Finite Group Databasebeta

← #33002732

Subgroups — #33002732

Order 18688.

396 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#17988694C211yesmaximal, characteristic
2C2×#17988410C211yesmaximal, characteristic
3C2×#17988403C211yesmaximal, characteristic
4#22610685C211yesmaximal, characteristic
5#22608036C211yesmaximal, characteristic
6#22608010C211yesmaximal, characteristic
7C73×#2141C211yesmaximal, characteristic
8C2×Q8×Dic73—21no—
9C2×#14556531C2211yescharacteristic
10#17988694—22no—
11#17988694—22no—
12C2×#14556527C2211yescharacteristic
13C2×#14556523C2211yescharacteristic
14#17988614—21no—
15C2×#14556515—21no—
16C2×#14556501C2211yescharacteristic
17#17988410—22no—
18#17988403—22no—
19#17988346—21no—
20C22×Q16×C73C2211yescharacteristic
21C2×M32×C73C2211yescharacteristic
22#427×C73C2211yescharacteristic
23#358×C73C414yescomplemented
24C2×#11924012—21no—
25C22×#10596831D411yescharacteristic
26Q8×Dic73—24no—
27Q8×Dic73—42no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—41no—
30C2×C4×Dic73—41no—
31C2×C4×Dic73D411yescharacteristic
32#14556531—21no—
33#14556531—21no—
34C2×#11924003—21no—
35#14556527—21no—
36#14556527—21no—
37#14556523—21no—
38#14556523D412yes—
39C4⋊#10596831—42no—
40#14556515—24no—
41#14556501—21no—
42#14556501—21no—
43C22×Q8×C73—21no—
44C2×Q16×C73—21no—
45C2×Q16×C73—21no—
46C2×Q16×C73—21no—
47C2×Q16×C73C2×C412yes—
48C2×Q16×C73D412yes—
49C2×Q16×C73Q812yes—
50C22×C8×C73C2311yescharacteristic
51#117×C73—21no—
52M32×C73—21no—
53M32×C73—21no—
54C2×C16×C73C2×C412yes—
55#107×C73C2×C412yes—
56#106×C73—21no—
57C22×Dic73—21no—
58C22×Dic73—41no—
59#11924012—41no—
60#11924012—41no—
61C2×#10596831—21no—
62C2×#10596831—21no—
63C2×#10596831—21no—
64C2×#10596831—41no—
65C4⋊Dic73—22no—
66C4⋊Dic73—44no—
67C4⋊Dic73—41no—
68C4×Dic73—21no—
69C4×Dic73—44no—
70C4×Dic73SD1612yes—
71#11924003—41no—
72C2×Q8×C73—22no—
73C2×Q8×C73—22no—
74C2×Q8×C73—22no—
75C2×Q8×C73—41no—
76C2×Q8×C73—41no—
77C2×Q8×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73C2×D411yescharacteristic
80Q16×C73—22no—
81Q16×C73—22no—
82Q16×C73—22no—
83Q16×C73C4⋊C414yes—
84C2×C8×C73C22×C411yescharacteristic
85C2×C8×C73C2×D411yescharacteristic
86C2×C8×C73C2×D411yescharacteristic
87C2×C8×C73C2×Q811yescharacteristic
88C2×C8×C73#4111yescharacteristic
89C2×C8×C73#4111yescharacteristic
90C4⋊C4×C73—22no—
91C22⋊C4×C73—41no—
92C42×C73—41no—
93C16×C73—22no—
94C23×C73AGL(1,ℤ/8)11yescharacteristic
95Q8×C73—24no—
96Q8×C73—42no—
97Q8×C73—42no—
98Q8×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73C4×D411yescharacteristic
104C2×C4×C73#12011yescharacteristic
105C2×C4×C73C22⋊Q811yescharacteristic
106C2×C4×C73C2×SD1611yescharacteristic
107C2×C4×C73#13511yescharacteristic
108C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
109C2×Dic73—22no—
110C2×Dic73—42no—
111C2×Dic73—42no—
112C2×Dic73—82no—
113C8×C73C2×C4⋊C411yescharacteristic
114C8×C73C4×D411yescharacteristic
115C8×C73C22⋊Q811yescharacteristic
116C8×C73#12311yescharacteristic
117#10596831—41no—
118#10596831—81no—
119C22×C73—22no—
120C22×C73#44211yescharacteristic
121C22×C73#45011yescharacteristic
122C22×C73#47811yescharacteristic
123C4×C73—44no—
124C4×C73—82no—
125C4×C73#38511yescharacteristic
126C4×C73C4×SD1611yescharacteristic
127C4×C73#44911yescharacteristic
128C4×C73#47611yescharacteristic
129Dic73—42no—
