Finite Group Databasebeta

← #32150488

Subgroups — #32150488

Order 18176.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D71C211yesmaximal, characteristic
2C2×#17986036C211yesmaximal, characteristic
3#427×D71C211yesmaximal, characteristic
4#22590129C211yesmaximal, characteristic, complemented
5#22590128C211yesmaximal, characteristic, complemented
6#22590127C212yesmaximal, complemented
7#22590126C211yesmaximal, characteristic, complemented
8#22590125C212yesmaximal, complemented
9#22590123C212yesmaximal, complemented
10#22590121C212yesmaximal, complemented
11C71×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D71C2211yescharacteristic
13C2×#14556086—21no—
14#117×D71—21no—
15C2×#14556068C2211yescharacteristic
16C2×#14556067C2212yes—
17M32×D71C418yescomplemented
18C2×C16×D71C2212yes—
19#17986037C2211yescharacteristic
20#17986036—21no—
21#17986036—21no—
22#17986036C2211yescharacteristic
23#17986036C2211yescharacteristic
24#17986036C2212yes—
25#17986036C2212yes—
26#17986035—21no—
27#17986034—21no—
28C2×#14556062C2211yescharacteristic
29#107×D71C2211yescharacteristic
30#107×D71C2211yescharacteristic
31C2×#14556059—21no—
32#106×D71—21no—
33#17985827C2212yescomplemented
34#17985826C2212yescomplemented
35#17985826C2212yescomplemented
36#17985825C2212yescomplemented
37#17985824C2212yescomplemented
38#17985824C2212yescomplemented
39C2×M32×C71C2211yescharacteristic
40#427×C71C2211yescharacteristic
41C2×#107×C71C2211yescharacteristic
42#367×C71C2212yescomplemented
43#366×C71C2212yescomplemented
44C22×C4×D71D411yescharacteristic
45C22×C71⋊C8D411yescharacteristic
46C2×C8×D71C2×C414yes—
47C2×C8×D71D411yescharacteristic
48C2×C8×D71D411yescharacteristic
49C2×C8×D71Q814yes—
50C2×C8×D71C2311yescharacteristic
51C2×C8×D71C2311yescharacteristic
52#14556087—21no—
53#14556086—21no—
54#14556086—21no—
55#14556086—22no—
56#14556086—22no—
57#14556086—41no—
58#14556086—41no—
59#14556085—41no—
60#14556084—41no—
61C2×C4⋊Dic71—21no—
62C4⋊C4×D71—21no—
63C4⋊C4×D71—21no—
64C2×(C2×D71)⋊C4—41no—
65C22⋊C4×D71—41no—
66C2×C4×Dic71—41no—
67C42×D71—41no—
68#14556068C2×C414yescomplemented
69#14556067C2×C418yescomplemented
70C2×C71⋊C16C2312yes—
71C16×D71C2×C418yescomplemented
72#14556062—21no—
73#14556062C2311yescharacteristic
74#14556062C2312yes—
75#14556062C2311yescharacteristic
76#14556061—21no—
77#14556061—21no—
78#14556060—21no—
79#14556059—21no—
80#14556059—21no—
81#14556059—22no—
82#14556059—22no—
83C22×C8×C71C2311yescharacteristic
84#117×C71—21no—
85C2×C4⋊C4×C71—21no—
86M32×C71C2×C414yescomplemented
87C2×C16×C71C2312yes—
88#107×C71—21no—
89#107×C71C2312yes—
90#107×C71C2311yescharacteristic
91#107×C71C2311yescharacteristic
92#106×C71—21no—
93C23×D71D811yescharacteristic
94C22×Dic71—41no—
95C22×Dic71D811yescharacteristic
96C2×C4×D71—41no—
97C2×C4×D71—41no—
98C2×C4×D71C4⋊C414yes—
99C2×C4×D71D811yescharacteristic
100C2×C4×D71D811yescharacteristic
101C2×C4×D71Q1614yes—
102C2×C4×D71C2×D411yescharacteristic
103C2×C4×D71C2×D411yescharacteristic
104C2×C71⋊C8C4⋊C414yes—
105C2×C71⋊C8C2×D411yescharacteristic
106C2×C71⋊C8C2×D411yescharacteristic
107C8×D71C4⋊C414yes—
108C8×D71C4⋊C414yes—
109C8×D71C22×C412yes—
110C8×D71C22×C412yes—
111C8×D71C2×Q812yes—
112C8×D71C2×Q812yes—
113C4⋊Dic71—22no—
114C4⋊Dic71—21no—
115C4⋊Dic71—21no—
116C4⋊Dic71—41no—
117Dic71⋊C4—41no—
118Dic71⋊C4—41no—
119C22⋊Dic71—41no—
120(C2×D71)⋊C4—41no—
121(C2×D71)⋊C4—41no—
122(C2×D71)⋊C4—42no—
123(C2×D71)⋊C4—42no—
