Finite Group Databasebeta

← #32999201

Subgroups — #32999201

Order 18688.

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

normal subgroups only

This group has 396 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#22620682C211yesmaximal, characteristic
2#22612116C212yesmaximal, complemented
3#22607939C212yesmaximal, complemented
4#22607855C211yesmaximal, characteristic, complemented
5C73×#2076C211yesmaximal, characteristic, complemented
6C2×#14556565—21no—
7#17988901—22no—
8#17988823—21no—
9#17988823C2212yes—
10#17988723—22no—
11#17988414—22no—
12#17988387—22no—
13C22⋊#11924003C2212yescomplemented
14C2×#107×C73C2211yescharacteristic, complemented
15C2×C22⋊C8×C73C2211yescharacteristic
16#380×C73—21no—
17#340×C73—21no—
18#340×C73C2211yescharacteristic, complemented
19C22×D292—21no—
20C22×C22⋊D73—41no—
21C2×D584—21no—
22C2×D584D412yes—
23#14556565—22no—
24#14556565—22no—
25#14556565—41no—
26#14556565—41no—
27C2×C4⋊Dic73—21no—
28C2×#11924005—41no—
29C2×#11924003—22no—
30#14556525—21no—
31#14556525D412yes—
32#14556513—22no—
33#14556513—22no—
34C23×C4×C73D411yescharacteristic
35C22×C8×C73D411yescharacteristic
36C22×C8×C73C2311yescharacteristic
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C22⋊C4×C73—41no—
40#107×C73—22no—
41C22⋊C8×C73—22no—
42#95×C73—41no—
43C23×D73—41no—
44C2×D292—22no—
45C2×D292—41no—
46C2×D292—81no—
47C2×D292—81no—
48C2×C22⋊D73—42no—
49C2×C22⋊D73—42no—
50C2×C22⋊D73—82no—
51C2×C22⋊D73—82no—
52C22×Dic73—41no—
53D584—42no—
54D584—81no—
55C4⋊Dic73—22no—
56C4⋊Dic73—41no—
57#11924005—42no—
58#11924005—42no—
59#11924003—42no—
60C24×C73D811yescharacteristic
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—41no—
64C22×C4×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73D811yescharacteristic
68C22×C4×C73C2×D411yescharacteristic
69C2×C8×C73—21no—
70C2×C8×C73—41no—
71C2×C8×C73—41no—
72C2×C8×C73C2×D412yes—
73C4⋊C4×C73—41no—
74C4⋊C4×C73—41no—
75C4⋊C4×C73—41no—
76C4⋊C4×C73—41no—
77C22⋊C4×C73—82no—
78C23×C73—21no—
79C23×C73—21no—
80C23×C73—41no—
81C23×C73—41no—
82C23×C73SD3212yes—
83C23×C73C2×D811yescharacteristic
84C22×D73—42no—
85C22×D73—82no—
86C22×D73—82no—
87C22×D73—82no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73—81no—
95C2×C4×C73—81no—
96C2×C4×C73—81no—
97C2×C4×C73—81no—
98C2×C4×C73—81no—
99C2×C4×C73—81no—
100C2×C4×C73—81no—
101C2×C4×C73#12011yescharacteristic
102C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
103C22⋊D73—84no—
104C22⋊D73—82no—
105C22⋊D73—82no—
106C2×Dic73—42no—
107C2×Dic73—82no—
108D292—42no—
109D292—42no—
110D292—82no—
111C8×C73—22no—
112C8×C73—41no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73—22no—
116C22×C73—42no—
117C22×C73—41no—
118C22×C73—41no—
119C22×C73—81no—
120C22×C73#44711yescharacteristic
121C22×C73C2×SD3211yescharacteristic
122C22×C73#50911yescharacteristic
123C2×D73—82no—
124C2×D73—82no—
125C2×D73—82no—
126C2×D73—164no—
127C4×C73—21no—
128C4×C73—41no—
129C4×C73—161no—
130C4×C73—161no—
131C4×C73#45511yescharacteristic
132C4×C73#45811yescharacteristic
133Dic73—82no—
134#12281—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—21no—
137C2×C73—42no—
138C2×C73#160711yescharacteristic
139C2×C73#218711yescharacteristic
140C2×C73#219011yescharacteristic
141D73—162no—
142#3186—731no—
143#2233—732no—
144#2076D7311yescharacteristic, complemented
145#1347—732no—
146#1314—731no—
147C73#1228111yescharacteristic, complemented, Sylow
148C2×#121—1461no—
149#480—1462no—
150#466—732no—
151#466—1461no—
152#447—1462no—
153C2×#107C2×D7311yescharacteristic, complemented
154C2×C22⋊C8C2×D7311yescharacteristic
155#380—21no—
156#367—1462no—
157#357—1462no—
158C22⋊C16—732no—
159#340—21no—
160#340C2×D7311yescharacteristic, complemented
161C22×D4—2921no—
162C22×D4—1461no—
163C23×C4C22⋊D7311yescharacteristic
164C2×D8—732no—
165C2×D8—1461no—
166C22×C8C22⋊D7311yescharacteristic
167C22×C8C22×D7311yescharacteristic
168#121—2921no—
169#121—2921no—
170#121—1462no—
171#121—1462no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—1461no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—2921no—
177C2×C16—1462no—
178#107—22no—
179#107—732no—
180#107—1461no—
181#102—1462no—
182#102—1462no—
183C22⋊C8—22no—
184#95—41no—
185C24—2921no—
186C24#1192401311yescharacteristic
187C2×D4—2921no—
188C2×D4—2922no—
189C2×D4—2922no—
190C2×D4—5841no—
191C2×D4—5841no—
192C2×D4—5842no—
193C2×D4—5842no—
194C2×D4—1462no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—2921no—
202C22×C4#1192401311yescharacteristic
203C22×C4C2×C22⋊D7311yescharacteristic
204D8—2922no—
205D8—5841no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D7312yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—2921no—
215C4⋊C4—1462no—
216C22⋊C4—82no—
217C22⋊C4—2922no—
218C22⋊C4—2922no—
219C16—2922no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—2922no—
225C23—5842no—
226C23—5842no—
227C23—5842no—
228C23#1455653412yes—
229C23C2×#1192401311yescharacteristic
230D4—2922no—
231D4—2922no—
232D4—5842no—
233D4—5844no—
234D4—5842no—
235D4—5842no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—2922no—
250C2×C4—5842no—
251C2×C4#1455655911yescharacteristic
252C2×C4#1455661611yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—5842no—
263C22—5842no—
264C22—5842no—
265C22—11684no—
266C22#1798872011yescharacteristic
267C22C2×#1455653411yescharacteristic
268C22#1798902211yescharacteristic
269C4—21no—
270C4—41no—
271C4—161no—
272C4—161no—
273C4—5842no—
274C4#1798876411yescharacteristic
275C4#1798877611yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—11682no—
280C2#2260891611yescharacteristic
281C2#2261185911yescharacteristic
282C2#2261187611yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#2076C2
socleC22×C73#447
prime radical#22620682C2
hypercenter#2076D73
hypocenterC73#12281
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2076

Complements

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

NH ≅ G/Ntype
2. #22612116C2semidirect
3. #22607939C2semidirect
4. #22607855C2semidirect
5. C73×#2076C2semidirect
13. C22⋊#11924003C22semidirect
14. C2×#107×C73C22semidirect
18. #340×C73C22semidirect
144. #2076D73semidirect
147. C73#12281semidirect
153. C2×#107C2×D73semidirect
160. #340C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226206822
#226121162
#226079392
#226078552
C73×#20762
#1228173