Finite Group Databasebeta

← #34009877

Subgroups — #34009877

Order 18688.

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

normal subgroups only

This group has 562 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
1C2×#17989430C212yesmaximal
2#22624875C214yesmaximal, complemented
3#22612545C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5#17989430—22no—
6#17989430C2214yes—
7C22×C22⋊Dic73C411yescharacteristic
8C22×C22⋊Dic73C2211yescharacteristic
9#17988462—22no—
10#17988462C412yes—
11#17988462C2212yes—
12#17988447—22no—
13C22×C22⋊C4×C73C411yescharacteristic
14C22×#10596833—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73C2×C412yes—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—22no—
20C2×C4⋊Dic73—22no—
21C2×C22⋊Dic73—21no—
22C2×C22⋊Dic73—21no—
23C2×C22⋊Dic73—42no—
24C2×C22⋊Dic73—42no—
25C2×C22⋊Dic73C2×C411yescharacteristic
26C2×C22⋊Dic73D412yes—
27C2×C22⋊Dic73C2311yescharacteristic
28#14556498—22no—
29#14556498—22no—
30C25×C73C2×C411yescharacteristic
31C23×C4×C73—21no—
32C2×C22⋊C4×C73—21no—
33C2×C22⋊C4×C73—22no—
34C2×C22⋊C4×C73C814yescomplemented
35C2×C22⋊C4×C73C2×C412yes—
36C2×#10596833—44no—
37C2×#10596833—84no—
38C22×Dic73—22no—
39C22×Dic73—22no—
40C22×Dic73—22no—
41C22×Dic73—42no—
42C22×Dic73—82no—
43C22×Dic73—82no—
44C22×Dic73C22⋊C412yes—
45C22×Dic73C2×D412yes—
46C4⋊Dic73—82no—
47C4⋊Dic73—82no—
48C22⋊Dic73—41no—
49C22⋊Dic73—42no—
50C22⋊Dic73—41no—
51C22⋊Dic73—42no—
52C22⋊Dic73—42no—
53C24×C73—22no—
54C24×C73—22no—
55C24×C73—82no—
56C24×C73C22⋊C411yescharacteristic
57C24×C73C22⋊C411yescharacteristic
58C24×C73C2×C812yes—
59C24×C73M1612yes—
60C24×C73C22×C411yescharacteristic
61C22×C4×C73—22no—
62C22×C4×C73—24no—
63C22×C4×C73—82no—
64C22⋊C4×C73—24no—
65C22⋊C4×C73—22no—
66C22⋊C4×C73C2×C814yescomplemented
67C23×C73—21no—
68C23×C73—21no—
69C23×C73—21no—
70C23×C73—21no—
71C23×C73—24no—
72C23×C73—24no—
73C23×C73—42no—
74C23×C73—42no—
75C23×C73—42no—
76C23×C73—82no—
77C23×C73—82no—
78C23×C73—82no—
79C23×C73—82no—
80C23×C73—84no—
81C23×C73C22⋊C812yes—
82C23×C73C22⋊C812yes—
83C23×C73#9912yes—
84C23×C73#10112yes—
85C23×C73C2×C22⋊C411yescharacteristic
86C23×C73C22×C811yescharacteristic
87C23×C73C2×M1611yescharacteristic
88C2×C4×C73—24no—
89C2×C4×C73—42no—
90C2×C4×C73—42no—
91C2×C4×C73—84no—
92C2×Dic73—22no—
93C2×Dic73—22no—
94C2×Dic73—44no—
95C2×Dic73—42no—
96C2×Dic73—42no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—82no—
101C2×Dic73—84no—
102#10596833—84no—
103C22×C73—22no—
104C22×C73—22no—
105C22×C73—22no—
106C22×C73—22no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73—42no—
110C22×C73—42no—
111C22×C73—42no—
112C22×C73—41no—
113C22×C73—41no—
114C22×C73—42no—
115C22×C73—42no—
116C22×C73—42no—
117C22×C73—42no—
118C22×C73—84no—
119C22×C73—84no—
120C22×C73#32414yes—
121C22×C73C2×C22⋊C811yescharacteristic
122C22×C73C2×#9911yescharacteristic
123C22×C73C2×#10111yescharacteristic
124C4×C73—44no—
125Dic73—42no—
126Dic73—42no—
127Dic73—82no—
128#11453—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73—24no—
