Finite Group Databasebeta

← #34009787

Subgroups — #34009787

Order 18688.

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

normal subgroups only

This group has 678 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
1#22625162C212yesmaximal, complemented
2#22624857C212yesmaximal, complemented
3#22624855C212yesmaximal
4#22612314C211yesmaximal, characteristic
5#17989603—22no—
6#17989432—22no—
7#17989430C2214yes—
8C2×#14556550C411yescharacteristic
9C2×#14556550C2211yescharacteristic
10#17988450C411yescharacteristic
11#17988450C2211yescharacteristic
12#17988447—22no—
13#17988443C411yescharacteristic
14#17988443C2211yescharacteristic
15C4×C22⋊Dic73—22no—
16C24×C4×C73C411yescharacteristic
17C4⋊#10596833—42no—
18C22⋊#10596833—24no—
19C4×#10596833—42no—
20#14556550—41no—
21#14556550—41no—
22#14556550—42no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73D412yes—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C2×C22⋊Dic73C2×C411yescharacteristic
28C2×C22⋊Dic73C2×C411yescharacteristic
29C2×C22⋊Dic73C2×C411yescharacteristic
30C2×C22⋊Dic73C2311yescharacteristic
31C2×C4×Dic73—21no—
32C2×C4×Dic73D412yes—
33#14556498—22no—
34#14556498—22no—
35C25×C73C2×C411yescharacteristic
36C23×C4×C73—21no—
37C23×C4×C73—21no—
38C23×C4×C73—22no—
39C23×C4×C73—41no—
40C23×C4×C73—41no—
41C23×C4×C73—42no—
42C23×C4×C73C814yescomplemented
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73C2×C411yescharacteristic
45C2×#10596833—44no—
46C22×Dic73—21no—
47C22×Dic73—21no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—42no—
51C4⋊Dic73—42no—
52C22⋊Dic73—41no—
53C22⋊Dic73—41no—
54C22⋊Dic73—44no—
55C22⋊Dic73—44no—
56C22⋊Dic73—41no—
57C22⋊Dic73—41no—
58C4×Dic73—22no—
59C4×Dic73—22no—
60C4×Dic73—42no—
61C24×C73—21no—
62C24×C73—21no—
63C24×C73—41no—
64C24×C73—41no—
65C24×C73—44no—
66C24×C73C2×C812yes—
67C24×C73C22×C411yescharacteristic
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—24no—
74C22×C4×C73—24no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73—44no—
78C22×C4×C73—44no—
79C22×C4×C73—44no—
80C22×C4×C73—44no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73—42no—
84C22×C4×C73—41no—
85C22×C4×C73—41no—
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C2×C812yes—
89C22×C4×C73C2×C812yes—
90C22×C4×C73M1612yes—
91C23×C73—24no—
92C23×C73—22no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—42no—
96C23×C73—44no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—44no—
101C23×C73—44no—
102C23×C73—48no—
103C23×C73—44no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73C2×C22⋊C411yescharacteristic
107C23×C73C22×C811yescharacteristic
108C23×C73C2×M1611yescharacteristic
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—48no—
117C2×C4×C73—42no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—44no—
121C2×C4×C73—44no—
122C2×C4×C73—42no—
123C2×C4×C73C22⋊C812yes—
124C2×C4×C73C22⋊C812yes—
125C2×C4×C73#9911yescharacteristic
126C2×C4×C73#9911yescharacteristic
127C2×C4×C73#10012yes—
128C2×Dic73—22no—
129C2×Dic73—22no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136#10596833—84no—
137C22×C73—22no—
138C22×C73—24no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73—44no—
144C22×C73—42no—
145C22×C73—42no—
146C22×C73—44no—
147C22×C73—44no—
148C22×C73—48no—
149C22×C73—44no—
150C22×C73—42no—
151C22×C73—42no—
