Finite Group Databasebeta

← #31914397

Subgroups — #31914397

Order 18176.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986237C211yesmaximal, characteristic
2C2×#107×D71C211yesmaximal, characteristic
3C2×#17985803C211yesmaximal, characteristic, complemented
4C2×(C2×D71)⋊C16C211yesmaximal, characteristic
5#22586432C211yesmaximal, characteristic, complemented
6#22586431C211yesmaximal, characteristic, complemented
7C2×#358×C71C211yesmaximal, characteristic, complemented
8C22×C8×D71C2211yescharacteristic
9C2×#14556104—21no—
10C2×C4⋊C4×D71—21no—
11C22×C71⋊C16C2211yescharacteristic
12#17986237—22no—
13#17986237C414yescomplemented
14C2×#14556062C2211yescharacteristic
15C2×#14556062C2211yescharacteristic
16C2×#14556061—21no—
17#107×D71—22no—
18#107×D71—22no—
19C2×#14556052—21no—
20#17985803—22no—
21(C2×D71)⋊C16—22no—
22C22×Q16×C71C2211yescharacteristic
23C22×C16×C71C2211yescharacteristic
24C2×#107×C71C2211yescharacteristic
25#358×C71—22no—
26C22×C4×D71—41no—
27C22×C4×D71D411yescharacteristic
28C22×C71⋊C8D411yescharacteristic
29C2×C8×D71—21no—
30C2×C8×D71—21no—
31C2×C8×D71—21no—
32C2×C8×D71—21no—
33C2×C8×D71C2×C412yes—
34C2×C8×D71D412yes—
35#14556104—24no—
36#14556104—42no—
37C2×C4⋊Dic71—21no—
38C2×C4⋊Dic71—21no—
39C2×Dic71⋊C4—41no—
40C2×Dic71⋊C4—41no—
41C4⋊C4×D71—24no—
42C4⋊C4×D71—42no—
43C2×(C2×D71)⋊C4—41no—
44C2×C71⋊C16—21no—
45C2×C71⋊C16—21no—
46C2×C71⋊C16—21no—
47#14556062—21no—
48#14556062—21no—
49#14556062C2×C412yescomplemented
50#14556062D412yes—
51#14556061—24no—
52#14556052—24no—
53C22×Q8×C71—21no—
54C2×Q16×C71—21no—
55C2×Q16×C71—21no—
56C2×Q16×C71—21no—
57C2×Q16×C71—21no—
58C2×Q16×C71—21no—
59C2×Q16×C71C2×C412yescomplemented
60C22×C8×C71C2311yescharacteristic
61C2×C4⋊C4×C71—21no—
62C2×C16×C71—21no—
63C2×C16×C71—21no—
64C2×C16×C71—21no—
65#107×C71—21no—
66#107×C71D412yes—
67C23×D71D811yescharacteristic
68C22×Dic71—41no—
69C22×Dic71—41no—
70C22×Dic71D811yescharacteristic
71C2×C4×D71—21no—
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74C2×C4×D71—21no—
75C2×C4×D71—44no—
76C2×C4×D71—82no—
77C2×C4×D71—82no—
78C2×C4×D71C22⋊C412yes—
79C2×C4×D71SD1612yes—
80C2×C71⋊C8—21no—
81C2×C71⋊C8—21no—
82C2×C71⋊C8C22⋊C412yes—
83C8×D71—22no—
84C8×D71—22no—
85C8×D71—22no—
86C8×D71—22no—
87C4⋊Dic71—22no—
88C4⋊Dic71—22no—
89C4⋊Dic71—41no—
90C4⋊Dic71—41no—
91Dic71⋊C4—44no—
92Dic71⋊C4—44no—
93(C2×D71)⋊C4—44no—
94C71⋊C16—41no—
95C2×Q8×C71—22no—
96C2×Q8×C71—41no—
97C2×Q8×C71—41no—
98C2×Q8×C71—41no—
99C2×Q8×C71—41no—
100C2×Q8×C71—41no—
101C22×C4×C71—41no—
102C22×C4×C71—41no—
103C22×C4×C71C2×D411yescharacteristic
104Q16×C71—22no—
105Q16×C71—22no—
106Q16×C71—41no—
107Q16×C71—41no—
108C2×C8×C71C22×C411yescharacteristic
109C2×C8×C71C2×D411yescharacteristic
110C2×C8×C71C2×D411yescharacteristic
111C2×C8×C71C2×D411yescharacteristic
112C2×C8×C71#4111yescharacteristic
113C2×C8×C71#4111yescharacteristic
114C4⋊C4×C71—22no—
115C4⋊C4×C71—41no—
116C16×C71—41no—
117C23×C71C2×D811yescharacteristic
118C22×D71—21no—
119C22×D71—21no—
120C22×D71—21no—
121C22×D71—21no—
122C22×D71#10212yes—
123C22×D71SD3212yes—
124C22×D71Q3212yes—
