Finite Group Databasebeta

← #32026522

Subgroups — #32026522

Order 18176.

874 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556061C211yesmaximal, characteristic
2C2×#17986027C211yesmaximal, characteristic
3#22590964C211yesmaximal, characteristic
4#22590963C211yesmaximal, characteristic
5#22589837C214yesmaximal, complemented
6#22589832C214yesmaximal, complemented
7#22588060C211yesmaximal, characteristic
8#22588056C211yesmaximal, characteristic
9C71×#1736C211yesmaximal, characteristic
10C23×C71⋊C8C2211yescharacteristic
11C2×#14556087C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C2×#14556061—24no—
15C2×#14556061C2212yes—
16C2×#14556061C2212yes—
17#17986027—24no—
18C2×Dic71⋊C8C2212yes—
19C2×Dic71⋊C8C2212yes—
20#17985898C2214yes—
21#17985895C2214yes—
22#17985874—22no—
23#17985872—22no—
24#17985869—21no—
25#17985868—21no—
26#17985865—21no—
27#17985864—21no—
28#17985762C2214yes—
29#17985761C2212yes—
30#17985761C2212yes—
31C22×M16×C71C2211yescharacteristic
32C2×#117×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34#340×C71C2214yes—
35C23×Dic71—21no—
36C22×C71⋊C8—22no—
37C22×C71⋊C8—22no—
38C22×C71⋊C8D414yes—
39C22×C71⋊C8C2311yescharacteristic
40C22×C71⋊C8C2311yescharacteristic
41#14556087—22no—
42#14556087—22no—
43#14556087—22no—
44#14556087Q814yes—
45C2×C4⋊Dic71—24no—
46C2×C4⋊Dic71—22no—
47C2×C4⋊Dic71—21no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71D411yescharacteristic
50C2×C4⋊Dic71D411yescharacteristic
51C2×C4⋊Dic71C2312yes—
52C2×C4⋊Dic71C2312yes—
53C2×Dic71⋊C4—24no—
54C2×C22⋊Dic71—21no—
55C2×C22⋊Dic71D412yes—
56C2×C4×Dic71—21no—
57C2×C4×Dic71C2312yes—
58#14556061—24no—
59#14556061—24no—
60Dic71⋊C8—24no—
61Dic71⋊C8—24no—
62C71⋊#95—24no—
63C71⋊#95—24no—
64C23×C4×C71—21no—
65C23×C4×C71C2311yescharacteristic
66C2×M16×C71—22no—
67C2×M16×C71—22no—
68C2×M16×C71—22no—
69C22×C8×C71C2312yes—
70#117×C71—22no—
71#117×C71Q814yes—
72C2×C4⋊C4×C71—22no—
73C2×C4⋊C4×C71—22no—
74C2×C4⋊C4×C71—21no—
75C2×C4⋊C4×C71C2312yes—
76C2×C4⋊C4×C71C2312yes—
77C2×C22⋊C4×C71—21no—
78C2×C42×C71—21no—
79C22×Dic71—22no—
80C22×Dic71—22no—
81C22×Dic71—42no—
82C22×Dic71—82no—
83C22×Dic71C2×D412yes—
84C22×Dic71C2×D412yes—
85C2×C71⋊C8—28no—
86C2×C71⋊C8—21no—
87C2×C71⋊C8—21no—
88C2×C71⋊C8—21no—
89C2×C71⋊C8—21no—
90C2×C71⋊C8C2×D412yes—
91C2×C71⋊C8C2×D412yes—
92C4⋊Dic71—22no—
93C4⋊Dic71—22no—
94C4⋊Dic71—22no—
95C4⋊Dic71—22no—
96C4⋊Dic71—22no—
97C4⋊Dic71—42no—
98C4⋊Dic71—84no—
99C4⋊Dic71C2×Q814yes—
100Dic71⋊C4—84no—
101C22⋊Dic71—24no—
102C22⋊Dic71—24no—
103C4×Dic71—24no—
104C4×Dic71—22no—
105C4×Dic71—22no—
106C24×C71C2×D411yescharacteristic
107C22×C4×C71—22no—
108C22×C4×C71—22no—
109C22×C4×C71—42no—
110C22×C4×C71—82no—
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2×D412yes—
113C22×C4×C71C2×Q812yes—
114C22×C4×C71C2×Q812yes—
115C22×C4×C71#4112yes—
116C22×C4×C71#4112yes—
117C22×C4×C71#4112yes—
118C22×C4×C71C2411yescharacteristic
119M16×C71—44no—
120C2×C8×C71—22no—
121C2×C8×C71—22no—
122C2×C8×C71—22no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—22no—
