Finite Group Databasebeta

← #32986611

Subgroups — #32986611

Order 18688.

492 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 85 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#22619875C212yesmaximal
2#22610000C211yesmaximal, characteristic
3#22608295C212yesmaximal
4#22607808C211yesmaximal, characteristic
5C73×#3189C211yesmaximal, characteristic
6C2×#14556550C2211yescharacteristic
7#17988925—24no—
8#17988920—22no—
9#17988915—22no—
10#17988627—22no—
11#17988596—22no—
12#17988551C2212yes—
13C2×C22⋊#10596831C2211yescharacteristic
14#17988341—22no—
15#17988313—24no—
16C2×C22⋊Q8×C73C2212yes—
17#484×C73—22no—
18#467×C73—22no—
19#451×C73—22no—
20C2×C22⋊C8×C73C2211yescharacteristic
21C2×#11924012—22no—
22C22×#10596831—21no—
23#14556550—22no—
24#14556550—22no—
25C2×C4⋊Dic73D411yescharacteristic
26C2×C4⋊Dic73D411yescharacteristic
27C2×C22⋊Dic73—21no—
28C2×C4×Dic73—21no—
29#14556524—24no—
30#14556522—24no—
31#14556515—24no—
32#14556515—24no—
33C22⋊#10596831—22no—
34C22⋊#10596831—22no—
35C22⋊#10596831D414yes—
36C22×Q8×C73D412yes—
37C23×C4×C73C2311yescharacteristic
38C2×Q16×C73—22no—
39C22×C8×C73D412yes—
40C22⋊Q8×C73—22no—
41C22⋊Q8×C73—22no—
42C22⋊Q8×C73—22no—
43C22⋊Q8×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73D412yes—
46C2×C22⋊C4×C73—22no—
47#107×C73—22no—
48#103×C73—24no—
49C22⋊C8×C73—22no—
50C22×Dic73—21no—
51C22×Dic73—21no—
52#11924012—44no—
53C2×#10596831—24no—
54C2×#10596831—42no—
55C2×#10596831—41no—
56C4⋊Dic73—22no—
57C4⋊Dic73—22no—
58C22⋊Dic73—42no—
59C4×Dic73—41no—
60C4×Dic73—41no—
61C24×C73#4111yescharacteristic
62C2×Q8×C73—22no—
63C2×Q8×C73—22no—
64C2×Q8×C73—44no—
65C22×C4×C73—22no—
66C22×C4×C73—22no—
67C22×C4×C73—42no—
68C22×C4×C73C2×D411yescharacteristic
69C22×C4×C73C2×D412yes—
70C22×C4×C73C2×D412yes—
71C22×C4×C73C2×D411yescharacteristic
72Q16×C73—82no—
73C2×C8×C73—22no—
74C2×C8×C73—42no—
75C4⋊C4×C73—22no—
76C4⋊C4×C73—22no—
77C4⋊C4×C73—42no—
78C4⋊C4×C73—42no—
79C22⋊C4×C73—42no—
80C22⋊C4×C73—42no—
81C23×C73—22no—
82C23×C73—41no—
83C23×C73—41no—
84C23×C73#12111yescharacteristic
85C23×C73#12112yes—
86Q8×C73—44no—
87Q8×C73—44no—
88C2×C4×C73—22no—
89C2×C4×C73—24no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73—44no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73—82no—
96C2×C4×C73#12012yes—
97C2×C4×C73#12011yescharacteristic
98C2×C4×C73#12711yescharacteristic
99C2×Dic73—42no—
100C2×Dic73—41no—
101C2×Dic73—41no—
102C2×Dic73—42no—
103C8×C73—42no—
104#10596831—44no—
105#10596831—81no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73—22no—
109C22×C73—22no—
110C22×C73—22no—
111C22×C73—82no—
112C22×C73#39211yescharacteristic
113C22×C73#47312yes—
114C4×C73—22no—
115C4×C73—21no—
116C4×C73—82no—
117C4×C73—82no—
118C4×C73#45812yes—
119Dic73—81no—
120Dic73—81no—
121#9979—731nomaximal, Sylow
122C2×C73—22no—
123C2×C73—42no—
124C2×C73#201711yescharacteristic
125C2×C73#202011yescharacteristic
126C2×C73#202211yescharacteristic
127#3189D7311yescharacteristic
128#3095—732no—
129#1818—731no—
130#1472—732no—
131#1293—731no—
132C73#997911yescharacteristic, complemented, Sylow
133C2×C22⋊Q8C2×D7312yes—
134C2×#117—731no—
135#485—1464no—
136#484—22no—
137#484—1462no—
138#483—1462no—
139#467—22no—
140#451—22no—
141#428—1462no—
142#419—1462no—
143#407—732no—
144C2×C22⋊C8—731no—
145C2×C22⋊C8C2×D7311yescharacteristic
146#336—1462no—
147#324—1464no—
148C22×Q8C22⋊D7312yes—
149C23×C4C22×D7311yescharacteristic
150C2×Q16—22no—
151C2×M16—1462no—
152C22×C8—1461no—
153C22×C8C22⋊D7312yes—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158#117—1462no—
159#117—1462no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—731no—
163C2×C4⋊C4C22⋊D7312yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—1461no—
166C2×C42—1461no—
167#107—22no—
168#107—1464no—
169#106—1464no—
170#103—24no—
171#103—1464no—
172#103—1464no—
173C22⋊C8—22no—
174C22⋊C8—734no—
175C22⋊C8—1462no—
176C22⋊C8—1462no—
177C24#1192402711yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—44no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—1461no—
185C22×C4—1461no—
186C22×C4D4×D7311yescharacteristic
187C22×C4D4×D7312yes—
188C22×C4C2×C22⋊D7312yes—
189C22×C4C2×C22⋊D7311yescharacteristic
190Q16—82no—
191M16—2924no—
192C2×C8—22no—
193C2×C8—42no—
194C2×C8—2922no—
195C2×C8—2921no—
196C2×C8—1464no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—1462no—
202C4⋊C4—1462no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—2922no—
206C42—2921no—
207C42—2921no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23#1455656211yescharacteristic
212C23#1455656312yes—
213Q8—44no—
214Q8—44no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—2922no—
224C2×C4—2921no—
225C2×C4—2921no—
226C2×C4—2922no—
227C2×C4#1455655912yes—
228C2×C4#1455656011yescharacteristic
229C2×C4#1455658711yescharacteristic
230C8—42no—
231C8—2924no—
232C8—5841no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—82no—
239C22#1798849511yescharacteristic
240C22#1798885912yes—
241C4—22no—
242C4—21no—
243C4—82no—
244C4—82no—
245C4—5841no—
246C4—5841no—
247C4#1798877712yes—
248C2—22no—
249C2—42no—
250C2#2261103411yescharacteristic
251C2#2261105011yescharacteristic
252C2#2261106011yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#3189C2
socleC22×C73#392
prime radicalC24×C73#41
hypercenter#3189D73
hypocenterC73#9979
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
132. C73#9979semidirect

Maximal subgroups

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

subgroupindex
#226198752
#226100002
#226082952
#226078082
C73×#31892
#997973