Finite Group Databasebeta

← #31999706

Subgroups — #31999706

Order 18176.

798 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986026C211yesmaximal, characteristic
2C2×#17985955C211yesmaximal, characteristic
3#22588941C218yesmaximal, complemented
4#22588067C211yesmaximal, characteristic
5#22588049C211yesmaximal, characteristic
6#22588049C211yesmaximal, characteristic
7#22588038C211yesmaximal, characteristic
8C71×#2911C211yesmaximal, characteristic
9C22×#11765849C2211yescharacteristic
10C22×#11765849C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13#17986026—21no—
14#17986026—21no—
15#17986026C2214yes—
16C2×C4⋊(C71⋊C8)—22no—
17#17985955—22no—
18#17985955C2214yes—
19C2×#14556039—21no—
20C2×C4×C71⋊C8—21no—
21#17985764C2214yes—
22#17985758C2214yes—
23#17985758C2214yes—
24#17985753C2214yes—
25C22×M16×C71C2211yescharacteristic
26C2×#117×C71C2211yescharacteristic
27C2×#117×C71C2211yescharacteristic
28#428×C71C414yescomplemented
29#428×C71C414yescomplemented
30#427×C71C2214yes—
31C2×#11765849—21no—
32C2×#11765849—21no—
33C2×#11765849—22no—
34C2×#11765849—22no—
35C2×#11765849—22no—
36C2×#11765849Q814yes—
37C2×#11765849C2312yes—
38C2×#11765849C2312yes—
39C22×C71⋊C8—21no—
40C22×C71⋊C8—21no—
41C22×C71⋊C8—22no—
42C22×C71⋊C8—22no—
43C22×C71⋊C8D411yescharacteristic
44C22×C71⋊C8D411yescharacteristic
45C22×C71⋊C8D412yes—
46C22×C71⋊C8C2312yes—
47#14556087—22no—
48#14556087—22no—
49C2×C4⋊Dic71—21no—
50C2×C22⋊Dic71—21no—
51C2×C4×Dic71C2312yes—
52C4⋊(C71⋊C8)—24no—
53C4⋊(C71⋊C8)—24no—
54#14556039—24no—
55C4×C71⋊C8—24no—
56C23×C4×C71C2311yescharacteristic
57C2×M16×C71—22no—
58C2×M16×C71—22no—
59C2×M16×C71C2×C412yes—
60C2×M16×C71C2×C412yes—
61C22×C8×C71C2312yes—
62#117×C71—21no—
63#117×C71—21no—
64#117×C71—21no—
65#117×C71—21no—
66#117×C71C2×C412yes—
67#117×C71C2×C412yes—
68#117×C71C2312yes—
69#117×C71C2312yes—
70C2×C4⋊C4×C71C2×C412yes—
71C2×C4⋊C4×C71C2×C412yes—
72C2×C22⋊C4×C71—21no—
73C2×C22⋊C4×C71—21no—
74C2×C42×C71—21no—
75C2×C42×C71—21no—
76#107×C71C2×C418yes—
77#106×C71C2×C418yes—
78C22×Dic71—22no—
79#11765849—24no—
80#11765849—22no—
81#11765849—22no—
82#11765849—22no—
83#11765849—42no—
84#11765849C2×Q814yes—
85C2×C71⋊C8—24no—
86C2×C71⋊C8—24no—
87C2×C71⋊C8—22no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—22no—
92C2×C71⋊C8—22no—
93C2×C71⋊C8—42no—
94C2×C71⋊C8C2×D412yes—
95C2×C71⋊C8C2×D412yes—
96C4⋊Dic71—42no—
97C22⋊Dic71—42no—
98C4×Dic71—22no—
99C4×Dic71—22no—
100C24×C71#4111yescharacteristic
101C22×C4×C71—22no—
102C22×C4×C71—22no—
103C22×C4×C71C22×C411yescharacteristic
104C22×C4×C71C22×C411yescharacteristic
105C22×C4×C71C2×D411yescharacteristic
106C22×C4×C71C2×D411yescharacteristic
107C22×C4×C71C2×Q812yes—
108C22×C4×C71#4111yescharacteristic
109C22×C4×C71#4112yes—
110C22×C4×C71#4112yes—
111C22×C4×C71#4112yes—
112C22×C4×C71C2411yescharacteristic
113M16×C71—24no—
114M16×C71—24no—
115C2×C8×C71—22no—
116C2×C8×C71—22no—
117C2×C8×C71C22×C414yes—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—22no—
120C4⋊C4×C71C22×C414yes—
121C4⋊C4×C71C22×C414yes—
122C22⋊C4×C71—22no—
123C22⋊C4×C71—22no—
