Finite Group Databasebeta

← #34008337

Subgroups — #34008337

Order 18688.

1294 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 220 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
1D73⋊#407C212yesmaximal
2C2×#17989456C214yesmaximal, complemented
3C23×#11924011C211yesmaximal, characteristic
4C2×D73⋊M16C2214yes—
5#17989559—22no—
6#17989558—24no—
7#17989557—22no—
8#17989555C418yes—
9#17989456—28no—
10C23×C4×D73C2211yescharacteristic
11C22×#11924011—24no—
12C22×#11924011—28no—
13C22×#11924011C412yes—
14C22×#11924011C2212yes—
15C23×#10596831C411yescharacteristic
16C23×C8×C73C411yescharacteristic
17C2×#11924038—24no—
18D73⋊M16—28no—
19D73⋊M16—28no—
20C2×#11924037—24no—
21C2×#11924036C2×C418yes—
22#14556648—28no—
23C22⋊#10596833—28no—
24C24×D73Q811yescharacteristic
25C23×Dic73D411yescharacteristic
26C22×C4×D73—24no—
27C22×C4×D73—24no—
28C22×C4×D73C2×C414yes—
29C22×C4×D73C2×C411yescharacteristic
30C22×C4×D73D414yes—
31C22×C4×D73D411yescharacteristic
32C22×C4×D73Q811yescharacteristic
33C22×C4×D73C2311yescharacteristic
34C2×#11924011—232no—
35C2×#11924011—24no—
36C2×#11924011—24no—
37C2×#11924011—24no—
38C2×#11924011—24no—
39C2×#11924011—24no—
40C2×#11924011D418yes—
41C22×#10596831—24no—
42C22×#10596831—22no—
43C22×#10596831C2×C412yes—
44C23×C4×C73C2×C411yescharacteristic
45C22×C8×C73—22no—
46C22×C8×C73—24no—
47C22×C8×C73C2×C412yes—
48#11924038—44no—
49#11924038—44no—
50#11924037—48no—
51#11924036—216no—
52C2×#10596833—28no—
53C23×D73—24no—
54C23×D73—44no—
55C23×D73C4⋊C414yes—
56C23×D73C4⋊C411yescharacteristic
57C23×D73C2×Q811yescharacteristic
58C22×Dic73—24no—
59C22×Dic73C22⋊C414yes—
60C22×Dic73C4⋊C411yescharacteristic
61C22×Dic73C2×D411yescharacteristic
62C2×C4×D73—24no—
63C2×C4×D73—22no—
64C2×C4×D73—22no—
65C2×C4×D73—24no—
66C2×C4×D73—216no—
67C2×C4×D73—28no—
68C2×C4×D73—28no—
69C2×C4×D73C4212yes—
70C2×C4×D73C22⋊C412yes—
71C2×C4×D73C22⋊C414yes—
72C2×C4×D73C4⋊C414yes—
73C2×C4×D73C22×C412yes—
74C2×C4×D73C2×D414yes—
75C2×C4×D73C2×D412yes—
76C2×C4×D73#4114yes—
77#11924011—216no—
78#11924011—216no—
79C2×#10596831—28no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831C22⋊C414yes—
83C24×C73C4⋊C411yescharacteristic
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73C4212yes—
87C22×C4×C73C22⋊C412yes—
88C22×C4×C73C4⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C2×C8×C73—28no—
91C2×C8×C73—22no—
92C2×C8×C73—22no—
93C2×C8×C73C22⋊C414yes—
94C23×C73—24no—
95C23×C73—22no—
96C23×C73#9512yes—
97C23×C73C2×C4⋊C411yescharacteristic
98C22×D73—28no—
99C22×D73—22no—
100C22×D73—28no—
101C22×D73—44no—
102C22×D73—42no—
103C22×D73—42no—
104C22×D73—416no—
105C22×D73#9512yes—
106C22×D73C2×C4⋊C412yes—
107C22×D73C22⋊Q814yes—
108C2×C4×C73—22no—
109C2×C4×C73—24no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73#9512yes—
113C2×C4×C73C2×C4211yescharacteristic
114C2×C4×C73C2×C22⋊C412yes—
115C2×C4×C73C2×C22⋊C411yescharacteristic
116C2×C4×C73#11712yes—
117C2×Dic73—22no—
118C2×Dic73—28no—
119C2×Dic73#9512yes—
120C2×Dic73C2×C22⋊C412yes—
121C2×Dic73#12114yes—
122C4×D73—28no—
123C4×D73—24no—
124C4×D73—22no—
125C4×D73—22no—
126C4×D73—28no—
127C4×D73C4×D418yes—
128C4×D73#12014yes—
129C4×D73#12314yes—
130#10596833—48no—
131C8×C73—24no—
132#10596831—24no—
133C22×C73—28no—
134C22×C73—22no—
135C22×C73—22no—
136C22×C73—22no—
137C22×C73#33814yescomplemented
138C22×C73C2×#9511yescharacteristic
139C22×C73#38012yes—
140C2×D73—24no—
141C2×D73—22no—
142C2×D73—24no—
143C2×D73—44no—
144C2×D73—416no—
145C2×D73—44no—
146C2×D73#38514yes—
147C4×C73—22no—
148C4×C73C4×C22⋊C412yes—
149C4×C73#37911yescharacteristic
150C4×C73#38111yescharacteristic
151Dic73—22no—
152Dic73#38614yes—
153C22⋊#338—731nomaximal, Sylow
