Finite Group Databasebeta

← #34010063

Subgroups — #34010063

Order 18688.

814 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 88 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#22625179C212yesmaximal, complemented
2#22624859C214yesmaximal, complemented
3C2×#17988490C211yesmaximal, characteristic
4#17989557—22no—
5#17989432—22no—
6#17989432C2214yescomplemented
7C22×C4⋊Dic73C2211yescharacteristic
8C22×C22⋊Dic73—22no—
9C22×C4×Dic73C411yescharacteristic
10#17988490—22no—
11#17988490C412yes—
12#17988490C2212yes—
13C23×D4×C73C411yescharacteristic
14C2×#11924037—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—41no—
20C2×C4⋊Dic73—41no—
21C2×C4⋊Dic73C2×C411yescharacteristic
22C2×C4⋊Dic73D412yes—
23C2×C4⋊Dic73C2311yescharacteristic
24C2×C22⋊Dic73—24no—
25C2×C22⋊Dic73—24no—
26C2×C22⋊Dic73—48no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73—42no—
29C2×C4×Dic73C2×C412yes—
30C25×C73—21no—
31C22×D4×C73—24no—
32C22×D4×C73—21no—
33C22×D4×C73—41no—
34C22×D4×C73—41no—
35C22×D4×C73—42no—
36C22×D4×C73C2×C412yes—
37C23×C4×C73C2×C411yescharacteristic
38#11924037—84no—
39C2×#10596833—44no—
40C22×Dic73—24no—
41C22×Dic73—22no—
42C22×Dic73—22no—
43C22×Dic73—22no—
44C22×Dic73—42no—
45C22×Dic73—42no—
46C22×Dic73—44no—
47C22×Dic73C22⋊C412yes—
48C22×Dic73C2×D412yes—
49C4⋊Dic73—41no—
50C4⋊Dic73—42no—
51C4⋊Dic73—41no—
52C22⋊Dic73—48no—
53C22⋊Dic73—48no—
54C4×Dic73—42no—
55C4×Dic73—42no—
56C24×C73—24no—
57C24×C73—22no—
58C24×C73—42no—
59C24×C73—44no—
60C24×C73—44no—
61C24×C73—41no—
62C24×C73—41no—
63C24×C73C22⋊C411yescharacteristic
64C2×D4×C73—24no—
65C2×D4×C73—22no—
66C2×D4×C73—42no—
67C2×D4×C73—42no—
68C2×D4×C73—48no—
69C2×D4×C73—48no—
70C2×D4×C73—42no—
71C2×D4×C73—42no—
72C2×D4×C73M1614yescomplemented
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22×C411yescharacteristic
79C23×C73—22no—
80C23×C73—22no—
81C23×C73—21no—
82C23×C73—21no—
83C23×C73—24no—
84C23×C73—44no—
85C23×C73—44no—
86C23×C73—416no—
87C23×C73—44no—
88C23×C73—44no—
89C23×C73—42no—
90C23×C73—42no—
91C23×C73—88no—
92C23×C73—82no—
93C23×C73—82no—
94C23×C73—82no—
95C23×C73—82no—
96C23×C73#9911yescharacteristic
97C23×C73#9911yescharacteristic
98C23×C73C2×C22⋊C411yescharacteristic
99D4×C73—44no—
100D4×C73—48no—
101D4×C73—44no—
102C2×C4×C73—24no—
103C2×C4×C73—24no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73#10012yes—
109C2×C4×C73C2×M1612yes—
110C2×Dic73—24no—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—48no—
114C2×Dic73—42no—
115C2×Dic73—44no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—82no—
120#10596833—84no—
121C22×C73—24no—
122C22×C73—24no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73—48no—
130C22×C73—44no—
131C22×C73—84no—
132C22×C73—88no—
133C22×C73—88no—
134C22×C73—82no—
135C22×C73—82no—
136C22×C73#35312yes—
137C22×C73#35512yes—
138C22×C73#40711yescharacteristic
139C22×C73C2×#9911yescharacteristic
140C22×C73C2×#10011yescharacteristic
141C4×C73—22no—
142C4×C73—22no—
143Dic73—44no—
144Dic73—42no—
145Dic73—42no—
146#11505—731nomaximal, Sylow
147C2×C73—21no—
148C2×C73—21no—
149C2×C73—22no—
150C2×C73—22no—
