Finite Group Databasebeta

← #31893172

Subgroups — #31893172

Order 18176.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596679C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3C2×#17985735C212yesmaximal
4#22586219C212yesmaximal
5C2×C4×C4⋊C4×C71C211yesmaximal, characteristic
6C2×#14556107C2211yescharacteristic
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985905C412yescomplemented
9#17985897C412yescomplemented
10#17985883—22no—
11#17985872—21no—
12#17985862—21no—
13#17985859C2211yescharacteristic
14#17985754—22no—
15#17985735—24no—
16C22×C4⋊C4×C71—21no—
17C22×C42×C71—21no—
18C22×C42×C71C2211yescharacteristic
19C4×C4⋊C4×C71—24no—
20C2×#95×C71C2212yes—
21C22×Dic142C2×C411yescharacteristic
22C22×C71⋊C8—22no—
23#14556107—22no—
24#14556107—22no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C22⋊Dic71—21no—
30C2×C22⋊Dic71—21no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C23×C4×C71—22no—
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C2×C4⋊C4×C71—28no—
42C2×C4⋊C4×C71—24no—
43C2×C42×C71—22no—
44C2×C42×C71—22no—
45C2×C42×C71—24no—
46C2×C42×C71—42no—
47C2×C42×C71—42no—
48C2×C42×C71C2×C412yes—
49C2×C42×C71D412yes—
50#95×C71—22no—
51#95×C71D414yes—
52C2×Dic142—22no—
53C2×Dic142—42no—
54C22×Dic71—21no—
55C22×Dic71—22no—
56C22×Dic71—21no—
57C2×C71⋊C8—44no—
58C2×C71⋊C8—42no—
59C4⋊Dic71—22no—
60C4⋊Dic71—42no—
61Dic71⋊C4—44no—
62Dic71⋊C4—42no—
63C22⋊Dic71—41no—
64C22⋊Dic71—41no—
65C22⋊Dic71—41no—
66C22⋊Dic71—41no—
67C24×C71C2×D411yescharacteristic
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—24no—
75C22×C4×C71—28no—
76C22×C4×C71—24no—
77C22×C4×C71—42no—
78C22×C4×C71—44no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C412yes—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71SD1612yes—
84C22×C4×C71Q1612yes—
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D412yes—
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4112yes—
89C4⋊C4×C71—216no—
90C42×C71—24no—
91C42×C71—24no—
92C42×C71—48no—
93C23×C71—22no—
94C23×C71—22no—
95C23×C71C2×C22⋊C411yescharacteristic
96C23×C71#12011yescharacteristic
97C23×C71#12311yescharacteristic
98C23×C71C2×SD1611yescharacteristic
99C23×C71C2×Q1611yescharacteristic
100C23×C71AGL(1,ℤ/8)11yescharacteristic
101C23×C71#13711yescharacteristic
102C2×C4×C71—24no—
103C2×C4×C71—24no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—28no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—44no—
118C2×C4×C71—48no—
119C2×C4×C71—48no—
120C2×C4×C71—44no—
121C2×C4×C71#10314yes—
122C2×C4×C71#10414yes—
123C2×C4×C71C4×D412yes—
124C2×C4×C71#12112yes—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—42no—
130C2×Dic71—42no—
131C2×Dic71—42no—
132Dic142—82no—
133C71⋊C8—82no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—22no—
142C22×C71#38611yescharacteristic
143C22×C71#41012yes—
144C22×C71C2×#10311yescharacteristic
145C22×C71#41711yescharacteristic
146C22×C71C2×#10412yes—
147C22×C71#45011yescharacteristic
148C22×C71#45111yescharacteristic
149C22×C71#48311yescharacteristic
150C22×C71#48411yescharacteristic
151C4×C71—24no—
152C4×C71—24no—
153C4×C71—22no—
154C4×C71—22no—
155C4×C71—48no—
156Dic71—81no—
157Dic71—81no—
158Dic71—82no—
159#9144—711nomaximal, Sylow
160C2×C71—21no—
161C2×C71—21no—
162C2×C71—21no—
163C2×C71—21no—
164C2×C71#150312yes—
165C2×C71#150912yes—
166C2×C71#187211yescharacteristic
