Finite Group Databasebeta

← #33095436

Subgroups — #33095436

Order 18688.

1130 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 235 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×#17989213C211yesmaximal, characteristic
2#22612527C212yesmaximal
3#22610914C218yesmaximal, complemented
4#22610005C211yesmaximal, characteristic
5#22610003C212yesmaximal
6C73×#1819C211yesmaximal, characteristic
7#17989213—24no—
8#17989213—24no—
9C2×#14556570C414yescomplemented
10C2×#14556568C414yescomplemented
11C2×#14556568C2214yes—
12C22×C22⋊Dic73C2211yescharacteristic
13C22×C22⋊Dic73C2211yescharacteristic
14#17988481—24no—
15#17988475—24no—
16#17988445C2214yes—
17#17988443C2212yes—
18C22⋊C4×Dic73C2214yes—
19#17988354C2214yes—
20#17988353C2218yes—
21C22×C22⋊C4×C73C2211yescharacteristic
22#379×C73C2212yes—
23#342×C73C2214yes—
24C22×Dic146C2×C414yes—
25C23×Dic73C2×C412yes—
26C23×Dic73C2312yes—
27#14556570—28no—
28#14556570—28no—
29#14556568—28no—
30#14556568—28no—
31#14556568—28no—
32#14556568—28no—
33C2×C4⋊Dic73C2×C414yes—
34C2×#11924007—24no—
35C2×#11924007C2×C418yes—
36C2×C22⋊Dic73—22no—
37C2×C22⋊Dic73—22no—
38C2×C22⋊Dic73—22no—
39C2×C22⋊Dic73—22no—
40C2×C22⋊Dic73C2×C412yes—
41C2×C22⋊Dic73C2×C412yes—
42C2×C22⋊Dic73C2×C412yes—
43C2×C22⋊Dic73D414yes—
44C2×C22⋊Dic73D414yes—
45C2×C22⋊Dic73C2314yes—
46C2×C22⋊Dic73C2312yes—
47C2×C4×Dic73—24no—
48#14556498—24no—
49#14556498—24no—
50C25×C73C2311yescharacteristic
51C23×C4×C73C2×C412yes—
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73—22no—
56C2×C22⋊C4×C73C2×C412yes—
57C2×C22⋊C4×C73C2314yes—
58C2×C22⋊C4×C73C2312yes—
59C2×Dic146—28no—
60C2×Dic146—84no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—24no—
64C22×Dic73—82no—
65C22×Dic73—82no—
66C22×Dic73C22×C414yes—
67C22×Dic73C22×C414yes—
68C22×Dic73C2×D414yes—
69C22×Dic73C2×D414yes—
70C4⋊Dic73—28no—
71#11924007—216no—
72#11924007—84no—
73C22⋊Dic73—24no—
74C22⋊Dic73—24no—
75C22⋊Dic73—28no—
76C22⋊Dic73—82no—
77C22⋊Dic73—82no—
78C22⋊Dic73C22⋊C414yes—
79C22⋊Dic73C22⋊C414yes—
80C22⋊Dic73C22⋊C418yes—
81C4×Dic73—84no—
82C24×C73—24no—
83C24×C73—82no—
84C24×C73C22×C411yescharacteristic
85C24×C73C22×C411yescharacteristic
86C24×C73C2×D412yes—
87C24×C73C2×D412yes—
88C24×C73C2411yescharacteristic
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—24no—
92C22×C4×C73—82no—
93C22×C4×C73C22×C414yes—
94C22⋊C4×C73—22no—
95C22⋊C4×C73—22no—
96C22⋊C4×C73—24no—
97C22⋊C4×C73—28no—
98C22⋊C4×C73—82no—
99C22⋊C4×C73—82no—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—24no—
103C23×C73—24no—
104C23×C73—42no—
105C23×C73—42no—
106C23×C73—42no—
107C23×C73—42no—
108C23×C73—42no—
109C23×C73—42no—
110C23×C73—42no—
111C23×C73—88no—
112C23×C73C2×C22⋊C411yescharacteristic
113C23×C73C2×C22⋊C411yescharacteristic
114C23×C73C2×C22⋊C412yes—
115C23×C73C23×C411yescharacteristic
116C23×C73C22×D412yes—
117C23×C732−1+414yes—
118C2×C4×C73—24no—
119C2×C4×C73—44no—
120C2×C4×C73—44no—
121C2×C4×C73—44no—
122C2×C4×C73—44no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×Dic73—24no—
126C2×Dic73—44no—
127C2×Dic73—44no—
128C2×Dic73—44no—
129C2×Dic73—44no—
130C2×Dic73—44no—
131C2×Dic73—44no—
132C2×Dic73—44no—
133C2×Dic73—82no—
134C2×Dic73—82no—
135C2×Dic73C2×C22⋊C418yes—
136Dic146—88no—
137C22×C73—22no—
138C22×C73—22no—
139C22×C73—22no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—22no—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—22no—
147C22×C73—44no—
148C22×C73—44no—
149C22×C73—44no—
150C22×C73—84no—