154C2×C73—24no—
155C2×C73—22no—
156C2×C73C22⋊#9511yescharacteristic
157C2×C73C2×#33812yes—
158D73—24no—
159D73—42no—
160C23×M16—731no—
161C2×#407—732no—
162C2×#338—734no—
163C73C22⋊#33811yescharacteristic, complemented, Sylow
164C24×C4—731no—
165C22×M16—734no—
166C22×M16—732no—
167C22×M16—732no—
168C22×M16—1468no—
169C22×M16—1464no—
170C23×C8—731no—
171C23×C8#6440811yescharacteristic
172#407—1462no—
173#407—1464no—
174#407—1462no—
175C2×C22⋊C8—738no—
176#338—1468no—
177C25—731no—
178C23×C4—731no—
179C23×C4—731no—
180C23×C4—734no—
181C23×C4—734no—
182C23×C4—731no—
183C23×C4—731no—
184C23×C4—731no—
185C23×C4—1464no—
186C23×C4—1464no—
187C23×C4C2×#6440811yescharacteristic
188C2×M16—738no—
189C2×M16—1464no—
190C2×M16—1464no—
191C2×M16—1464no—
192C2×M16—1464no—
193C2×M16—14632no—
194C2×M16—1464no—
195C2×M16—1468no—
196C2×M16—1468no—
197C2×M16—1464no—
198C2×M16—1464no—
199C22×C8—22no—
200C22×C8—24no—
201C22×C8—732no—
202C22×C8—738no—
203C22×C8—1464no—
204C22×C8—1462no—
205C22×C8C2×#6440812yes—
206C22⋊C8—1468no—
207C22⋊C8—1468no—
208C24—2924no—
209C24—731no—
210C24—731no—
211C24—734no—
212C24—1464no—
213C24C4⋊#6440811yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—731no—
217C22×C4—732no—
218C22×C4—734no—
219C22×C4—732no—
220C22×C4—734no—
221C22×C4—734no—
222C22×C4—731no—
223C22×C4—734no—
224C22×C4—734no—
225C22×C4—732no—
226C22×C4—732no—
227C22×C4—1468no—
228C22×C4—1464no—
229C22×C4—14616no—
230C22×C4—1468no—
231C22×C4—1464no—
232C22×C4—1464no—
233C22×C4—1462no—
234C22×C4—1462no—
235C22×C4C4×#6440812yes—
236C22×C4C22⋊#6440812yes—
237C22×C4C4⋊#6440811yescharacteristic
238C22×C4C22×#6440811yescharacteristic
239M16—2928no—
240M16—2924no—
241M16—2924no—
242M16—14616no—
243M16—14616no—
244C2×C8—28no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—734no—
248C2×C8—1468no—
249C2×C8—14616no—
250C2×C8—1468no—
251C2×C8—1462no—
252C2×C8—1462no—
253C2×C8C22⋊#6440814yes—
254C23—24no—
255C23—22no—
256C23—29216no—
257C23—2924no—
258C23—2922no—
259C23—2922no—
260C23—734no—
261C23—732no—
262C23—732no—
263C23—1468no—
264C23—1468no—
265C23—1462no—
266C23(C2×#64408)⋊C412yes—
267C23C2×C4⋊#6440811yescharacteristic
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—732no—
273C2×C4—734no—
274C2×C4—738no—
275C2×C4—734no—
276C2×C4—734no—
277C2×C4—732no—
278C2×C4—1468no—
279C2×C4—1468no—
280C2×C4—1468no—
281C2×C4—1462no—
282C2×C4—1464no—
283C2×C4—1462no—
284C2×C4—1462no—
285C2×C4(C2×#64408)⋊C412yes—
286C2×C4C2×C4×#6440811yescharacteristic
287C2×C4C2×C22⋊#6440812yes—
288C2×C4C2×C22⋊#6440811yescharacteristic
289C2×C4#1455667012yes—
290C8—24no—
291C8—2928no—
292C8—1464no—
293C22—28no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—2924no—
298C22—29216no—
299C22—2924no—
300C22—734no—
301C22—1464no—
302C22—1462no—
303C22—1464no—
304C22#1798945614yescomplemented
305C22C2×(C2×#64408)⋊C411yescharacteristic
306C22#1798952912yes—
307C4—22no—
308C4—734no—
309C4—1462no—
310C4C4×C22⋊#6440812yes—
311C4#1798952811yescharacteristic
312C4#1798953011yescharacteristic
313C2—24no—
314C2—22no—
315C2—2922no—
316C2—1464no—
317C2#2262510511yescharacteristic
318C2C2×#1798945612yes—
subgroupG/H
centerC2×C4C2×C4×#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC23×C8×C73C4
socleC22×C73C2×#95
prime radicalC24×D73Q8
cosocleC22×C4×D73C23
hypercenterC23×C8#64408
hypocenterC73C22⋊#338
LCSG ▷ C2×C4×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C2×#17989456C2semidirect
137. C22×C73#338semidirect
163. C73C22⋊#338semidirect
304. C22#17989456semidirect

Maximal subgroups

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

subgroupindex
D73⋊#4072
C2×#179894562
C23×#119240112
C22⋊#33873