8D29211yescharacteristic
164#41—5844no—
165C2×Q8—2921no—
166C2×D4—2921no—
167C2×D4—2921no—
168C2×D4—2922no—
169C2×D4—1462no—
170C22×C4—21no—
171C22×C4—2922no—
172C22×C4C2×C22⋊D7311yescharacteristic
173D8—2922no—
174D8—2922no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—41no—
180C2×C8#1192400511yescharacteristic
181C2×C8#1192400511yescharacteristic
182C2×C8C2×C4×D7311yescharacteristic
183C2×C8C2×D29211yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—2921no—
186C4⋊C4—1462no—
187C22⋊C4—41no—
188C22⋊C4—2922no—
189C22⋊C4—2922no—
190C42—21no—
191C42—41no—
192C42#1192400511yescharacteristic
193C42#1192400511yescharacteristic
194C16—2922no—
195C23—2922no—
196C23#1455661611yescharacteristic
197Q8—5842no—
198D4—2922no—
199D4—2922no—
200D4—5842no—
201D4—5842no—
202D4—5844no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—2922no—
210C2×C4—5844no—
211C2×C4C2×#1192400511yescharacteristic
212C2×C4#1455661611yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—5842no—
221C22—5842no—
222C22—5842no—
223C22#1798859111yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—161no—
228C4—5842no—
229C4#1798856011yescharacteristic
230C4#1798860511yescharacteristic
231C2—21no—
232C2—41no—
233C2—11682no—
234C2#2260847711yescharacteristic
subgroupG/H
centerC2#22608477
commutatorC2×C8×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#1841C2
socleC2×C73#1514
prime radical#22620669C2
cosocleC2×M16×C73C23
hypercenter#1841D73
hypocenterC73#11837
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #22611743C2semidirect
3. #22607921C2semidirect
4. #22607915C2semidirect
5. C73×#1841C2semidirect
7. #17988985C4semidirect
8. #17988964C4semidirect
13. #17988700C4semidirect
120. #1841D73semidirect
123. C73#11837semidirect

Maximal subgroups

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

subgroupindex
#226206692
#226117432
#226079212
#226079152
C73×#18412
#1183773