Finite Group Databasebeta

← #31997813

Subgroups — #31997813

Order 18176.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596762C211yesmaximal, characteristic
2#22588488C212yesmaximal, complemented
3#22588464C211yesmaximal, characteristic, complemented
4#22588442C212yesmaximal
5#22588387C211yesmaximal, characteristic
6#22588383C211yesmaximal, characteristic
7#22588222C211yesmaximal, characteristic, complemented
8#22588221C211yesmaximal, characteristic, complemented
9#22588139C212yesmaximal
10#22588094C212yesmaximal, complemented
11C71×#1833C211yesmaximal, characteristic, complemented
12C2×#14556136C2211yescharacteristic
13C2×#14556134C2211yescharacteristic
14#17986467C2211yescharacteristic
15C2×#14556064C2211yescharacteristic
16#17986037C2211yescharacteristic
17#17985981C2212yes—
18#17985972C2212yes—
19#17985957C2212yes—
20#17985956C412yescomplemented
21#17985955C2212yes—
22#17985955C2211yescharacteristic
23#17985953C412yescomplemented
24#17985951C412yescomplemented
25M16×Dic71C2211yescharacteristic
26C2×C8×Dic71C2211yescharacteristic
27C4×C8×D71C412yescomplemented
28#17985770C2212yes—
29#17985760C2212yes—
30#17985758C2212yes—
31C71⋊#338C2212yes—
32#17985755C2212yes—
33#17985755C2212yes—
34C2×#108×C71C2211yescharacteristic
35#427×C71C2211yescharacteristic
36#405×C71C2211yescharacteristic
37#344×C71C2212yes—
38H(3,ℤ/4)×C71C2212yes—
39C2×#11765865C2311yescharacteristic
40#14556136—21no—
41#14556136—21no—
42#14556136—21no—
43#14556136—21no—
44#14556134—21no—
45#14556134—21no—
46#14556134—21no—
47#14556134—21no—
48C2×#11765849C2312yes—
49C2×#11765849C2311yescharacteristic
50C2×#11765848C2×C412yes—
51C2×#11765848C2×C412yes—
52C22×C71⋊C8C2311yescharacteristic
53C2×C8×D71C2×C412yes—
54C2×C8×D71C2×C412yes—
55#14556093—21no—
56#14556090—21no—
57(C22⋊D71)⋊C4—22no—
58#14556087C2312yes—
59#14556085C2×C412yes—
60C2×C4×Dic71C2311yescharacteristic
61C42×D71C2×C412yes—
62#14556064—21no—
63#14556064—21no—
64#14556062—21no—
65#14556060—21no—
66#14556046—22no—
67#14556044—22no—
68#14556042—24no—
69#14556039—22no—
70#14556039C2×C412yes—
71#14556038—22no—
72#14556038C2×C412yes—
73C4×C71⋊C8—22no—
74C4×C71⋊C8C2×C412yes—
75C8×Dic71—22no—
76C8×Dic71C2×C411yescharacteristic
77C8×Dic71C2×C412yes—
78C8×Dic71C2×C411yescharacteristic
79C8×Dic71C2×C411yescharacteristic
80C8×Dic71C2×C411yescharacteristic
81C2×M16×C71C2312yes—
82C2×M16×C71C2311yescharacteristic
83C22×C8×C71C2311yescharacteristic
84#117×C71C2312yes—
85#117×C71C2311yescharacteristic
86#108×C71—21no—
87#108×C71—21no—
88#107×C71—21no—
89#106×C71—21no—
90#97×C71C2×C412yes—
91C4×C8×C71C2×C412yes—
92#11765865—21no—
93#11765865—21no—
94#11765865—21no—
95#11765865—21no—
96C2×Dic142#4111yescharacteristic
97C2×D284#4111yescharacteristic
98C2×C22⋊D71#4112yes—
99C22×Dic71—21no—
100C2×C4×D71—22no—
101C2×C4×D71C22×C412yes—
102#11765849—21no—
103#11765849—21no—
104#11765849—22no—
105#11765849—22no—
106#11765848—22no—
107#11765848—22no—
108#11765848—22no—
109#11765848—22no—
110C2×C71⋊C8—22no—
111C2×C71⋊C8C22×C412yes—
112C2×C71⋊C8C22×C411yescharacteristic
113C2×C71⋊C8C22×C411yescharacteristic
114C2×C71⋊C8#4111yescharacteristic
115C2×C71⋊C8#4111yescharacteristic
116C2×C71⋊C8#4111yescharacteristic
117C2×C71⋊C8#4111yescharacteristic
118C8×D71—22no—
119C8×D71—22no—
120C8×D71—22no—
121C8×D71—22no—
122C4⋊Dic71—22no—
123Dic71⋊C4—22no—
124C22⋊Dic71—22no—
125(C2×D71)⋊C4—22no—
126C4×Dic71—21no—
127C4×Dic71—22no—
128C4×Dic71—22no—
