Finite Group Databasebeta

← #33095479

Subgroups — #33095479

Order 18688.

1074 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988640C211yesmaximal, characteristic
2#22612338C211yesmaximal, characteristic
3#22610893C212yesmaximal, complemented
4#22610893C212yesmaximal, complemented
5#22610889C214yesmaximal, complemented
6#22610026C214yesmaximal
7C2×#533×C73C211yesmaximal, characteristic
8C2×#14556550—21no—
9#17988640—24no—
10C2×C4⋊#10596831C2212yes—
11C2×C22⋊#10596831C2212yes—
12C2×C4×#10596831C2212yes—
13#17988457C2212yes—
14#17988453—21no—
15#17988447—21no—
16#17988447C2212yes—
17C4×C4⋊Dic73—21no—
18C4×C4⋊Dic73C2212yes—
19#17988349C2218yes—
20#17988349C2218yes—
21#533×C73—24no—
22#533×C73C418yescomplemented
23C2×C4⋊Q8×C73C2211yescharacteristic
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×#125×C73C2212yes—
26C2×C22⋊Q8×C73—21no—
27C2×C22⋊Q8×C73—21no—
28C2×C4×Q8×C73—21no—
29C2×C4×Q8×C73—21no—
30C2×#117×C73C2212yes—
31C22×C42×C73C2211yescharacteristic
32C22×#10596831C2314yes—
33#14556550—44no—
34C2×C4⋊Dic73—22no—
35C2×C4⋊Dic73—22no—
36C2×C4⋊Dic73C2314yes—
37C2×C22⋊Dic73—22no—
38C2×C4×Dic73—22no—
39C4⋊#10596831—24no—
40C22⋊#10596831—24no—
41C4×#10596831—24no—
42#14556498—22no—
43#14556498—22no—
44C22×Q8×C73—21no—
45C22×Q8×C73—21no—
46C23×C4×C73D411yescharacteristic
47C23×C4×C73D411yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C4⋊Q8×C73—22no—
50C4⋊Q8×C73—22no—
51C4⋊Q8×C73C2×C414yes—
52C4⋊Q8×C73C2×C414yes—
53#125×C73—24no—
54#125×C73C2×C418yes—
55C22⋊Q8×C73—28no—
56C22⋊Q8×C73—28no—
57C4×Q8×C73—28no—
58C4×Q8×C73—28no—
59#117×C73—24no—
60#117×C73C2×C418yes—
61C2×C4⋊C4×C73—22no—
62C2×C4⋊C4×C73—22no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73C2312yes—
66C2×C4⋊C4×C73C2312yes—
67C2×C22⋊C4×C73—22no—
68C2×C42×C73—22no—
69C2×C42×C73C2×C414yes—
70C2×C42×C73D411yescharacteristic
71C2×C42×C73D411yescharacteristic
72C2×C42×C73D414yes—
73C2×C42×C73C2311yescharacteristic
74C2×C42×C73C2311yescharacteristic
75C22×Dic73—24no—
76C2×#10596831—24no—
77C2×#10596831—28no—
78C4⋊Dic73—28no—
79C4⋊Dic73—44no—
80C4⋊Dic73—44no—
81C22⋊Dic73—44no—
82C4×Dic73—44no—
83C24×C73C2×D411yescharacteristic
84C2×Q8×C73—24no—
85C2×Q8×C73—24no—
86C2×Q8×C73—44no—
87C2×Q8×C73—42no—
88C2×Q8×C73—42no—
89C22×C4×C73—24no—
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—24no—
93C22×C4×C73—22no—
94C22×C4×C73—22no—
95C22×C4×C73C22⋊C414yes—
96C22×C4×C73C22⋊C414yes—
97C22×C4×C73C22×C412yes—
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D411yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D412yes—
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D411yescharacteristic
104C22×C4×C73C2411yescharacteristic
105C4⋊C4×C73—28no—
106C4⋊C4×C73—28no—
107C4⋊C4×C73—24no—
108C4⋊C4×C73—24no—
109C4⋊C4×C73—22no—
110C4⋊C4×C73—22no—
111C4⋊C4×C73C22×C414yes—
112C4⋊C4×C73C22×C414yes—
113C22⋊C4×C73—28no—
114C42×C73—28no—
115C42×C73C22⋊C414yes—
116C42×C73C22⋊C414yes—
117C42×C73C22×C412yes—
118C42×C73C22×C412yes—
119C42×C73C2×D412yes—
120C42×C73C2×D412yes—
121C23×C73—22no—
122C23×C73—22no—
123C23×C73—21no—
124C23×C73—21no—
125C23×C73C2×C22⋊C412yes—
126C23×C73C22×D411yescharacteristic
127Q8×C73—48no—
128Q8×C73—48no—
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—24no—
134C2×C4×C73—24no—
135C2×C4×C73—28no—
136C2×C4×C73—28no—
137C2×C4×C73—24no—
138C2×C4×C73—24no—
139C2×C4×C73—44no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73C2×C22⋊C412yes—
143C2×C4×C73C2×C22⋊C412yes—
144C2×C4×C73C2×C22⋊C412yes—
145C2×C4×C73C2×C22⋊C412yes—
146C2×C4×C73C2×C22⋊C412yes—
147C2×C4×C73#13514yes—
148C2×C4×C73AGL(1,ℤ/8)12yes—
149C2×C4×C73#13712yes—
150C2×C4×C73C23×C411yescharacteristic
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C732−1+414yes—
153C2×Dic73—44no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156#10596831—44no—
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—22no—
160C22×C73—42no—
161C22×C73—42no—
162C22×C73C22×C22⋊C411yescharacteristic
163C22×C73#53612yes—
164C22×C73C2×#13512yes—
165C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
166C22×C73C2×#13711yescharacteristic
167C4×C73—22no—
168C4×C73—22no—
169C4×C73—22no—
170C4×C73—44no—
171C4×C73—44no—
172C4×C73#41614yes—
173C4×C73#41714yes—
174C4×C73#51912yes—
175C4×C73#53612yes—
