Finite Group Databasebeta

← #31924463

Subgroups — #31924463

Order 18176.

1590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985936C211yesmaximal, characteristic
2C2×#17985897C213yesmaximal, complemented
3C2×#17985890C213yesmaximal
4#22586607C213yesmaximal, complemented
5#22586605C211yesmaximal, characteristic, complemented
6#22586586C213yesmaximal
7C4×#95×C71C211yesmaximal, characteristic, complemented
8C22×C4⋊Dic71C2213yes—
9C22×C4⋊Dic71C2213yes—
10C22×Dic71⋊C4C2213yes—
11C22×(C2×D71)⋊C4C2211yescharacteristic
12C22×(C2×D71)⋊C4C2213yes—
13C22×(C2×D71)⋊C4C2213yes—
14#17985936C418yescomplemented
15#17985897—212no—
16#17985890—212no—
17C2×C71⋊#95C2213yes—
18C2×C71⋊#95C2213yes—
19C2×C71⋊#95C2216yes—
20C22×C42×C71C2213yes—
21C2×#95×C71C2213yes—
22C2×#95×C71C2211yescharacteristic
23C24×D71C2311yescharacteristic
24C23×Dic71C2311yescharacteristic
25C23×Dic71C2313yes—
26C23×Dic71C2313yes—
27C2×C4⋊Dic71—26no—
28C2×C4⋊Dic71—26no—
29C2×C4⋊Dic71—26no—
30C2×C4⋊Dic71—46no—
31C2×C4⋊Dic71C2×C4112yes—
32C2×Dic71⋊C4—26no—
33C2×Dic71⋊C4—212no—
34C2×(C2×D71)⋊C4—26no—
35C2×(C2×D71)⋊C4—26no—
36C2×(C2×D71)⋊C4—26no—
37C2×(C2×D71)⋊C4—26no—
38C2×(C2×D71)⋊C4—46no—
39C2×(C2×D71)⋊C4C2×C4112yes—
40C71⋊#95—26no—
41C71⋊#95—26no—
42C71⋊#95—212no—
43C23×C4×C71C2313yes—
44C23×C4×C71C2313yes—
45C23×C4×C71C2311yescharacteristic
46C2×C42×C71—46no—
47C2×C42×C71D4112yes—
48#95×C71—26no—
49#95×C71C2×C414yes—
50C23×D71—23no—
51C23×D71—23no—
52C23×D71—23no—
53C23×D71—21no—
54C23×D71—23no—
55C23×D71—21no—
56C23×D71C22×C412yes—
57C22×Dic71—23no—
58C22×Dic71—26no—
59C22×Dic71—23no—
60C22×Dic71—26no—
61C22×Dic71—23no—
62C22×Dic71—23no—
63C22×Dic71—26no—
64C22×Dic71—26no—
65C22×Dic71—23no—
66C22×Dic71—21no—
67C22×Dic71—23no—
68C22×Dic71—23no—
69C22×Dic71C22×C416yes—
70C4⋊Dic71—224no—
71C4⋊Dic71—412no—
72Dic71⋊C4—412no—
73(C2×D71)⋊C4—224no—
74(C2×D71)⋊C4—412no—
75(C2×D71)⋊C4—44no—
76C24×C71C2411yescharacteristic
77C22×C4×C71—23no—
78C22×C4×C71—26no—
79C22×C4×C71—26no—
80C22×C4×C71—21no—
81C22×C4×C71—26no—
82C22×C4×C71—26no—
83C22×C4×C71—23no—
84C22×C4×C71—23no—
85C22×C4×C71C22×C416yes—
86C22×C4×C71C2×D416yes—
87C22×C4×C71C2×D416yes—
88C22×C4×C71C2×D416yes—
89C22×C4×C71#4116yes—
90C42×C71—412no—
91C23×C71C23×C411yescharacteristic
92C23×C71C22×D413yes—
93C23×C71C2×#4113yes—
94C23×C71H(5,𝔽2)13yes—
95C23×C712−1+411yescharacteristic
96C23×C712−1+413yes—
97C23×C712−1+411yescharacteristic
98C22×D71—26no—
99C22×D71—22no—
100C22×D71—26no—
101C22×D71—41no—
102C22×D71—43no—
103C22×D71—43no—
104C22×D71—43no—
105C22×D71—43no—