125C4⋊C4×C71—22no—
126C4⋊C4×C71—42no—
127C4⋊C4×C71C2×Q814yes—
128C22⋊C4×C71—24no—
129C42×C71—24no—
130C23×C71—22no—
131C23×C71—22no—
132C23×C71#12112yes—
133C23×C71C22⋊Q812yes—
134C23×C71C22⋊Q812yes—
135C23×C71C22×D411yescharacteristic
136C2×C4×C71—24no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—44no—
141C2×C4×C71—84no—
142C2×C4×C71#12112yes—
143C2×C4×C71C22⋊Q812yes—
144C2×C4×C71C22⋊Q812yes—
145C2×C4×C71#12314yes—
146C2×C4×C71#12514yes—
147C2×C4×C71#12514yes—
148C2×C4×C71C4⋊Q814yes—
149C2×C4×C71C22×D411yescharacteristic
150C2×C4×C71C22×Q811yescharacteristic
151C2×C4×C71C22×Q811yescharacteristic
152C2×C4×C71C2×#4111yescharacteristic
153C2×C4×C71C2×#4111yescharacteristic
154C2×C4×C71C2×#4111yescharacteristic
155C2×Dic71—22no—
156C2×Dic71—22no—
157C2×Dic71—24no—
158C2×Dic71—24no—
159C2×Dic71—24no—
160C2×Dic71—42no—
161C2×Dic71—42no—
162C2×Dic71—42no—
163C2×Dic71—44no—
164C2×Dic71—84no—
165C8×C71—42no—
166C71⋊C8—22no—
167C71⋊C8—22no—
168C22×C71—24no—
169C22×C71—24no—
170C22×C71—22no—
171C22×C71—22no—
172C22×C71#39814yes—
173C22×C71C2×#12111yescharacteristic
174C22×C71C2×C22⋊Q811yescharacteristic
175C22×C71C2×C22⋊Q811yescharacteristic
176C22×C71#57612yes—
177C22×C71#57812yes—
178C4×C71—44no—
179C4×C71—82no—
180C4×C71#39814yes—
181C4×C71C2×#12311yescharacteristic
182C4×C71C2×#12511yescharacteristic
183C4×C71C2×#12511yescharacteristic
184C4×C71C2×C4⋊Q811yescharacteristic
185Dic71—44no—
186Dic71—42no—
187Dic71—42no—
188Dic71—82no—
189#23667—711nomaximal, Sylow
190C2×C71—24no—
191C2×C71C2×#39811yescharacteristic
192C2×C71#305811yescharacteristic
193C2×C71#305911yescharacteristic
194C2×C71#308312yes—
195C2×C71#308312yes—
196C22×#107—711no—
197C2×#424—711no—
198#2317—711no—
199#2317—711no—
200#2087—714no—
201#2086—714no—
202#1736—711no—
203#1736D7111yescharacteristic
204#1735—711no—
205C71#2366711yescharacteristic, complemented, Sylow
206C22×M16C2×D7111yescharacteristic
207C23×C8—711no—
208C2×#117—711no—
209C2×#117—711no—
210C2×#117C2×D7111yescharacteristic
211C22×C4⋊C4—711no—
212C22×C4⋊C4C2×D7111yescharacteristic
213C2×#107—712no—
214C2×#107—712no—
215C2×#107—1424no—
216#424—1424no—
217C2×C4⋊C8—712no—
218C2×C4⋊C8—712no—
219#389—714no—
220#389—714no—
221#385—1422no—
222#385—1422no—
223#384—1421no—
224#384—1421no—
225#382—1421no—
226#382—1421no—
227#340—714no—
228#340—712no—
229#340—712no—
230#340C2×D7114yes—
231C23×C4—21no—
232C23×C4—1421no—
233C23×C4C22×D7111yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C22×C8—714no—
238C22×C8—711no—
239C22×C8—711no—
240C22×C8—1422no—
241C22×C8—1422no—
242C22×C8C22×D7112yes—
243#117—22no—
244#117—714no—
245#117—1422no—
246#117—1422no—
247#117—1422no—
248#117Dic14214yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—712no—
253C2×C4⋊C4—711no—
254C2×C4⋊C4—711no—
255C2×C4⋊C4—712no—
256C2×C4⋊C4—1421no—
257C2×C4⋊C4—1422no—
258C2×C4⋊C4—1422no—
259C2×C4⋊C4—1424no—
260C2×C4⋊C4—1424no—
261C2×C4⋊C4C22×D7112yes—
262C2×C4⋊C4C22×D7112yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—712no—