124C4×Dic71—41no—
125C4×Dic71—41no—
126C4×Dic71—41no—
127C4×Dic71—42no—
128C4×Dic71—42no—
129C71⋊C16C22×C414yes—
130C22×C4×C71—41no—
131C22×C4×C71C2×D411yescharacteristic
132C2×C8×C71C22×C412yes—
133C2×C8×C71C2×D411yescharacteristic
134C2×C8×C71C2×Q812yes—
135C2×C8×C71C2411yescharacteristic
136C4⋊C4×C71—21no—
137C4⋊C4×C71—21no—
138C4⋊C4×C71—22no—
139C4⋊C4×C71—41no—
140C22⋊C4×C71—41no—
141C42×C71—41no—
142C16×C71C22×C414yes—
143C23×C71C2×D811yescharacteristic
144C22×D71—41no—
145C22×D71—41no—
146C22×D71#10714yes—
147C22×D71C2×D811yescharacteristic
148C22×D71C2×D811yescharacteristic
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—42no—
152C2×C4×C71—82no—
153C2×C4×C71C2×C4⋊C412yes—
154C2×C4×C71C2×D811yescharacteristic
155C2×C4×C71C2×Q1612yes—
156C2×C4×C71C22×D411yescharacteristic
157C2×Dic71—42no—
158C2×Dic71—41no—
159C2×Dic71—41no—
160C2×Dic71—82no—
161C2×Dic71#10714yes—
162C2×Dic71C2×D811yescharacteristic
163C2×Dic71C2×D811yescharacteristic
164C4×D71—82no—
165C4×D71—82no—
166C4×D71#10714yes—
167C4×D71#10714yes—
168C4×D71C2×C4⋊C412yes—
169C4×D71C2×C4⋊C412yes—
170C4×D71C2×Q1612yes—
171C4×D71C2×Q1612yes—
172C8×C71C2×C4⋊C412yes—
173C8×C71C23×C411yescharacteristic
174C8×C71C22×Q811yescharacteristic
175C71⋊C8C2×C4⋊C412yes—
176C71⋊C8C2×C4⋊C412yes—
177C22×C71—21no—
178C22×C71—21no—
179C22×C71C2×#10712yes—
180C22×C71C22×D811yescharacteristic
181C2×D71—21no—
182C2×D71—21no—
183C2×D71—21no—
184C2×D71—21no—
185C2×D71—44no—
186C2×D71C2×#10712yes—
187C2×D71C2×#10712yes—
188C4×C71—81no—
189C4×C71—81no—
190C4×C71—82no—
191C4×C71C2×#10712yes—
192C4×C71C22×C4⋊C411yescharacteristic
193C4×C71C22×Q1611yescharacteristic
194Dic71—81no—
195Dic71—81no—
196Dic71—82no—
197Dic71C2×#10712yes—
198Dic71C2×#10712yes—
199#33409—711nomaximal, Sylow
200C2×C71—22no—
201C2×C71C22×#10711yescharacteristic
202C2×C71#341411yescharacteristic
203C2×C71#341411yescharacteristic
204D71—22no—
205D71—22no—
206C22×M32—711no—
207C2×#427—711no—
208C2×#427—711no—
209#2159—711no—
210#2159—711no—
211#2159—711no—
212#2159—712no—
213#2159—712no—
214#2159D7111yescharacteristic, complemented
215#2158—712no—
216#2158—712no—
217C71#3340911yescharacteristic, complemented, Sylow
218C23×C8—711no—
219C2×#117—1421no—
220C2×#117—1421no—
221C2×M32—711no—
222C2×M32—718no—
223C2×M32—712no—
224C2×M32C2×D7111yescharacteristic
225C22×C16—712no—
226#427—711no—
227#427—711no—
228#427—712no—
229#427—711no—
230#427—712no—
231#427—1421no—
232#427—1421no—
233#427—1421no—
234#427—1421no—
235#427C2×D7111yescharacteristic
236C2×#107—711no—
237C2×#107—711no—
238C2×#107—711no—
239C2×#107C2×D7111yescharacteristic
240C2×#106—1421no—
241C2×#106—1421no—
242#367—712no—
243#367—712no—
244#367—712no—
245#367C2×D7112yescomplemented
246#366—712no—
247#366—712no—
248#366—712no—
249#366C2×D7112yescomplemented
250C23×C4—711no—
251C22×C8—711no—
252C22×C8—714no—
253C22×C8—711no—
254C22×C8—711no—
255C22×C8—714no—
256C22×C8—711no—
257C22×C8—711no—
258C22×C8C22×D7111yescharacteristic
259#117—21no—
260#117—2841no—
261#117—2841no—
262#117—2841no—
263#117—2841no—
264#117—1421no—
265#117—1421no—
266#117—1422no—
267#117—1422no—
268#117—1421no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—1421no—
271C2×C4⋊C4—1421no—
272C2×C4⋊C4—1421no—
273C2×C22⋊C4—2841no—
274C2×C22⋊C4—2841no—
275C2×C42—2841no—
276C2×C42—2841no—
277M32—714no—
278M32—718no—
279M32C4×D7114yescomplemented
280C2×C16—712no—
281C2×C16—718no—
