Finite Group Databasebeta

← #32164793

Subgroups — #32164793

Order 18176.

1598 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22594317C214yesmaximal, complemented
2#22594316C214yesmaximal, complemented
3#22594312C218yesmaximal
4#22591647C218yesmaximal, complemented
5#22590824C211yesmaximal, characteristic
6#22590823C211yesmaximal, characteristic
7#22590822C211yesmaximal, characteristic
8#22590743C211yesmaximal, characteristic
9#22590743C211yesmaximal, characteristic
10C2×#17985845C211yesmaximal, characteristic
11C71×#2296C211yesmaximal, characteristic
12C2×#14556087C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14C22×C4×Dic71C2211yescharacteristic
15C22×C4×Dic71C2211yescharacteristic
16#17985940C2218yes—
17#17985940C2218yes—
18#17985939C2218yes—
19#17985937C2218yes—
20#17985933C2218yes—
21#17985932C2218yes—
22#17985930C2218yes—
23#17985929C2218yes—
24#17985898C2214yes—
25#17985897C2214yes—
26#17985895C2214yes—
27#17985874C2214yes—
28#17985872C2214yes—
29#17985871C2214yes—
30#17985868C2212yes—
31#17985868C2212yes—
32#17985864C2212yes—
33#17985864C2212yes—
34#17985857C2214yes—
35#17985857C2214yes—
36C4×Dic71⋊C4C2214yes—
37C4×Dic71⋊C4C2214yes—
38C4×Dic71⋊C4C2214yes—
39C4×Dic71⋊C4C2214yes—
40C4×C4⋊Dic71C2214yes—
41#17985845C2212yes—
42#17985845C2212yes—
43#17985845C2214yes—
44C2×C71⋊#95C2214yes—
45C2×#117×C71C2211yescharacteristic
46C2×#117×C71C2211yescharacteristic
47C22×C42×C71C2211yescharacteristic
48#389×C71C2214yes—
49#385×C71C2214yes—
50C4×C4⋊C4×C71C2214yes—
51C23×Dic71C2312yes—
52C23×Dic71C2312yes—
53#14556087—24no—
54#14556087—24no—
55C2×C4⋊Dic71C2312yes—
56C2×C4⋊Dic71C2312yes—
57C2×C4⋊Dic71C2318yes—
58C2×Dic71⋊C4C2318yes—
59C2×Dic71⋊C4C2318yes—
60C2×Dic71⋊C4C2318yes—
61C2×Dic71⋊C4C2318yes—
62C2×C22⋊Dic71C2312yes—
63C2×C22⋊Dic71C2312yes—
64C2×C4×Dic71—24no—
65C2×C4×Dic71Q818yes—
66C2×C4×Dic71C2312yes—
67C2×C4×Dic71C2312yes—
68C2×C4×Dic71C2312yes—
69C2×C4×Dic71C2312yes—
70C2×C4×Dic71C2312yes—
71C2×C4×Dic71C2311yescharacteristic
72C2×C4×Dic71C2311yescharacteristic
73C71⋊#95C2314yes—
74C71⋊#95C2314yes—
75C71⋊#95C2314yes—
76C71⋊#95C2314yes—
77C71⋊#95C2318yes—
78C71⋊#95C2318yes—
79C71⋊#95C2318yes—
80C71⋊#95C2318yes—
81C71⋊#95C2318yes—
82C23×C4×C71C2312yes—
83C23×C4×C71C2311yescharacteristic
84#117×C71—24no—
85#117×C71—24no—
86C2×C4⋊C4×C71C2312yes—
87C2×C4⋊C4×C71C2312yes—
88C2×C4⋊C4×C71C2318yes—
89C2×C22⋊C4×C71C2312yes—
90C2×C22⋊C4×C71C2312yes—
91C2×C42×C71—24no—
92C2×C42×C71C2312yes—
93C2×C42×C71C2312yes—
94C2×C42×C71C2312yes—
95C2×C42×C71C2312yes—
96#95×C71C2318yes—
97C22×Dic71—24no—
98C22×Dic71—44no—
99C22×Dic71—44no—
100C22×Dic71C2×Q818yes—
101C22×Dic71C2414yes—
102C22×Dic71C2414yes—
103C22×Dic71C2414yes—
104C22×Dic71C2412yes—
105C22×Dic71C2412yes—
106C4⋊Dic71—24no—
107C4⋊Dic71—24no—
108C4⋊Dic71—48no—
109Dic71⋊C4—48no—
110Dic71⋊C4—48no—
111Dic71⋊C4—48no—
112Dic71⋊C4—48no—
113C22⋊Dic71—24no—
114C22⋊Dic71—24no—
115C4×Dic71—24no—
116C4×Dic71—24no—
117C4×Dic71—24no—
118C4×Dic71—24no—
119C4×Dic71C2×Q818yes—