125Q8×C71—42no—
126Q8×C71—42no—
127Q8×C71—81no—
128Q8×C71—81no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71—81no—
134C2×C4×C71—81no—
135C2×C4×C71C2×C22⋊C411yescharacteristic
136C2×C4×C71#12011yescharacteristic
137C2×C4×C71#12311yescharacteristic
138C2×C4×C71C2×SD1611yescharacteristic
139C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C71#13711yescharacteristic
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—42no—
144C2×Dic71—42no—
145C2×Dic71—81no—
146C2×Dic71—81no—
147C2×Dic71—81no—
148C2×Dic71—81no—
149C2×Dic71#10212yes—
150C4×D71—22no—
151C4×D71—22no—
152C4×D71—22no—
153C4×D71—22no—
154C4×D71—88no—
155C8×C71C4×D411yescharacteristic
156C8×C71C4×D411yescharacteristic
157C8×C71#12111yescharacteristic
158C8×C71#12111yescharacteristic
159C71⋊C8—22no—
160C22×C71C2×#10211yescharacteristic
161C22×C71#44711yescharacteristic
162C22×C71#48011yescharacteristic
163C22×C71C2×SD3211yescharacteristic
164C22×C71C2×Q3211yescharacteristic
165C22×C71#50911yescharacteristic
166C22×C71#51011yescharacteristic
167C2×D71—22no—
168C2×D71—22no—
169C2×D71—22no—
170C2×D71—22no—
171C2×D71—22no—
172C2×D71—22no—
173C2×D71#35814yescomplemented
174C4×C71—82no—
175C4×C71—82no—
176C4×C71#38611yescharacteristic
177C4×C71#41711yescharacteristic
178C4×C71#44811yescharacteristic
179C4×C71#48111yescharacteristic
180Dic71—22no—
181Dic71—82no—
182Dic71—82no—
183#12993—711nomaximal, Sylow
184C2×C71#187411yescharacteristic
185C2×C71C2×#35811yescharacteristic
186C2×C71#213911yescharacteristic
187C2×C71#218711yescharacteristic
188C2×C71#218811yescharacteristic
189C2×C71#223511yescharacteristic
190C2×C71#223611yescharacteristic
191D71—24no—
192C2×#467—711no—
193C22×#107—711no—
194C2×#358—711no—
195C2×#358D7111yescharacteristic, complemented
196C2×C22⋊C16—711no—
197#1380—711no—
198#1380—711no—
199C71#1299311yescharacteristic, complemented, Sylow
200C22×Q16C2×D7111yescharacteristic
201C23×C8—711no—
202C2×C22⋊Q8—1421no—
203C22×C4⋊C4—1421no—
204C22×C16—711no—
205C22×C16C2×D7111yescharacteristic
206#467—714no—
207#467—1422no—
208C2×#107—711no—
209C2×#107—711no—
210C2×#107—1422no—
211C2×#107—1421no—
212C2×#107—1422no—
213C2×#107C2×D7111yescharacteristic
214C2×#103—1421no—
215#358—22no—
216#358—1422no—
217C22⋊C16—1422no—
218C22×Q8—21no—
219C23×C4—2841no—
220C23×C4—711no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16C4×D7112yescomplemented
227C22×C8—711no—
228C22×C8—712no—
229C22×C8—712no—
230C22×C8—1421no—
231C22×C8—1421no—
232C22×C8—1421no—
233C22×C8—1421no—
234C22×C8C22×D7111yescharacteristic
235C22⋊Q8—2842no—
236C22⋊Q8—1424no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—2842no—
239C2×C4⋊C4—2841no—
240C2×C4⋊C4—2841no—
241C2×C4⋊C4—1424no—
242C2×C4⋊C4—1421no—
243C2×C4⋊C4—1421no—
244C2×C22⋊C4—2841no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1421no—
249C2×C16—1421no—
250C2×C16—1421no—
251#107—21no—
252#107—712no—
253#107—712no—
254#107—1424no—
255#107—1421no—
256#107—1421no—
257#107C22⋊D7112yes—
258#103—1424no—
259C24—711no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—2844no—
269C22×C4—2841no—
270C22×C4—2841no—
