Finite Group Databasebeta

← #31903256

Subgroups — #31903256

Order 18176.

842 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11765849C211yesmaximal, characteristic
2#22590747C211yesmaximal, characteristic
3C2×#17985755C214yesmaximal
4C71×#2285C211yesmaximal, characteristic
5C22×#11765849C2212yes—
6C2×#14556087C2212yes—
7C4×#11765849—21no—
8C4×#11765849—21no—
9C4×#11765849—21no—
10C4×#11765849—21no—
11C2×#14556039C412yescomplemented
12C2×C4×C71⋊C8C412yescomplemented
13#17985869—21no—
14#17985865—21no—
15#17985859—22no—
16C4×C4⋊Dic71C414yescomplemented
17#17985755—28no—
18C2×#117×C71C2212yes—
19C22×C42×C71C2211yescharacteristic
20#384×C71—21no—
21#382×C71—21no—
22#380×C71—22no—
23C4×C4⋊C4×C71C414yescomplemented
24C2×#11765849—22no—
25C2×#11765849—22no—
26C2×#11765849—22no—
27C2×#11765849—22no—
28C2×#11765849D414yes—
29C2×#11765849Q814yes—
30C22×C71⋊C8C2×C414yes—
31#14556087—24no—
32#14556087—24no—
33C2×C4⋊Dic71—24no—
34C2×C4⋊Dic71—22no—
35C2×C22⋊Dic71—22no—
36C2×C4×Dic71C2×C414yes—
37#14556039—22no—
38#14556039—22no—
39C4×C71⋊C8—22no—
40C4×C71⋊C8—22no—
41C71⋊#95—24no—
42C23×C4×C71D412yes—
43C23×C4×C71C2311yescharacteristic
44#117×C71—24no—
45#117×C71—24no—
46C2×C4⋊C4×C71—24no—
47C2×C4⋊C4×C71—22no—
48C2×C22⋊C4×C71—22no—
49C2×C42×C71C2×C414yes—
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71C2×C411yescharacteristic
52C2×C42×C71C2×C412yes—
53C2×C42×C71C2×C412yes—
54C2×C42×C71D411yescharacteristic
55C2×C42×C71D411yescharacteristic
56C2×C42×C71D411yescharacteristic
57C2×C42×C71D411yescharacteristic
58C2×C42×C71Q812yes—
59#95×C71—24no—
60C22×Dic71—24no—
61#11765849—24no—
62#11765849—24no—
63#11765849—24no—
64#11765849—24no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—24no—
67C2×C71⋊C8C4⋊C418yescomplemented
68C4⋊Dic71—48no—
69C4⋊Dic71—44no—
70C22⋊Dic71—42no—
71C22⋊Dic71—42no—
72C4×Dic71—28no—
73C24×C71C2×D411yescharacteristic
74C22×C4×C71—24no—
75C22×C4×C71—44no—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22⋊C412yes—
80C22×C4×C71C4⋊C412yes—
81C22×C4×C71C4⋊C412yes—
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×Q812yes—
89C22×C4×C71C2×Q811yescharacteristic
90C22×C4×C71#4112yes—
91C22×C4×C71#4112yes—
92C4⋊C4×C71—48no—
93C4⋊C4×C71—44no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C42×C71—28no—
97C42×C71C4212yes—
98C42×C71C4212yes—
99C42×C71C22⋊C411yescharacteristic
100C42×C71C22⋊C411yescharacteristic
101C42×C71C22⋊C411yescharacteristic
102C42×C71C22⋊C411yescharacteristic
103C42×C71C4⋊C412yes—
104C42×C71C4⋊C412yes—
105C42×C71C4⋊C412yes—
106C42×C71C4⋊C412yes—
107C23×C71—42no—
108C23×C71C2×C22⋊C411yescharacteristic
109C23×C71C2×C22⋊C411yescharacteristic
110C23×C71C2×C4⋊C411yescharacteristic
111C23×C71#12411yescharacteristic
112C23×C71#12411yescharacteristic
113C23×C71#12711yescharacteristic
114C23×C71C4⋊Q811yescharacteristic
115C2×C4×C71—24no—
116C2×C4×C71—24no—
117C2×C4×C71—24no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—48no—
122C2×C4×C71—44no—
123C2×C4×C71#9512yes—
