Finite Group Databasebeta

← #34001428

Subgroups — #34001428

Order 18688.

630 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 124 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#22625821C212yesmaximal
2#22624752C214yesmaximal, complemented
3C2×#17988492C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5#17989453—22no—
6#17989453C2214yes—
7C23×Dic146C411yescharacteristic
8C22×C4×Dic73C2211yescharacteristic
9#17988492—22no—
10#17988492C412yes—
11#17988492C2212yes—
12C2×#14556498—22no—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15C22⋊#10596833—44no—
16C22×Dic146—21no—
17C22×Dic146—22no—
18C22×Dic146—42no—
19C22×Dic146—42no—
20C22×Dic146C814yescomplemented
21C22×Dic146C2×C412yes—
22C23×Dic73—21no—
23C23×Dic73D412yes—
24C2×C4×Dic73—42no—
25C2×C4×Dic73C2×C411yescharacteristic
26C2×C4×Dic73D412yes—
27C2×C4×Dic73C2311yescharacteristic
28#14556498—24no—
29#14556498—24no—
30C23×C4×C73—21no—
31C23×C4×C73C2×C411yescharacteristic
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—42no—
34C2×C4⋊C4×C73C2×C412yes—
35C2×#10596833—44no—
36C2×#10596833—44no—
37C2×#10596833—44no—
38C2×Dic146—24no—
39C2×Dic146—22no—
40C2×Dic146—48no—
41C2×Dic146—42no—
42C2×Dic146—42no—
43C2×Dic146—48no—
44C2×Dic146—42no—
45C2×Dic146—42no—
46C2×Dic146C2×C814yes—
47C22×Dic73—24no—
48C22×Dic73—24no—
49C22×Dic73—22no—
50C22×Dic73—24no—
51C22×Dic73—42no—
52C22×Dic73—42no—
53C22×Dic73—42no—
54C22×Dic73C22⋊C412yes—
55C22×Dic73C2×D412yes—
56C4×Dic73—41no—
57C4×Dic73—42no—
58C4×Dic73—41no—
59C24×C73C22⋊C411yescharacteristic
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—44no—
64C22×C4×C73—42no—
65C22×C4×C73C22⋊C411yescharacteristic
66C22×C4×C73C2×C812yes—
67C22×C4×C73M1612yes—
68C22×C4×C73C22×C411yescharacteristic
69C4⋊C4×C73—42no—
70C4⋊C4×C73—42no—
71C23×C73—22no—
72C23×C73—22no—
73C23×C73C22⋊C812yes—
74C23×C73#9912yes—
75C23×C73#10012yes—
76C23×C73C2×C22⋊C411yescharacteristic
77C2×C4×C73—24no—
78C2×C4×C73—24no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—42no—
84C2×C4×C73—44no—
85C2×C4×C73—84no—
86C2×C4×C73C22⋊C812yes—
87C2×C4×C73C22×C811yescharacteristic
88C2×C4×C73C2×M1611yescharacteristic
89C2×Dic73—22no—
90C2×Dic73—22no—
91C2×Dic73—24no—
92C2×Dic73—22no—
93C2×Dic73—22no—
94C2×Dic73—22no—
95C2×Dic73—48no—
96C2×Dic73—48no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73#13714yes—
100Dic146—48no—
101Dic146—44no—
102Dic146—44no—
103#10596833—84no—
104C22×C73—22no—
105C22×C73—24no—
106C22×C73—24no—
107C22×C73—22no—
108C22×C73#32314yes—
109C22×C73C2×C22⋊C811yescharacteristic
110C22×C73C2×#9911yescharacteristic
111C22×C73C2×#10011yescharacteristic
112C22×C73#41912yes—
113C22×C73#42112yes—
114C4×C73—22no—
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—82no—
118Dic73—24no—
119Dic73—44no—
120Dic73—42no—
121#7595—731nomaximal, Sylow
122C2×C73—24no—
123C2×C73C2×#32311yescharacteristic
124C2×C73#147912yes—
125C2×C73#151112yes—
126C2×C73#153311yescharacteristic
127C2×C73#155811yescharacteristic
128C2×#391—731no—
129#1817—732no—
130#1272—734no—
131C73#759511yescharacteristic, complemented, Sylow
132C23×Q8—731no—
133C22×C4⋊C4#6440811yescharacteristic
134C22×C42—731no—
135C2×C22⋊C8—1462no—
136#391—732no—
137#391—732no—
138#391—1462no—
139C2×#95—1462no—
140#336—734no—
141#336—1462no—
142C22×Q8—2922no—
143C22×Q8—2922no—
144C22×Q8—734no—
145C22×Q8—732no—
146C22×Q8—1461no—
147C22×Q8—1462no—
148C23×C4—21no—
149C23×C4—732no—
150C23×C4—1461no—
151C23×C4C2×#6440811yescharacteristic
152C22×C8—1464no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C2×#6440812yes—
156C2×C42—2922no—
157C2×C42—732no—
158C2×C42—731no—
159C2×C42—731no—
160C22⋊C8—2924no—
161#95—1464no—
162#95—1464no—
163C24C22⋊#6440811yescharacteristic
164C2×Q8—2928no—
165C2×Q8—2922no—
166C2×Q8—2922no—
167C2×Q8—2928no—
168C2×Q8—2922no—
169C2×Q8—2922no—
170C2×Q8—734no—
171C2×Q8—1462no—
172C2×Q8—1464no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—2922no—
179C22×C4—2922no—
180C22×C4—2922no—
181C22×C4—732no—
182C22×C4—732no—
183C22×C4—1464no—
184C22×C4—1464no—
185C22×C4—1464no—
186C22×C4—1462no—
187C22×C4C22⋊#6440811yescharacteristic
188C22×C4#1192403612yes—
189C22×C4#1192403812yes—
190C22×C4C22×#6440811yescharacteristic
191C2×C8—2924no—
192C2×C8—2924no—
193C2×C8—2924no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C42—2922no—
197C42—2921no—
198C42—2921no—
199C23—22no—
200C23—22no—
201C23#1455664812yes—
202C23#1455665012yes—
203C23#1455665112yes—
204C23C2×C22⋊#6440811yescharacteristic
205Q8—2924no—
206Q8—2924no—
207Q8—2928no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—84no—
217C2×C4—2928no—
218C2×C4—2922no—
219C2×C4—2922no—
220C2×C4—2928no—
221C2×C4—734no—
222C2×C4—1464no—
223C2×C4—1462no—
224C2×C4—1462no—
225C2×C4—1462no—
226C2×C4—1462no—
227C2×C4—1462no—
228C2×C4#1455664812yes—
229C2×C4C2×#1192403611yescharacteristic
230C2×C4C2×#1192403811yescharacteristic
231C8—5844no—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22#1798943114yes—
237C22C2×#1455664811yescharacteristic
238C22C2×#1455665011yescharacteristic
239C22C2×#1455665111yescharacteristic
240C22#1798958012yes—
241C22#1798958512yes—
242C4—21no—
243C4—21no—
244C4—22no—
245C4—82no—
246C4—2922no—
247C4—2924no—
248C4—1464no—
249C2—24no—
250C2C2×#1798943111yescharacteristic
251C2#2262523512yes—
252C2#2262533912yes—
253C2#2262541011yescharacteristic
254C2#2262546411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#7595
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624752C2semidirect
20. C22×Dic146C8semidirect
131. C73#7595semidirect

Maximal subgroups

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

subgroupindex
#226258212
#226247522
C2×#179884922
#759573