271C22×C4—5682no—
272C22×C4—5682no—
273C22×C4—711no—
274C22×C4—712no—
275C22×C4—712no—
276C22×C4—1421no—
277C22×C4—1421no—
278C22×C4—1421no—
279C22×C4—1421no—
280C22×C4D4×D7111yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285C2×C8—712no—
286C2×C8—1421no—
287C2×C8—1421no—
288C2×C8—1422no—
289C2×C8—1422no—
290C2×C8—1422no—
291C2×C8—1422no—
292C2×C8C2×C4×D7111yescharacteristic
293C2×C8D4×D7111yescharacteristic
294C2×C8D4×D7111yescharacteristic
295C2×C8C2×C22⋊D7111yescharacteristic
296C2×C8#1176586411yescharacteristic
297C2×C8#1176586511yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—2844no—
301C4⋊C4—2844no—
302C4⋊C4—2841no—
303C4⋊C4—2841no—
304C4⋊C4—1422no—
305C4⋊C4—1422no—
306C22⋊C4—2844no—
307C16—41no—
308C16—2841no—
309C23—712no—
310C23—712no—
311C23—712no—
312C23—1421no—
313C23—1421no—
314C23—1421no—
315C23—1421no—
316C23D8×D7111yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—81no—
320Q8—81no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—2842no—
328C2×C4—2842no—
329C2×C4—5688no—
330C2×C4—5681no—
331C2×C4—5681no—
332C2×C4—5681no—
333C2×C4—5681no—
334C2×C4—712no—
335C2×C4—1421no—
336C2×C4—1421no—
337C2×C4—1422no—
338C2×C4—1422no—
339C2×C4—1422no—
340C2×C4—1422no—
341C2×C4C22⋊C4×D7111yescharacteristic
342C2×C4#1455609611yescharacteristic
343C2×C4#1455611011yescharacteristic
344C2×C4SD16×D7111yescharacteristic
345C2×C4#1455615211yescharacteristic
346C2×C4#1455615711yescharacteristic
347C8—1422no—
348C8(C22⋊D71)⋊C411yescharacteristic
349C8C4×C22⋊D7111yescharacteristic
350C8#1455609911yescharacteristic
351C8#1455610111yescharacteristic
352C22—714no—
353C22—1422no—
354C22—1422no—
355C22—1422no—
356C22—1422no—
357C22—1422no—
358C22—1422no—
359C22#102×D7111yescharacteristic
360C22#1798613111yescharacteristic
361C22#1798631111yescharacteristic
362C22SD32×D7111yescharacteristic
363C22Q32×D7111yescharacteristic
364C22#1798643611yescharacteristic
365C22#1798644111yescharacteristic
366C4—82no—
367C4—82no—
368C4—5682no—
369C4—5682no—
370C4—1422no—
371C4#1798587511yescharacteristic
372C4#1798599711yescharacteristic
373C4#1798613611yescharacteristic
374C4#1798631611yescharacteristic
375C2—1424no—
376C2#2258865411yescharacteristic
377C2#358×D7111yescharacteristic
378C2#2259005111yescharacteristic
379C2#2259026711yescharacteristic
380C2#2259027211yescharacteristic
381C2#2259053611yescharacteristic
382C2#2259054111yescharacteristic
subgroupG/H
centerC23D8×D71
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC2×#358×C71C2
socleC23×C71C2×D8
prime radicalC23×D71D8
cosocleC22×C8×C71C23
hypercenterC2×#358D71
hypocenterC71#12993
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#17985803C2semidirect
5. #22586432C2semidirect
6. #22586431C2semidirect
7. C2×#358×C71C2semidirect
13. #17986237C4semidirect
49. #14556062C2×C4semidirect
59. C2×Q16×C71C2×C4semidirect
173. C2×D71#358semidirect
195. C2×#358D71semidirect
199. C71#12993semidirect
226. C2×Q16C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862372
C2×#107×D712
C2×#179858032
C2×(C2×D71)⋊C162
#225864322
#225864312
C2×#358×C712
#1299371