Finite Group Databasebeta

← #32000768

Subgroups — #32000768

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×#17986027C211yesmaximal, characteristic
2C2×#17986026C211yesmaximal, characteristic
3C2×#17986022C211yesmaximal, characteristic
4C2×#17985954C211yesmaximal, characteristic
5#22590822C211yesmaximal, characteristic
6#22589062C214yesmaximal, complemented
7#22589057C214yesmaximal, complemented
8C2×#17985753C211yesmaximal, characteristic
9C71×#1729C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C22×C4×Dic71C2211yescharacteristic
14#17986027—24no—
15#17986026—24no—
16#17986022—24no—
17C2×C4⋊(C71⋊C8)C2214yes—
18C2×Dic71⋊C8C2214yes—
19C2×Dic71⋊C8C2214yes—
20#17985954—24no—
21C2×#14556038C2212yes—
22C2×C8×Dic71C2212yes—
23#17985895C412yes—
24#17985895C412yes—
25#17985871C414yes—
26C4×Dic71⋊C4C2214yes—
27#17985753C2212yes—
28#17985753C2212yes—
29C22×M16×C71C2211yescharacteristic
30C23×C8×C71C2211yescharacteristic
31C2×#117×C71C2211yescharacteristic
32#336×C71C2212yes—
33#336×C71C2212yes—
34C23×Dic71C2×C412yes—
35C2×#11765849—22no—
36C2×#11765849—22no—
37C2×#11765849D414yes—
38C22×C71⋊C8—22no—
39C22×C71⋊C8—22no—
40C22×C71⋊C8—22no—
41C22×C71⋊C8C2312yes—
42C22×C71⋊C8C2312yes—
43#14556087D414yes—
44#14556087Q814yes—
45C2×C4⋊Dic71C2×C412yes—
46C2×Dic71⋊C4C2×C414yes—
47C2×Dic71⋊C4C2×C414yes—
48C2×C22⋊Dic71C2×C412yes—
49C2×C4×Dic71—22no—
50C2×C4×Dic71—22no—
51C2×C4×Dic71C2×C411yescharacteristic
52C2×C4×Dic71C2×C411yescharacteristic
53C2×C4×Dic71C2312yes—
54C2×C4×Dic71C2312yes—
55C4⋊(C71⋊C8)—28no—
56Dic71⋊C8—28no—
57Dic71⋊C8—28no—
58#14556038—24no—
59C8×Dic71—24no—
60C71⋊#95C2×C414yes—
61C71⋊#95C2×C414yes—
62C23×C4×C71C2311yescharacteristic
63C2×M16×C71—22no—
64C2×M16×C71—22no—
65C2×M16×C71—22no—
66C22×C8×C71—22no—
67C22×C8×C71—22no—
68C22×C8×C71Q814yes—
69C22×C8×C71C2312yes—
70C22×C8×C71C2311yescharacteristic
71C22×C8×C71C2311yescharacteristic
72#117×C71—22no—
73#117×C71—22no—
74C2×C4⋊C4×C71C2×C412yes—
75C2×C22⋊C4×C71C2×C412yes—
76C2×C42×C71C2312yes—
77C22×Dic71—24no—
78C22×Dic71—24no—
79C22×Dic71—42no—
80C22×Dic71C22×C412yes—
81C22×Dic71C22×C414yes—
82C22×Dic71C22×C412yes—
83#11765849—28no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8—22no—
86C2×C71⋊C8—22no—
87C2×C71⋊C8—22no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8—44no—
90C2×C71⋊C8C2×D414yes—
91C4⋊Dic71C4⋊C418yes—
92Dic71⋊C4—28no—
93Dic71⋊C4—44no—
94C22⋊Dic71C4⋊C418yes—
95C4×Dic71—22no—
96C4×Dic71—22no—
97C4×Dic71—22no—
98C4×Dic71—22no—
99C4×Dic71C2×D414yes—
100C4×Dic71C2×Q814yes—
101C24×C71C22×C411yescharacteristic
102C22×C4×C71C22×C414yes—
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
111M16×C71—44no—
112C2×C8×C71—28no—
113C2×C8×C71—21no—
114C2×C8×C71—21no—
115C2×C8×C71—21no—
116C2×C8×C71—21no—
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71—22no—
120C2×C8×C71C2×Q812yes—
121C2×C8×C71C2×Q812yes—
122C4⋊C4×C71—24no—
123C22⋊C4×C71—24no—
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—24no—
133C2×C4×C71—24no—
134C2×C4×C71—24no—
135C2×C4×C71C2×C4⋊C412yes—
136C2×C4×C71C4×D412yes—
137C2×C4×C71C4×Q812yes—
138C2×C4×C71#12114yes—
139C2×C4×C71C22⋊Q814yes—
140C2×C4×C71#12514yes—
141C2×C4×C71C4⋊Q814yes—
142C2×C4×C71C22×D411yescharacteristic
143C2×C4×C71C22×D411yescharacteristic
144C2×C4×C71C22×Q811yescharacteristic
145C2×C4×C71C22×Q811yescharacteristic
146C2×C4×C71C2×#4111yescharacteristic
147C2×C4×C71C2×#4111yescharacteristic
148C2×Dic71—22no—
149C2×Dic71—24no—
