Finite Group Databasebeta

← #33008392

Subgroups — #33008392

Order 18688.

1710 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988453C211yesmaximal, characteristic
2C2×C4×C4⋊Dic73C211yesmaximal, characteristic
3#22608202C212yesmaximal
4#22608199C212yesmaximal
5#22608163C214yesmaximal
6#22608161C214yesmaximal
7C22×C4×Q8×C73C211yesmaximal, characteristic
8C22×C4⋊Dic73C2212yes—
9C22×C4⋊Dic73C2212yes—
10C22×C4×Dic73C2212yes—
11#17988453—24no—
12C4×C4⋊Dic73—24no—
13C2×#14556498C2218yes—
14C2×#14556498C2212yes—
15C2×#14556498C2212yes—
16C2×#14556498C2212yes—
17C2×#14556498C2218yes—
18C23×Q8×C73C2211yescharacteristic
19C2×C4×Q8×C73—24no—
20C2×C4×Q8×C73—24no—
21C2×C4×Q8×C73C418yescomplemented
22C22×C4⋊C4×C73C2212yes—
23C22×C4⋊C4×C73C2211yescharacteristic
24C22×C42×C73C2212yes—
25C22×C42×C73C2211yescharacteristic
26C23×Dic73C2314yes—
27C23×Dic73C2314yes—
28C2×C4⋊Dic73—24no—
29C2×C4⋊Dic73—24no—
30C2×C4⋊Dic73—24no—
31C2×C4⋊Dic73—28no—
32C2×C4⋊Dic73Q818yescomplemented
33C2×C4×Dic73—24no—
34C2×C4×Dic73—44no—
35#14556498—24no—
36#14556498—24no—
37#14556498—24no—
38#14556498—216no—
39#14556498—48no—
40C22×Q8×C73—22no—
41C22×Q8×C73—22no—
42C22×Q8×C73—28no—
43C22×Q8×C73C2×C414yes—
44C23×C4×C73C2312yes—
45C23×C4×C73C2311yescharacteristic
46C23×C4×C73C2311yescharacteristic
47C23×C4×C73C2312yes—
48C23×C4×C73C2311yescharacteristic
49C4×Q8×C73—232no—
50C2×C4⋊C4×C73—24no—
51C2×C4⋊C4×C73—24no—
52C2×C4⋊C4×C73—22no—
53C2×C4⋊C4×C73—22no—
54C2×C4⋊C4×C73C2×C414yes—
55C2×C4⋊C4×C73C2×C418yes—
56C2×C42×C73—24no—
57C2×C42×C73—24no—
58C2×C42×C73C2×C418yes—
59C2×C42×C73C2×C414yes—
60C2×C42×C73D414yes—
61C2×C42×C73D414yes—
62C22×Dic73—24no—
63C22×Dic73—28no—
64C22×Dic73—24no—
65C22×Dic73—28no—
66C22×Dic73—24no—
67C22×Dic73—28no—
68C22×Dic73—28no—
69C22×Dic73—24no—
70C22×Dic73—24no—
71C22×Dic73C2×Q818yes—
72C4⋊Dic73—216no—
73C4⋊Dic73—48no—
74C4×Dic73—48no—
75C24×C73C2411yescharacteristic
76C2×Q8×C73—216no—
77C2×Q8×C73—216no—
78C2×Q8×C73—28no—
79C2×Q8×C73—216no—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—24no—
83C22×C4×C73—24no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—22no—
88C22×C4×C73—24no—
89C22×C4×C73—24no—
90C22×C4×C73C22×C414yes—
91C22×C4×C73C22×C412yes—
92C22×C4×C73C22×C412yes—
93C22×C4×C73C22×C414yes—
94C22×C4×C73C22×C412yes—
95C22×C4×C73C2×D412yes—
96C22×C4×C73C2×D412yes—
97C22×C4×C73C2×D412yes—
98C22×C4×C73C2×D412yes—
99C22×C4×C73C2×D412yes—
100C22×C4×C73C2×D412yes—
101C22×C4×C73C2×Q814yes—
102C22×C4×C73#4118yes—
103C22×C4×C73#4114yes—
104C4⋊C4×C73—216no—
105C4⋊C4×C73—28no—
106C42×C73—216no—
107C42×C73C22⋊C4116yes—
108C23×C73C23×C411yescharacteristic
109C23×C73C22×D411yescharacteristic
110C23×C73C22×D411yescharacteristic
111C23×C73C22×Q812yes—
112C23×C73C2×#4114yes—
113C23×C73C2×#4112yes—