265C2×C22⋊C4—1421no—
266C2×C42—21no—
267C2×C42—712no—
268C2×C42—1421no—
269#107—1424no—
270#107—1424no—
271C4⋊C8—1424no—
272C4⋊C8—1424no—
273#95—1424no—
274#95—1424no—
275C24D4×D7111yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—2842no—
281C22×C4—5682no—
282C22×C4—712no—
283C22×C4—712no—
284C22×C4—1422no—
285C22×C4—1422no—
286C22×C4D4×D7112yes—
287C22×C4D4×D7111yescharacteristic
288C22×C4Q8×D7112yes—
289C22×C4C2×Dic14212yes—
290C22×C4#1176586412yes—
291C22×C4#1176586512yes—
292C22×C4#1176586512yes—
293C22×C4C23×D7111yescharacteristic
294M16—44no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—712no—
299C2×C8—712no—
300C2×C8—1421no—
301C2×C8—1421no—
302C2×C8—1421no—
303C2×C8—1421no—
304C2×C8—1428no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—2842no—
310C4⋊C4—5684no—
311C4⋊C4—5684no—
312C4⋊C4—714no—
313C4⋊C4—1422no—
314C4⋊C4—1422no—
315C4⋊C4—1422no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4C2×Dic14214yes—
319C22⋊C4—24no—
320C22⋊C4—1424no—
321C22⋊C4—1424no—
322C42—24no—
323C42—1422no—
324C42—1422no—
325C42—1424no—
326C23—22no—
327C23—22no—
328C23#1455610112yes—
329C23#1455610512yes—
330C23#1455610612yes—
331C23C2×D4×D7111yescharacteristic
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—2842no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4—2844no—
342C2×C4—5684no—
343C2×C4—1422no—
344C2×C4—1422no—
345C2×C4—1424no—
346C2×C4—1424no—
347C2×C4—1424no—
348C2×C4#1455610112yes—
349C2×C4#1455610512yes—
350C2×C4#1455610612yes—
351C2×C4#1455611014yes—
352C2×C4#1455611814yes—
353C2×C4#1455612014yes—
354C2×C4#1455612714yes—
355C2×C4C2×D4×D7111yescharacteristic
356C2×C4C2×Q8×D7111yescharacteristic
357C2×C4C22×Dic14211yescharacteristic
358C2×C4C2×#1176586411yescharacteristic
359C2×C4C2×#1176586511yescharacteristic
360C2×C4C2×#1176586511yescharacteristic
361C8—42no—
362C8—1422no—
363C8—1422no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22#1798593314yes—
369C22C2×#1455610111yescharacteristic
370C22C2×#1455610511yescharacteristic
371C22C2×#1455610611yescharacteristic
372C22#1798680312yes—
373C22#1798681512yes—
374C4—44no—
375C4—82no—
376C4—2844no—
377C4—2842no—
378C4—2842no—
379C4—5682no—
380C4#1798593314yes—
381C4C2×#1455611011yescharacteristic
382C4C2×#1455611811yescharacteristic
383C4C2×#1455612011yescharacteristic
384C4C2×#1455612711yescharacteristic
385C2—24no—
386C2C2×#1798593311yescharacteristic
387C2#2259798711yescharacteristic
388C2#2259799411yescharacteristic
389C2#2259819012yes—
390C2#2259819512yes—
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#1736C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#1736D71
hypocenterC71#23667
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
5. #22589837C2semidirect
6. #22589832C2semidirect
205. C71#23667semidirect

Maximal subgroups

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

subgroupindex
C22×#145560612
C2×#179860272
#225909642
#225909632
#225898372
#225898322
#225880602
#225880562
C71×#17362
#2366771