Finite Group Databasebeta

← #31892512

Subgroups — #31892512

Order 18176.

660 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
1#22590958C211yesmaximal, characteristic
2#22588426C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586219C212yesmaximal
5C2×#398×C71C211yesmaximal, characteristic
6C2×#14556087—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985898—24no—
9#17985859—21no—
10#17985859C2212yes—
11#17985761—24no—
12#17985725—22no—
13C22×C4⋊C4×C71—21no—
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#398×C71—22no—
17#398×C71C414yescomplemented
18C2×#95×C71—21no—
19C2×#95×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21#14556087—42no—
22#14556087—42no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71D414yes—
26C2×C22⋊Dic71—22no—
27C2×C4×Dic71—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C23×C4×C71—21no—
33C23×C4×C71—21no—
34C23×C4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—24no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—42no—
44C2×C4⋊C4×C71—42no—
45C2×C4⋊C4×C71C2×C412yes—
46C2×C4⋊C4×C71C2×C412yes—
47C2×C4⋊C4×C71D412yes—
48#95×C71—24no—
49#95×C71—21no—
50#95×C71C2×C412yes—
51C22×Dic71—24no—
52C2×C71⋊C8—44no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—44no—
55C4⋊Dic71—44no—
56C4⋊Dic71—82no—
57C22⋊Dic71—42no—
58C22⋊Dic71—42no—
59C4×Dic71—44no—
60C24×C71C2×D411yescharacteristic
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—24no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71—41no—
73C22×C4×C71—41no—
74C22×C4×C71—42no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71C22⋊C412yes—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71D812yes—
84C22×C4×C71SD1612yes—
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D412yes—
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4112yes—
89C4⋊C4×C71—24no—
90C4⋊C4×C71—24no—
91C4⋊C4×C71—24no—
92C4⋊C4×C71—48no—
93C4⋊C4×C71—42no—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71C2×C22⋊C411yescharacteristic
97C23×C71#12011yescharacteristic
98C23×C71#12311yescharacteristic
99C23×C71C2×D811yescharacteristic
100C23×C71C2×SD1611yescharacteristic
101C23×C71AGL(1,ℤ/8)11yescharacteristic
102C23×C71#13711yescharacteristic
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—24no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—24no—
111C2×C4×C71—22no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—44no—
116C2×C4×C71—44no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—81no—
125C2×C4×C71—81no—
126C2×C4×C71—81no—
127C2×C4×C71—81no—
128C2×C4×C71—81no—
129C2×C4×C71#10214yes—
130C2×C4×C71C4×D412yes—
131C2×C4×C71#12112yes—
132C2×Dic71—44no—
133C2×Dic71—44no—
134C2×Dic71—44no—
135C71⋊C8—82no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71#38611yescharacteristic
145C22×C71C2×#10211yescharacteristic
146C22×C71#41711yescharacteristic
147C22×C71#44711yescharacteristic
148C22×C71#44811yescharacteristic
149C22×C71#45612yes—
150C22×C71#45812yes—
151C22×C71#48011yescharacteristic
152C22×C71#48111yescharacteristic
153C4×C71—24no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—44no—
157C4×C71—42no—
158C4×C71—82no—
159Dic71—84no—
160#9027—711nomaximal, Sylow
161C2×C71—21no—
162C2×C71—21no—
163C2×C71—21no—
164C2×C71—21no—
165C2×C71#161012yes—
166C2×C71#161112yes—
167C2×C71#187411yescharacteristic
168C2×C71#188611yescharacteristic
169C2×C71#189011yescharacteristic
170C2×#398D7111yescharacteristic
171#2316—711no—
172#1823—712no—
173C2×#324—711no—
174#1294—712no—
175C71#902711yescharacteristic, complemented, Sylow
176C2×#117—1421no—
177C22×C4⋊C4—21no—
178C22×C4⋊C4C2×D7111yescharacteristic
179C22×C4⋊C4C2×D7111yescharacteristic
180C2×C22⋊C8—712no—
181#398—22no—
182#398Dic7114yescomplemented
183#389—1424no—
184#380—712no—
185#380—1421no—
186C2×#95—21no—
187C2×#95C2×D7111yescharacteristic
188#340—1424no—
189#324—1422no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C22⋊D7111yescharacteristic
193C23×C4C22⋊D7111yescharacteristic
194C23×C4C22×D7111yescharacteristic
195C22×C8—1422no—
196#117—2842no—
197#117—2842no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—714no—
207C2×C4⋊C4—1422no—
208C2×C4⋊C4—1422no—
209C2×C4⋊C4C2×Dic7112yes—
210C2×C4⋊C4C2×Dic7112yes—
211C2×C4⋊C4C22⋊D7112yes—
212C2×C22⋊C4—1422no—
213C2×C42—1422no—
214C22⋊C8—1422no—
215C22⋊C8—1422no—
216#95—24no—
217#95—21no—
218#95—1422no—
219#95—1422no—
220#95C2×Dic7112yes—
221C24C2×C22⋊D7111yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1424no—
243C22×C4C22⋊Dic7112yes—
244C22×C4C22⋊Dic7112yes—
245C22×C4#1176585012yes—
246C22×C4#1176585312yes—
247C22×C4C22×Dic7111yescharacteristic
248C22×C4D4×D7112yes—
249C22×C4C2×C22⋊D7111yescharacteristic
250C22×C4#1176586412yes—
251C2×C8—2844no—
252C2×C8—2842no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—42no—
258C4⋊C4—2844no—
259C4⋊C4—2844no—
260C4⋊C4—5682no—
261C22⋊C4—2842no—
262C22⋊C4—2842no—
263C42—2844no—
264C23—22no—
265C23—22no—
266C23C2×C22⋊Dic7111yescharacteristic
267C23#1455609611yescharacteristic
268C23#1455610911yescharacteristic
269C23C2×#1176585011yescharacteristic
270C23C2×#1176585311yescharacteristic
271C23#1455615311yescharacteristic
272C23#1455615811yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—2844no—
300C2×C4—2844no—
301C2×C4—2844no—
302C2×C4#1455604914yes—
303C2×C4D4×Dic7112yes—
304C2×C4#1455610012yes—
305C8—5682no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22#1798587911yescharacteristic
315C22C2×#1455604911yescharacteristic
316C22#1798600011yescharacteristic
317C22#1798613211yescharacteristic
318C22#1798613711yescharacteristic
319C22#1798618112yes—
320C22#1798618812yes—
321C22#1798631011yescharacteristic
322C22#1798631511yescharacteristic
323C4—22no—
324C4—22no—
325C4—24no—
326C4—44no—
327C4—42no—
328C4—82no—
329C4—5684no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2258734212yes—
335C2#2258734712yes—
336C2#2258865811yescharacteristic
337C2#2258873411yescharacteristic
338C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#398×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#398D71
hypocenterC71#9027
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
17. #398×C71C4semidirect
175. C71#9027semidirect
182. #398Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909582
#225884262
C2×#179857252
#225862192
C2×#398×C712
#902771