114C23×C73H(5,𝔽2)12yes—
115C23×C732−1+412yes—
116Q8×C73—232no—
117C2×C4×C73—24no—
118C2×C4×C73—24no—
119C2×C4×C73—24no—
120C2×C4×C73—24no—
121C2×C4×C73—24no—
122C2×C4×C73—24no—
123C2×C4×C73—28no—
124C2×C4×C73—28no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—24no—
128C2×C4×C73—24no—
129C2×C4×C73—28no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—44no—
133C2×C4×C73C2×C22⋊C414yes—
134C2×C4×C73C2×C22⋊C414yes—
135C2×C4×C73C2×C22⋊C414yes—
136C2×C4×C73#11718yes—
137C2×C4×C73C4×Q818yes—
138C2×C4×C73C22⋊Q818yes—
139C2×C4×C73#12414yes—
140C2×C4×C73C4⋊Q814yes—
141C2×Dic73—28no—
142C2×Dic73—216no—
143C2×Dic73—44no—
144C2×Dic73—44no—
145C2×Dic73—48no—
146C2×Dic73—44no—
147C2×Dic73—44no—
148C2×Dic73—44no—
149C2×Dic73—48no—
150C2×Dic73—48no—
151C22×C73C22×C22⋊C411yescharacteristic
152C22×C73C2×#11712yes—
153C22×C73C2×C4×Q812yes—
154C22×C73#52012yes—
155C22×C73C2×C22⋊Q812yes—
156C22×C73#52512yes—
157C22×C73#52512yes—
158C22×C73C2×#12411yescharacteristic
159C22×C73C2×C4⋊Q811yescharacteristic
160C22×C73#53511yescharacteristic
161C22×C73#53611yescharacteristic
162C22×C73#53712yes—
163C22×C73#55414yes—
164C22×C73#55614yes—
165C22×C73#55714yes—
166C22×C73#55914yes—
167C4×C73—24no—
168C4×C73—24no—
169C4×C73—28no—
170C4×C73—44no—
171C4×C73#39118yes—
172Dic73—48no—
173Dic73—84no—
174#13925—731nomaximal, Sylow
175C2×C73#228011yescharacteristic
176C2×C73C2×#39111yescharacteristic
177C2×C73#231111yescharacteristic
178C2×C73#235514yes—
179C2×C73#258012yes—
180C2×C73#258112yes—
181C2×C73#258412yes—
182C2×C73#259212yes—
183C22×C4×Q8D7311yescharacteristic
184C2×#382—731no—
185C2×C4×C4⋊C4—731no—
186#1446—732no—
187#1445—732no—
188#1436—734no—
189C4⋊#95—734no—
190C73#1392511yescharacteristic, complemented, Sylow
191C23×Q8C2×D7311yescharacteristic
192C2×C4×Q8—24no—
193C2×C4×Q8—24no—
194C2×C4×Q8Dic7318yescomplemented
195C22×C4⋊C4—732no—
196C22×C4⋊C4—732no—
197C22×C4⋊C4C2×D7312yes—
198C22×C4⋊C4C2×D7311yescharacteristic
199C22×C42—732no—
200C22×C42C2×D7312yes—
201C22×C42C2×D7311yescharacteristic
202#382—1464no—
203C4×C4⋊C4—1464no—
204C2×#95—738no—
205C2×#95—738no—
206C2×#95—732no—
207C2×#95—732no—
208C2×#95—732no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—28no—
212C22×Q8C2×Dic7314yes—
213C23×C4—734no—
214C23×C4—734no—
215C23×C4C22×D7312yes—
216C23×C4C22×D7311yescharacteristic
217C23×C4C22×D7311yescharacteristic
218C23×C4C22×D7312yes—
219C23×C4C22×D7311yescharacteristic
220C4×Q8—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—738no—
226C2×C4⋊C4—1464no—
227C2×C4⋊C4—1468no—
228C2×C4⋊C4—1464no—
229C2×C4⋊C4—1464no—
230C2×C4⋊C4C2×Dic7314yes—
231C2×C4⋊C4C2×Dic7318yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—2924no—
235C2×C42—1464no—
236C2×C42C2×Dic7318yes—
237C2×C42C2×Dic7314yes—
238C2×C42C22⋊D7314yes—
