Finite Group Databasebeta

← #31890912

Subgroups — #31890912

Order 18176.

514 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590697C211yesmaximal, characteristic
2C2×#17985725C211yesmaximal, characteristic
3#22586187C212yesmaximal
4#22586186C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C2×#14556087C411yescharacteristic
7C2×#14556087C411yescharacteristic
8C2×C22⋊(C71⋊C8)C2212yes—
9C4⋊C4×Dic71—21no—
10C4⋊C4×Dic71—21no—
11C4⋊C4×Dic71—21no—
12C4⋊C4×Dic71—21no—
13C4×C22⋊Dic71C411yescharacteristic
14C4×C22⋊Dic71C411yescharacteristic
15C4×C22⋊Dic71C2211yescharacteristic
16C4×C22⋊Dic71C2211yescharacteristic
17#17985725—21no—
18#17985725—21no—
19C22×C4⋊C4×C71C2211yescharacteristic
20C2×C22⋊C8×C71C2212yes—
21#324×C71—21no—
22#324×C71—21no—
23C22×C71⋊C8—22no—
24#14556087—22no—
25#14556087—22no—
26#14556087—22no—
27#14556087C814yescomplemented
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×C22⋊Dic71C2×C411yescharacteristic
31C2×C22⋊Dic71C2×C411yescharacteristic
32C2×C22⋊Dic71C2×C411yescharacteristic
33C2×C22⋊Dic71C2×C411yescharacteristic
34C2×C4×Dic71—21no—
35C2×C4×Dic71—21no—
36C2×C4×Dic71—21no—
37C2×C4×Dic71—21no—
38C2×C4×Dic71—21no—
39C2×C4×Dic71—21no—
40C22⋊(C71⋊C8)—22no—
41C22⋊(C71⋊C8)—22no—
42C71⋊#95—21no—
43C71⋊#95—21no—
44C71⋊#95—21no—
45C71⋊#95—21no—
46C23×C4×C71C2×C411yescharacteristic
47C23×C4×C71C2×C411yescharacteristic
48C23×C4×C71C2311yescharacteristic
49C22×C8×C71—22no—
50C2×C4⋊C4×C71—22no—
51C2×C4⋊C4×C71—22no—
52C2×C4⋊C4×C71D411yescharacteristic
53C2×C4⋊C4×C71D411yescharacteristic
54C2×C4⋊C4×C71D411yescharacteristic
55C2×C4⋊C4×C71D411yescharacteristic
56C22⋊C8×C71—22no—
57C22⋊C8×C71—22no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C22×Dic71—21no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—82no—
64C4⋊Dic71—24no—
65C4⋊Dic71—24no—
66C22⋊Dic71—21no—
67C22⋊Dic71—21no—
68C22⋊Dic71—21no—
69C22⋊Dic71—21no—
70C22⋊Dic71C2×C812yes—
71C22⋊Dic71C2×C812yes—
72C22⋊Dic71M1612yes—
73C22⋊Dic71M1612yes—
74C4×Dic71—24no—
75C4×Dic71—24no—
76C4×Dic71—81no—
77C4×Dic71—81no—
78C4×Dic71—81no—
79C4×Dic71—81no—
80C24×C71C22×C411yescharacteristic
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—81no—
88C22×C4×C71—81no—
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C2×C812yes—
94C22×C4×C71M1612yes—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71#4112yes—
98C22×C4×C71#4112yes—
99C2×C8×C71—42no—
100C2×C8×C71—82no—
101C4⋊C4×C71—22no—
102C4⋊C4×C71—22no—
103C4⋊C4×C71—22no—
104C4⋊C4×C71—22no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71C2×C22⋊C411yescharacteristic
108C23×C71#11711yescharacteristic
109C23×C71#11711yescharacteristic
110C23×C71C22×C811yescharacteristic
111C23×C71C2×M1611yescharacteristic
112C23×C71C2×M1611yescharacteristic
113C23×C71C2×M1611yescharacteristic
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—42no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—42no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—81no—
131C2×C4×C71—81no—
132C2×C4×C71C22⋊C812yes—
133C2×C4×C71C22⋊C812yes—
134C2×C4×C71#12312yes—
135C2×C4×C71#12312yes—
136C2×Dic71—22no—
137C2×Dic71—22no—
138C2×Dic71—22no—
139C2×Dic71—22no—
140C2×Dic71—41no—
141C2×Dic71—41no—
142C2×Dic71—41no—
143C2×Dic71—41no—
144C2×Dic71—41no—
145C2×Dic71—41no—
146C2×Dic71—41no—
147C2×Dic71—41no—
148C8×C71—82no—
149C71⋊C8—82no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71#38111yescharacteristic
161C22×C71C2×C22⋊C811yescharacteristic
162C22×C71#40711yescharacteristic
163C22×C71#43111yescharacteristic
164C22×C71#43111yescharacteristic
165C22×C71#43211yescharacteristic
166C22×C71#43211yescharacteristic
167C4×C71—22no—
168C4×C71—22no—
169C4×C71—22no—
170C4×C71—22no—
171C4×C71—41no—
172C4×C71—41no—
173Dic71—42no—
174Dic71—42no—
175Dic71—42no—
176Dic71—42no—
177#8735—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—21no—
