Finite Group Databasebeta

← #32998447

Subgroups — #32998447

Order 18688.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#22611864C212yesmaximal
2#22611254C211yesmaximal, characteristic
3#22607943C212yesmaximal
4#22607871C211yesmaximal, characteristic
5C73×#3249C211yesmaximal, characteristic
6C22×Dic292C2211yescharacteristic
7C2×#14556537—22no—
8C2×#14556516—21no—
9#17988493—21no—
10#17988390—22no—
11C22⋊#11924003C2212yes—
12#17988350C2211yescharacteristic
13#533×C73—21no—
14#477×C73—22no—
15#467×C73—21no—
16#467×C73C2212yes—
17#462×C73—22no—
18C2×C4⋊C8×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×Dic292—22no—
21C2×Dic292—42no—
22C2×C4⋊Dic73—21no—
23#14556537—44no—
24C2×#11924003—22no—
25#14556516—42no—
26#14556498—41no—
27#14556498—41no—
28C2×Q16×C73—21no—
29C2×Q16×C73D412yes—
30C22×C8×C73D411yescharacteristic
31C22×C8×C73C2311yescharacteristic
32C4⋊Q8×C73—21no—
33C4⋊Q8×C73—21no—
34#125×C73—41no—
35C22⋊Q8×C73—22no—
36C22⋊Q8×C73—41no—
37C4×Q8×C73—22no—
38C4×Q8×C73—41no—
39#117×C73—41no—
40C2×C42×C73D411yescharacteristic
41#107×C73—21no—
42#107×C73D412yes—
43C4⋊C8×C73—22no—
44#103×C73—22no—
45#103×C73—22no—
46C2×Dic146—22no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C22×Dic73—41no—
53C22×Dic73—41no—
54Dic292—24no—
55Dic292—42no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924003—42no—
59C2×Q8×C73—22no—
60C2×Q8×C73—41no—
61C22×C4×C73D811yescharacteristic
62C22×C4×C73D811yescharacteristic
63C22×C4×C73C2×D411yescharacteristic
64Q16×C73—42no—
65Q16×C73—81no—
66C2×C8×C73—21no—
67C2×C8×C73—41no—
68C2×C8×C73—41no—
69C2×C8×C73C2×D412yes—
70C4⋊C4×C73—22no—
71C4⋊C4×C73—41no—
72C4⋊C4×C73—41no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—41no—
75C4⋊C4×C73—42no—
76C22⋊C4×C73—42no—
77C42×C73—21no—
78C42×C73—21no—
79C42×C73—42no—
80C23×C73C2×D811yescharacteristic
81Q8×C73—42no—
82Q8×C73—42no—
83Q8×C73—82no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—41no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73Q3212yescomplemented
93C2×C4×C73#12011yescharacteristic
94C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
95C2×Dic73—42no—
96C2×Dic73—81no—
97C2×Dic73—81no—
98C2×Dic73—81no—
99C2×Dic73—81no—
100C2×Dic73—81no—
101Dic146—42no—
102Dic146—42no—
103Dic146—81no—
104Dic146—81no—
105C8×C73—22no—
106C8×C73—41no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73#44711yescharacteristic
110C22×C73C2×Q3211yescharacteristic
111C22×C73#51011yescharacteristic
112C4×C73—21no—
113C4×C73—22no—
114C4×C73—41no—
115C4×C73—82no—
116C4×C73—82no—
117C4×C73#45411yescharacteristic
118C4×C73B(4,𝔽2)11yescharacteristic
119Dic73—82no—
120Dic73—161no—
121#12162—731nomaximal, Sylow
122C2×C73—21no—
123C2×C73—21no—
124C2×C73#159811yescharacteristic
125C2×C73#218811yescharacteristic
126C2×C73#219111yescharacteristic
127#3249D7311yescharacteristic
128#2188—732no—
129#2056—731no—
130#1348—732no—
131#1322—731no—
132C73#1216211yescharacteristic, complemented, Sylow
133C22×Q16—731no—
134#533—21no—
135C2×Q32—1462no—
136#477—22no—
137#467—21no—
138#467C2×D7312yes—
139#462—22no—
140C2×C4⋊C8C2×D7311yescharacteristic
141C2×#103—1461no—
142#391—1461no—
143#358—1462no—
144C22⋊C16—732no—
145#340—731no—
146C22×Q8—1461no—
147C2×Q16—21no—
148C2×Q16—2922no—
149C2×Q16—1462no—
150C2×Q16C22⋊D7312yes—
151C22×C8C22⋊D7311yescharacteristic
152C22×C8C22×D7311yescharacteristic
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155#125—41no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—41no—
160#117—41no—
161C2×C4⋊C4—1461no—
162C2×C42C22⋊D7311yescharacteristic
163Q32—2924no—
164C2×C16—1462no—
165#107—21no—
166#107C22⋊D7312yes—
167C4⋊C8—22no—
168#103—22no—
169#103—22no—
170#103—2922no—
171#95—2921no—
172#95—2921no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—2921no—
176C2×Q8—2921no—
177C2×Q8—2921no—
178C2×Q8—2921no—
179C2×Q8—2921no—
180C2×Q8—1462no—
181C22×C4—2921no—
182C22×C4—2921no—
183C22×C4#1192401311yescharacteristic
184C22×C4#1192401311yescharacteristic
185C22×C4C2×C22⋊D7311yescharacteristic
186Q16—42no—
187Q16—81no—
188Q16—2922no—
189Q16—1464no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8C2×C22⋊D7312yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—2921no—
201C4⋊C4—2921no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—42no—
206C16—2922no—
207C23C2×#1192401311yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—82no—
211Q8—2922no—
212Q8—2922no—
213Q8—5841no—
214Q8—5841no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—2922no—
224C2×C4—5841no—
225C2×C4—5841no—
226C2×C4—5841no—
227C2×C4—5841no—
228C2×C4—5841no—
229C2×C4#1455653712yescomplemented
230C2×C4#1455656011yescharacteristic
231C2×C4#1455661711yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#1798872111yescharacteristic
237C22C2×#1455653711yescharacteristic
238C22#1798902811yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4—5842no—
245C4—11681no—
246C4#1798876011yescharacteristic
247C4#1798877411yescharacteristic
248C2—21no—
249C2—21no—
250C2#2260886611yescharacteristic
251C2#2261186511yescharacteristic
252C2#2261188211yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#3249C2
socleC22×C73#447
prime radicalC23×C73C2×D8
hypercenter#3249D73
hypocenterC73#12162
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
92. C2×C4×C73Q32semidirect
132. C73#12162semidirect
229. C2×C4#14556537semidirect

Maximal subgroups

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

subgroupindex
#226118642
#226112542
#226079432
#226078712
C73×#32492
#1216273