124C2×C4×C71#9512yes—
125C2×C4×C71#9512yes—
126C2×C4×C71#9512yes—
127C2×C4×C71C2×C4211yescharacteristic
128C2×C4×C71C2×C22⋊C411yescharacteristic
129C2×C4×C71C2×C4⋊C412yes—
130C2×C4×C71C2×C4⋊C411yescharacteristic
131C2×C4×C71C2×C4⋊C411yescharacteristic
132C2×C4×C71#11712yes—
133C2×C4×C71C4×D412yes—
134C2×C4×C71C4×D412yes—
135C2×C4×C71C4×Q812yes—
136C2×C4×C71C4×Q812yes—
137C2×C4×C71#12112yes—
138C2×C4×C71#12112yes—
139C2×C4×C71C22⋊Q812yes—
140C2×C4×C71C22⋊Q812yes—
141C2×C4×C71#12512yes—
142C2×C4×C71C4⋊Q812yes—
143C2×Dic71—48no—
144C2×Dic71—44no—
145C71⋊C8—28no—
146C22×C71—22no—
147C22×C71—22no—
148C22×C71—22no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71C2×#9511yescharacteristic
154C22×C71#38211yescharacteristic
155C22×C71#38411yescharacteristic
156C22×C71#39011yescharacteristic
157C22×C71#39011yescharacteristic
158C22×C71#39111yescharacteristic
159C22×C71#39111yescharacteristic
160C4×C71—22no—
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—84no—
164C4×C71H(3,ℤ/4)14yescomplemented
165C4×C71C4×C4⋊C412yes—
166C4×C71#38012yes—
167C4×C71#38912yes—
168C4×C71#38912yes—
169Dic71—84no—
170#10837—711nomaximal, Sylow
171C2×C71—21no—
172C2×C71—21no—
173C2×C71—21no—
174C2×C71—21no—
175C2×C71—21no—
176C2×C71—21no—
177C2×C71C4⋊#9511yescharacteristic
178C2×C71C2×H(3,ℤ/4)12yes—
179C2×C4×M16—711no—
180#2285—711no—
181#2285D7111yescharacteristic
182C2×H(3,ℤ/4)—714no—
183C71#1083711yescharacteristic, complemented, Sylow
184C22×M16—712no—
185C2×#117—712no—
186C2×#117C2×D7112yes—
187C22×C42C2×D7111yescharacteristic
188C4×M16—1421no—
189C4×M16—1421no—
190C4×M16—1421no—
191C4×M16—1421no—
192C2×#97—712no—
193C2×C4×C8—712no—
194#384—21no—
195#384—1421no—
196#382—21no—
197#382—1421no—
198#380—22no—
199#380—1422no—
200C4×C4⋊C4—714no—
201C4×C4⋊C4Dic7114yescomplemented
202H(3,ℤ/4)—1428no—
203C23×C4C22⋊D7112yes—
204C23×C4C22×D7111yescharacteristic
205C2×M16—714no—
206C2×M16—714no—
207C2×M16—1422no—
208C2×M16—1422no—
209C2×M16—1422no—
210C2×M16—1422no—
211C22×C8—714no—
212#117—24no—
213#117—24no—
214#117—1424no—
215#117—1424no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—1424no—
219C2×C4⋊C4—1422no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—1422no—
222C2×C42—714no—
223C2×C42Dic14212yes—
224C2×C42C4×D7111yescharacteristic
225C2×C42C4×D7111yescharacteristic
226C2×C42C4×D7112yes—
227C2×C42D28411yescharacteristic
228C2×C42D28411yescharacteristic
229C2×C42C2×Dic7114yes—
230C2×C42C2×Dic7112yes—
231C2×C42C22⋊D7111yescharacteristic
232C2×C42C22⋊D7111yescharacteristic
233#97—1422no—
234#97—1422no—
235C4×C8—1422no—
236C4×C8—1422no—
237#95—24no—
238#95—1424no—
239C24C2×C22⋊D7111yescharacteristic
240C22×C4—24no—
241C22×C4—44no—
242C22×C4—1424no—
243C22×C4(C2×D71)⋊C412yes—
244C22×C4(C2×D71)⋊C412yes—
245C22×C4C22⋊Dic7112yes—
246C22×C4C22⋊Dic7112yes—
247C22×C4Dic71⋊C412yes—
248C22×C4Dic71⋊C412yes—
249C22×C4C2×C4×D7111yescharacteristic
250C22×C4C2×C4×D7111yescharacteristic
251C22×C4C22×Dic7111yescharacteristic
