Finite Group Databasebeta

← #32997485

Subgroups — #32997485

Order 18688.

352 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 85 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#22620511C211yesmaximal, characteristic
2#22611685C211yesmaximal, characteristic, complemented
3#22608688C211yesmaximal, characteristic, complemented
4#22607989C211yesmaximal, characteristic, complemented
5#22607978C211yesmaximal, characteristic, complemented
6#22607976C211yesmaximal, characteristic, complemented
7C73×#1365C211yesmaximal, characteristic, complemented
8#17989138—21no—
9C2×#14556530—21no—
10#17988964C2211yescharacteristic
11#17988935—21no—
12#17988824—21no—
13#17988824C412yes—
14#17988669—21no—
15#17988669C412yes—
16#17988609—21no—
17C4×C8×D73C2211yescharacteristic
18C4⋊#11924003C2211yescharacteristic
19#17988405—21no—
20#17988340C2211yescharacteristic
21#17988339—21no—
22#500×C73C2211yescharacteristic
23C4⋊C16×C73C2211yescharacteristic
24#335×C73C2211yescharacteristic
25C2×#11924029—41no—
26C2×D584—21no—
27C2×D584C2×C412yes—
28C2×#11924011—41no—
29C2×C8×D73C2×C411yescharacteristic
30C2×C8×D73D411yescharacteristic
31#14556588—21no—
32#14556580—41no—
33#14556566—22no—
34#14556566—41no—
35#14556553—22no—
36#14556553—41no—
37C42×D73D411yescharacteristic
38#14556530—42no—
39C2×#11924003—21no—
40C4⋊#10596831—21no—
41#14556520—41no—
42#14556511—22no—
43#14556511—22no—
44C4×#10596831D411yescharacteristic
45C8×Dic73C2×C411yescharacteristic
46C8×Dic73D411yescharacteristic
47C4⋊Q8×C73—21no—
48C2×C16×C73—21no—
49#107×C73—21no—
50#107×C73C2×C412yes—
51C4⋊C8×C73—21no—
52C4×C8×C73C2311yescharacteristic
53#11924029—84no—
54C2×D292—22no—
55C2×D292—42no—
56C2×D292—41no—
57C2×D292—41no—
58C2×C4×D73—42no—
59C2×C4×D73C22⋊C411yescharacteristic
60C2×C4×D73D811yescharacteristic
61C2×C4×D73SD1611yescharacteristic
62D584—42no—
63D584—41no—
64D584—41no—
65#11924011—82no—
66C2×#10596831—41no—
67C2×#10596831C22⋊C411yescharacteristic
68C2×#10596831C22⋊C411yescharacteristic
69C8×D73—21no—
70C8×D73—21no—
71C8×D73—41no—
72#11924005—42no—
73#11924005—42no—
74C4×Dic73C22⋊C411yescharacteristic
75C4×Dic73D811yescharacteristic
76C4×Dic73SD1611yescharacteristic
77#11924003—41no—
78C2×Q8×C73—41no—
79C2×C8×C73—41no—
80C2×C8×C73C22×C411yescharacteristic
81C2×C8×C73C2×D411yescharacteristic
82C4⋊C4×C73—22no—
83C4⋊C4×C73—41no—
84C42×C73C2×D411yescharacteristic
85C16×C73—41no—
86C22×D73—42no—
87C22×D73#10211yescharacteristic
88Q8×C73—82no—
89C2×C4×C73—42no—
90C2×C4×C73C2×C22⋊C411yescharacteristic
91C2×C4×C73C2×D811yescharacteristic
92C2×C4×C73C2×SD1611yescharacteristic
93C2×Dic73#10211yescharacteristic
94C2×Dic73#10211yescharacteristic
95C2×Dic73#10211yescharacteristic
96D292—42no—
97D292—42no—
98D292—82no—
99D292—84no—
100D292—81no—
101D292—81no—
102C4×D73—21no—
103C4×D73—21no—
104C4×D73—21no—
105C4×D73—21no—
106C4×D73—22no—
107C4×D73—84no—
108C8×C73—21no—
109C8×C73—81no—
110C8×C73AGL(1,ℤ/8)11yescharacteristic
111C8×C73#13711yescharacteristic
112#10596831—21no—
113#10596831—21no—
114#10596831—81no—
115C22×C73C2×#10211yescharacteristic
116C2×D73—21no—
117C2×D73—21no—
118C2×D73—21no—
119C2×D73—82no—
120C2×D73—82no—
121C2×D73—82no—
122C4×C73—21no—
123C4×C73—82no—
124C4×C73#41311yescharacteristic
125C4×C73#41711yescharacteristic
126C4×C73#50911yescharacteristic
127C4×C73#51011yescharacteristic
128Dic73—21no—
129Dic73—21no—
130Dic73#36112yes—