130Dic73—82no—
131#12957—731nomaximal, Sylow
132C2×C73—21no—
133C2×C73—21no—
134C2×C73#186411yescharacteristic
135C2×C73#218012yes—
136C2×#441—731no—
137C2×#365—731no—
138C2×#362—731no—
139#2141D7311yescharacteristic
140#1946—731no—
141#1386—731no—
142#1375—731no—
143C73#1295711yescharacteristic, complemented, Sylow
144C22×Q16C2×D7311yescharacteristic
145C2×C4×Q8—1461no—
146C2×M32—731no—
147C2×M32C2×D7311yescharacteristic
148#441—1462no—
149#441—1462no—
150C2×#108—731no—
151#427C2×D7311yescharacteristic
152C2×#106—731no—
153#424—1461no—
154C2×#103—1461no—
155C2×#97—731no—
156#365—1462no—
157#362—1462no—
158#358Dic7314yescomplemented
159#339—1461no—
160C22×Q8—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16Dic14612yes—
165C2×Q16D29212yes—
166C2×Q16C2×Dic7312yes—
167C2×M16—1461no—
168C22×C8—731no—
169C22×C8C22×D7311yescharacteristic
170C4×Q8—2922no—
171C4×Q8—1464no—
172#117—21no—
173C2×C4⋊C4—2921no—
174C2×C4⋊C4—1461no—
175C2×C42—2921no—
176C2×C42—731no—
177M32—21no—
178M32—21no—
179M32—1461no—
180M32—1461no—
181C2×C16—1461no—
182C2×C16C2×Dic7312yes—
183#108—1461no—
184#108—1461no—
185#107C2×Dic7312yes—
186#106—21no—
187#106—732no—
188#106—1461no—
189C4⋊C8—2922no—
190#103—1464no—
191#97—1461no—
192#97—1461no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C22×C4—41no—
200C22×C4—2921no—
201C22×C4—1461no—
202C22×C4D4×D7311yescharacteristic
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16C4⋊Dic7314yes—
207M16—2921no—
208M16—2921no—
209C2×C8—2921no—
210C2×C8—1461no—
211C2×C8—1461no—
212C2×C8—1461no—
213C2×C8C22×Dic7311yescharacteristic
214C2×C8D4×D7311yescharacteristic
215C2×C8C2×D29211yescharacteristic
216C2×C8C2×Dic14611yescharacteristic
217C2×C8#1192402711yescharacteristic
218C2×C8#1192402711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—2924no—
221C4⋊C4—2921no—
222C4⋊C4—1462no—
223C22⋊C4—41no—
224C42—41no—
225C42—2924no—
226C42—732no—
227C42—1461no—
228C16—22no—
229C16—2921no—
230C23#1455661511yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—2922no—
240C2×C4—2922no—
241C2×C4—5842no—
242C2×C4—1462no—
243C2×C4D4×Dic7311yescharacteristic
244C2×C4#1455656111yescharacteristic
245C2×C4#1455656811yescharacteristic
246C2×C4SD16×D7311yescharacteristic
247C2×C4#1455661211yescharacteristic
248C2×C4#1455661511yescharacteristic
249C8—2921no—
250C8—5841no—
251C8C2×C4⋊Dic7311yescharacteristic
252C8D4×Dic7311yescharacteristic
253C8#1455656811yescharacteristic
254C8#1455657511yescharacteristic
255C22—22no—
256C22#1798869911yescharacteristic
257C22#1798873911yescharacteristic
258C22#1798888911yescharacteristic
259C4—44no—
260C4—82no—
261C4—2922no—
262C4—5842no—
263C4#1798846211yescharacteristic
264C4SD16×Dic7311yescharacteristic
265C4#1798873311yescharacteristic
266C4#1798887511yescharacteristic
267C2—21no—
268C2—21no—
269C2#2261019411yescharacteristic
270C2#2261182312yes—
subgroupG/H
centerC22#17988699
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2141C2
socleC22×C73#442
prime radicalC23×C73AGL(1,ℤ/8)
cosocleC22×C8×C73C23
hypercenter#2141D73
hypocenterC73#12957
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
23. #358×C73C4semidirect
143. C73#12957semidirect
158. #358Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886942
C2×#179884102
C2×#179884032
#226106852
#226080362
#226080102
C73×#21412
#1295773