Finite Group Databasebeta

← #31908061

Subgroups — #31908061

Order 18176.

590 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556043C211yesmaximal, characteristic
2#22586681C211yesmaximal, characteristic
3#22586361C214yesmaximal, complemented
4C71×#2409C211yesmaximal, characteristic
5#17986056—21no—
6C2×#14556043—21no—
7C4×C22⋊Dic71—21no—
8#17985766C2212yes—
9#17985723—21no—
10#17985723C2212yes—
11C23×Q8×C71C411yescharacteristic, complemented
12C2×C22⋊Q8×C71—21no—
13C2×C4×D4×C71C2211yescharacteristic
14#395×C71C412yescomplemented
15#393×C71C2212yes—
16#391×C71—21no—
17#386×C71—21no—
18#385×C71—21no—
19#380×C71C411yescharacteristic, complemented
20C2×C22⋊Dic71—22no—
21C2×C4×Dic71—41no—
22C4⋊(C71⋊C8)—41no—
23#14556043—24no—
24C22⋊(C71⋊C8)—22no—
25#14556039—41no—
26C71⋊#95—41no—
27C22×Q8×C71—21no—
28C22×Q8×C71—41no—
29C22×Q8×C71—41no—
30C22×Q8×C71—44no—
31C22×Q8×C71C2×C412yes—
32C22×D4×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C22⋊Q8×C71—44no—
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71D411yescharacteristic
45C2×C22⋊C4×C71C2311yescharacteristic
46C2×C42×C71D411yescharacteristic
47#95×C71—22no—
48#95×C71—22no—
49#95×C71C2×C412yes—
50C22×Dic71—42no—
51C2×C71⋊C8—42no—
52C22⋊Dic71—24no—
53C22⋊Dic71—42no—
54C4×Dic71—82no—
55C24×C71C22⋊C411yescharacteristic
56C24×C71C2×D411yescharacteristic
57C2×Q8×C71—24no—
58C2×Q8×C71—44no—
59C2×Q8×C71—42no—
60C2×Q8×C71—42no—
61C2×Q8×C71—44no—
62C2×Q8×C71—48no—
63C2×Q8×C71—44no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—41no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—44no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—44no—
87C22⋊C4×C71—41no—
88C22⋊C4×C71—41no—
89C22⋊C4×C71—44no—
90C22⋊C4×C71—42no—
91C42×C71—21no—
92C42×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71#10412yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71#13712yes—
104Q8×C71—44no—
105Q8×C71—44no—
106Q8×C71—44no—
107Q8×C71—44no—
108D4×C71—44no—
109C2×C4×C71—22no—
110C2×C4×C71—24no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—48no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—81no—
130C2×C4×C71#9911yescharacteristic
131C2×C4×C71#9911yescharacteristic
132C2×Dic71—44no—
133C2×Dic71—82no—
134C2×Dic71—82no—
135C71⋊C8—82no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—42no—
141C22×C71—42no—
142C22×C71—42no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—81no—
148C22×C71C2×#9911yescharacteristic
149C22×C71#41911yescharacteristic
150C22×C71C2×#10411yescharacteristic
151C4×C71—21no—
152C4×C71—44no—
153C4×C71—41no—
154C4×C71—41no—
155C4×C71—41no—
156C4×C71—41no—
157C4×C71—82no—
158C4×C71AΣL(1,𝔽16)12yes—
159Dic71—84no—
160#11785—711nomaximal, Sylow
161C2×C71—21no—
162C2×C71—21no—
163C2×C71—42no—
164C2×C71—42no—
165C2×C71#150811yescharacteristic
166C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
167C2×C71#212911yescharacteristic
168#2409D7111yescharacteristic
169C4×#99—711no—
170#1466—711no—
171#1351—714no—
172C71#1178511yescharacteristic, complemented, Sylow
173C23×Q8Dic7111yescharacteristic, complemented
174C2×C22⋊Q8—21no—
175C2×C4×D4C2×D7111yescharacteristic
176#432—1421no—
177C2×#99—1421no—
178#395Dic7112yescomplemented
179#393C2×D7112yes—
180#391—21no—
181#386—21no—
182#385—21no—
183#380Dic7111yescharacteristic, complemented
184C4×C22⋊C4—1421no—
185#342—712no—
186#323—712no—
187#323—1421no—
188C22×Q8—21no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—44no—
192C22×Q8C2×Dic7112yes—
193C22×D4C22⋊D7111yescharacteristic
194C23×C4—21no—
195C23×C4C2×Dic7111yescharacteristic
196C23×C4C22⋊D7111yescharacteristic
197C22⋊Q8—44no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic7111yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—1422no—
206C2×C22⋊C4C22⋊D7111yescharacteristic
207C2×C22⋊C4C22×D7111yescharacteristic
208C2×C42—2841no—
209C2×C42C22⋊D7111yescharacteristic
210C4⋊C8—2841no—
211#99—1424no—
212C22⋊C8—1422no—
213#97—2841no—
214#95—22no—
215#95—22no—
216#95—2841no—
217#95C2×Dic7112yes—
218C24C22⋊Dic7111yescharacteristic
219C24C2×C22⋊D7111yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—2842no—
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22×Dic7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248C2×C8—2842no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—2842no—
257C22⋊C4—1424no—
258C42—21no—
259C42—21no—
260C42—5682no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1455605512yes—
270C23C2×C22⋊Dic7111yescharacteristic
271C23#1455615912yes—
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276D4—44no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—2844no—
299C2×C4—5682no—
300C2×C4—5682no—
301C2×C4#1455604311yescharacteristic
302C2×C4#1455604311yescharacteristic
303C8—5682no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1455604311yescharacteristic
317C22#1798600811yescharacteristic
318C22C2×#1455605511yescharacteristic
319C4—21no—
320C4—44no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—5684no—
327C4#1798578612yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#2258686011yescharacteristic
333C2C2×#1798578611yescharacteristic
334C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2409C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2409D71
hypocenterC71#11785
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22586361C2semidirect
11. C23×Q8×C71C4semidirect
14. #395×C71C4semidirect
19. #380×C71C4semidirect
172. C71#11785semidirect
173. C23×Q8Dic71semidirect
178. #395Dic71semidirect
183. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866812
#225863612
C71×#24092
#1178571