Finite Group Databasebeta

← #31914896

Subgroups — #31914896

Order 18176.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986416C211yesmaximal, characteristic, complemented
2C2×#17985812C211yesmaximal, characteristic, complemented
3C2×#17985800C211yesmaximal, characteristic, complemented
4#22588404C211yesmaximal, characteristic, complemented
5#22586450C211yesmaximal, characteristic, complemented
6#22586433C211yesmaximal, characteristic, complemented
7C2×#472×C71C211yesmaximal, characteristic
8#17986416—22no—
9#17986416—22no—
10C2×#14556068—21no—
11C2×#14556068C2211yescharacteristic
12C22×C71⋊C16—21no—
13C22×C71⋊C16C2211yescharacteristic
14C2×#14556064C2211yescharacteristic
15C2×#14556062C2211yescharacteristic
16#17986000—21no—
17#17985812—22no—
18#17985800—21no—
19#17985800—21no—
20#17985767—21no—
21C2×AGL(1,ℤ/8)×C71—21no—
22C22×D8×C71C2211yescharacteristic
23C2×#131×C71C2211yescharacteristic
24#472×C71—22no—
25#472×C71C414yescomplemented
26C2×#108×C71C2211yescharacteristic
27C2×#100×C71—21no—
28C2×#11765849—21no—
29C2×C4⋊Dic71—21no—
30#14556068—24no—
31#14556068—21no—
32#14556068D412yes—
33C2×C71⋊C16—24no—
34C2×C71⋊C16—21no—
35C2×C71⋊C16—21no—
36C2×C71⋊C16—21no—
37C2×C71⋊C16D411yescharacteristic
38C2×C71⋊C16D411yescharacteristic
39#14556064—21no—
40#14556064—21no—
41#14556062—21no—
42#14556062—21no—
43#14556052—41no—
44#14556049—41no—
45C2×#41×C71D411yescharacteristic
46C22×D4×C71—21no—
47AGL(1,ℤ/8)×C71—24no—
48AGL(1,ℤ/8)×C71—42no—
49C2×SD16×C71—41no—
50C2×D8×C71—21no—
51C2×D8×C71—21no—
52C2×D8×C71—21no—
53C2×D8×C71—21no—
54C2×D8×C71—21no—
55C2×D8×C71—41no—
56C2×D8×C71C2×C412yes—
57#131×C71—21no—
58#131×C71—21no—
59#131×C71C2×C412yes—
60#131×C71D412yes—
61C2×M16×C71—21no—
62C2×M16×C71—21no—
63C2×M16×C71D411yescharacteristic
64C22×C8×C71—21no—
65C22×C8×C71C2311yescharacteristic
66#108×C71—21no—
67#108×C71C2×C412yes—
68#100×C71—24no—
69C22×Dic71—41no—
70#11765849—41no—
71#11765849—41no—
72C2×C71⋊C8—41no—
73C4⋊Dic71—41no—
74C4⋊Dic71—41no—
75C71⋊C16—21no—
76C71⋊C16—22no—
77C71⋊C16—21no—
78C24×C71—41no—
79#41×C71—21no—
80#41×C71—21no—
81#41×C71C22⋊C412yes—
82#41×C71SD1612yes—
83C2×Q8×C71D811yescharacteristic
84C2×D4×C71—21no—
85C2×D4×C71—22no—
86C2×D4×C71—41no—
87C2×D4×C71—41no—
88C2×D4×C71—41no—
89C2×D4×C71—41no—
90C2×D4×C71—41no—
91C2×D4×C71D811yescharacteristic
92C22×C4×C71—21no—
93C22×C4×C71C2×D411yescharacteristic
94SD16×C71—44no—
95D8×C71—22no—
96D8×C71—22no—
97D8×C71—44no—
98D8×C71—41no—
99D8×C71—41no—
100M16×C71—21no—
101M16×C71—22no—
102M16×C71—22no—
103M16×C71—22no—
104M16×C71—41no—
105M16×C71C22⋊C412yes—
106C2×C8×C71—21no—
107C2×C8×C71—22no—
108C2×C8×C71—22no—
109C2×C8×C71—41no—
110C2×C8×C71C22×C411yescharacteristic
111C2×C8×C71C2×D411yescharacteristic
112C2×C8×C71C2×D411yescharacteristic
113C2×C8×C71C2×D411yescharacteristic
114C2×C8×C71#4111yescharacteristic
115C2×C8×C71#4111yescharacteristic
116C23×C71—21no—
117C23×C71—42no—
118C23×C71—81no—
119C23×C71—81no—
120C23×C71—161no—
121C23×C71—161no—
122C23×C71#13511yescharacteristic
123Q8×C71—21no—
124Q8×C71#10212yes—
125D4×C71—22no—
126D4×C71—22no—
127D4×C71—21no—
128D4×C71—42no—
129D4×C71—42no—
130D4×C71—81no—
131D4×C71—81no—
132D4×C71#10212yes—
133C2×C4×C71—21no—
134C2×C4×C71—22no—
135C2×C4×C71—22no—
136C2×C4×C71C2×C22⋊C411yescharacteristic
137C2×C4×C71#12111yescharacteristic
138C2×C4×C71#12111yescharacteristic
139C2×C4×C71C2×D811yescharacteristic
140C2×C4×C71C2×SD1611yescharacteristic
141C2×C4×C71#13511yescharacteristic
142C2×Dic71—81no—
143C2×Dic71—161no—
144C8×C71—22no—
145C8×C71—42no—
146C8×C71C4×D411yescharacteristic
147C8×C71C4×D411yescharacteristic
148C8×C71#12011yescharacteristic
149C8×C71#12311yescharacteristic
150C71⋊C8—81no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—22no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—82no—
157C22×C71—162no—
158C22×C71—162no—
159C22×C71—161no—
160C22×C71—161no—
161C22×C71#41611yescharacteristic
162C22×C71#46511yescharacteristic
163C22×C71#46611yescharacteristic
164C4×C71—22no—
165C4×C71#38611yescharacteristic
166C4×C71C2×#10211yescharacteristic