120C4×Dic71C2×Q814yes—
121C4×Dic71C2×Q814yes—
122C24×C71C2411yescharacteristic
123C22×C4×C71—24no—
124C22×C4×C71—44no—
125C22×C4×C71C2×Q814yes—
126C22×C4×C71#4114yes—
127C22×C4×C71#4114yes—
128C22×C4×C71C2414yes—
129C22×C4×C71C2414yes—
130C22×C4×C71C2414yes—
131C22×C4×C71C2411yescharacteristic
132C22×C4×C71C2411yescharacteristic
133C4⋊C4×C71—24no—
134C4⋊C4×C71—24no—
135C4⋊C4×C71—48no—
136C22⋊C4×C71—24no—
137C22⋊C4×C71—24no—
138C42×C71—24no—
139C42×C71—24no—
140C42×C71—24no—
141C42×C71—24no—
142C23×C71—24no—
143C23×C71C22×Q812yes—
144C23×C71C2×#4112yes—
145C23×C71C2×#4112yes—
146C23×C71C2511yescharacteristic
147C2×C4×C71—24no—
148C2×C4×C71—24no—
149C2×C4×C71—24no—
150C2×C4×C71—24no—
151C2×C4×C71—28no—
152C2×C4×C71—28no—
153C2×C4×C71—24no—
154C2×C4×C71—48no—
155C2×C4×C71#125116yes—
156C2×C4×C71C22×Q812yes—
157C2×C4×C71C22×Q812yes—
158C2×C4×C71C2×#4112yes—
159C2×C4×C71C2×#4112yes—
160C2×C4×C71C2×#4112yes—
161C2×C4×C71C2×#4112yes—
162C2×Dic71—24no—
163C2×Dic71—24no—
164C2×Dic71—24no—
165C2×Dic71—24no—
166C2×Dic71—24no—
167C2×Dic71—28no—
168C2×Dic71—28no—
169C2×Dic71—24no—
170C2×Dic71—24no—
171C2×Dic71—48no—
172C2×Dic71—48no—
173C2×Dic71C22×Q814yes—
174C2×Dic71C22×Q814yes—
175C22×C71—24no—
176C22×C71—24no—
177C22×C71—24no—
178C22×C71C2×#12514yes—
179C22×C71C23×Q811yescharacteristic
180C22×C71C22×#4111yescharacteristic
181C22×C71C22×#4111yescharacteristic
182C22×C71#58514yes—
183C4×C71—44no—
184C4×C71—44no—
185C4×C71—44no—
186C4×C71C2×#12514yes—
187C4×C71C2×#12514yes—
188Dic71—24no—
189Dic71—24no—
190Dic71—44no—
191Dic71—44no—
192Dic71—44no—
193#34638—711nomaximal, Sylow
194C2×C71—24no—
195C2×C71C22×#12511yescharacteristic
196C2×C71#345512yes—
197C2×C71#345512yes—
198C2×C71#345812yes—
199#2647—714no—
200#2647—714no—
201#2646—718no—
202#2402—718no—
203#2296—711no—
204#2296—711no—
205#2296—711no—
206#2296D7111yescharacteristic
207#2285—711no—
208#2285—711no—
209C2×#376—711no—
210C71#3463811yescharacteristic, complemented, Sylow
211C2×#117—711no—
212C2×#117—711no—
213C2×#117C2×D7111yescharacteristic
214C2×#117C2×D7111yescharacteristic
215C22×C42—711no—
216C22×C42—711no—
217C22×C42C2×D7111yescharacteristic
218#400—718no—
219#400—718no—
220#400—718no—
221#400—718no—
222#398—718no—
223#398—718no—
224#398—718no—
225#398—718no—
226#389—714no—
227#389—714no—
228#389—714no—
229#389C2×D7114yes—
230#385—714no—
231#385—714no—
232#385—714no—
233#385C2×D7114yes—
234#384—712no—
235#384—712no—
236#382—712no—
237#382—712no—
238#380—714no—
239#380—714no—
240C4×C4⋊C4—714no—
241C4×C4⋊C4—714no—
242C4×C4⋊C4—714no—
243C4×C4⋊C4—714no—
244C4×C4⋊C4—714no—
245C4×C4⋊C4C2×D7114yes—
246#376—712no—
247#376—712no—
248#376—714no—
249C2×#95—714no—
250C23×C4—712no—
251C23×C4—712no—
252C23×C4C22×D7112yes—
253C23×C4C22×D7111yescharacteristic
254#117—24no—
255#117—24no—
256#117—1424no—
257#117—1424no—
258C2×C4⋊C4—718no—
259C2×C4⋊C4—718no—
260C2×C4⋊C4—718no—
261C2×C4⋊C4—718no—
262C2×C4⋊C4—718no—
263C2×C4⋊C4—712no—
264C2×C4⋊C4—712no—
265C2×C4⋊C4C22×D7112yes—