152C22×C73C2×C22⋊C811yescharacteristic
153C22×C73C2×#9911yescharacteristic
154C22×C73C2×#10011yescharacteristic
155C4×C73—22no—
156C4×C73—42no—
157C4×C73#32312yes—
158C4×C73#32312yes—
159Dic73—42no—
160Dic73—42no—
161Dic73—81no—
162Dic73—81no—
163#11425—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—44no—
167C2×C73—41no—
168C2×C73—41no—
169C2×C73C2×#32311yescharacteristic
170C2×C73#212012yes—
171#2280—731no—
172#1458—732no—
173#1317—732no—
174#1316—732no—
175C73#1142511yescharacteristic, complemented, Sylow
176C24×C4#6440811yescharacteristic
177C2×#117—731no—
178C2×#117—731no—
179#431—1462no—
180#381—731no—
181#381—731no—
182#380—1462no—
183#379—731no—
184#379—731no—
185C4×C22⋊C4—1462no—
186#324—1462no—
187#323—734no—
188C25C2×#6440811yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4#1059683314yescomplemented
196C23×C4C2×#6440811yescharacteristic
197C23×C4C2×#6440811yescharacteristic
198#117—2922no—
199#117—2921no—
200#117—2921no—
201C2×C4⋊C4—732no—
202C2×C4⋊C4—1461no—
203C2×C22⋊C4—731no—
204C2×C22⋊C4—731no—
205C2×C22⋊C4—731no—
206C2×C22⋊C4—731no—
207C2×C22⋊C4—1462no—
208C2×C22⋊C4—1462no—
209C2×C42—732no—
210C2×C42—1461no—
211C4⋊C8—2922no—
212C22⋊C8—1464no—
213C4×C8—2922no—
214#95—1462no—
215#95—1462no—
216C24—21no—
217C24—21no—
218C24—41no—
219C24—41no—
220C24—44no—
221C24C2×#1059683312yes—
222C24C22×#6440811yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—732no—
242C22×C4—732no—
243C22×C4—1461no—
244C22×C4—1461no—
245C22×C4C22⋊#6440811yescharacteristic
246C22×C4C22⋊#6440811yescharacteristic
247C22×C4C2×#1059683312yes—
248C22×C4C2×#1059683312yes—
249C22×C4#1192403712yes—
250C2×C8—2924no—
251C4⋊C4—2922no—
252C4⋊C4—2922no—
253C22⋊C4—2924no—
254C22⋊C4—2924no—
255C22⋊C4—2921no—
256C22⋊C4—2921no—
257C22⋊C4—2921no—
258C22⋊C4—2921no—
259C42—2922no—
260C42—1462no—
261C42—1462no—
262C23—24no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—42no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—48no—
274C23—44no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#6440811yescharacteristic
278C23C22×#1059683311yescharacteristic
279C23C2×#1192403711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—2922no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—1462no—
301C2×C4—1462no—
302C2×C4C22⋊#1059683312yes—
303C2×C4C22⋊#1059683312yes—
304C2×C4#1455664911yescharacteristic
305C2×C4#1455664911yescharacteristic
306C2×C4#1455665112yes—
307C8—5844no—
308C22—22no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22C2×C22⋊#1059683311yescharacteristic
324C22C2×#1455664911yescharacteristic
325C22C2×#1455665111yescharacteristic
326C4—22no—
327C4—42no—
328C4—2922no—
329C4—2922no—
330C4—5841no—
331C4—5841no—
332C4#1798943012yes—
333C4#1798943012yes—
334C2—21no—
335C2—21no—
336C2—44no—
337C2—41no—
338C2—41no—
339C2C2×#1798943011yescharacteristic
340C2#2262618212yes—
subgroupG/H
centerC2×C4#14556649
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC24×C4×C73C4
socleC22×C73C2×#99
prime radicalC25×C73C2×C4
cosocleC2×C22⋊Dic73C23
hypercenterC24×C4#64408
hypocenterC73#11425
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #22625162C2semidirect
2. #22624857C2semidirect
42. C23×C4×C73C8semidirect
175. C73#11425semidirect
195. C23×C4#10596833semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248572
#226248552
#226123142
#1142573