132C2×C73—42no—
133C2×C73—41no—
134C2×C73—41no—
135C2×C73C2×#32411yescharacteristic
136C2×C73#212112yes—
137#2315—731no—
138C2×#323—732no—
139#1326—734no—
140C73#1145311yescharacteristic, complemented, Sylow
141C22×C22⋊C4—731no—
142C22×C22⋊C4—731no—
143C22×C22⋊C4#6440811yescharacteristic
144C2×C22⋊C8—1462no—
145#385—732no—
146#385—732no—
147#385—1462no—
148#380—1462no—
149#323—734no—
150#323—1462no—
151C25C2×#6440811yescharacteristic
152C23×C4—21no—
153C23×C4—732no—
154C23×C4—732no—
155C22×C8—2922no—
156C2×C4⋊C4—1462no—
157C2×C4⋊C4—1462no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—2922no—
161C2×C22⋊C4—2922no—
162C2×C22⋊C4—732no—
163C2×C22⋊C4—731no—
164C2×C22⋊C4—731no—
165C2×C22⋊C4—1461no—
166C2×C22⋊C4—1461no—
167C2×C22⋊C4#1059683314yescomplemented
168C2×C22⋊C4C2×#6440812yes—
169C22⋊C8—2922no—
170C22⋊C8—1464no—
171#95—1462no—
172#95—1462no—
173C24—22no—
174C24—22no—
175C24—82no—
176C24C22⋊#6440811yescharacteristic
177C24C22⋊#6440811yescharacteristic
178C24C2×#1059683312yes—
179C24#1192403712yes—
180C24C22×#6440811yescharacteristic
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—82no—
184C22×C4—2922no—
185C22×C4—5842no—
186C22×C4—5842no—
187C22×C4—732no—
188C22×C4—732no—
189C22×C4—1462no—
190C22×C4—1462no—
191C22×C4—1462no—
192C2×C8—2924no—
193C2×C8—5844no—
194C4⋊C4—5842no—
195C4⋊C4—5842no—
196C22⋊C4—22no—
197C22⋊C4—24no—
198C22⋊C4—2922no—
199C22⋊C4—2922no—
200C22⋊C4—2921no—
201C22⋊C4—2922no—
202C22⋊C4—2921no—
203C22⋊C4C2×#1059683314yescomplemented
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23C22⋊#1059683312yes—
219C23C22⋊#1059683312yes—
220C23#1455665012yes—
221C23#1455665212yes—
222C23C2×C22⋊#6440811yescharacteristic
223C23C22×#1059683311yescharacteristic
224C23C2×#1192403711yescharacteristic
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—2924no—
230C2×C4—2922no—
231C2×C4—2922no—
232C2×C4—2922no—
233C2×C4—2922no—
234C2×C4—2922no—
235C2×C4—5842no—
236C2×C4—5844no—
237C2×C4—1462no—
238C2×C4—1462no—
239C8—5844no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22#1798943214yes—
258C22C2×C22⋊#1059683311yescharacteristic
259C22C2×#1455665011yescharacteristic
260C22C2×#1455665211yescharacteristic
261C4—44no—
262C4—2922no—
263C4—2922no—
264C4—5842no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—41no—
269C2—41no—
270C2—42no—
271C2C2×#1798943211yescharacteristic
272C2#2262618812yes—
subgroupG/H
centerC22C2×#14556650
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C22⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC25×C73C2×C4
cosocleC2×C22⋊Dic73C23
hypercenterC22×C22⋊C4#64408
hypocenterC73#11453
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624875C2semidirect
34. C2×C22⋊C4×C73C8semidirect
66. C22⋊C4×C73C2×C8semidirect
140. C73#11453semidirect
167. C2×C22⋊C4#10596833semidirect
203. C22⋊C4C2×#10596833semidirect

Maximal subgroups

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

subgroupindex
C2×#179894302
#226248752
#226125452
#1145373