106C22×D71—43no—
107C22×D71—46no—
108C22×D71—43no—
109C22×D71—43no—
110C2×C4×C71—212no—
111C2×C4×C71—26no—
112C2×C4×C71—26no—
113C2×C4×C71—26no—
114C2×C4×C71—26no—
115C2×C4×C71—212no—
116C2×C4×C71—26no—
117C2×C4×C71—26no—
118C2×C4×C71—26no—
119C2×C4×C71—46no—
120C2×C4×C71—46no—
121C2×C4×C71C4×D4112yes—
122C2×C4×C71#12714yes—
123C2×Dic71—26no—
124C2×Dic71—212no—
125C2×Dic71—43no—
126C2×Dic71—43no—
127C2×Dic71—46no—
128C2×Dic71—41no—
129C2×Dic71—43no—
130C2×Dic71—43no—
131C2×Dic71—46no—
132C2×Dic71—43no—
133C2×Dic71—43no—
134C2×Dic71—43no—
135C2×Dic71—46no—
136C22×C71C2×C4×D413yes—
137C22×C71#51911yescharacteristic
138C22×C71#52013yes—
139C22×C71C2×#12711yescharacteristic
140C22×C71#53213yes—
141C22×C71#53413yes—
142C22×C71#53613yes—
143C22×C71#53613yes—
144C22×C71#53713yes—
145C22×C71#53916yes—
146C22×C71#54113yes—
147C22×C71#54213yes—
148C2×D71—42no—
149C2×D71—46no—
150C2×D71—46no—
151C2×D71—81no—
152C2×D71—81no—
153C2×D71—83no—
154C2×D71—83no—
155C4×C71—212no—
156C4×C71—24no—
157C4×C71—46no—
158Dic71—46no—
159Dic71—83no—
160Dic71—81no—
161#14322—711nomaximal, Sylow
162C2×C71C4×#12711yescharacteristic
163C2×C71#232213yes—
164C2×C71#232313yes—
165C2×C71#265611yescharacteristic
166C2×C71#265811yescharacteristic
167C2×C71#266313yes—
168C2×C71#266413yes—
169D71—82no—
170C2×#399—711no—
171C2×#389—713no—
172C2×#388—713no—
173#1445—713no—
174#1444—711no—
175#1439—713no—
176C4×#95D7111yescharacteristic, complemented
177C71#1432211yescharacteristic, complemented, Sylow
178C22×C4⋊C4—713no—
179C22×C4⋊C4—713no—
180C22×C4⋊C4—713no—
181C22×C22⋊C4—713no—
182C22×C22⋊C4—713no—
183C22×C22⋊C4—711no—
184C22×C42C2×D7113yes—
185#399—718no—
186#389—14212no—
187#388—14212no—
188C2×#95—716no—
189C2×#95—713no—
190C2×#95—713no—
191C2×#95C2×D7113yes—
192C2×#95C2×D7111yescharacteristic
193C25—711no—
194C23×C4—713no—
195C23×C4—713no—
196C23×C4—711no—
197C23×C4C22×D7113yes—
198C23×C4C22×D7113yes—
199C23×C4C22×D7111yescharacteristic
200C2×C4⋊C4—2846no—
201C2×C4⋊C4—7112no—
202C2×C4⋊C4—1426no—
203C2×C4⋊C4—14212no—
204C2×C4⋊C4—1426no—
205C2×C4⋊C4—1426no—
206C2×C4⋊C4—1426no—
207C2×C22⋊C4—2846no—
208C2×C22⋊C4—7112no—
209C2×C22⋊C4—1426no—
210C2×C22⋊C4—1426no—
211C2×C22⋊C4—1426no—
212C2×C22⋊C4—1426no—
213C2×C42—46no—
214C2×C42D284112yes—
215#95—26no—
216#95—1426no—
217#95—1426no—
218#95—14212no—
219#95C4×D7114yes—
220C24—712no—
221C24—1423no—
222C24—1423no—
223C24—1423no—
224C24—1423no—
225C24—1421no—
226C24—1421no—
227C24C23×D7111yescharacteristic
