Finite Group Databasebeta

← #32000764

Subgroups — #32000764

Order 18176.

1002 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986025C211yesmaximal, characteristic
2C2×#17986025C211yesmaximal, characteristic
3C2×#17986023C211yesmaximal, characteristic
4C2×#17985954C211yesmaximal, characteristic
5#22590823C211yesmaximal, characteristic
6#22589062C214yesmaximal, complemented
7#22589057C214yesmaximal, complemented
8#22588036C211yesmaximal, characteristic
9C2×#336×C71C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C22×#11765849C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14#17986025—24no—
15#17986025—24no—
16#17986023—24no—
17C2×C4⋊(C71⋊C8)C2214yes—
18C2×Dic71⋊C8C2214yes—
19C2×Dic71⋊C8C2214yes—
20#17985954—24no—
21C2×#14556038C2212yes—
22C2×C8×Dic71C2212yes—
23#17985897C412yes—
24#17985897C412yes—
25#17985872C414yes—
26C4×Dic71⋊C4C2214yes—
27#17985753C2212yes—
28#17985753C2212yes—
29C23×C8×C71C2211yescharacteristic
30C23×C8×C71C2211yescharacteristic
31C22×C42×C71C2211yescharacteristic
32#336×C71C2212yes—
33#336×C71C2212yes—
34C2×#11765849—22no—
35C2×#11765849—22no—
36C2×#11765849—22no—
37C2×#11765849—22no—
38C2×#11765849—22no—
39C2×#11765849D414yes—
40C22×C71⋊C8C2312yes—
41C22×C71⋊C8C2312yes—
42#14556087—22no—
43#14556087—22no—
44#14556087D414yes—
45#14556087Q814yes—
46C2×C4⋊Dic71C2×C412yes—
47C2×C4⋊Dic71C2×C412yes—
48C2×Dic71⋊C4C2×C414yes—
49C2×Dic71⋊C4C2×C414yes—
50C2×C22⋊Dic71C2×C412yes—
51C2×C22⋊Dic71C2×C412yes—
52C2×C4×Dic71C2312yes—
53C2×C4×Dic71C2312yes—
54C4⋊(C71⋊C8)—28no—
55Dic71⋊C8—28no—
56Dic71⋊C8—28no—
57#14556038—24no—
58C8×Dic71—24no—
59C71⋊#95C2×C414yes—
60C71⋊#95C2×C414yes—
61C23×C4×C71C2×C412yes—
62C23×C4×C71C2311yescharacteristic
63C22×C8×C71—22no—
64C22×C8×C71—22no—
65C22×C8×C71—22no—
66C22×C8×C71—22no—
67C22×C8×C71—22no—
68C22×C8×C71Q814yes—
69C22×C8×C71C2312yes—
70C22×C8×C71C2311yescharacteristic
71C22×C8×C71C2311yescharacteristic
72C2×C42×C71—22no—
73C2×C42×C71—22no—
74C2×C42×C71C2×C411yescharacteristic
75C2×C42×C71C2×C411yescharacteristic
76C2×C42×C71C2312yes—
77C22×Dic71C22×C414yes—
78C22×Dic71C22×C414yes—
79#11765849—28no—
80#11765849—44no—
81C2×C71⋊C8—22no—
82C2×C71⋊C8—22no—
83C2×C71⋊C8—22no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8—22no—
86C2×C71⋊C8C2×D414yes—
87C4⋊Dic71—24no—
88C4⋊Dic71C4⋊C418yes—
89Dic71⋊C4—28no—
90Dic71⋊C4—44no—
91C22⋊Dic71—24no—
92C22⋊Dic71C4⋊C418yes—
93C4×Dic71—22no—
94C4×Dic71—22no—
95C4×Dic71C2×D414yes—
96C4×Dic71C2×Q814yes—
97C24×C71C22×C411yescharacteristic
98C22×C4×C71—24no—
99C22×C4×C71—24no—
100C22×C4×C71—42no—
101C22×C4×C71C22×C412yes—
102C22×C4×C71C22×C412yes—
103C22×C4×C71C22×C411yescharacteristic
104C22×C4×C71C2×D412yes—
105C22×C4×C71C2×D412yes—
106C22×C4×C71C2×Q812yes—
107C22×C4×C71C2×Q812yes—
108C22×C4×C71#4112yes—
109C22×C4×C71#4112yes—
110C22×C4×C71C2411yescharacteristic
111C2×C8×C71—28no—
112C2×C8×C71—21no—
113C2×C8×C71—21no—
114C2×C8×C71—21no—
115C2×C8×C71—21no—
116C2×C8×C71—22no—