124C22⋊C4×C71—22no—
125C22⋊C4×C71—22no—
126C42×C71—22no—
127C42×C71—22no—
128C42×C71—22no—
129C42×C71—22no—
130C23×C71—42no—
131C23×C71C4×D411yescharacteristic
132C23×C71C4×D411yescharacteristic
133C23×C71#12512yes—
134C23×C71#12612yes—
135C23×C71C2×#4111yescharacteristic
136C2×C4×C71—24no—
137C2×C4×C71—24no—
138C2×C4×C71—44no—
139C2×C4×C71—44no—
140C2×C4×C71#11712yes—
141C2×C4×C71#11712yes—
142C2×C4×C71C4×D411yescharacteristic
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71C4×Q812yes—
145C2×C4×C71C4×Q812yes—
146C2×C4×C71C22⋊Q812yes—
147C2×C4×C71C22⋊Q812yes—
148C2×C4×C71#12312yes—
149C2×C4×C71#12312yes—
150C2×C4×C71#12512yes—
151C2×C4×C71#12612yes—
152C2×C4×C71C23×C411yescharacteristic
153C2×C4×C71C22×D411yescharacteristic
154C2×C4×C71C22×Q811yescharacteristic
155C2×C4×C71C2×#4111yescharacteristic
156C2×C4×C71C2×#4111yescharacteristic
157C2×C4×C71C2×#4111yescharacteristic
158C2×Dic71—44no—
159C2×Dic71—42no—
160C8×C71—24no—
161C71⋊C8—24no—
162C71⋊C8—22no—
163C71⋊C8—22no—
164C71⋊C8—42no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71—22no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71#38712yes—
171C22×C71#38712yes—
172C22×C71C2×C4×D411yescharacteristic
173C22×C71C2×#12511yescharacteristic
174C22×C71C2×#12611yescharacteristic
175C4×C71—44no—
176C4×C71—44no—
177C4×C71#38712yes—
178C4×C71#38712yes—
179C4×C71C2×#11711yescharacteristic
180C4×C71C2×C4×Q811yescharacteristic
181C4×C71C2×C22⋊Q811yescharacteristic
182C4×C71C2×#12311yescharacteristic
183Dic71—82no—
184#21041—711nomaximal, Sylow
185C2×C71—21no—
186C2×C71—21no—
187C2×C71—22no—
188C2×C71—22no—
189C2×C71C2×#38711yescharacteristic
190C2×C71#295512yes—
191#2911D7111yescharacteristic
192C2×#424—711no—
193C2×#405—711no—
194#1922—718no—
195#1738—711no—
196#1733—711no—
197#1733—711no—
198#1729—711no—
199C71#2104111yescharacteristic, complemented, Sylow
200C22×M16—711no—
201C22×M16—711no—
202C22×M16C2×D7111yescharacteristic
203C23×C8—711no—
204C2×#117—711no—
205C2×#117C2×D7111yescharacteristic
206C2×#117C2×D7111yescharacteristic
207#428Dic7114yescomplemented
208#428Dic7114yescomplemented
209#427C2×D7114yes—
210#424—714no—
211#424—1421no—
212#424—1421no—
213C2×C4⋊C8—1422no—
214#405—714no—
215#405—1422no—
216C2×#97—1421no—
217C2×C4×C8—1421no—
218#341—714no—
219#339—714no—
220#339—714no—
221#336—714no—
222C23×C4C22×D7111yescharacteristic
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—714no—
226C2×M16—712no—
227C2×M16—712no—
228C2×M16—1422no—
229C2×M16—1422no—
230C2×M16—1421no—
231C2×M16—1421no—
232C2×M16—1422no—
233C2×M16C2×Dic7112yes—
234C2×M16C2×Dic7112yes—
235C22×C8—711no—
236C22×C8—711no—
237C22×C8—712no—
238C22×C8—712no—
239C22×C8—1421no—
240C22×C8—1421no—
241C22×C8—1422no—
242C22×C8—1422no—
243C22×C8C22×D7112yes—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—1422no—
249#117—1422no—
250#117C2×Dic7112yes—
251#117C2×Dic7112yes—
252#117C22×D7112yes—
253#117C22×D7112yes—
254C2×C4⋊C4—1421no—
255C2×C4⋊C4C2×Dic7112yes—
256C2×C4⋊C4C2×Dic7112yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—1421no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—712no—