176Dic73—84no—
177#21835—731nomaximal, Sylow
178C2×C73—42no—
179C2×C73#231011yescharacteristic
180C2×C73C2×#41611yescharacteristic
181C2×C73C2×#41711yescharacteristic
182C2×C73#322812yes—
183C2×C73#323112yes—
184C2×#533D7311yescharacteristic
185C2×#431—731no—
186#2285—731no—
187#1990—732no—
188#1990—732no—
189#1989—734no—
190#1825—734no—
191C73#2183511yescharacteristic, complemented, Sylow
192#533—24no—
193#533Dic7318yescomplemented
194C2×C4⋊Q8C2×D7311yescharacteristic
195C2×C4⋊Q8C2×D7311yescharacteristic
196C2×#125C2×D7312yes—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×C4×Q8—21no—
200C2×C4×Q8—21no—
201C2×#117—1461no—
202C2×#117C2×D7312yes—
203C22×C42C2×D7311yescharacteristic
204#431—1464no—
205C2×C4⋊C8—732no—
206C2×C22⋊C8—732no—
207C2×C4×C8—732no—
208#384—732no—
209#382—1461no—
210#380—732no—
211#380—1461no—
212C4×C4⋊C4—732no—
213C4×C4⋊C4—1461no—
214#340—738no—
215#340—738no—
216C22×Q8—21no—
217C22×Q8—21no—
218C23×C4C22⋊D7311yescharacteristic
219C23×C4C22⋊D7311yescharacteristic
220C23×C4C22×D7311yescharacteristic
221C22×C8—734no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8C2×Dic7314yes—
225C4⋊Q8C2×Dic7314yes—
226#125—24no—
227#125C2×Dic7318yes—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230C4×Q8—28no—
231C4×Q8—28no—
232#117—24no—
233#117—2924no—
234#117C2×Dic7318yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—734no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C4⋊C4C22×D7312yes—
243C2×C4⋊C4C22×D7312yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—1462no—
246C2×C42—22no—
247C2×C42—1462no—
248C2×C42C2×Dic7314yes—
249C2×C42C22⋊D7311yescharacteristic
250C2×C42C22⋊D7311yescharacteristic
251C2×C42C22⋊D7314yes—
252C2×C42C22×D7311yescharacteristic
253C2×C42C22×D7311yescharacteristic
254C4⋊C8—1464no—
255C22⋊C8—1464no—
256C4×C8—1464no—
257#95—1462no—
258#95—1462no—
259C24C2×C22⋊D7311yescharacteristic
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—42no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—1464no—
272C22×C4C22⋊Dic7314yes—
273C22×C4C22⋊Dic7314yes—
274C22×C4C22×Dic7312yes—
275C22×C4C2×C22⋊D7311yescharacteristic
276C22×C4C2×C22⋊D7311yescharacteristic
277C22×C4C2×C22⋊D7311yescharacteristic
278C22×C4C2×C22⋊D7312yes—
279C22×C4C2×C22⋊D7311yescharacteristic
280C22×C4C2×C22⋊D7311yescharacteristic
281C22×C4C23×D7311yescharacteristic
282C2×C8—1464no—
283C2×C8—1468no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—2924no—
291C4⋊C4—2924no—
292C4⋊C4—1468no—
293C4⋊C4C22×Dic7314yes—
294C4⋊C4C22×Dic7314yes—
295C22⋊C4—28no—
296C22⋊C4—2924no—
297C42—28no—
298C42—2924no—
299C42C22⋊Dic7314yes—
300C42C22⋊Dic7314yes—
301C42C22×Dic7312yes—
302C42C22×Dic7312yes—
303C42C2×C22⋊D7312yes—
304C42C2×C22⋊D7312yes—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23C2×C22⋊Dic7312yes—
310C23C22×C22⋊D7311yescharacteristic
311Q8—48no—
312Q8—48no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—2924no—
327C2×C4—2924no—
328C2×C4—2924no—
329C2×C4C2×C22⋊Dic7312yes—
330C2×C4C2×C22⋊Dic7312yes—
331C2×C4C2×C22⋊Dic7312yes—
332C2×C4C2×C22⋊Dic7312yes—
333C2×C4C2×C22⋊Dic7312yes—
334C2×C4#1455661114yes—
335C2×C4#1455661612yes—
336C2×C4#1455662112yes—
337C2×C4C23×Dic7311yescharacteristic
338C2×C4C22×C22⋊D7311yescharacteristic
339C2×C4#1455663914yes—
340C8—2924no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22C22×C22⋊Dic7311yescharacteristic
347C22#1798916112yes—
348C22C2×#1455661112yes—
349C22C2×#1455661611yescharacteristic
350C22C2×#1455662111yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—44no—
355C4—44no—
356C4—5844no—
357C4#1798858314yes—
358C4#1798858814yes—
359C4#1798906512yes—
360C4#1798916112yes—
361C2—42no—
362C2#2261251711yescharacteristic
363C2C2×#1798858311yescharacteristic
364C2C2×#1798858811yescharacteristic
365C2#2262106712yes—
366C2#2262109012yes—
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#533×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#533D73
hypocenterC73#21835
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
3. #22610893C2semidirect
4. #22610893C2semidirect
5. #22610889C2semidirect
22. #533×C73C4semidirect
191. C73#21835semidirect
193. #533Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886402
#226123382
#226108932
#226108892
#226100262
C2×#533×C732
#2183573