Finite Group Databasebeta

← #32991832

Subgroups — #32991832

Order 18688.

998 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#10596831C211yesmaximal, characteristic
2C2×C4×C4⋊Dic73C211yesmaximal, characteristic
3C2×#17988349C214yesmaximal
4C2×#382×C73C211yesmaximal, characteristic
5C23×#10596831C2212yes—
6C22×C4⋊Dic73C2212yes—
7C22×C4×Dic73—21no—
8C2×C4⋊#10596831—22no—
9C2×C4⋊#10596831—22no—
10C2×C4⋊#10596831C414yescomplemented
11C4×C4⋊Dic73—22no—
12C4×C4⋊Dic73C414yescomplemented
13C2×#14556498—21no—
14#17988349—28no—
15C22×C4⋊C4×C73C2212yes—
16C22×C42×C73C2211yescharacteristic
17#382×C73—22no—
18#382×C73C414yescomplemented
19C2×#95×C73—21no—
20C2×#95×C73—21no—
21C23×Dic73—22no—
22C22×#10596831—22no—
23C22×#10596831—22no—
24C22×#10596831—24no—
25C22×#10596831—24no—
26C22×#10596831C2×C414yes—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73—22no—
30C2×C4⋊Dic73C2×C414yes—
31C2×C4⋊Dic73D414yes—
32C2×C4⋊Dic73Q814yes—
33C2×C4×Dic73—24no—
34C2×C4×Dic73—44no—
35C4⋊#10596831—28no—
36#14556498—24no—
37C23×C4×C73—22no—
38C23×C4×C73D411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41C2×C4⋊C4×C73—24no—
42C2×C4⋊C4×C73—24no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73C2×C414yes—
45C2×C42×C73C2×C412yes—
46C2×C42×C73C2×C412yes—
47C2×C42×C73C2×C412yes—
48C2×C42×C73D412yes—
49C2×C42×C73D412yes—
50C2×C42×C73Q812yes—
51#95×C73—24no—
52#95×C73—24no—
53C22×Dic73—24no—
54C22×Dic73—24no—
55C22×Dic73—24no—
56C22×Dic73—44no—
57C22×Dic73—44no—
58C2×#10596831—24no—
59C2×#10596831—28no—
60C2×#10596831—44no—
61C2×#10596831—42no—
62C2×#10596831—42no—
63C4⋊Dic73—24no—
64C4⋊Dic73—24no—
65C4⋊Dic73—24no—
66C4⋊Dic73C4⋊C418yes—
67C4×Dic73—48no—
68C24×C73C2×D411yescharacteristic
69C22×C4×C73—24no—
70C22×C4×C73—22no—
71C22×C4×C73—22no—
72C22×C4×C73—24no—
73C22×C4×C73—44no—
74C22×C4×C73—44no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C22⋊C412yes—
80C22×C4×C73C22⋊C412yes—
81C22×C4×C73C4⋊C412yes—
82C22×C4×C73C4⋊C412yes—
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C22×C411yescharacteristic
86C22×C4×C73C2×D412yes—
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×Q812yes—
90C22×C4×C73C2×Q811yescharacteristic
91C22×C4×C73#4112yes—
92C22×C4×C73#4112yes—
93C4⋊C4×C73—28no—
94C4⋊C4×C73—42no—
95C4⋊C4×C73—42no—
96C42×C73C4214yes—
97C42×C73C22⋊C414yes—
98C42×C73C4⋊C414yes—
99C42×C73C4⋊C414yes—
100C23×C73C2×C22⋊C411yescharacteristic
101C23×C73C2×C22⋊C411yescharacteristic
102C23×C73C2×C4⋊C411yescharacteristic
103C23×C73#12112yes—
104C23×C73C22⋊Q812yes—
105C23×C73#13514yes—
106C23×C73AGL(1,ℤ/8)12yes—
107C23×C73#13712yes—
108C2×C4×C73—24no—
109C2×C4×C73—24no—
110C2×C4×C73—24no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—24no—
114C2×C4×C73—24no—
115C2×C4×C73—44no—
116C2×C4×C73—44no—
117C2×C4×C73—44no—
118C2×C4×C73—84no—
119C2×C4×C73—84no—
120C2×C4×C73#9514yes—
121C2×C4×C73#9514yes—
122C2×C4×C73C2×C4211yescharacteristic
123C2×C4×C73C2×C22⋊C411yescharacteristic