228C22×C4—23no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—21no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—716no—
237C22×C4—1423no—
238C22×C4—1423no—
239C22×C4—1426no—
240C22×C4—1426no—
241C22×C4—1423no—
242C22×C4—1423no—
243C22×C4—1421no—
244C22×C4—1423no—
245C22×C4—1423no—
246C22×C4—1426no—
247C22×C4—1423no—
248C22×C4—1426no—
249C22×C4C2×C4×D7116yes—
250C22×C4C2×D28416yes—
251C22×C4C2×D28416yes—
252C22×C4C2×D28416yes—
253C22×C4#1176586516yes—
254C4⋊C4—28412no—
255C4⋊C4—28412no—
256C4⋊C4—14224no—
257C22⋊C4—28412no—
258C22⋊C4—2844no—
259C22⋊C4—14224no—
260C42—412no—
261C23—2843no—
262C23—2843no—
263C23—2843no—
264C23—2843no—
265C23—2846no—
266C23—2843no—
267C23—2843no—
268C23—2841no—
269C23—2843no—
270C23—1426no—
271C23—1422no—
272C23—1426no—
273C23C22×C4×D7111yescharacteristic
274C23C22×D28413yes—
275C23C2×#1176586513yes—
276C23#1455617313yes—
277C23#1455617513yes—
278C23#1455617511yescharacteristic
279C23#1455617611yescharacteristic
280C2×C4—26no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—212no—
285C2×C4—212no—
286C2×C4—26no—
287C2×C4—26no—
288C2×C4—26no—
289C2×C4—46no—
290C2×C4—46no—
291C2×C4—2843no—
292C2×C4—2843no—
293C2×C4—2846no—
294C2×C4—2843no—
295C2×C4—2846no—
296C2×C4—2843no—
297C2×C4—2843no—
298C2×C4—2843no—
299C2×C4—2846no—
300C2×C4—2841no—
301C2×C4—2843no—
302C2×C4—1426no—
303C2×C4—14212no—
304C2×C4C4×D284112yes—
305C2×C4#1455612514yes—
306C22—2846no—
307C22—2846no—
308C22—2842no—
309C22—5681no—
310C22—5681no—
311C22—5683no—
312C22—5683no—
313C22C2×C4×D28413yes—
314C22#1798647511yescharacteristic
315C22#1798647813yes—
316C22C2×#1455612511yescharacteristic
317C22#1798654913yes—
318C22#1798656113yes—
319C22#1798657513yes—
320C22#1798657513yes—
321C22#1798658013yes—
322C22#1798659116yes—
323C22#1798660713yes—
324C22#1798661413yes—
325C4—212no—
326C4—24no—
327C4—46no—
328C4—2846no—
329C4—5683no—
330C4—5681no—
331C2—5682no—
332C2C4×#1455612511yescharacteristic
333C2#2259101113yes—
334C2#2259102313yes—
335C2#2259444211yescharacteristic
336C2#2259445011yescharacteristic
337C2#2259448813yes—
338C2#2259449913yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC4×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4×#95D71
hypocenterC71#14322
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#17985897C2semidirect
4. #22586607C2semidirect
5. #22586605C2semidirect
7. C4×#95×C71C2semidirect
14. #17985936C4semidirect
176. C4×#95D71semidirect
177. C71#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#179859362
C2×#179858972
C2×#179858902
#225866072
#225866052
#225865862
C4×#95×C712
#1432271