Finite Group Databasebeta

← #34005262

Subgroups — #34005262

Order 18688.

514 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 112 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#22625787C212yesmaximal
2#22624809C212yesmaximal, complemented
3#22624809C212yesmaximal, complemented
4#22612334C211yesmaximal, characteristic
5C2×#14556658C2214yes—
6C2×C22⋊#10596833—22no—
7#17989453—22no—
8C2×#14556550C411yescharacteristic
9C22×C4×Dic73C2211yescharacteristic
10#17988456C2211yescharacteristic
11#17988456C2211yescharacteristic
12#17988452C411yescharacteristic
13#17988452C411yescharacteristic
14#17988445—22no—
15C4×#11924007—22no—
16C2×#117×C73C411yescharacteristic
17C22×#10596833—24no—
18#14556658—28no—
19C22⋊#10596833—44no—
20C23×Dic73D412yes—
21#14556550—42no—
22C2×C4⋊Dic73C2×C411yescharacteristic
23C2×C4⋊Dic73C2×C411yescharacteristic
24C2×#11924007—22no—
25C2×#11924007—22no—
26C2×C22⋊Dic73—21no—
27C2×C4×Dic73—24no—
28C2×C4×Dic73—21no—
29C2×C4×Dic73C2×C411yescharacteristic
30C2×C4×Dic73D412yes—
31C2×C4×Dic73C2311yescharacteristic
32#14556498—22no—
33#14556498—22no—
34C23×C4×C73C2×C411yescharacteristic
35#117×C73—42no—
36C2×C4⋊C4×C73C2×C411yescharacteristic
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C22⋊C4×C73—21no—
39C2×C42×C73—21no—
40C2×#10596833—44no—
41C2×#10596833—44no—
42C2×#10596833—44no—
43C22×Dic73—24no—
44C22×Dic73—24no—
45C22×Dic73—22no—
46C22×Dic73—22no—
47C22×Dic73—21no—
48C22×Dic73—21no—
49C22×Dic73C22⋊C412yes—
50C22×Dic73C2×D412yes—
51C4⋊Dic73—41no—
52C4⋊Dic73—41no—
53#11924007—44no—
54#11924007—42no—
55#11924007—42no—
56C22⋊Dic73—42no—
57C4×Dic73—22no—
58C4×Dic73—24no—
59C4×Dic73—42no—
60C4×Dic73#4114yes—
61C24×C73C22⋊C411yescharacteristic
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—22no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22×C411yescharacteristic
69C4⋊C4×C73—41no—
70C4⋊C4×C73—41no—
71C22⋊C4×C73—42no—
72C42×C73—42no—
73C23×C73—21no—
74C23×C73—21no—
75C23×C73—22no—
76C23×C73—22no—
77C23×C73#10112yes—
78C23×C73C2×C22⋊C411yescharacteristic
79C2×C4×C73—28no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—41no—
85C2×C4×C73—41no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73#10012yes—
89C2×C4×C73#11712yes—
90C2×Dic73—22no—
91C2×Dic73—22no—
92C2×Dic73—22no—
93C2×Dic73—44no—
94C2×Dic73—44no—
95C2×Dic73—41no—
96C2×Dic73—41no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73#12314yes—
100C2×Dic73#13514yes—
101C2×Dic73AGL(1,ℤ/8)12yes—
102C2×Dic73#13712yes—
103#10596833—84no—
104C22×C73—24no—
105C22×C73—22no—
106C22×C73—41no—
107C22×C73—41no—
108C22×C73—42no—
109C22×C73#38111yescharacteristic
110C22×C73C2×#10011yescharacteristic
111C22×C73C2×#10111yescharacteristic
112C22×C73#41612yes—
113C22×C73#41712yes—
114C4×C73—22no—
115C4×C73—22no—
116C4×C73—81no—
117C4×C73—81no—
118Dic73—42no—
119Dic73—81no—
120Dic73—81no—
121Dic73#48214yes—
122Dic73#48514yes—
123#8728—731nomaximal, Sylow
124C2×C73—42no—
125C2×C73#154311yescharacteristic
126C2×C73#155212yes—
127C2×C73#156011yescharacteristic
128C2×C73#182111yescharacteristic
129C2×C73#182512yes—
130#2285—731no—
131#1797—732no—
132#1296—732no—
133#1296—732no—
134C73#872811yescharacteristic, complemented, Sylow
135C2×#117—731no—
136C2×#117#6440811yescharacteristic
137C22×C42—731no—
138C2×C4⋊C8—734no—
139C2×C22⋊C8—1462no—
140#384—731no—
141#384—731no—
142#382—731no—
143#382—731no—
144#380—1462no—
145C4×C4⋊C4—1462no—
146#336—1462no—
147C23×C4—732no—
148C23×C4C2×#6440811yescharacteristic
149C22×C8—1464no—
150#117—42no—
151#117—2922no—
152C2×C4⋊C4—731no—
153C2×C4⋊C4—731no—
154C2×C4⋊C4—1462no—
155C2×C4⋊C4—1462no—
156C2×C4⋊C4C2×#6440811yescharacteristic
157C2×C4⋊C4C2×#6440811yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—1461no—
160C2×C42—21no—
161C2×C42—731no—
162C2×C42—732no—
163C2×C42—731no—
164C2×C42—1461no—
165C2×C42—1464no—
166C4⋊C8—1468no—
167C22⋊C8—2924no—
168#95—1462no—
169#95—1462no—
170C24C22⋊#6440811yescharacteristic
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—732no—
177C22×C4—732no—
178C22×C4—1461no—
179C22×C4—1461no—
180C22×C4—1462no—
181C22×C4—1464no—
182C22×C4—1464no—
183C22×C4—1462no—
184C22×C4C22⋊#6440811yescharacteristic
185C22×C4C22×#6440811yescharacteristic
186C2×C8—2924no—
187C2×C8—2924no—
188C2×C8—2924no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—2921no—
192C4⋊C4—2921no—
193C4⋊C4—2924no—
194C4⋊C4—2922no—
195C4⋊C4—2922no—
196C22⋊C4—42no—
197C22⋊C4—2922no—
198C42—42no—
199C42—2922no—
200C42—734no—
201C42—1464no—
202C42—1462no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23#1455665212yes—
208C23C2×C22⋊#6440811yescharacteristic
209C2×C4—28no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—2924no—
219C2×C4—2924no—
220C2×C4—2921no—
221C2×C4—2921no—
222C2×C4—2922no—
223C2×C4—2922no—
224C2×C4—734no—
225C2×C4—734no—
226C2×C4—732no—
227C2×C4—732no—
228C2×C4—1462no—
229C2×C4—1462no—
230C2×C4—1462no—
231C2×C4#1455665112yes—
232C2×C4#1455667012yes—
233C8—5844no—
234C22—22no—
235C22—24no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22#1798953011yescharacteristic
240C22C2×#1455665111yescharacteristic
241C22C2×#1455665211yescharacteristic
242C22#1798957712yes—
243C22#1798957812yes—
244C4—22no—
245C4—22no—
246C4—81no—
247C4—81no—
248C4—2922no—
249C4—5841no—
250C4—5841no—
251C4—734no—
252C4—734no—
253C2—42no—
254C2#2262543411yescharacteristic
255C2#2262545212yes—
256C2#2262546811yescharacteristic
257C2#2262583011yescharacteristic
258C2#2262583612yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#8728
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624809C2semidirect
3. #22624809C2semidirect
134. C73#8728semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226248092
#226123342
#872873