239C2×C42C22⋊D7314yes—
240#95—2928no—
241#95—1464no—
242#95—1464no—
243#95—1464no—
244#95—14616no—
245C24C23×D7311yescharacteristic
246C2×Q8—216no—
247C2×Q8—216no—
248C2×Q8—28no—
249C2×Q8—216no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—738no—
261C22×C4—1464no—
262C22×C4—1468no—
263C22×C4—1468no—
264C22×C4—1464no—
265C22×C4—1464no—
266C22×C4—1464no—
267C22×C4—1468no—
268C22×C4—1464no—
269C22×C4—1468no—
270C22×C4C22×Dic7314yes—
271C22×C4C22×Dic7312yes—
272C22×C4C22×Dic7312yes—
273C22×C4C22×Dic7314yes—
274C22×C4C22×Dic7312yes—
275C22×C4C2×C22⋊D7312yes—
276C22×C4C2×C22⋊D7312yes—
277C22×C4C2×C22⋊D7312yes—
278C22×C4C2×C22⋊D7312yes—
279C22×C4C2×C22⋊D7312yes—
280C22×C4C2×C22⋊D7312yes—
281C22×C4Q8×D7314yes—
282C22×C4#1192402818yes—
283C22×C4#1192402914yes—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—2928no—
287C4⋊C4—14616no—
288C42—216no—
289C42—2928no—
290C42C22⋊Dic73116yes—
291C23C23×Dic7311yescharacteristic
292C23C22×C22⋊D7311yescharacteristic
293C23C22×C22⋊D7311yescharacteristic
294C23C2×Q8×D7312yes—
295C23C2×#1192402814yes—
296C23C2×#1192402912yes—
297C23#1455663712yes—
298C23#1455663812yes—
299Q8—232no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—2924no—
317C2×C4—2928no—
318C2×C4—2924no—
319C2×C4—2924no—
320C2×C4—2928no—
321C2×C4—2928no—
322C2×C4—2924no—
323C2×C4—2924no—
324C2×C4—14616no—
325C2×C4—1468no—
326C2×C4C2×C22⋊Dic7314yes—
327C2×C4C2×C22⋊Dic7314yes—
328C2×C4C2×C22⋊Dic7314yes—
329C2×C4#1455655018yes—
330C2×C4Q8×Dic7318yes—
331C2×C4#1455656718yes—
332C2×C4#1455658014yes—
333C2×C4#1455658914yes—
334C22C22×C22⋊Dic7311yescharacteristic
335C22C2×#1455655012yes—
336C22C2×Q8×Dic7312yes—
337C22#1798907012yes—
338C22C2×#1455656712yes—
339C22#1798909812yes—
340C22#1798909812yes—
341C22C2×#1455658011yescharacteristic
342C22C2×#1455658911yescharacteristic
343C22#1798915111yescharacteristic
344C22#1798915811yescharacteristic
345C22#1798916512yes—
346C22#1798929714yes—
347C22#1798931014yes—
348C22#1798931514yes—
349C22#1798932214yes—
350C4—24no—
351C4—24no—
352C4—28no—
353C4—44no—
354C4—2928no—
355C4—5844no—
356C4#1798849418yes—
357C2#2261231411yescharacteristic
358C2C2×#1798849411yescharacteristic
359C2#2261252311yescharacteristic
360C2#2261288814yes—
361C2#2261528912yes—
362C2#2261529712yes—
363C2#2261531912yes—
364C2#2261541912yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC22×C4×Q8×C73C2
socle, prime radical, cosocleC24×C73C24
hypercenterC22×C4×Q8D73
hypocenterC73#13925
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
21. C2×C4×Q8×C73C4semidirect
32. C2×C4⋊Dic73Q8semidirect
190. C73#13925semidirect
194. C2×C4×Q8Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884532
C2×C4×C4⋊Dic732
#226082022
#226081992
#226081632
#226081612
C22×C4×Q8×C732
#1392573