129C4×Dic71C22×C411yescharacteristic
130C4×Dic71C22×C411yescharacteristic
131C22×C4×C71C2411yescharacteristic
132M16×C71—21no—
133M16×C71—21no—
134M16×C71—22no—
135M16×C71—22no—
136C2×C8×C71—22no—
137C2×C8×C71C22×C412yes—
138C2×C8×C71C22×C411yescharacteristic
139C2×C8×C71C22×C411yescharacteristic
140C2×C8×C71#4111yescharacteristic
141C2×C8×C71#4111yescharacteristic
142C2×C8×C71#4111yescharacteristic
143C2×C8×C71#4111yescharacteristic
144C4⋊C4×C71—21no—
145C4⋊C4×C71—22no—
146C22⋊C4×C71—21no—
147C22⋊C4×C71—22no—
148C42×C71—22no—
149C42×C71C22×C412yes—
150C23×C71C2×#4111yescharacteristic
151C22×D71#11712yes—
152C2×C4×C71—22no—
153C2×C4×C71—42no—
154C2×C4×C71C23×C411yescharacteristic
155C2×C4×C71C2×#4111yescharacteristic
156C2×C4×C71C2×#4111yescharacteristic
157C2×C4×C71C2×#4111yescharacteristic
158C2×C4×C71C2×#4111yescharacteristic
159C2×C4×C71C2×#4111yescharacteristic
160C22⋊D71—22no—
161C22⋊D71—22no—
162C2×Dic71—21no—
163C2×Dic71—21no—
164C2×Dic71—42no—
165C2×Dic71—42no—
166C2×Dic71#11712yes—
167D284—21no—
168D284—21no—
169C4×D71—22no—
170C4×D71—22no—
171C4×D71—44no—
172Dic142—21no—
173Dic142—21no—
174C8×C71—22no—
175C8×C71—42no—
176C8×C71#11711yescharacteristic
177C8×C71#11711yescharacteristic
178C8×C71#11711yescharacteristic
179C8×C71#11711yescharacteristic
180C71⋊C8—22no—
181C71⋊C8—42no—
182C71⋊C8#11711yescharacteristic
183C71⋊C8#11711yescharacteristic
184C71⋊C8#11711yescharacteristic
185C71⋊C8#11711yescharacteristic
186C22×C71—41no—
187C22×C71C2×#11711yescharacteristic
188C22×C71#52911yescharacteristic
189C22×C71#52911yescharacteristic
190C2×D71—22no—
191C2×D71—42no—
192C4×C71—42no—
193C4×C71—82no—
194C4×C71C2×#11711yescharacteristic
195C4×C71C2×#11711yescharacteristic
196C4×C71#52911yescharacteristic
197C4×C71#52911yescharacteristic
198Dic71—22no—
199Dic71—22no—
200Dic71—41no—
201Dic71—82no—
202#20787—711nomaximal, Sylow
203C2×C71—21no—
204C2×C71—21no—
205C2×C71—21no—
206C2×C71#228311yescharacteristic
207D71—42no—
208C4×#131—711no—
209#1841—712no—
210#1833—711no—
211#1833D7111yescharacteristic, complemented
212#1827—712no—
213#1813—711no—
214#1812—711no—
215#1772—711no—
216#1772—711no—
217#1757—712no—
218#1747—712no—
219C71#2078711yescharacteristic, complemented, Sylow
220C2×#131—711no—
221C2×#131—711no—
222C4×#41—711no—
223C2×#108—711no—
224C2×#108C2×D7111yescharacteristic
225#427—711no—
226#427C2×D7111yescharacteristic
227#413—712no—
228#410—712no—
229#405—711no—
230#405—712no—
231#405—712no—
232#405—712no—
233#405—712no—
234#405C2×D7111yescharacteristic
235C4×M16—711no—
236C4×M16—712no—
237C2×C4×C8—711no—
238C2×C4×C8—712no—
239#344—712no—
240#344C2×D7112yes—
241#339—712no—
242#339—712no—
243#338—712no—
244H(3,ℤ/4)—712no—
245H(3,ℤ/4)—712no—
246H(3,ℤ/4)C2×D7112yes—
247C2×#41—711no—
248#131—1421no—
249#131—1421no—
250#131—1421no—
251#131—1421no—
252#131—1421no—
253#131—1421no—
254#131—1421no—
255#131—1421no—
256C2×M16—711no—
257C2×M16—712no—
258C2×M16—712no—
259C2×M16—712no—
260C2×M16C22×D7112yes—
261C2×M16C22×D7111yescharacteristic
262C22×C8—711no—
263C22×C8—712no—
264C22×C8—712no—
265C22×C8C22×D7111yescharacteristic
266C4×Q8—1421no—
267C4×D4—1422no—
268C4×D4—1421no—
269#117—712no—
270#117—712no—
271#117C22×D7112yes—
272#117C22×D7111yescharacteristic
273C2×C42—711no—
274C2×C42—712no—
275#108—21no—
276#108—21no—
277#108—1421no—
278#108—1421no—
279#107—21no—
280#107—1421no—
281#106—21no—
282#106—1421no—
283#101—1422no—
284#100—1422no—