167C2×C71#189611yescharacteristic
168C2×C71#189911yescharacteristic
169#2855—711no—
170C2×C4×C4⋊C4D7111yescharacteristic
171#1817—711no—
172C2×#328—712no—
173#1294—712no—
174C71#914411yescharacteristic, complemented, Sylow
175C2×C22⋊Q8—711no—
176C22×C4⋊C4—21no—
177C22×C42—21no—
178C22×C42C2×D7111yescharacteristic
179C2×C22⋊C8—712no—
180#391—712no—
181#389—712no—
182#387—1422no—
183#385—1421no—
184#381—1421no—
185#380—711no—
186C4×C4⋊C4—24no—
187C2×#95C2×D7112yes—
188#336—1422no—
189#328—1424no—
190C22×Q8—711no—
191C23×C4—22no—
192C23×C4C22⋊D7111yescharacteristic
193C23×C4C22⋊D7111yescharacteristic
194C23×C4C22×D7111yescharacteristic
195C22×C8—1422no—
196C22⋊Q8—1422no—
197C22⋊Q8—1422no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—712no—
201C2×C4⋊C4—711no—
202C2×C4⋊C4—1422no—
203C2×C4⋊C4—1421no—
204C2×C22⋊C4—1421no—
205C2×C22⋊C4—1421no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C4×D7112yes—
212C2×C42D28412yes—
213C22⋊C8—1422no—
214C22⋊C8—1422no—
215#95—22no—
216#95—1422no—
217#95—1422no—
218#95—1422no—
219#95—1421no—
220#95C22⋊D7114yes—
221C24C2×C22⋊D7111yescharacteristic
222C2×Q8—2842no—
223C2×Q8—1422no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1422no—
238C22×C4—1421no—
239C22×C4—1421no—
240C22×C4(C2×D71)⋊C412yes—
241C22×C4(C2×D71)⋊C412yes—
242C22×C4#1176585312yes—
243C22×C4#1176585512yes—
244C22×C4C2×C4×D7111yescharacteristic
245C22×C4C2×C22⋊D7112yes—
246C22×C4C2×D28411yescharacteristic
247C22×C4#1176586512yes—
248C2×C8—2844no—
249C2×C8—2842no—
250C4⋊C4—216no—
251C4⋊C4—2842no—
252C4⋊C4—2844no—
253C4⋊C4—2842no—
254C4⋊C4—1422no—
255C22⋊C4—2841no—
256C22⋊C4—2841no—
257C22⋊C4—2841no—
258C22⋊C4—2841no—
259C42—24no—
260C42—24no—
261C42—48no—
262C23—22no—
263C23—22no—
264C23C2×(C2×D71)⋊C411yescharacteristic
265C23#1455609711yescharacteristic
266C23#1455611111yescharacteristic
267C23C2×#1176585311yescharacteristic
268C23C2×#1176585511yescharacteristic
269C23#1455615411yescharacteristic
270C23#1455615911yescharacteristic
271Q8—5682no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—2841no—
292C2×C4—2841no—
293C2×C4—2842no—
294C2×C4—2842no—
295C2×C4—2842no—
296C2×C4—2842no—
297C2×C4—2842no—
298C2×C4#1455605114yes—
299C2×C4#1455605614yes—
300C2×C4C4×C22⋊D7112yes—
301C2×C4#1455610212yes—
302C8—5682no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1798587811yescharacteristic
312C22#1798597212yes—
313C22C2×#1455605111yescharacteristic
314C22#1798599811yescharacteristic
315C22C2×#1455605612yes—
316C22#1798615011yescharacteristic
317C22#1798615511yescharacteristic
318C22#1798632911yescharacteristic
319C22#1798633411yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—48no—
325C4—5681no—
326C4—5681no—
327C4—5682no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2258683812yes—
333C2#2258686312yes—
334C2#2258864711yescharacteristic
335C2#2258878311yescharacteristic
336C2#2258879811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×C4×C4⋊C4×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×C4×C4⋊C4D71
hypocenterC71#9144
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #17985905C4semidirect
9. #17985897C4semidirect
174. C71#9144semidirect

Maximal subgroups

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

subgroupindex
#225966792
#225884082
C2×#179857352
#225862192
C2×C4×C4⋊C4×C712
#914471