Finite Group Databasebeta

← #31892502

Subgroups — #31892502

Order 18176.

768 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590954C211yesmaximal, characteristic
2#22588426C212yesmaximal, complemented
3C2×#17985723C211yesmaximal, characteristic
4#22586219C212yesmaximal, complemented
5C2×#397×C71C211yesmaximal, characteristic
6C22×C22⋊Dic71—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985874—24no—
9#17985859—21no—
10#17985859C2212yes—
11#17985761—24no—
12#17985723—22no—
13C22×C4⋊C4×C71C2211yescharacteristic
14C22×C22⋊C4×C71—21no—
15C22×C22⋊C4×C71C2211yescharacteristic
16#397×C71—22no—
17#397×C71C414yescomplemented
18C2×#95×C71—21no—
19C2×#95×C71C2211yescharacteristic
20C23×Dic71—22no—
21C22×C71⋊C8—22no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71D414yes—
24C2×C22⋊Dic71—22no—
25C2×C22⋊Dic71—22no—
26C2×C22⋊Dic71—42no—
27C2×C22⋊Dic71—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C25×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71—21no—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C412yescomplemented
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—24no—
44C2×C22⋊C4×C71—42no—
45C2×C22⋊C4×C71—42no—
46C2×C22⋊C4×C71C2×C412yes—
47C2×C22⋊C4×C71D412yes—
48#95×C71—24no—
49#95×C71—21no—
50#95×C71C2×C412yescomplemented
51C22×Dic71—24no—
52C22×Dic71—44no—
53C22×Dic71—82no—
54C22×Dic71—82no—
55C2×C71⋊C8—44no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—44no—
58C4⋊Dic71—82no—
59C22⋊Dic71—44no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71—21no—
63C24×C71—21no—
64C24×C71—21no—
65C24×C71—21no—
66C24×C71—42no—
67C24×C71—42no—
68C24×C71C22⋊C412yes—
69C24×C71D812yes—
70C24×C71SD1612yes—
71C24×C71C2×D411yescharacteristic
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71C22⋊C412yes—
89C22×C4×C71C22×C411yescharacteristic
90C22×C4×C71C2×D412yes—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71#4112yes—
93C4⋊C4×C71—24no—
94C4⋊C4×C71—42no—
95C22⋊C4×C71—24no—
96C22⋊C4×C71—24no—
97C22⋊C4×C71—48no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—22no—
104C23×C71—22no—
105C23×C71—22no—
106C23×C71—42no—
107C23×C71—42no—
108C23×C71—44no—
109C23×C71—44no—
110C23×C71—42no—
111C23×C71—42no—
112C23×C71—82no—
113C23×C71—82no—
114C23×C71—81no—
115C23×C71—81no—
116C23×C71#10214yes—
117C23×C71C2×C22⋊C411yescharacteristic
118C23×C71#12011yescharacteristic
119C23×C71#12311yescharacteristic
120C23×C71C2×D811yescharacteristic
121C23×C71C2×SD1611yescharacteristic
122C23×C71AGL(1,ℤ/8)11yescharacteristic
123C23×C71#13711yescharacteristic
124C2×C4×C71—24no—
125C2×C4×C71—22no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—81no—
138C2×C4×C71—82no—
139C2×C4×C71—82no—
140C2×C4×C71—81no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71—81no—
144C2×C4×C71C4×D412yes—
145C2×C4×C71#12112yes—
146C2×Dic71—44no—
147C2×Dic71—44no—
148C2×Dic71—44no—
149C2×Dic71—88no—
150C71⋊C8—82no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—24no—
160C22×C71—44no—
161C22×C71—44no—
162C22×C71—44no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—82no—
166C22×C71—82no—
167C22×C71—82no—
168C22×C71#38611yescharacteristic
169C22×C71C2×#10211yescharacteristic
170C22×C71#41711yescharacteristic
171C22×C71#44711yescharacteristic
172C22×C71#44811yescharacteristic
173C22×C71#45512yes—
174C22×C71#45812yes—
175C22×C71#48011yescharacteristic
176C22×C71#48111yescharacteristic
177C4×C71—22no—
178C4×C71—22no—
179C4×C71—42no—
180C4×C71—82no—
181C4×C71—82no—
182Dic71—84no—
183#9025—711nomaximal, Sylow
184C2×C71—21no—