285#99—1424no—
286#97—712no—
287#97—712no—
288#97—1422no—
289#97—1422no—
290#97C4×D7112yes—
291C4×C8—711no—
292C4×C8—711no—
293C4×C8—712no—
294C4×C8—711no—
295C4×C8—711no—
296C4×C8—712no—
297C4×C8—1422no—
298C4×C8—1422no—
299C4×C8C4×D7112yes—
300#41—1421no—
301#41—1421no—
302#41—1421no—
303#41—1421no—
304C2×Q8—711no—
305C2×D4—712no—
306C2×D4—711no—
307C22×C4—712no—
308C22×C4—1422no—
309C22×C4—1421no—
310C22×C4C23×D7111yescharacteristic
311M16—21no—
312M16—21no—
313M16—22no—
314M16—22no—
315M16—1422no—
316M16—1422no—
317M16—1422no—
318M16—1422no—
319M16—1421no—
320M16—1421no—
321M16—1422no—
322M16—1422no—
323C2×C8—22no—
324C2×C8—711no—
325C2×C8—711no—
326C2×C8—711no—
327C2×C8—711no—
328C2×C8—711no—
329C2×C8—711no—
330C2×C8—712no—
331C2×C8—1422no—
332C2×C8—1422no—
333C2×C8—1422no—
334C2×C8—1422no—
335C2×C8—1422no—
336C2×C8C2×C4×D7112yes—
337C2×C8C2×C4×D7111yescharacteristic
338C2×C8C2×C4×D7111yescharacteristic
339C2×C8#1176586411yescharacteristic
340C2×C8#1176586411yescharacteristic
341C2×C8#1176586611yescharacteristic
342C2×C8#1176586611yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—1422no—
346C4⋊C4—1422no—
347C22⋊C4—21no—
348C22⋊C4—22no—
349C22⋊C4—1422no—
350C22⋊C4—1422no—
351C42—22no—
352C42—711no—
353C42—711no—
354C42—1422no—
355C42—1422no—
356C42—1421no—
357C42C2×C4×D7112yes—
358C23—712no—
359C23#41×D7111yescharacteristic
360Q8—1421no—
361Q8—1421no—
362D4—1422no—
363D4—1422no—
364D4—1421no—
365D4—1421no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—2844no—
369C2×C4—2842no—
370C2×C4—2842no—
371C2×C4—712no—
372C2×C4—1421no—
373C2×C4—1421no—
374C2×C4—1422no—
375C2×C4—1422no—
376C2×C4C22×C4×D7111yescharacteristic
377C2×C4#41×D7111yescharacteristic
378C2×C4#41×D7111yescharacteristic
379C2×C4#41×D7111yescharacteristic
380C2×C4C2×#1176586411yescharacteristic
381C2×C4C2×#1176586611yescharacteristic
382C8—22no—
383C8—42no—
384C8—2842no—
385C8—711no—
386C8—711no—
387C8—711no—
388C8—711no—
389C8—1422no—
390C8#1455608611yescharacteristic
391C8#1455608611yescharacteristic
392C8#1455608611yescharacteristic
393C8#1455608611yescharacteristic
394C22—41no—
395C22—2842no—
396C22—1422no—
397C22#117×D7111yescharacteristic
398C22#1798652811yescharacteristic
399C22#1798653411yescharacteristic
400C4—42no—
401C4—82no—
402C4—2841no—
403C4—5682no—
404C4—1422no—
405C4—1422no—
406C4#117×D7111yescharacteristic
407C4C2×#1455608611yescharacteristic
408C4#1798652811yescharacteristic
409C4#1798653411yescharacteristic
410C2—21no—
411C2—21no—
412C2—21no—
413C2—2842no—
414C2#2259073311yescharacteristic
subgroupG/H
centerC4#117×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1833C2
socleC2×C71#2283
prime radicalC2×#11765865C23
cosocleC22×C4×C71C24
hypercenter#1833D71
hypocenterC71#20787
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ #117 ◁ #1833

Complements

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

NH ≅ G/Ntype
2. #22588488C2semidirect
3. #22588464C2semidirect
7. #22588222C2semidirect
8. #22588221C2semidirect
10. #22588094C2semidirect
11. C71×#1833C2semidirect
20. #17985956C4semidirect
23. #17985953C4semidirect
24. #17985951C4semidirect
27. C4×C8×D71C4semidirect
211. #1833D71semidirect
219. C71#20787semidirect

Maximal subgroups

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

subgroupindex
#225967622
#225884882
#225884642
#225884422
#225883872
#225883832
#225882222
#225882212
#225881392
#225880942
C71×#18332
#2078771