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71—44no—
120C2×C8×C71C2×Q812yes—
121C2×C8×C71C2×Q812yes—
122C42×C71—22no—
123C42×C71—22no—
124C42×C71—22no—
125C42×C71—22no—
126C23×C71—22no—
127C23×C71—22no—
128C23×C71C2×C4⋊C412yes—
129C23×C71C4×D412yes—
130C23×C71C4×Q812yes—
131C23×C71C23×C411yescharacteristic
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—22no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—48no—
138C2×C4×C71C2×C4⋊C412yes—
139C2×C4×C71C4×D412yes—
140C2×C4×C71C4×Q812yes—
141C2×C4×C71#12114yes—
142C2×C4×C71C22⋊Q814yes—
143C2×C4×C71#12514yes—
144C2×C4×C71C4⋊Q814yes—
145C2×C4×C71#13114yes—
146C2×C4×C71C22×D411yescharacteristic
147C2×C4×C71C22×D411yescharacteristic
148C2×C4×C71C22×Q811yescharacteristic
149C2×C4×C71C22×Q811yescharacteristic
150C2×C4×C71C2×#4111yescharacteristic
151C2×C4×C71C2×#4111yescharacteristic
152C2×Dic71—24no—
153C2×Dic71—24no—
154C2×Dic71—24no—
155C2×Dic71—24no—
156C2×Dic71—24no—
157C2×Dic71C2×C4⋊C418yes—
158C8×C71—22no—
159C8×C71—22no—
160C8×C71—42no—
161C71⋊C8—24no—
162C71⋊C8—42no—
163C22×C71—24no—
164C22×C71—24no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71#38914yes—
168C22×C71C22×C4⋊C411yescharacteristic
169C22×C71C2×C4×D411yescharacteristic
170C22×C71C2×C4×Q811yescharacteristic
171C22×C71C2×#13112yes—
172C22×C71#56812yes—
173C4×C71—24no—
174C4×C71—42no—
175C4×C71#38914yes—
176C4×C71C2×#12111yescharacteristic
177C4×C71C2×C22⋊Q811yescharacteristic
178C4×C71C2×#12511yescharacteristic
179C4×C71C2×C4⋊Q811yescharacteristic
180Dic71—28no—
181Dic71—44no—
182#21148—711nomaximal, Sylow
183C2×C71—24no—
184C2×C71C2×#38911yescharacteristic
185C2×C71#290612yes—
186C2×C71#293312yes—
187C2×C71#296011yescharacteristic
188C2×C71#296111yescharacteristic
189C2×#424—711no—
190C2×#424—711no—
191C2×#423—711no—
192C2×#405—711no—
193#2296—711no—
194#1940—714no—
195#1939—714no—
196#1729—711no—
197C2×#336D7111yescharacteristic
198C71#2114811yescharacteristic, complemented, Sylow
199C22×M16—711no—
200C22×M16—711no—
201C23×C8C2×D7111yescharacteristic
202C23×C8C2×D7111yescharacteristic
203C2×#117—711no—
204C2×#117—711no—
205C22×C42C2×D7111yescharacteristic
206#424—1424no—
207#424—1424no—
208#423—1424no—
209C2×C4⋊C8—714no—
210C2×C4⋊C8—714no—
211C2×C4⋊C8—714no—
212#405—1424no—
213C2×#97—712no—
214C2×C4×C8—712no—
215#389—712no—
216#389—712no—
217#385—714no—
218C4×C4⋊C4—714no—
219#336—712no—
220#336—712no—
221#336C2×D7112yes—
222#336C2×D7112yes—
223C23×C4C4×D7112yes—
224C23×C4C22×D7111yescharacteristic
225C2×M16—714no—
226C2×M16—1422no—
227C2×M16—1422no—
228C2×M16—1422no—
229C2×M16—1422no—
230C2×M16—1422no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—712no—
237C22×C8—712no—
238C22×C8Dic14214yes—
239C22×C8C22×D7112yes—
240C22×C8C22×D7111yescharacteristic
241C22×C8C22×D7111yescharacteristic
242#117—714no—
243#117—714no—
244#117—1422no—
245#117—1422no—
246C2×C4⋊C4—714no—
247C2×C4⋊C4—712no—
248C2×C4⋊C4—712no—
249C2×C4⋊C4—714no—
250C2×C22⋊C4—712no—