182C2×C71—21no—
183C2×C71—21no—
184C2×C71#181711yescharacteristic
185C2×C71#182811yescharacteristic
186C2×C71#183011yescharacteristic
187#2274—711no—
188C2×#324—711no—
189C2×#324D7111yescharacteristic
190#1282—712no—
191#1282—712no—
192C71#873511yescharacteristic, complemented, Sylow
193C2×#117—711no—
194C2×#117—711no—
195C22×C4⋊C4C2×D7111yescharacteristic
196C2×C22⋊C8—712no—
197C2×C22⋊C8C2×D7112yes—
198C4×C4⋊C4—1421no—
199C4×C4⋊C4—1421no—
200C4×C4⋊C4—1421no—
201C4×C4⋊C4—1421no—
202C4×C22⋊C4—711no—
203C4×C22⋊C4—711no—
204C4×C22⋊C4—711no—
205C4×C22⋊C4—711no—
206#324—21no—
207#324—21no—
208#324—1421no—
209#324—1421no—
210C23×C4C4×D7111yescharacteristic
211C23×C4C4×D7111yescharacteristic
212C23×C4C22×D7111yescharacteristic
213C22×C8—22no—
214C22×C8—1422no—
215#117—714no—
216#117—1422no—
217#117—1422no—
218#117—1422no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—1421no—
222C2×C4⋊C4—1421no—
223C2×C4⋊C4D28411yescharacteristic
224C2×C4⋊C4D28411yescharacteristic
225C2×C4⋊C4C22⋊D7111yescharacteristic
226C2×C4⋊C4C22⋊D7111yescharacteristic
227C2×C22⋊C4—711no—
228C2×C22⋊C4—711no—
229C2×C22⋊C4—711no—
230C2×C22⋊C4—711no—
231C2×C42—1421no—
232C2×C42—1421no—
233C2×C42—1421no—
234C2×C42—1421no—
235C2×C42—1421no—
236C2×C42—1421no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1422no—
240C22⋊C8—1422no—
241#95—1421no—
242#95—1421no—
243#95—1421no—
244#95—1421no—
245C24C2×C4×D7111yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—1421no—
255C22×C4—1421no—
256C22×C4—1421no—
257C22×C4—1421no—
258C22×C4(C2×D71)⋊C411yescharacteristic
259C22×C4(C2×D71)⋊C411yescharacteristic
260C22×C4(C2×D71)⋊C411yescharacteristic
261C22×C4(C2×D71)⋊C411yescharacteristic
262C22×C4C8×D7112yes—
263C22×C4#1176584812yes—
264C22×C4C2×C22⋊D7111yescharacteristic
265C22×C4C2×D28411yescharacteristic
266C22×C4#1176586412yes—
267C22×C4#1176586412yes—
268C2×C8—42no—
269C2×C8—82no—
270C2×C8—2842no—
271C2×C8—5682no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—1424no—
277C4⋊C4—1424no—
278C22⋊C4—712no—
279C22⋊C4—712no—
280C22⋊C4—712no—
281C22⋊C4—712no—
282C22⋊C4—1421no—
283C22⋊C4—1421no—
284C22⋊C4—1421no—
285C22⋊C4—1421no—
286C42—5681no—
287C42—5681no—
288C42—5681no—
289C42—5681no—
290C42—1424no—
291C42—1424no—
292C23—41no—
293C23—41no—
294C23C2×(C2×D71)⋊C411yescharacteristic
295C23#1455608411yescharacteristic
296C23#1455608411yescharacteristic
297C23C2×C8×D7111yescharacteristic
298C23M16×D7111yescharacteristic
299C23M16×D7111yescharacteristic
300C23C2×#1176584811yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—2841no—
320C2×C4—2841no—
321C2×C4—2841no—
322C2×C4—2841no—
323C2×C4—2841no—
324C2×C4—2841no—
325C2×C4—2841no—
326C2×C4—2841no—
327C2×C4—1422no—
328C2×C4—1422no—
329C2×C4—1422no—
330C2×C4—1422no—
331C2×C4(C2×D71)⋊C812yes—
332C2×C4(C2×D71)⋊C812yes—
333C2×C4#1455610912yes—
334C2×C4#1455611212yes—
335C8—82no—
336C8—5682no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#1798586011yescharacteristic
348C22C2×(C2×D71)⋊C811yescharacteristic
349C22#1798596211yescharacteristic
350C22#1798604911yescharacteristic
351C22#1798604911yescharacteristic
352C22#1798605311yescharacteristic
353C22#1798605311yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—2842no—
361C4—2842no—
362C4—2842no—
363C4—2842no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2258840611yescharacteristic
371C2#2258844611yescharacteristic
372C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC23×C71C22×C8
FrattiniC23×C4C22×D71
FittingC2×#324×C71C2
socleC22×C71#381
prime radicalC24×C71C22×C4
cosocleC23×C4×C71C23
hypercenterC2×#324D71
hypocenterC71#8735
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
27. #14556087C8semidirect
192. C71#8735semidirect

Maximal subgroups

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

subgroupindex
#225906972
C2×#179857252
#225861872
#225861862
C2×#324×C712
#873571