Finite Group Databasebeta

← #32999492

Subgroups — #32999492

Order 18688.

378 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612180C211yesmaximal, characteristic, complemented
2#22611966C211yesmaximal, characteristic, complemented
3#22609291C211yesmaximal, characteristic, complemented
4#22607977C211yesmaximal, characteristic, complemented
5Q8⋊#11924003C211yesmaximal, characteristic, complemented
6#22607892C211yesmaximal, characteristic, complemented
7C73×#3391C211yesmaximal, characteristic, complemented
8C2×#14556532—22no—
9C2×#14556532—21no—
10#17988964C2211yescharacteristic
11#17988787—21no—
12C4⋊#11924003C2211yescharacteristic
13#17988387—21no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988330—21no—
17#550×C73—21no—
18#493×C73—21no—
19#493×C73C2211yescharacteristic
20#459×C73—22no—
21#459×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×D8×C73—21no—
24C4×D8×C73C2211yescharacteristic
25C2×#11924015—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556588—21no—
29#14556532—42no—
30#14556532—44no—
31C2×#11924003—21no—
32C2×#11924003—21no—
33C4⋊#10596831—21no—
34#14556511—41no—
35C2×#41×C73—41no—
36C2×D8×C73—22no—
37C2×D8×C73—21no—
38C2×D8×C73D411yescharacteristic
39C2×D8×C73D412yes—
40#127×C73—22no—
41#127×C73—21no—
42#121×C73—42no—
43#121×C73—41no—
44C4×Q8×C73D411yescharacteristic
45C4×D4×C73—22no—
46C4×D4×C73—21no—
47#107×C73D411yescharacteristic
48C4⋊C8×C73—21no—
49C4⋊C8×C73D411yescharacteristic
50#102×C73—22no—
51#102×C73—21no—
52C4×C8×C73—21no—
53C4×C8×C73C2311yescharacteristic
54C2×D292—41no—
55C2×D292—41no—
56C2×D292—41no—
57#11924015—82no—
58D584—42no—
59D584—41no—
60D584—41no—
61C2×#10596831—41no—
62#11924003—22no—
63#11924003—41no—
64#41×C73—82no—
65#41×C73—82no—
66C2×Q8×C73D811yescharacteristic
67C2×D4×C73—21no—
68C2×D4×C73—22no—
69C2×D4×C73—42no—
70C2×D4×C73—42no—
71C2×D4×C73—41no—
72C2×D4×C73—41no—
73C22×C4×C73—42no—
74C22×C4×C73—41no—
75D8×C73—21no—
76D8×C73—21no—
77D8×C73—42no—
78D8×C73—41no—
79D8×C73—41no—
80D8×C73—82no—
81C2×C8×C73—21no—
82C2×C8×C73C2×D411yescharacteristic
83C2×C8×C73C2×D411yescharacteristic
84C4⋊C4×C73—21no—
85C4⋊C4×C73—21no—
86C4⋊C4×C73D811yescharacteristic
87C22⋊C4×C73—42no—
88C22⋊C4×C73—41no—
89C42×C73—21no—
90C42×C73C2×D411yescharacteristic
91C23×C73—42no—
92C23×C73—41no—
93C22×D73—81no—
94Q8×C73—21no—
95Q8×C73D1612yescomplemented
96D4×C73—41no—
97D4×C73—42no—
98D4×C73—42no—
99D4×C73—41no—
100D4×C73—84no—
101D4×C73—81no—
102D4×C73—81no—
103D4×C73—161no—
104D4×C73—162no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—41no—
108C2×C4×C73—84no—
109C2×C4×C73—81no—
110C2×C4×C73—81no—
111C2×C4×C73#12011yescharacteristic
112C2×C4×C73C2×D811yescharacteristic
113C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
114D292—81no—
115D292—81no—
116D292—81no—
117D292—81no—
118D292—161no—
119C8×C73—21no—
120C8×C73—41no—
121C8×C73#13511yescharacteristic
122C8×C73AGL(1,ℤ/8)11yescharacteristic
123#10596831—81no—
124C22×C73—81no—
125C22×C73—82no—
126C22×C73—82no—
127C22×C73—81no—
128C22×C73—82no—
129C22×C73—81no—
130C22×C73#44711yescharacteristic
131C2×D73—161no—
132C2×D73—161no—
133C2×D73—161no—
134C4×C73—22no—
135C4×C73—21no—
136C4×C73—41no—
137C4×C73—81no—
138C4×C73#44911yescharacteristic
139C4×C73#45311yescharacteristic
140C4×C73C2×D1611yescharacteristic
141C4×C73#50911yescharacteristic
142#12328—731nomaximal, Sylow
143C2×C73—161no—
144C2×C73—162no—
145C2×C73#162111yescharacteristic