263#107C2×Dic7118yes—
264#106C2×Dic7118yes—
265C4⋊C8—1424no—
266C4⋊C8—1424no—
267#97—1424no—
268C4×C8—1424no—
269C24#1176586411yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—1422no—
273C22×C4C22×Dic7111yescharacteristic
274C22×C4C22×Dic7111yescharacteristic
275C22×C4D4×D7111yescharacteristic
276C22×C4D4×D7111yescharacteristic
277C22×C4Q8×D7112yes—
278C22×C4#1176586411yescharacteristic
279C22×C4#1176586512yes—
280C22×C4#1176586512yes—
281C22×C4#1176586612yes—
282C22×C4C23×D7111yescharacteristic
283M16—24no—
284M16—24no—
285M16—2842no—
286M16—714no—
287M16—1422no—
288M16—1422no—
289M16—1422no—
290M16—1424no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—2842no—
294C2×C8—712no—
295C2×C8—712no—
296C2×C8—1422no—
297C2×C8—1422no—
298C2×C8—1422no—
299C2×C8—1422no—
300C2×C8—1424no—
301C2×C8—1424no—
302C2×C8—1422no—
303C2×C8—1422no—
304C2×C8C22×Dic7114yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—2842no—
308C4⋊C4C22×Dic7114yes—
309C4⋊C4C22×Dic7114yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—2842no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—1422no—
320C42—1422no—
321C23—42no—
322C23D4×Dic7111yescharacteristic
323C23D4×Dic7111yescharacteristic
324C23#1455611812yes—
325C23#1455612312yes—
326C23C2×#1176586411yescharacteristic
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—2844no—
332C2×C4—2842no—
333C2×C4#1455608712yes—
334C2×C4#1455608712yes—
335C2×C4D4×Dic7111yescharacteristic
336C2×C4D4×Dic7111yescharacteristic
337C2×C4Q8×Dic7112yes—
338C2×C4Q8×Dic7112yes—
339C2×C4#1455610612yes—
340C2×C4#1455610612yes—
341C2×C4#1455611312yes—
342C2×C4#1455611312yes—
343C2×C4#1455611812yes—
344C2×C4#1455612312yes—
345C2×C4C23×Dic7111yescharacteristic
346C2×C4C2×D4×D7111yescharacteristic
347C2×C4C2×Q8×D7111yescharacteristic
348C2×C4C2×#1176586511yescharacteristic
349C2×C4C2×#1176586511yescharacteristic
350C2×C4C2×#1176586611yescharacteristic
351C8—24no—
352C8—2842no—
353C8—1424no—
354C8—1422no—
355C8—1422no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22#1798588512yes—
362C22#1798588512yes—
363C22C2×D4×Dic7111yescharacteristic
364C22C2×#1455611811yescharacteristic
365C22C2×#1455612311yescharacteristic
366C4—44no—
367C4—44no—
368C4—5682no—
369C4#1798588512yes—
370C4#1798588512yes—
371C4C2×#1455608711yescharacteristic
372C4C2×Q8×Dic7111yescharacteristic
373C4C2×#1455610611yescharacteristic
374C4C2×#1455611311yescharacteristic
375C2—21no—
376C2—22no—
377C2—22no—
378C2—21no—
379C2C2×#1798588511yescharacteristic
380C2#2259724312yes—
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2911C2
socleC22×C71C2×C4×D4
prime radicalC24×C71#41
cosocleC22×C4×C71C24
hypercenter#2911D71
hypocenterC71#21041
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
3. #22588941C2semidirect
28. #428×C71C4semidirect
29. #428×C71C4semidirect
199. C71#21041semidirect
207. #428Dic71semidirect
208. #428Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860262
C2×#179859552
#225889412
#225880672
#225880492
#225880382
C71×#29112
#2104171