131#11986—731nomaximal, Sylow
132C2×C73#154711yescharacteristic
133C2×C73#213911yescharacteristic
134C2×C73C2×#36111yescharacteristic
135D73—41no—
136D73—162no—
137#3169—731no—
138#2150—731no—
139#1567—731no—
140#1365—731no—
141#1365D7311yescharacteristic, complemented
142#1361—731no—
143#1361—731no—
144C73#1198611yescharacteristic, complemented, Sylow
145#532—1461no—
146C2×M32—1461no—
147#500C2×D7311yescharacteristic
148#493—731no—
149#486—1461no—
150#466—732no—
151#466—1461no—
152C4×D8—732no—
153C4×D8—1461no—
154#423—1461no—
155C2×C4×C8—731no—
156C4⋊C16—731no—
157C4⋊C16—1461no—
158C4⋊C16C2×D7311yescharacteristic
159#335—731no—
160#335—1461no—
161#335C2×D7311yescharacteristic
162C2×#41—2921no—
163C2×D8—732no—
164C2×D8—1461no—
165C2×M16—2921no—
166C22×C8—731no—
167C22×C8—731no—
168C4⋊Q8—21no—
169#127—1461no—
170#124—2921no—
171#121—2921no—
172#121—1462no—
173C4×D4—2921no—
174C4×D4—1462no—
175C2×C42—731no—
176M32—2922no—
177C2×C16—21no—
178C2×C16—1461no—
179#107—21no—
180#107C4×D7312yes—
181C4⋊C8—21no—
182C4⋊C8—2921no—
183C4⋊C8—1461no—
184#102—1462no—
185#102—1462no—
186C4×C8—731no—
187C4×C8—731no—
188C4×C8—731no—
189C4×C8C22×D7311yescharacteristic
190#41—5844no—
191C2×Q8—41no—
192C2×D4—2922no—
193C2×D4—2921no—
194C2×D4—2921no—
195C2×D4—1462no—
196C22×C4—2922no—
197C22×C4—731no—
198C22×C4—731no—
199C22×C4—731no—
200D8—2922no—
201D8—2921no—
202D8—2921no—
203M16—5842no—
204C2×C8—41no—
205C2×C8—2921no—
206C2×C8—2921no—
207C2×C8—731no—
208C2×C8—731no—
209C2×C8—1461no—
210C2×C8—1461no—
211C2×C8C2×C4×D7311yescharacteristic
212C2×C8D4×D7311yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—41no—
215C22⋊C4—2922no—
216C22⋊C4—2922no—
217C42—731no—
218C42—731no—
219C42—731no—
220C42D4×D7311yescharacteristic
221C16—41no—
222C16—2921no—
223C23—2922no—
224C23—731no—
225Q8—82no—
226D4—2922no—
227D4—2922no—
228D4—5842no—
229D4—5844no—
230D4—5841no—
231D4—5841no—
232C2×C4—42no—
233C2×C4—5844no—
234C2×C4—731no—
235C2×C4—731no—
236C2×C4—731no—
237C2×C4—1462no—
238C2×C4—1461no—
239C2×C4—1461no—
240C2×C4—1461no—
241C2×C4—1461no—
242C2×C4C22⋊C4×D7311yescharacteristic
243C2×C4D8×D7311yescharacteristic
244C2×C4SD16×D7311yescharacteristic
245C8—21no—
246C8—81no—
247C8—5841no—
248C8—1461no—
249C8—1461no—
250C8#1455661811yescharacteristic
251C8#1455662311yescharacteristic
252C22—5842no—
253C22—5842no—
254C22—5842no—
255C22—1461no—
256C22—1461no—
257C22—1461no—
258C22#102×D7311yescharacteristic
259C4—21no—
260C4—82no—
261C4—732no—
262C4—1461no—
263C4—1461no—
264C4#1798857011yescharacteristic
265C4#1798858711yescharacteristic
266C4#1798902411yescharacteristic
267C4#1798902911yescharacteristic
268C2—2921no—
269C2—11682no—
270C2#2260860711yescharacteristic
271C2#2261164211yescharacteristic
272C2#361×D7311yescharacteristic
subgroupG/H
centerC22#102×D73
commutatorC2×C8×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#1365C2
socleC22×C73C2×#102
prime radical#22620511C2
cosocleC4×C8×C73C23
hypercenter#1365D73
hypocenterC73#11986
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
2. #22611685C2semidirect
3. #22608688C2semidirect
4. #22607989C2semidirect
5. #22607978C2semidirect
6. #22607976C2semidirect
7. C73×#1365C2semidirect
141. #1365D73semidirect
144. C73#11986semidirect

Maximal subgroups

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

subgroupindex
#226205112
#226116852
#226086882
#226079892
#226079782
#226079762
C73×#13652
#1198673