146C2×C73#218411yescharacteristic
147C2×C73#219211yescharacteristic
148D73—321no—
149#3391D7311yescharacteristic, complemented
150#2246—731no—
151#2206—731no—
152#1668—731no—
153#1361—731no—
154Q8⋊C16—731no—
155#1331—731no—
156C73#1232811yescharacteristic, complemented, Sylow
157#550—21no—
158C2×D16—1461no—
159C2×D16—1462no—
160#493—21no—
161#493—731no—
162#493C2×D7311yescharacteristic
163#460—1461no—
164#459—22no—
165#459—21no—
166C8×Q8C2×D7311yescharacteristic
167C4×D8—21no—
168C4×D8C2×D7311yescharacteristic
169C4⋊C16—731no—
170#357—1461no—
171C4×C16—731no—
172#335—731no—
173#331—1461no—
174C2×#41—41no—
175C2×SD16—2921no—
176C2×D8—22no—
177C2×D8—21no—
178C2×D8—1461no—
179C2×D8—1461no—
180C2×D8C22⋊D7311yescharacteristic
181C2×D8C22⋊D7312yes—
182#127—22no—
183#127—21no—
184#127—1461no—
185#121—42no—
186#121—41no—
187C4×Q8C22⋊D7311yescharacteristic
188C4×D4—22no—
189C4×D4—21no—
190D16—2922no—
191D16—2924no—
192C2×C16—1461no—
193C2×C16—1461no—
194#107C22⋊D7311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—1461no—
197C4⋊C8C22⋊D7311yescharacteristic
198#102—22no—
199#102—21no—
200#102—2921no—
201C4×C8—21no—
202C4×C8C22×D7311yescharacteristic
203#41—82no—
204#41—82no—
205C2×Q8#1192401311yescharacteristic
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—2921no—
213C2×D4—2921no—
214C2×D4—2921no—
215C22×C4—42no—
216C22×C4—41no—
217SD16—5842no—
218D8—21no—
219D8—21no—
220D8—42no—
221D8—41no—
222D8—41no—
223D8—82no—
224D8—2922no—
225D8—2921no—
226D8—2921no—
227C2×C8—21no—
228C2×C8—2921no—
229C2×C8C2×C22⋊D7311yescharacteristic
230C2×C8C2×C22⋊D7311yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4#1192401311yescharacteristic
234C22⋊C4—42no—
235C22⋊C4—41no—
236C42—21no—
237C42C2×C22⋊D7311yescharacteristic
238C16—2921no—
239C16—1462no—
240C23—42no—
241C23—41no—
242C23—5841no—
243Q8—21no—
244Q8#1455653212yescomplemented
245D4—41no—
246D4—42no—
247D4—42no—
248D4—41no—
249D4—84no—
250D4—81no—
251D4—81no—
252D4—161no—
253D4—162no—
254D4—5841no—
255D4—5841no—
256D4—5841no—
257D4—5841no—
258D4—11681no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#1455656011yescharacteristic
266C2×C4C2×#1192401311yescharacteristic
267C2×C4#1455661711yescharacteristic
268C8—21no—
269C8—41no—
270C8—5841no—
271C8#1455661111yescharacteristic
272C8#1455661611yescharacteristic
273C22—81no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—82no—
278C22—81no—
279C22—11681no—
280C22—11681no—
281C22—11681no—
282C22#1798872111yescharacteristic
283C4—22no—
284C4—21no—
285C4—41no—
286C4—81no—
287C4#1798873211yescharacteristic
288C4#1798875511yescharacteristic
289C4C2×#1455653211yescharacteristic
290C4#1798902311yescharacteristic
291C2—161no—
292C2—162no—
293C2—23361no—
294C2#2260899611yescharacteristic
295C2#2261184311yescharacteristic
296C2#2261188811yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3391C2
socleC22×C73#447
hypercenter#3391D73
hypocenterC73#12328
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. #22612180C2semidirect
2. #22611966C2semidirect
3. #22609291C2semidirect
4. #22607977C2semidirect
5. Q8⋊#11924003C2semidirect
6. #22607892C2semidirect
7. C73×#3391C2semidirect
95. Q8×C73D16semidirect
149. #3391D73semidirect
156. C73#12328semidirect
244. Q8#14556532semidirect

Maximal subgroups

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

subgroupindex
#226121802
#226119662
#226092912
#226079772
Q8⋊#119240032
#226078922
C73×#33912
#1232873