124C2×C4×C73C2×C4⋊C412yes—
125C2×C4×C73C2×C4⋊C411yescharacteristic
126C2×C4×C73C2×C4⋊C411yescharacteristic
127C2×C4×C73#11712yes—
128C2×C4×C73C4×D412yes—
129C2×C4×C73C4×D412yes—
130C2×C4×C73C4×Q812yes—
131C2×C4×C73C4×Q812yes—
132C2×C4×C73#12112yes—
133C2×C4×C73C22⋊Q812yes—
134C2×C4×C73#12411yescharacteristic
135C2×C4×C73#12411yescharacteristic
136C2×C4×C73#12512yes—
137C2×C4×C73#12711yescharacteristic
138C2×C4×C73C4⋊Q811yescharacteristic
139C2×C4×C73C4⋊Q812yes—
140C2×Dic73—28no—
141C2×Dic73—44no—
142C2×Dic73—44no—
143C2×Dic73—48no—
144C2×Dic73—48no—
145#10596831—44no—
146C22×C73C2×#9511yescharacteristic
147C22×C73#38012yes—
148C22×C73#38912yes—
149C22×C73#39011yescharacteristic
150C22×C73#39111yescharacteristic
151C22×C73#41612yes—
152C22×C73#41612yes—
153C22×C73#41712yes—
154C22×C73#41811yescharacteristic
155C22×C73#41911yescharacteristic
156C22×C73#42712yes—
157C22×C73#42812yes—
158C22×C73#46314yes—
159C22×C73#46414yes—
160C22×C73#47814yes—
161C22×C73#47914yes—
162C4×C73—24no—
163C4×C73—24no—
164C4×C73—84no—
165C4×C73C4×C4⋊C412yes—
166C4×C73#38211yescharacteristic
167C4×C73#38411yescharacteristic
168C4×C73#38912yes—
169C4×C73#39011yescharacteristic
170C4×C73#39111yescharacteristic
171Dic73—48no—
172#10850—731nomaximal, Sylow
173C2×C73C4⋊#9511yescharacteristic
174C2×C73#173511yescharacteristic
175C2×C73#173611yescharacteristic
176C2×C73#180412yes—
177C2×C73#180512yes—
178C2×C73#194914yes—
179C2×C73#198312yes—
180C2×C73#198912yes—
181C22×C4⋊C8—731no—
182C2×#382D7311yescharacteristic
183C2×C4×C4⋊C4—731no—
184C2×#340—734no—
185C73#1085011yescharacteristic, complemented, Sylow
186C23×C8—732no—
187C22×C4⋊C4—732no—
188C22×C4⋊C4C2×D7312yes—
189C22×C42—1461no—
190C22×C42C2×D7311yescharacteristic
191C2×C4⋊C8—734no—
192C2×C4⋊C8—1462no—
193C2×C4⋊C8—1462no—
194#382—22no—
195#382Dic7314yescomplemented
196C4×C4⋊C4—734no—
197C4×C4⋊C4—1462no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—1461no—
201#340—1468no—
202C23×C4—22no—
203C23×C4—1462no—
204C23×C4C22⋊D7311yescharacteristic
205C23×C4C22⋊D7311yescharacteristic
206C23×C4C22×D7311yescharacteristic
207C22×C8—734no—
208C22×C8—1464no—
209C22×C8—1462no—
210C22×C8—1462no—
211C22×C8—1464no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—734no—
216C2×C4⋊C4—734no—
217C2×C4⋊C4—734no—
218C2×C4⋊C4—1462no—
219C2×C4⋊C4—1462no—
220C2×C4⋊C4—1462no—
221C2×C4⋊C4C2×Dic7314yes—
222C2×C42—2924no—
223C2×C42—1464no—
224C2×C42Dic14612yes—
225C2×C42C4×D7312yes—
226C2×C42C4×D7312yes—
227C2×C42D29212yes—
228C2×C42C2×Dic7312yes—
229C2×C42C22⋊D7312yes—
230C4⋊C8—1468no—
231#95—24no—
232#95—24no—
233#95—1464no—
234C24C2×C22⋊D7311yescharacteristic
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—2924no—
244C22×C4—2924no—
245C22×C4—1464no—
246C22×C4—1464no—
247C22×C4—1464no—
248C22×C4#1192400512yes—
249C22×C4#1192400512yes—
250C22×C4C22⋊Dic7312yes—
251C22×C4C22⋊Dic7312yes—
252C22×C4#1192400712yes—
253C22×C4#1192400712yes—
254C22×C4C2×C4×D7311yescharacteristic