185C2×C71—21no—
186C2×C71—21no—
187C2×C71—21no—
188C2×C71—44no—
189C2×C71#160712yes—
190C2×C71#160812yes—
191C2×C71#187411yescharacteristic
192C2×C71#188511yescharacteristic
193C2×C71#189011yescharacteristic
194C2×#397D7111yescharacteristic
195#2315—711no—
196#1823—712no—
197C2×#323—711no—
198#1294—712no—
199C71#902511yescharacteristic, complemented, Sylow
200C22×C4⋊C4C2×D7111yescharacteristic
201C22×C22⋊C4—21no—
202C22×C22⋊C4—1421no—
203C22×C22⋊C4C2×D7111yescharacteristic
204C2×C22⋊C8—712no—
205#397—22no—
206#397Dic7114yescomplemented
207#385—1424no—
208#380—712no—
209#380—1421no—
210C2×#95—21no—
211C2×#95C2×D7111yescharacteristic
212#340—1424no—
213#323—1422no—
214C25C22⋊D7111yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—1422no—
218C23×C4C22⋊D7111yescharacteristic
219C23×C4C22×D7111yescharacteristic
220C22×C8—1422no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—714no—
225C2×C4⋊C4—1422no—
226C2×C4⋊C4C2×Dic7112yescomplemented
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—2842no—
233C2×C22⋊C4—2842no—
234C2×C22⋊C4—1422no—
235C2×C22⋊C4—1422no—
236C2×C22⋊C4C2×Dic7112yes—
237C2×C22⋊C4C22⋊D7112yes—
238C22⋊C8—1422no—
239C22⋊C8—1422no—
240#95—24no—
241#95—21no—
242#95—1422no—
243#95—1422no—
244#95C2×Dic7112yescomplemented
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24C22⋊Dic7112yes—
252C24#1176585012yes—
253C24#1176585312yes—
254C24C2×C22⋊D7111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2844no—
272C22×C4—5682no—
273C22×C4—5682no—
274C22×C4—1424no—
275C22×C4C22⋊Dic7112yes—
276C22×C4C22×Dic7111yescharacteristic
277C22×C4D4×D7112yes—
278C22×C4C2×C22⋊D7111yescharacteristic
279C22×C4#1176586412yes—
280C2×C8—2844no—
281C2×C8—2842no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—2844no—
285C4⋊C4—5682no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—48no—
289C22⋊C4—2844no—
290C22⋊C4—2842no—
291C22⋊C4—2842no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—82no—
309C23—82no—
310C23#1455604914yes—
311C23C2×C22⋊Dic7111yescharacteristic
312C23#1455609611yescharacteristic
313C23#1455610911yescharacteristic
314C23C2×#1176585011yescharacteristic
315C23C2×#1176585311yescharacteristic
316C23#1455615311yescharacteristic
317C23#1455615811yescharacteristic
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—2844no—
339C2×C4—2844no—
340C2×C4—2844no—
341C2×C4—5688no—
342C2×C4D4×Dic7112yes—
343C2×C4#1455610012yes—
344C8—5682no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#1798587911yescharacteristic
363C22C2×#1455604911yescharacteristic
364C22#1798600011yescharacteristic
365C22#1798613211yescharacteristic
366C22#1798613711yescharacteristic
367C22#1798617612yes—
368C22#1798618812yes—
369C22#1798631011yescharacteristic
370C22#1798631511yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—5684no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2258732512yes—
383C2#2258733012yes—
384C2#2258865811yescharacteristic
385C2#2258872911yescharacteristic
386C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#397×C71C2
socleC23×C71#120
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenterC2×#397D71
hypocenterC71#9025
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #22588426C2semidirect
4. #22586219C2semidirect
17. #397×C71C4semidirect
40. C2×C4⋊C4×C71C2×C4semidirect
50. #95×C71C2×C4semidirect
199. C71#9025semidirect
206. #397Dic71semidirect
226. C2×C4⋊C4C2×Dic71semidirect
244. #95C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909542
#225884262
C2×#179857232
#225862192
C2×#397×C712
#902571