Finite Group Databasebeta

← #32998441

Subgroups — #32998441

Order 18688.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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#22621241C211yesmaximal, characteristic
2#22611848C212yesmaximal, complemented
3#22607939C212yesmaximal, complemented
4#22607871C211yesmaximal, characteristic, complemented
5C73×#2056C211yesmaximal, characteristic, complemented
6#17989137—21no—
7C2×#14556534—22no—
8#17988868—22no—
9#17988823—21no—
10#17988823C2212yes—
11#17988783—22no—
12#17988391—22no—
13C22⋊#11924003C2212yes—
14C22×Q16×C73C2211yescharacteristic
15C2×C4⋊C8×C73C2211yescharacteristic
16C2×#103×C73—21no—
17#391×C73—21no—
18#340×C73C2211yescharacteristic
19C2×#11924028—41no—
20C2×D584—21no—
21C2×D584D412yes—
22#14556591—21no—
23#14556588—21no—
24#14556579—41no—
25#14556565—22no—
26#14556565—41no—
27C4×D292—22no—
28C4×D292—41no—
29#14556534—44no—
30C2×#11924003—22no—
31#14556525—21no—
32#14556525D412yes—
33#14556513—22no—
34#14556513—22no—
35C22×Q8×C73—21no—
36C2×Q16×C73—22no—
37C2×Q16×C73—42no—
38C22×C8×C73D411yescharacteristic
39C22×C8×C73C2311yescharacteristic
40C2×C4⋊C4×C73—21no—
41C2×C42×C73D411yescharacteristic
42C4⋊C8×C73—22no—
43#103×C73—42no—
44#95×C73—41no—
45#95×C73—41no—
46#11924028—82no—
47#11924028—82no—
48C2×Dic146—41no—
49C2×D292—22no—
50C2×D292—41no—
51C2×D292—41no—
52C2×C22⋊D73—42no—
53C2×C4×D73—42no—
54D584—42no—
55D584—81no—
56C4⋊Dic73—22no—
57C4⋊Dic73—41no—
58#11924005—42no—
59#11924005—42no—
60#11924003—42no—
61C2×Q8×C73—22no—
62C2×Q8×C73—41no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×Q8×C73—41no—
67C22×C4×C73—41no—
68C22×C4×C73—41no—
69C22×C4×C73D811yescharacteristic
70C22×C4×C73D811yescharacteristic
71C22×C4×C73C2×D411yescharacteristic
72Q16×C73—24no—
73Q16×C73—42no—
74C2×C8×C73—21no—
75C2×C8×C73—41no—
76C2×C8×C73—41no—
77C2×C8×C73C2×D412yes—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C42×C73—21no—
81C42×C73—21no—
82C23×C73C2×D811yescharacteristic
83C22×D73—42no—
84Q8×C73—42no—
85Q8×C73—42no—
86Q8×C73—81no—
87Q8×C73—81no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73—42no—
95C2×C4×C73—81no—
96C2×C4×C73—81no—
97C2×C4×C73—81no—
98C2×C4×C73—81no—
99C2×C4×C73—81no—
100C2×C4×C73SD3212yescomplemented
101C2×C4×C73#12011yescharacteristic
102C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
103C22⋊D73—82no—
104C22⋊D73—82no—
105C2×Dic73—42no—
106D292—42no—
107D292—42no—
108D292—82no—
109D292—82no—
110C4×D73—84no—
111Dic146—82no—
112C8×C73—22no—
113C8×C73—41no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73#44711yescharacteristic
117C22×C73C2×SD3211yescharacteristic
118C22×C73#50911yescharacteristic
119C2×D73—82no—
120C2×D73—82no—
121C2×D73—82no—
122C4×C73—21no—
123C4×C73—22no—
124C4×C73—41no—
125C4×C73—82no—
126C4×C73—161no—
127C4×C73#45411yescharacteristic
128C4×C73B(4,𝔽2)11yescharacteristic
129Dic73—82no—
130#12161—731nomaximal, Sylow
131C2×C73—21no—
132C2×C73—21no—
133C2×C73#159811yescharacteristic
134C2×C73#218711yescharacteristic
135C2×C73#219111yescharacteristic
136D73—162no—
137#3245—731no—
138#2185—732no—
139#2056D7311yescharacteristic, complemented
140#1347—732no—
141#1322—731no—
142C73#1216111yescharacteristic, complemented, Sylow
143C22×Q16C2×D7311yescharacteristic
144#532—1461no—
145C2×SD32—1462no—
146#474—1462no—
147#466—732no—
148#466—1461no—
149#459—1462no—
150C2×C4⋊C8C2×D7311yescharacteristic
151C2×#103—21no—
152#391—21no—
153#358—1462no—
154C22⋊C16—732no—
155#340C2×D7311yescharacteristic
156C2×#41—2921no—
157C22×Q8—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×D8—732no—
161C2×D8—1461no—
162C22×C8C22⋊D7311yescharacteristic
163C22×C8C22×D7311yescharacteristic
164C4⋊Q8—1461no—
165#127—1461no—
166#124—2921no—
167#121—2921no—
168#121—1462no—
169C4×D4—2921no—
170C4×D4—1462no—
171C2×C4⋊C4—21no—
172C2×C42C22⋊D7311yescharacteristic
173SD32—2924no—
174C2×C16—1462no—
175#107—732no—
176#107—1461no—
177C4⋊C8—22no—
178#103—42no—
179#102—1462no—
180#102—1462no—
181#95—41no—
182#95—41no—
183#41—5842no—
184#41—5842no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—2921no—
192C2×D4—2921no—
193C2×D4—2921no—
194C2×D4—2922no—
195C2×D4—1462no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—2922no—
199C22×C4#1192401311yescharacteristic
200C22×C4#1192401311yescharacteristic
201C22×C4C2×C22⋊D7311yescharacteristic
202Q16—24no—
203Q16—42no—
204D8—2922no—
205D8—5841no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D7312yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—2921no—
213C4⋊C4—1462no—
214C22⋊C4—2922no—
215C22⋊C4—2922no—
216C42—21no—
217C42—21no—
218C16—2922no—
219C23—2922no—
220C23C2×#1192401311yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—81no—
224Q8—81no—
225Q8—5842no—
226D4—2922no—
227D4—2922no—
228D4—5842no—
229D4—5842no—
230D4—5842no—
231D4—5842no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—2922no—
245C2×C4—5844no—
246C2×C4#1455653412yescomplemented
247C2×C4#1455655911yescharacteristic
248C2×C4#1455661611yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—5842no—
254C22—5842no—
255C22—5842no—
256C22#1798872011yescharacteristic
257C22C2×#1455653411yescharacteristic
258C22#1798902211yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—161no—
264C4—5842no—
265C4#1798875911yescharacteristic
266C4#1798877311yescharacteristic
267C2—21no—
268C2—21no—
269C2—11682no—
270C2#2260886511yescharacteristic
271C2#2261185911yescharacteristic
272C2#2261188111yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#2056C2
socleC22×C73#447
prime radical#22621241C2
hypercenter#2056D73
hypocenterC73#12161
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. #22611848C2semidirect
3. #22607939C2semidirect
4. #22607871C2semidirect
5. C73×#2056C2semidirect
100. C2×C4×C73SD32semidirect
139. #2056D73semidirect
142. C73#12161semidirect
246. C2×C4#14556534semidirect

Maximal subgroups

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

subgroupindex
#226212412
#226118482
#226079392
#226078712
C73×#20562
#1216173