255C22×C4C2×C4×D7311yescharacteristic
256C22×C4C22×Dic7311yescharacteristic
257C22×C4D4×D7312yes—
258C22×C4C2×C22⋊D7311yescharacteristic
259C22×C4C2×D29211yescharacteristic
260C22×C4Q8×D7312yes—
261C22×C4C2×Dic14611yescharacteristic
262C22×C4#1192402712yes—
263C22×C4#1192402912yes—
264C2×C8—2924no—
265C2×C8—2922no—
266C2×C8—2922no—
267C2×C8—1464no—
268C2×C8—1468no—
269C4⋊C4—28no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—738no—
273C4⋊C4—1464no—
274C4⋊C4—1464no—
275C4⋊C4—1464no—
276C42—2928no—
277C42C4×Dic7314yes—
278C42#1192400514yes—
279C42#1192400714yes—
280C42C4⋊Dic7314yes—
281C23C2×#1192400511yescharacteristic
282C23C2×C22⋊Dic7311yescharacteristic
283C23C2×#1192400711yescharacteristic
284C23#1455656212yes—
285C23#1455656712yes—
286C23#1455661114yes—
287C23#1455661612yes—
288C23#1455662112yes—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—2928no—
302C2×C4—2928no—
303C2×C4—2924no—
304C2×C4—2924no—
305C2×C4—1468no—
306C2×C4#1455649814yes—
307C2×C4#1455649814yes—
308C2×C4C2×C4×Dic7311yescharacteristic
309C2×C4C2×#1192400511yescharacteristic
310C2×C4C4⋊C4×D7312yes—
311C2×C4C2×#1192400711yescharacteristic
312C2×C4C2×C4⋊Dic7311yescharacteristic
313C2×C4#1455654912yes—
314C2×C4#1455655312yes—
315C2×C4D4×Dic7312yes—
316C2×C4#1455655612yes—
317C2×C4Q8×Dic7312yes—
318C2×C4#1455656612yes—
319C2×C4#1455657112yes—
320C2×C4#1455657811yescharacteristic
321C2×C4#1455658011yescharacteristic
322C2×C4#1455658212yes—
323C2×C4#1455658711yescharacteristic
324C2×C4#1455658911yescharacteristic
325C2×C4#1455659012yes—
326C8—2924no—
327C22C2×#1455649811yescharacteristic
328C22#1798844412yes—
329C22#1798848312yes—
330C22#1798849011yescharacteristic
331C22#1798849411yescharacteristic
332C22#1798858112yes—
333C22#1798858312yes—
334C22#1798858812yes—
335C22#1798859111yescharacteristic
336C22#1798859511yescharacteristic
337C22#1798862312yes—
338C22#1798863012yes—
339C22#1798879914yes—
340C22#1798880614yes—
341C22#1798888414yes—
342C22#1798889114yes—
343C4—24no—
344C4—24no—
345C4—84no—
346C4—2928no—
347C4C4⋊C4×Dic7312yes—
348C4#1798845211yescharacteristic
349C4#1798845611yescharacteristic
350C4#1798848612yes—
351C4#1798848811yescharacteristic
352C4#1798849211yescharacteristic
353C2#2260816011yescharacteristic
354C2#2260964711yescharacteristic
355C2#2260965211yescharacteristic
356C2#2260993812yes—
357C2#2260994312yes—
358C2#2261069714yes—
359C2#2261086412yes—
360C2#2261088912yes—
subgroupG/H
centerC24C2×C22⋊D73
commutatorC2×C4×C73C2×C42
FrattiniC23×C4C22×D73
FittingC2×#382×C73C2
socle, prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenterC2×#382D73
hypocenterC73#10850
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
10. C2×C4⋊#10596831C4semidirect
12. C4×C4⋊Dic73C4semidirect
18. #382×C73C4semidirect
185. C73#10850semidirect
195. #382Dic73semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#105968312
C2×C4×C4⋊Dic732
C2×#179883492
C2×#382×C732
#1085073