251C2×C22⋊C4—712no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—712no—
255C2×C42—712no—
256C2×C42C4×D7111yescharacteristic
257C2×C42C4×D7111yescharacteristic
258C2×C42C22×D7112yes—
259C4⋊C8—1428no—
260C4⋊C8—1428no—
261C4⋊C8—1428no—
262#97—1424no—
263C4×C8—1424no—
264#95—714no—
265#95—714no—
266C24C2×C4×D7111yescharacteristic
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—714no—
271C22×C4—714no—
272C22×C4C2×C4×D7112yes—
273C22×C4C2×C4×D7112yes—
274C22×C4C2×C4×D7111yescharacteristic
275C22×C4D4×D7112yes—
276C22×C4D4×D7112yes—
277C22×C4Q8×D7112yes—
278C22×C4C2×Dic14212yes—
279C22×C4#1176586412yes—
280C22×C4#1176586512yes—
281C22×C4C23×D7111yescharacteristic
282M16—2844no—
283M16—1428no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—28no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—44no—
293C2×C8—714no—
294C2×C8—1422no—
295C2×C8—1422no—
296C2×C8—1422no—
297C2×C8—1422no—
298C2×C8—1422no—
299C2×C8C2×Dic14212yes—
300C2×C8C2×Dic14212yes—
301C4⋊C4—2844no—
302C4⋊C4—718no—
303C4⋊C4—1424no—
304C4⋊C4—1428no—
305C22⋊C4—718no—
306C22⋊C4—1424no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—714no—
312C42—714no—
313C42—1422no—
314C42—1422no—
315C23—22no—
316C23—22no—
317C23C4⋊C4×D7112yes—
318C23(C22⋊D71)⋊C412yes—
319C23C4×Dic14212yes—
320C23C22×C4×D7111yescharacteristic
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—718no—
328C2×C4—1424no—
329C2×C4—1424no—
330C2×C4—1424no—
331C2×C4—1424no—
332C2×C4—1424no—
333C2×C4C4⋊C4×D7112yes—
334C2×C4(C22⋊D71)⋊C412yes—
335C2×C4C4×Dic14212yes—
336C2×C4#1455610114yes—
337C2×C4#1455610514yes—
338C2×C4#1455611814yes—
339C2×C4#1455612714yes—
340C2×C4#1455613414yes—
341C2×C4C2×D4×D7111yescharacteristic
342C2×C4C2×D4×D7111yescharacteristic
343C2×C4C2×Q8×D7111yescharacteristic
344C2×C4C22×Dic14211yescharacteristic
345C2×C4C2×#1176586411yescharacteristic
346C2×C4C2×#1176586511yescharacteristic
347C8—22no—
348C8—22no—
349C8—42no—
350C8—2842no—
351C8—1424no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22#1798589614yes—
357C22C2×C4⋊C4×D7111yescharacteristic
358C22C2×(C22⋊D71)⋊C411yescharacteristic
359C22C2×C4×Dic14211yescharacteristic
360C22C2×#1455613412yes—
361C22#1798676512yes—
362C4—24no—
363C4—42no—
364C4—2844no—
365C4—1428no—
366C4#1798589614yes—
367C4C2×#1455610111yescharacteristic
368C4C2×#1455610511yescharacteristic
369C4C2×#1455611811yescharacteristic
370C4C2×#1455612711yescharacteristic
371C2—24no—
372C2C2×#1798589611yescharacteristic
373C2#2259690512yes—
374C2#2259706112yes—
375C2#2259727411yescharacteristic
376C2#2259728011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC2×#336×C71C2
prime radicalC24×C71C22×C4
cosocleC22×C4×C71C24
hypercenterC2×#336D71
hypocenterC71#21148
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
6. #22589062C2semidirect
7. #22589057C2semidirect
198. C71#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#179860252
C2×#179860232
C2×#179859542
#225908232
#225890622
#225890572
#225880362
C2×#336×C712
#2114871