167C4×C71#46511yescharacteristic
168C4×C71#46611yescharacteristic
169Dic71—161no—
170#13073—711nomaximal, Sylow
171C2×C71—21no—
172C2×C71—21no—
173C2×C71—42no—
174C2×C71—162no—
175C2×C71#189311yescharacteristic
176C2×C71#222111yescharacteristic
177C2×C71#222311yescharacteristic
178C2×#504—711no—
179C2×#472D7111yescharacteristic
180C2×#361—711no—
181C2×#357—711no—
182#1816—711no—
183#1387—711no—
184#1381—711no—
185C71#1307311yescharacteristic, complemented, Sylow
186C2×AGL(1,ℤ/8)—21no—
187C22×D8C2×D7111yescharacteristic
188C2×#131C2×D7111yescharacteristic
189#504—1422no—
190#504—1422no—
191C2×M32—711no—
192C2×M32—1421no—
193C22×C16—711no—
194C22×C16—1421no—
195#472—22no—
196#472Dic7114yescomplemented
197C2×#108—711no—
198C2×#108C2×D7111yescharacteristic
199C2×#107—711no—
200#417—1421no—
201C2×#100—21no—
202#361—1422no—
203#357—1421no—
204#357—1421no—
205#343—1421no—
206C2×#41C22⋊D7111yescharacteristic
207C22×D4—21no—
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—41no—
217C2×D8C2×Dic7112yes—
218#131—21no—
219#131—21no—
220#131C2×Dic7112yes—
221#131C22⋊D7112yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1421no—
225C2×M16C22⋊D7111yescharacteristic
226C22×C8—21no—
227C22×C8C22×D7111yescharacteristic
228C2×C4⋊C4—1421no—
229M32—712no—
230M32—1424no—
231M32—1421no—
232C2×C16—711no—
233C2×C16—711no—
234C2×C16—1424no—
235C2×C16—1421no—
236C2×C16—1421no—
237C2×C16—1421no—
238#108—21no—
239#108—1421no—
240#108—1421no—
241#108C2×Dic7112yes—
242#107—1421no—
243#107—1421no—
244#103—2841no—
245#102—2841no—
246#100—24no—
247C24—41no—
248#41—21no—
249#41—21no—
250#41C22⋊Dic7112yes—
251#41#1176585312yes—
252C2×Q8#1176585011yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4#1176585011yescharacteristic
261C22×C4—21no—
262C22×C4—2841no—
263C22×C4C2×C22⋊D7111yescharacteristic
264SD16—44no—
265D8—22no—
266D8—22no—
267D8—44no—
268D8—41no—
269D8—41no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—2841no—
276M16—2841no—
277M16C22⋊Dic7112yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—2841no—
283C2×C8C22×Dic7111yescharacteristic
284C2×C8D4×D7111yescharacteristic
285C2×C8D4×D7111yescharacteristic
286C2×C8C2×C22⋊D7111yescharacteristic
287C2×C8#1176586411yescharacteristic
288C2×C8#1176586411yescharacteristic
289C4⋊C4—2841no—
290C4⋊C4—2841no—
291C16—1421no—
292C16—1421no—
293C16—1422no—
294C23—21no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—161no—
299C23—161no—
300C23#1455614811yescharacteristic
301Q8—21no—
302Q8#1455604912yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4#1455604912yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—5681no—
315C2×C4—11361no—
316C2×C4C2×C22⋊Dic7111yescharacteristic
317C2×C4#1455610011yescharacteristic
318C2×C4#1455610011yescharacteristic
319C2×C4C2×#1176585011yescharacteristic
320C2×C4C2×#1176585311yescharacteristic
321C2×C4#1455614811yescharacteristic
322C8—22no—
323C8—42no—
324C8—5681no—
325C8D4×Dic7111yescharacteristic
326C8D4×Dic7111yescharacteristic
327C8#1455609611yescharacteristic
328C8#1455610911yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1798599511yescharacteristic
340C22#1798622611yescharacteristic
341C22#1798623311yescharacteristic
342C4—22no—
343C4—11361no—
344C4#1798587911yescharacteristic
345C4C2×#1455604911yescharacteristic
346C4#1798622611yescharacteristic
347C4#1798623311yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#2258876911yescharacteristic
353C2#2259045811yescharacteristic
354C2#2259046811yescharacteristic
subgroupG/H
centerC22#17986233
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
Fitting, prime radicalC2×#472×C71C2
socleC22×C71#466
cosocleC22×C8×C71C23
hypercenterC2×#472D71
hypocenterC71#13073
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C2×#17986416C2semidirect
2. C2×#17985812C2semidirect
3. C2×#17985800C2semidirect
4. #22588404C2semidirect
5. #22586450C2semidirect
6. #22586433C2semidirect
25. #472×C71C4semidirect
185. C71#13073semidirect
196. #472Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864162
C2×#179858122
C2×#179858002
#225884042
#225864502
#225864332
C2×#472×C712
#1307371