Finite Group Databasebeta

← #34004124

Subgroups — #34004124

Order 18688.

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

normal subgroups only

This group has 874 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#22625822C212yesmaximal, complemented
2C2×#17989433C212yesmaximal, complemented
3C2×#17989431C212yesmaximal
4C22×#14556586C211yesmaximal, characteristic
5C2×C22⋊#10596833C2214yes—
6#17989453—24no—
7#17989433—24no—
8#17989431—24no—
9C2×#14556586—24no—
10C2×#14556586—24no—
11C2×#14556586—24no—
12C22×C4⋊Dic73C411yescharacteristic
13C22×#11924007C2211yescharacteristic
14C22×#11924005C411yescharacteristic
15C22×#11924005C411yescharacteristic
16C22×#11924005C2211yescharacteristic
17C22×C4×Dic73C2211yescharacteristic
18C22×C4⋊C4×C73C411yescharacteristic
19C22×#10596833—24no—
20C22⋊#10596833—24no—
21C22⋊#10596833—24no—
22C24×D73C2×C411yescharacteristic
23C23×Dic73C2×C411yescharacteristic
24C23×Dic73C2×C411yescharacteristic
25C23×Dic73C2311yescharacteristic
26#14556586—232no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73—22no—
30C2×#11924007—22no—
31C2×#11924007—22no—
32C2×#11924007D414yes—
33C2×#11924005—22no—
34C2×#11924005—22no—
35C2×#11924005—22no—
36C2×#11924005—22no—
37C2×#11924005—22no—
38C2×#11924005—22no—
39C2×#11924005—22no—
40C2×#11924005—22no—
41C2×#11924005D414yes—
42C2×C4×Dic73—22no—
43C2×C4×Dic73—22no—
44C2×C4×Dic73D414yes—
45C23×C4×C73C2×C411yescharacteristic
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73—22no—
49C2×C4⋊C4×C73—22no—
50C2×C4⋊C4×C73—22no—
51C2×#10596833—48no—
52C2×#10596833—44no—
53C23×D73—21no—
54C23×D73—21no—
55C23×D73—24no—
56C23×D73—42no—
57C23×D73—42no—
58C23×D73C22⋊C412yes—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—24no—
64C22×Dic73—24no—
65C22×Dic73—42no—
66C22×Dic73C22⋊C412yes—
67C22×Dic73C22⋊C412yes—
68C22×Dic73C2×D412yes—
69C22×Dic73C2×D412yes—
70C22×Dic73C2×D412yes—
71C4⋊Dic73—28no—
72#11924007—28no—
73#11924005—28no—
74#11924005—28no—
75#11924005—28no—
76C4×Dic73—28no—
77C24×C73C22×C411yescharacteristic
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—42no—
85C22×C4×C73—42no—
86C22×C4×C73—42no—
87C22×C4×C73C22⋊C412yes—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22⋊C412yes—
90C4⋊C4×C73—28no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73C2×C22⋊C411yescharacteristic
94C23×C73C2×C22⋊C411yescharacteristic
95C23×C73C2×C22⋊C411yescharacteristic
96C23×C73#13114yes—
97C22×D73—22no—
98C22×D73—24no—
99C22×D73—42no—
100C22×D73—42no—
101C22×D73—42no—
102C22×D73—42no—
103C22×D73—42no—
104C22×D73—42no—
105C22×D73—42no—
106C22×D73—42no—
107C22×D73—88no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—24no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×Dic73—24no—
121C2×Dic73—22no—
122C2×Dic73—22no—
123C2×Dic73—24no—
124C2×Dic73—24no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73#12014yes—
132#10596833—84no—
133C22×C73—28no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73#37911yescharacteristic
139C22×C73#40812yes—
140C22×C73#40812yes—
141C22×C73#40812yes—
142C22×C73#41012yes—
143C22×C73#41312yes—
144C2×D73—42no—
145C2×D73—42no—
146C2×D73—42no—
147C2×D73—84no—
148C2×D73—84no—
149C2×D73—84no—
150C4×C73—42no—
151C4×C73—42no—
152C4×C73—42no—
153Dic73—24no—
154Dic73—42no—
155Dic73—42no—
