Finite Group Databasebeta

← #32999039

Subgroups — #32999039

Order 18688.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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#22611276C211yesmaximal, characteristic
2#22608904C212yesmaximal
3#22607961C212yesmaximal
4#22607877C211yesmaximal, characteristic
5C73×#2551C211yesmaximal, characteristic
6#17989218C2211yescharacteristic
7#17988743—22no—
8#17988627—21no—
9#17988356C2211yescharacteristic
10#17988347—21no—
11#17988323—22no—
12#17988313C2212yes—
13C2×C4⋊Q8×C73—22no—
14#395×C73—21no—
15#395×C73C2212yes—
16#389×C73—22no—
17#384×C73C2211yescharacteristic
18C4×C4⋊C4×C73—21no—
19C2×#11924018—42no—
20C2×#11924012—21no—
21#14556590—41no—
22#14556582—41no—
23#14556570—22no—
24#14556550—21no—
25#14556524—41no—
26#14556522—41no—
27#14556515—42no—
28C22⋊#10596831—22no—
29C22×Q8×C73—21no—
30C22×Q8×C73D412yes—
31C4⋊Q8×C73—48no—
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73—22no—
36C2×C4⋊C4×C73—22no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C4⋊C4×C73C2311yescharacteristic
39C2×C42×C73—22no—
40C2×C42×C73D411yescharacteristic
41#95×C73—21no—
42#95×C73—22no—
43#95×C73D412yes—
44C2×Dic146—22no—
45#11924018—84no—
46#11924012—22no—
47#11924012—41no—
48C2×#10596831—41no—
49C2×#10596831—42no—
50C4⋊Dic73—22no—
51C4⋊Dic73—21no—
52C4⋊Dic73—21no—
53#11924007—42no—
54C22⋊Dic73—41no—
55C4×Dic73—41no—
56C2×Q8×C73—44no—
57C2×Q8×C73—44no—
58C2×Q8×C73—42no—
59C2×Q8×C73—81no—
60C22×C4×C73—21no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—21no—
64C22×C4×C73C2×D411yescharacteristic
65C22×C4×C73C2×D412yes—
66C4⋊C4×C73—24no—
67C4⋊C4×C73—24no—
68C4⋊C4×C73—41no—
69C4⋊C4×C73—41no—
70C4⋊C4×C73—44no—
71C4⋊C4×C73—44no—
72C4⋊C4×C73—44no—
73C42×C73—21no—
74C42×C73—44no—
75C42×C73Q1612yescomplemented
76C23×C73#12011yescharacteristic
77Q8×C73—44no—
78Q8×C73—82no—
79Q8×C73—84no—
80C2×C4×C73—22no—
81C2×C4×C73—22no—
82C2×C4×C73—22no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—24no—
86C2×C4×C73—24no—
87C2×C4×C73—22no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73—42no—
93C2×C4×C73—42no—
94C2×C4×C73—41no—
95C2×C4×C73C2×Q1611yescharacteristic
96C2×C4×C73#13711yescharacteristic
97C2×Dic73—41no—
98C2×Dic73—41no—
99Dic146—82no—
100#10596831—42no—
101#10596831—82no—
102C22×C73—21no—
103C22×C73—21no—
104C22×C73#45111yescharacteristic
105C22×C73#45611yescharacteristic
106C22×C73B(4,𝔽2)11yescharacteristic
107C4×C73—24no—
108C4×C73—21no—
109C4×C73—21no—
110C4×C73—44no—
111C4×C73—42no—
112C4×C73—44no—
113C4×C73—81no—
114Dic73—81no—
115Dic73—81no—
116#12256—731nomaximal, Sylow
117C2×C73—21no—
118C2×C73—21no—
119C2×C73#160511yescharacteristic
120C2×C73#219611yescharacteristic
121C2×C73#220511yescharacteristic
122#2551D7311yescharacteristic
123#2060—731no—
124#1605—732no—
125#1354—732no—
126#1325—731no—
127C73#1225611yescharacteristic, complemented, Sylow
128#544—731no—
129C2×C4⋊Q8—22no—
130#451—1462no—
131#428—1461no—
132#395—21no—
133#395C2×D7312yes—
134#389—22no—
135#384C2×D7311yescharacteristic
136C4×C4⋊C4—21no—
137#343—731no—
138#339—1461no—
139#328—1462no—
140#324—732no—
141C22×Q8—21no—
142C22×Q8C22⋊D7312yes—
143C2×Q16—2922no—
144C2×M16—1461no—
145C4⋊Q8—48no—
146C4⋊Q8—2921no—
147#125—2921no—
148C22⋊Q8—1462no—
149#117—1461no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D7311yescharacteristic
156C2×C4⋊C4C22×D7311yescharacteristic
157C2×C42—22no—
158C2×C42C22⋊D7311yescharacteristic
159#107—2921no—
160#106—2921no—
161#103—2922no—
162C22⋊C8—1462no—
163#95—21no—
164#95—22no—
165#95C22⋊D7312yes—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—1462no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D7311yescharacteristic
176C22×C4C2×C22⋊D7312yes—
177Q16—5844no—
178M16—2921no—
179M16—1462no—
180C2×C8—2921no—
181C2×C8—2922no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—2922no—
190C4⋊C4—1462no—
191C4⋊C4—1461no—
192C4⋊C4—1461no—
193C22⋊C4—2921no—
194C42—21no—
195C42—44no—
196C42—2921no—
197C42#1192401812yescomplemented
198C23#1455656011yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—5842no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—2921no—
219C2×C4—2921no—
220C2×C4C2×#1192401811yescharacteristic
221C2×C4#1455662211yescharacteristic
222C8—2922no—
223C8—5842no—
224C22—21no—
225C22—21no—
226C22#1798874311yescharacteristic
227C22#1798877011yescharacteristic
228C22#1798877411yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—5841no—
237C4—5841no—
238C2—21no—
239C2—21no—
240C2#2260890511yescharacteristic
241C2#2261191511yescharacteristic
242C2#2261196111yescharacteristic
subgroupG/H
centerC22#17988774
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2551C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2551D73
hypocenterC73#12256
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
75. C42×C73Q16semidirect
127. C73#12256semidirect
197. C42#11924018semidirect

Maximal subgroups

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

subgroupindex
#226112762
#226089042
#226079612
#226078772
C73×#25512
#1225673