266C2×C4⋊C4C22×D7112yes—
267C2×C4⋊C4C22×D7118yes—
268C2×C22⋊C4—712no—
269C2×C22⋊C4—712no—
270C2×C22⋊C4C22×D7112yes—
271C2×C22⋊C4C22×D7112yes—
272C2×C42—24no—
273C2×C42—718no—
274C2×C42—711no—
275C2×C42—711no—
276C2×C42—712no—
277C2×C42—712no—
278C2×C42—712no—
279C2×C42—712no—
280C2×C42—712no—
281C2×C42—1424no—
282C2×C42C22×D7112yes—
283C2×C42C22×D7112yes—
284C2×C42C22×D7112yes—
285C2×C42C22×D7112yes—
286#95—714no—
287#95—714no—
288#95—714no—
289#95—714no—
290#95—718no—
291#95—718no—
292#95—718no—
293#95—718no—
294#95—718no—
295#95C22×D7118yes—
296C24C23×D7111yescharacteristic
297C22×C4—24no—
298C22×C4—44no—
299C22×C4—2844no—
300C22×C4—2844no—
301C22×C4—718no—
302C22×C4—712no—
303C22×C4—712no—
304C22×C4—714no—
305C22×C4—714no—
306C22×C4—714no—
307C22×C4—1424no—
308C22×C4Q8×D7114yes—
309C22×C4#1176586414yes—
310C22×C4#1176586514yes—
311C22×C4C23×D7114yes—
312C22×C4C23×D7114yes—
313C22×C4C23×D7114yes—
314C22×C4C23×D7111yescharacteristic
315C22×C4C23×D7111yescharacteristic
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—48no—
319C4⋊C4—2848no—
320C4⋊C4—2848no—
321C4⋊C4—2848no—
322C4⋊C4—2848no—
323C4⋊C4—2848no—
324C4⋊C4—1424no—
325C4⋊C4—1424no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—1424no—
329C22⋊C4—1424no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—714no—
335C42—714no—
336C42—718no—
337C42—1424no—
338C42—1424no—
339C42—1424no—
340C42—1424no—
341C23—24no—
342C23C2×Q8×D7112yes—
343C23C2×#1176586412yes—
344C23C2×#1176586512yes—
345C23C24×D7111yescharacteristic
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—48no—
354C2×C4—2848no—
355C2×C4—2848no—
356C2×C4—714no—
357C2×C4—714no—
358C2×C4—1424no—
359C2×C4—1424no—
360C2×C4—1424no—
361C2×C4—1428no—
362C2×C4—1428no—
363C2×C4—1424no—
364C2×C4—1424no—
365C2×C4—1424no—
366C2×C4—1424no—
367C2×C4#14556118116yes—
368C2×C4C2×Q8×D7112yes—
369C2×C4C2×Q8×D7112yes—
370C2×C4C2×#1176586412yes—
371C2×C4C2×#1176586412yes—
372C2×C4C2×#1176586512yes—
373C2×C4C2×#1176586512yes—
374C22—24no—
375C22—24no—
376C22—24no—
377C22C2×#1455611814yes—
378C22C22×Q8×D7111yescharacteristic
379C22C22×#1176586411yescharacteristic
380C22C22×#1176586511yescharacteristic
381C22#1798683514yes—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—2844no—
386C4—2844no—
387C4—2844no—
388C4—1424no—
389C4—1424no—
390C4C2×#1455611814yes—
391C4C2×#1455611814yes—
392C2—24no—
393C2C22×#1455611811yescharacteristic
394C2#2260172512yes—
395C2#2260172812yes—
396C2#2260174312yes—
subgroupG/H
centerC22×C4C23×D71
commutator, socle, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC71×#2296C2
prime radicalC24×C71C24
hypercenter#2296D71
hypocenterC71#34638
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
1. #22594317C2semidirect
2. #22594316C2semidirect
4. #22591647C2semidirect
210. C71#34638semidirect

Maximal subgroups

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

subgroupindex
#225943172
#225943162
#225943122
#225916472
#225908242
#225908232
#225908222
#225907432
C2×#179858452
C71×#22962
#3463871