156#8309—731nomaximal, Sylow
157C2×C73—22no—
158C2×C73—22no—
159C2×C73#147314yes—
160C2×C73#178011yescharacteristic
161C2×C73#178211yescharacteristic
162C2×C73#178511yescharacteristic
163D73—82no—
164C22×#126—731no—
165#1817—732no—
166C2×#324—732no—
167C2×#323—732no—
168C73#830911yescharacteristic, complemented, Sylow
169C2×#126—1464no—
170C2×#126—1464no—
171C2×#126—1464no—
172C22×C4⋊C4—731no—
173C22×C4⋊C4—731no—
174C22×C4⋊C4#6440811yescharacteristic
175C22×C22⋊C4—731no—
176C22×C22⋊C4—731no—
177C22×C22⋊C4—731no—
178C22×C42—731no—
179C2×C22⋊C8—734no—
180#336—1464no—
181#324—1464no—
182#323—1464no—
183C25—731no—
184C23×C4—731no—
185C23×C4—731no—
186C23×C4—731no—
187C23×C4C2×#6440811yescharacteristic
188C23×C4C2×#6440811yescharacteristic
189C23×C4C2×#6440811yescharacteristic
190C22×C8—1464no—
191#126—14632no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—734no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4—1462no—
198C2×C4⋊C4—1462no—
199C2×C4⋊C4—1462no—
200C2×C4⋊C4—1462no—
201C2×C22⋊C4—734no—
202C2×C22⋊C4—1462no—
203C2×C22⋊C4—1462no—
204C2×C22⋊C4—1462no—
205C2×C22⋊C4—1462no—
206C2×C22⋊C4—1462no—
207C2×C22⋊C4—1462no—
208C2×C22⋊C4—1462no—
209C2×C22⋊C4—1462no—
210C2×C42—734no—
211C2×C42—1462no—
212C2×C42—1462no—
213C22⋊C8—1464no—
214C22⋊C8—1464no—
215C24—2922no—
216C24—2922no—
217C24—732no—
218C24—1461no—
219C24—1461no—
220C24—1464no—
221C24C22×#6440811yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—2922no—
232C22×C4—732no—
233C22×C4—732no—
234C22×C4—732no—
235C22×C4—732no—
236C22×C4—732no—
237C22×C4—1464no—
238C22×C4—1464no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4C22⋊#6440812yes—
244C22×C4C22⋊#6440812yes—
245C22×C4C22⋊#6440812yes—
246C2×C8—2924no—
247C2×C8—2928no—
248C4⋊C4—28no—
249C4⋊C4—1468no—
250C4⋊C4—1468no—
251C22⋊C4—1468no—
252C22⋊C4—1468no—
253C22⋊C4—1468no—
254C42—1468no—
255C23—22no—
256C23—22no—
257C23—2922no—
258C23—2922no—
259C23—2922no—
260C23—2922no—
261C23—2922no—
262C23—2922no—
263C23—2922no—
264C23—2922no—
265C23—5848no—
266C23—1462no—
267C23—1464no—
268C23C2×C22⋊#6440811yescharacteristic
269C23C2×C22⋊#6440811yescharacteristic
270C23C2×C22⋊#6440811yescharacteristic
271C23#1455667914yes—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4—734no—
291C2×C4—1462no—
292C2×C4—1462no—
293C2×C4—1464no—
294C2×C4—1464no—
295C2×C4—1464no—
296C8—5844no—
297C22—28no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—2922no—
303C22—2922no—
304C22—2922no—
305C22—5844no—
306C22—5844no—
307C22—5844no—
308C22#1798952811yescharacteristic
309C22#1798956212yes—
310C22#1798956212yes—
311C22#1798956212yes—
312C22#1798956612yes—
313C22#1798957412yes—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—2922no—
318C4—2922no—
319C4—1464no—
320C2—22no—
321C2—22no—
322C2—5842no—
323C2#2262521414yes—
324C2#2262575611yescharacteristic
325C2#2262576011yescharacteristic
326C2#2262576811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×D73C2×C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8309
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22625822C2semidirect
2. C2×#17989433C2semidirect
168. C73#8309semidirect

Maximal subgroups

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

subgroupindex
#226258222
C2×#179894332
C2×#179894312
C22×#145565862
#830973