252C22×C4D4×D7112yes—
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4C2×D28411yescharacteristic
255C22×C4Q8×D7112yes—
256C22×C4C2×Dic14211yescharacteristic
257C22×C4#1176586412yes—
258C22×C4#1176586612yes—
259M16—1424no—
260M16—1424no—
261M16—1424no—
262M16—1424no—
263C2×C8—718no—
264C2×C8—1424no—
265C2×C8—1424no—
266C4⋊C4—48no—
267C4⋊C4—44no—
268C4⋊C4—2848no—
269C4⋊C4—2844no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—2842no—
273C22⋊C4—2842no—
274C42—28no—
275C42—1428no—
276C42C4×Dic7112yes—
277C42C4×Dic7112yes—
278C42(C2×D71)⋊C411yescharacteristic
279C42(C2×D71)⋊C411yescharacteristic
280C42(C2×D71)⋊C411yescharacteristic
281C42(C2×D71)⋊C411yescharacteristic
282C42Dic71⋊C412yes—
283C42Dic71⋊C412yes—
284C42C4⋊Dic7112yes—
285C42C4⋊Dic7112yes—
286C23—42no—
287C23C2×(C2×D71)⋊C411yescharacteristic
288C23C2×C22⋊Dic7111yescharacteristic
289C23C2×Dic71⋊C411yescharacteristic
290C23#1455611511yescharacteristic
291C23#1455611711yescharacteristic
292C23#1455612411yescharacteristic
293C23#1455612611yescharacteristic
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—2848no—
303C2×C4—2844no—
304C2×C4C71⋊#9512yes—
305C2×C4C71⋊#9512yes—
306C2×C4C71⋊#9512yes—
307C2×C4C71⋊#9512yes—
308C2×C4C2×C4×Dic7111yescharacteristic
309C2×C4C2×(C2×D71)⋊C411yescharacteristic
310C2×C4C4⋊C4×D7112yes—
311C2×C4C2×Dic71⋊C411yescharacteristic
312C2×C4C2×C4⋊Dic7111yescharacteristic
313C2×C4#1455608612yes—
314C2×C4#1455609012yes—
315C2×C4D4×Dic7112yes—
316C2×C4#1455609312yes—
317C2×C4Q8×Dic7112yes—
318C2×C4#1455609912yes—
319C2×C4#1455610312yes—
320C2×C4#1455610412yes—
321C2×C4#1455610812yes—
322C2×C4#1455611912yes—
323C2×C4#1455612712yes—
324C8—1428no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×C71⋊#9511yescharacteristic
333C22#1798586411yescharacteristic
334C22#1798586811yescharacteristic
335C22#1798590011yescharacteristic
336C22#1798590211yescharacteristic
337C22#1798590411yescharacteristic
338C22#1798590611yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—84no—
343C4—5684no—
344C4#1798575514yescomplemented
345C4C4⋊C4×Dic7112yes—
346C4#1798585612yes—
347C4#1798589512yes—
348C4#1798589812yes—
349C2—21no—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2#2258656911yescharacteristic
356C2C2×#1798575512yes—
subgroupG/H
centerC2×C4C2×C4×Dic71
commutatorC2×C4×C71C2×C42
FrattiniC23×C4C22×D71
FittingC71×#2285C2
socleC22×C71C2×#95
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2285D71
hypocenterC71#10837
LCSG ▷ C2×C4×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #2285

Complements

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

NH ≅ G/Ntype
11. C2×#14556039C4semidirect
12. C2×C4×C71⋊C8C4semidirect
16. C4×C4⋊Dic71C4semidirect
23. C4×C4⋊C4×C71C4semidirect
67. C2×C71⋊C8C4⋊C4semidirect
164. C4×C71H(3,ℤ/4)semidirect
183. C71#10837semidirect
201. C4×C4⋊C4Dic71semidirect
344. C4#17985755semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117658492
#225907472
C2×#179857552
C71×#22852
#1083771