282C2×C16C22×D7112yes—
283#107—21no—
284#107—711no—
285#107—712no—
286#107—711no—
287#107—1421no—
288#107—1421no—
289#107—1421no—
290#107C22×D7111yescharacteristic
291#107C22×D7112yes—
292#107C22×D7111yescharacteristic
293#106—21no—
294#106—1421no—
295#106—1421no—
296#106—1422no—
297#106—1422no—
298#106—1421no—
299C24—711no—
300C22×C4—41no—
301C22×C4—2841no—
302C22×C4—2841no—
303C22×C4—2841no—
304C22×C4—711no—
305C22×C4—711no—
306C22×C4—711no—
307C22×C4—714no—
308C22×C4—711no—
309C22×C4—711no—
310C22×C4—714no—
311C22×C4D4×D7111yescharacteristic
312C2×C8—711no—
313C2×C8—711no—
314C2×C8—714no—
315C2×C8—712no—
316C2×C8—712no—
317C2×C8—712no—
318C2×C8—712no—
319C2×C8—714no—
320C2×C8—714no—
321C2×C8C2×C4×D7112yes—
322C2×C8D4×D7111yescharacteristic
323C2×C8Q8×D7112yes—
324C2×C8C23×D7111yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—2841no—
330C4⋊C4—2841no—
331C4⋊C4—2841no—
332C4⋊C4—1422no—
333C4⋊C4—1421no—
334C4⋊C4—1421no—
335C22⋊C4—41no—
336C22⋊C4—2841no—
337C22⋊C4—2841no—
338C22⋊C4—2842no—
339C22⋊C4—2842no—
340C22⋊C4—2841no—
341C42—41no—
342C42—2841no—
343C42—2841no—
344C42—2842no—
345C42—2842no—
346C42—2841no—
347C16—714no—
348C16C2×C4×D7114yes—
349C23—2841no—
350C23—2841no—
351C23—714no—
352C23—711no—
353C23—711no—
354C23D8×D7111yescharacteristic
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—2841no—
360C2×C4—2842no—
361C2×C4—2841no—
362C2×C4—5682no—
363C2×C4—5682no—
364C2×C4—5682no—
365C2×C4—711no—
366C2×C4—711no—
367C2×C4—714no—
368C2×C4—712no—
369C2×C4—712no—
370C2×C4—712no—
371C2×C4—712no—
372C2×C4—714no—
373C2×C4—714no—
374C2×C4C4⋊C4×D7112yes—
375C2×C4D8×D7111yescharacteristic
376C2×C4Q16×D7112yes—
377C2×C4C2×D4×D7111yescharacteristic
378C8—712no—
379C8—712no—
380C8C4⋊C4×D7112yes—
381C8C22×C4×D7111yescharacteristic
382C8C2×Q8×D7111yescharacteristic
383C22—21no—
384C22—21no—
385C22—2844no—
386C22—712no—
387C22—712no—
388C22—1421no—
389C22—1421no—
390C22—1421no—
391C22—1421no—
392C22#107×D7112yes—
393C22C2×D8×D7111yescharacteristic
394C4—82no—
395C4—81no—
396C4—81no—
397C4—5681no—
398C4—5681no—
399C4—5682no—
400C4—712no—
401C4—712no—
402C4#107×D7112yes—
403C4C2×C4⋊C4×D7111yescharacteristic
404C4C2×Q16×D7111yescharacteristic
405C2—22no—
406C2—1422no—
407C2—1422no—
408C2C2×#107×D7111yescharacteristic
409C2#2260147211yescharacteristic
410C2#2260147511yescharacteristic
subgroupG/H
centerC22C2×D8×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2159C2
socleC22×C71C22×D8
prime radicalC23×D71D8
cosocleC2×C8×C71C24
hypercenter#2159D71
hypocenterC71#33409
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #22590129C2semidirect
5. #22590128C2semidirect
6. #22590127C2semidirect
7. #22590126C2semidirect
8. #22590125C2semidirect
9. #22590123C2semidirect
10. #22590121C2semidirect
11. C71×#2159C2semidirect
17. M32×D71C4semidirect
33. #17985827C22semidirect
34. #17985826C22semidirect
35. #17985826C22semidirect
36. #17985825C22semidirect
37. #17985824C22semidirect
38. #17985824C22semidirect
42. #367×C71C22semidirect
43. #366×C71C22semidirect
68. #14556068C2×C4semidirect
69. #14556067C2×C4semidirect
71. C16×D71C2×C4semidirect
86. M32×C71C2×C4semidirect
214. #2159D71semidirect
217. C71#33409semidirect
245. #367C2×D71semidirect
249. #366C2×D71semidirect
279. M32C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D712
C2×#179860362
#427×D712
#225901292
#225901282
#225901272
#225901262
#225901252
#225901232
#225901212
C71×#21592
#3340971