151C2×C73—84no—
152C2×C73#146211yescharacteristic
153C2×C73C2×#35311yescharacteristic
154C2×C73C2×#35511yescharacteristic
155C2×#390—731no—
156#1463—732no—
157#1318—734no—
158C73#1150511yescharacteristic, complemented, Sylow
159C23×D4#6440811yescharacteristic
160C22×C4⋊C4—731no—
161C22×C22⋊C4—1462no—
162C22×C42—731no—
163#407—1462no—
164#390—732no—
165#390—732no—
166#390—1462no—
167#324—734no—
168#324—1462no—
169C25—21no—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—42no—
175C22×D4C2×#6440812yes—
176C23×C4—732no—
177C23×C4—1461no—
178C23×C4C2×#6440811yescharacteristic
179C2×M16—2922no—
180C2×C4⋊C4—2921no—
181C2×C4⋊C4—2921no—
182C2×C4⋊C4—732no—
183C2×C4⋊C4—731no—
184C2×C4⋊C4—731no—
185C2×C22⋊C4—2928no—
186C2×C22⋊C4—1464no—
187C2×C22⋊C4—1464no—
188C2×C42—2922no—
189C2×C42—732no—
190C2×C42—1461no—
191C22⋊C8—2922no—
192C22⋊C8—1464no—
193C24—24no—
194C24—22no—
195C24—44no—
196C24—42no—
197C24—44no—
198C24—41no—
199C24—41no—
200C24C22⋊#6440811yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4#1192403714yescomplemented
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—2922no—
215C22×C4—2922no—
216C22×C4—2924no—
217C22×C4—732no—
218C22×C4—732no—
219C22×C4—1464no—
220C22×C4—1462no—
221C22×C4—1462no—
222C22×C4—1462no—
223C22×C4C22⋊#6440811yescharacteristic
224C22×C4C22×#6440811yescharacteristic
225M16—5844no—
226C2×C8—2924no—
227C4⋊C4—2922no—
228C4⋊C4—2921no—
229C4⋊C4—2921no—
230C22⋊C4—2928no—
231C22⋊C4—2928no—
232C42—2922no—
233C42—2922no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—24no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—416no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—88no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23#1455664911yescharacteristic
252C23#1455664911yescharacteristic
253C23C2×C22⋊#6440811yescharacteristic
254D4—44no—
255D4—44no—
256D4—48no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—2928no—
264C2×C4—2922no—
265C2×C4—2924no—
266C2×C4—2922no—
267C2×C4—2922no—
268C2×C4—2922no—
269C2×C4—5842no—
270C2×C4—1464no—
271C2×C4—1462no—
272C2×C4—1462no—
273C2×C4#1455665112yes—
274C2×C4C2×#1192403712yes—
275C8—5844no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—48no—
285C22—44no—
286C22—84no—
287C22—88no—
288C22—88no—
289C22—82no—
290C22—82no—
291C22#1798949112yes—
292C22#1798949512yes—
293C22#1798955711yescharacteristic
294C22C2×#1455664911yescharacteristic
295C22C2×#1455665111yescharacteristic
296C4—22no—
297C4—22no—
298C4—2924no—
299C4—2922no—
300C4—2922no—
301C2—21no—
302C2—21no—
303C2—22no—
304C2—22no—
305C2—84no—
306C2#2262517611yescharacteristic
307C2C2×#1798949111yescharacteristic
308C2C2×#1798949511yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting, prime radicalC23×D4×C73C4
socleC22×C73C2×#99
cosocleC2×C4⋊Dic73C23
hypercenterC23×D4#64408
hypocenterC73#11505
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #22625179C2semidirect
2. #22624859C2semidirect
6. #17989432C22semidirect
72. C2×D4×C73M16semidirect
158. C73#11505semidirect
209. C2×D4#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251792
#226248592
C2×#179884902
#1150573