150C2×Dic71—24no—
151C2×Dic71—22no—
152C2×Dic71—22no—
153C2×Dic71—22no—
154C2×Dic71—22no—
155C2×Dic71—48no—
156C2×Dic71C2×C4⋊C418yes—
157C2×Dic71#13114yes—
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—44no—
174C4×C71#38914yes—
175C4×C71C2×#12111yescharacteristic
176C4×C71C2×C22⋊Q811yescharacteristic
177C4×C71C2×#12511yescharacteristic
178C4×C71C2×C4⋊Q811yescharacteristic
179Dic71—28no—
180Dic71—24no—
181Dic71—42no—
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#1729D7111yescharacteristic
197C2×#336—711no—
198C71#2114811yescharacteristic, complemented, Sylow
199C22×M16—711no—
200C22×M16C2×D7111yescharacteristic
201C23×C8—711no—
202C23×C8C2×D7111yescharacteristic
203C2×#117—711no—
204C2×#117C2×D7111yescharacteristic
205C22×C42—711no—
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×C4—712no—
224C23×C4C22×D7111yescharacteristic
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—714no—
229C2×M16—1422no—
230C2×M16—1422no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—712no—
234C22×C8—712no—
235C22×C8—1422no—
236C22×C8—1422no—
237C22×C8—1422no—
238C22×C8Dic14214yes—
239C22×C8C22×D7112yes—
240C22×C8C22×D7111yescharacteristic
241C22×C8C22×D7111yescharacteristic
242#117—22no—
243#117—22no—
244#117—714no—
245#117—714no—
246C2×C4⋊C4—714no—
247C2×C4⋊C4—712no—
248C2×C4⋊C4—714no—
249C2×C4⋊C4C4×D7112yes—
250C2×C22⋊C4—712no—
251C2×C22⋊C4C4×D7112yes—
252C2×C42—712no—
253C2×C42—711no—
254C2×C42—711no—
255C2×C42—712no—
256C2×C42—1422no—
257C2×C42—1422no—
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—2842no—
268C22×C4—712no—
269C22×C4—714no—
270C22×C4—712no—
271C22×C4—1424no—
272C22×C4—1424no—
273C22×C4C2×C4×D7114yes—
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—44no—
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—2844no—
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—24no—
302C4⋊C4—2844no—
303C4⋊C4—718no—
304C4⋊C4—1428no—
305C22⋊C4—24no—
306C22⋊C4—718no—
307C42—22no—
308C42—22no—
309C42—714no—
310C42—714no—
311C42—1422no—
312C42—1422no—
313C42—1422no—
314C42—1422no—
315C23—22no—
316C23—22no—
317C23C4⋊C4×D7112yes—
318C23(C22⋊D71)⋊C412yes—
319C23C4×Dic14212yes—
320C23C22×C4×D7111yescharacteristic
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—2848no—
325C2×C4—718no—
326C2×C4—714no—
327C2×C4—1422no—
328C2×C4—1422no—
329C2×C4—1422no—
330C2×C4—1422no—
331C2×C4—1424no—
332C2×C4—1424no—
333C2×C4—1422no—
334C2×C4C4⋊C4×D7112yes—
335C2×C4(C22⋊D71)⋊C412yes—
336C2×C4C4×Dic14212yes—
337C2×C4#1455610114yes—
338C2×C4#1455610514yes—
339C2×C4#1455611814yes—
340C2×C4#1455612714yes—
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
360C22#131×D7112yes—
361C22#1798676412yes—
362C4—44no—
363C4—2842no—
364C4—1428no—
365C4—1424no—
366C4#1798589614yes—
367C4C2×#1455610111yescharacteristic
368C4C2×#1455610511yescharacteristic
369C4C2×#1455611811yescharacteristic
370C4C2×#1455612711yescharacteristic
371C2—24no—
372C2C2×#1798589611yescharacteristic
373C2#2259690712yes—
374C2#2259707012yes—
375C2#2259727211yescharacteristic
376C2#2259727811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1729C2
prime radicalC24×C71C22×C4
cosocleC22×C4×C71C24
hypercenter#1729D71
hypocenterC71#21148
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1729

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×#179860272
C2×#179860262
C2×#179860222
C2×#179859542
#225908222
#225890622
#225890572
C2×#179857532
C71×#17292
#2114871