151C22×C73—84no—
152C22×C73C22×C22⋊C411yescharacteristic
153C22×C73#51912yes—
154C22×C73#53614yes—
155C4×C73—44no—
156C4×C73—84no—
157Dic73—28no—
158Dic73—44no—
159Dic73—84no—
160#21827—731nomaximal, Sylow
161C2×C73—24no—
162C2×C73—22no—
163C2×C73—41no—
164C2×C73—41no—
165C2×C73—42no—
166C2×C73#231011yescharacteristic
167C2×C73#288312yes—
168C2×#543—731no—
169#2312—732no—
170#1996—738no—
171#1819—732no—
172#1819—731no—
173#1819D7311yescharacteristic
174C73#2182711yescharacteristic, complemented, Sylow
175#543—1464no—
176#543—1464no—
177C2×C22⋊Q8—734no—
178C2×C22⋊Q8—734no—
179C2×C22⋊Q8—734no—
180C22×C22⋊C4—731no—
181C22×C22⋊C4—731no—
182C22×C22⋊C4C2×D7311yescharacteristic
183#388—1464no—
184#388—1464no—
185#380—734no—
186#379—732no—
187#379C2×D7312yes—
188C4×C22⋊C4—734no—
189#342—738no—
190#342—734no—
191#342C2×D7314yes—
192C25C22×D7311yescharacteristic
193C22×Q8—734no—
194C23×C4—732no—
195C23×C4—732no—
196C23×C4C4×D7312yes—
197C22⋊Q8—1468no—
198C22⋊Q8—1468no—
199C22⋊Q8—1468no—
200C22⋊Q8—1468no—
201C22⋊Q8—1468no—
202C22⋊Q8—1468no—
203C2×C4⋊C4—738no—
204C2×C4⋊C4—734no—
205C2×C4⋊C4—1464no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—732no—
211C2×C22⋊C4—732no—
212C2×C22⋊C4—734no—
213C2×C22⋊C4—734no—
214C2×C22⋊C4—732no—
215C2×C22⋊C4—732no—
216C2×C22⋊C4—734no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4—1462no—
219C2×C22⋊C4—1462no—
220C2×C22⋊C4—1462no—
221C2×C22⋊C4C4×D7312yes—
222C2×C22⋊C4C22×D7314yes—
223C2×C22⋊C4C22×D7312yes—
224C2×C42—1464no—
225#95—1464no—
226#95—1464no—
227C24—24no—
228C24—82no—
229C24C2×C4×D7311yescharacteristic
230C24C2×C4×D7311yescharacteristic
231C24D4×D7312yes—
232C24D4×D7312yes—
233C24C23×D7311yescharacteristic
234C2×Q8—5844no—
235C2×Q8—1468no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—82no—
240C22×C4—5842no—
241C22×C4—5842no—
242C22×C4—734no—
243C22×C4—734no—
244C22×C4—734no—
245C22×C4—734no—
246C22×C4—1464no—
247C22×C4—1462no—
248C22×C4—1462no—
249C22×C4C2×C4×D7314yes—
250C4⋊C4—5844no—
251C4⋊C4—14616no—
252C4⋊C4—1468no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—5842no—
260C22⋊C4—5842no—
261C22⋊C4—734no—
262C22⋊C4—734no—
263C22⋊C4—738no—
264C22⋊C4—1468no—
265C22⋊C4—1464no—
266C22⋊C4—1464no—
267C42—5844no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—88no—
280C23C22⋊C4×D7311yescharacteristic
281C23C22⋊C4×D7311yescharacteristic
282C23C22⋊C4×D7312yes—
283C23C22×C4×D7311yescharacteristic
284C23C2×D4×D7312yes—
285C23#1455663814yes—
286Q8—5848no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—2924no—
295C2×C4—2924no—
296C2×C4—2924no—
297C2×C4—2924no—
298C2×C4—2924no—
299C2×C4—2924no—
300C2×C4—2924no—
301C2×C4—5842no—
302C2×C4—5842no—
303C2×C4—738no—
304C2×C4—1464no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C2×C22⋊C4×D7311yescharacteristic
321C22#1798906412yes—
322C22#1798915914yes—
323C4—44no—
324C4—84no—
325C4—2924no—
326C4—5844no—
327C4—1468no—
328C2—24no—
329C2—22no—
330C2—42no—
331C2—41no—
332C2—41no—
333C2#2261251611yescharacteristic
334C2#2261838912yes—
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#1819C2
socleC22×C73C22×C22⋊C4
prime radicalC25×C73C23
cosocleC24×C73C24
hypercenter#1819D73
hypocenterC73#21827
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #22610914C2semidirect
9. C2×#14556570C4semidirect
10. C2×#14556568C4semidirect
174. C73#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#179892132
#226125272
#226109142
#226100052
#226100032
C73×#18192
#2182773