Finite Group Databasebeta

← #31892527

Subgroups — #31892527

Order 18176.

484 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#22597425C211yesmaximal, characteristic
2C4×#14556047C211yesmaximal, characteristic
3#22586787C212yesmaximal
4#22586233C212yesmaximal
5C71×#2399C211yesmaximal, characteristic
6#17986130—24no—
7#17986056—21no—
8#17986056C2212yes—
9#17986052—21no—
10C2×#14556047—21no—
11C4×#11765849C2212yes—
12#17985768—21no—
13Q8⋊(C71⋊C8)—24no—
14C2×C4×Q8×C71C2211yescharacteristic
15C2×C4×Q8×C71C2211yescharacteristic
16#398×C71—21no—
17#398×C71C411yescharacteristic
18#398×C71C411yescharacteristic
19#395×C71—21no—
20#395×C71C411yescharacteristic
21#395×C71C411yescharacteristic
22C4×C4⋊C4×C71—21no—
23C4×C4⋊C4×C71—21no—
24#376×C71C2211yescharacteristic
25C2×#11765849—22no—
26C4⋊(C71⋊C8)—22no—
27C4⋊(C71⋊C8)—22no—
28C4⋊(C71⋊C8)D414yes—
29#14556047—24no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32#14556039—22no—
33#14556039—22no—
34C4×C71⋊C8—22no—
35C4×C71⋊C8—22no—
36C22×Q8×C71C2×C411yescharacteristic
37C22×Q8×C71D411yescharacteristic
38C4×Q8×C71—22no—
39C4×Q8×C71—22no—
40C4×Q8×C71—22no—
41C4×Q8×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C4⋊C4×C71C2×C411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C42×C71—21no—
52C2×C42×C71—21no—
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C2×C42×C71C2311yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63#11765849—24no—
64#11765849—42no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—42no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C2×Q8×C71—22no—
70C2×Q8×C71—41no—
71C2×Q8×C71—42no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—22no—
84C4⋊C4×C71—22no—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—42no—
92C42×C71—24no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71C2×D412yes—
100C42×C71#4112yes—
101C23×C71C2×C22⋊C411yescharacteristic
102Q8×C71—24no—
103Q8×C71—44no—
104Q8×C71—42no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71#10412yes—
131C2×C4×C71C4×D412yes—
132C2×C4×C71#12011yescharacteristic
133C2×C4×C71#12112yes—
134C2×C4×C71#12311yescharacteristic
135C71⋊C8—44no—
136C71⋊C8—44no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71#38611yescharacteristic
140C22×C71C2×#10411yescharacteristic
141C22×C71#42111yescharacteristic
142C4×C71—22no—
143C4×C71—22no—
144C4×C71—21no—
145C4×C71—22no—
146C4×C71—42no—
147C4×C71—42no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—42no—
152C4×C71—81no—
153C4×C71#45612yes—
154#9030—711nomaximal, Sylow
155C2×C71—21no—
156C2×C71—21no—
157C2×C71#187111yescharacteristic
158C2×C71#188611yescharacteristic
159C2×C71#189111yescharacteristic
160#2983—711no—
161#2399D7111yescharacteristic
162C4×#101—711no—
163#1491—712no—
164#1300—712no—
165C71#903011yescharacteristic, complemented, Sylow
166C2×C4×Q8C2×D7111yescharacteristic
167C2×C4×Q8C2×D7111yescharacteristic
168#446—1424no—
169#432—712no—
170#432—1421no—
171#431—1421no—
172C2×#101—1421no—
173C4×M16—712no—
174#398—21no—
175#398Dic7111yescharacteristic
176#398Dic7111yescharacteristic
177#395—21no—
178#395Dic7111yescharacteristic
179#395Dic7111yescharacteristic
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182#376C2×D7111yescharacteristic
183#343—1421no—
184Q8⋊C8—1424no—
185C22×Q8C2×Dic7111yescharacteristic
186C22×Q8C22⋊D7111yescharacteristic
187C2×M16—1422no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic7111yescharacteristic
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C4⋊C4C2×Dic7111yescharacteristic
200C2×C4⋊C4C22⋊D7111yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊D7111yescharacteristic
204C2×C42C22⋊D7111yescharacteristic
205C2×C42C22×D7111yescharacteristic
206C4⋊C8—714no—
207C4⋊C8—1422no—
208C4⋊C8—1422no—
209#101—1424no—
210C22⋊C8—1422no—
211#97—1422no—
212#97—1422no—
213#97—1422no—
214C4×C8—1422no—
215C4×C8—1422no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×Dic7111yescharacteristic
222#95C2×Dic7111yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C22⋊Dic7111yescharacteristic
236C22×C4C22×Dic7111yescharacteristic
237C22×C4C2×C22⋊D7111yescharacteristic
238C22×C4C2×C22⋊D7111yescharacteristic
239M16—2842no—
240M16—1424no—
241C2×C8—2842no—
242C2×C8—1424no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C42—24no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42D4×D7112yes—
260C42#1176586412yes—
261C23C2×C22⋊Dic7111yescharacteristic
262Q8—24no—
263Q8—44no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#1455605512yes—
291C2×C4D4×Dic7112yes—
292C2×C4#1455609611yescharacteristic
293C2×C4#1455610012yes—
294C2×C4#1455610911yescharacteristic
295C8—2844no—
296C8—2844no—
297C22—21no—
298C22—21no—
299C22#1798587911yescharacteristic
300C22C2×#1455605511yescharacteristic
301C22#1798601611yescharacteristic
302C4—21no—
303C4—22no—
304C4—22no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4#1798618112yes—
314C2—21no—
315C2—21no—
316C2#2258864311yescharacteristic
317C2#2258873411yescharacteristic
318C2#2258875911yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2399C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2399D71
hypocenterC71#9030
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
165. C71#9030semidirect

Maximal subgroups

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

subgroupindex
#225974252
C4×#145560472
#225867872
#225862332
C71×#23992
#903071