Finite Group Databasebeta

← #31895447

Subgroups — #31895447

Order 18176.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 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#22596978C211yesmaximal, characteristic
2#22588423C211yesmaximal, characteristic
3#22586670C212yesmaximal
4#22586248C212yesmaximal
5C71×#2484C211yesmaximal, characteristic
6#17986060—24no—
7#17986023—21no—
8#17985963C2212yes—
9#17985963C2212yes—
10C4×#11765849—21no—
11#17985768—21no—
12#17985768—21no—
13#17985725—24no—
14C2×#117×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#400×C71—21no—
17#398×C71—21no—
18#387×C71—21no—
19#387×C71—21no—
20#385×C71C411yescharacteristic
21#385×C71C411yescharacteristic
22#381×C71C411yescharacteristic
23#381×C71C411yescharacteristic
24#381×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×#11765849—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)D414yes—
32#14556039—22no—
33C4×C71⋊C8—22no—
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37#117×C71—22no—
38#117×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71C2×C411yescharacteristic
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C42×C71D411yescharacteristic
52C2×C42×C71D411yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#11765849—42no—
62#11765849—42no—
63#11765849—42no—
64#11765849—42no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—42no—
67C24×C71C22×C411yescharacteristic
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—42no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C2×D412yes—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—41no—
94C22⋊C4×C71—22no—
95C22⋊C4×C71—22no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—21no—
101C42×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71C2×C22⋊C411yescharacteristic
107C23×C71C4×D412yes—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—22no—
113C2×C4×C71—24no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—42no—
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—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—82no—
137C2×C4×C71#12011yescharacteristic
138C2×C4×C71#12112yes—
139C2×C4×C71#12311yescharacteristic
140C71⋊C8—44no—
141C71⋊C8—82no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—22no—
147C22×C71—42no—
148C22×C71—42no—
149C22×C71#38611yescharacteristic
150C22×C71#41011yescharacteristic
151C22×C71#41011yescharacteristic
152C4×C71—21no—
153C4×C71—22no—
154C4×C71—21no—
155C4×C71—42no—
156C4×C71—41no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71—81no—
163#9514—711nomaximal, Sylow
164C2×C71—21no—
165C2×C71—21no—
166C2×C71—42no—
167C2×C71#189111yescharacteristic
168C2×C71#189911yescharacteristic
169C2×C71#191511yescharacteristic
170#2921—711no—
171#2484D7111yescharacteristic
172#1822—711no—
173#1463—712no—
174#1306—712no—
175C71#951411yescharacteristic, complemented, Sylow
176C2×#117C2×D7111yescharacteristic
177C22×C4⋊C4C2×D7111yescharacteristic
178#433—1424no—
179#423—1421no—
180#407—712no—
181#407—712no—
182C4×M16—1421no—
183#400—21no—
184#398—21no—
185#387—21no—
186#387—21no—
187#385Dic7111yescharacteristic
188#385Dic7111yescharacteristic
189#381Dic7111yescharacteristic
190#381Dic7111yescharacteristic
191#381C2×D7111yescharacteristic
192#343—1421no—
193#343—1421no—
194#324—1424no—
195C23×C4C2×Dic7111yescharacteristic
196C23×C4C2×Dic7111yescharacteristic
197C23×C4C22×D7111yescharacteristic
198C2×M16—1422no—
199C2×M16—1422no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊D7111yescharacteristic
209C2×C4⋊C4C22⋊D7111yescharacteristic
210C2×C22⋊C4C2×Dic7111yescharacteristic
211C2×C22⋊C4C2×Dic7111yescharacteristic
212C2×C22⋊C4C2×Dic7111yescharacteristic
213C2×C22⋊C4C2×Dic7111yescharacteristic
214C2×C42C22⋊D7111yescharacteristic
215C2×C42C22⋊D7111yescharacteristic
216C4⋊C8—1424no—
217C22⋊C8—714no—
218C22⋊C8—1422no—
219C22⋊C8—1422no—
220C22⋊C8—1422no—
221#97—1422no—
222C4×C8—1422no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24C22×Dic7111yescharacteristic
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4D4×D7112yes—
248C22×C4C2×C22⋊D7111yescharacteristic
249C22×C4C2×C22⋊D7111yescharacteristic
250C22×C4#1176586412yes—
251M16—2842no—
252M16—2842no—
253M16—2842no—
254M16—2842no—
255C2×C8—2842no—
256C2×C8—1424no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C22⋊Dic7111yescharacteristic
277C23D4×Dic7112yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4#1455609611yescharacteristic
308C2×C4#1455610012yes—
309C2×C4#1455610911yescharacteristic
310C8—2844no—
311C8—5682no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#1798587911yescharacteristic
320C22#1798597311yescharacteristic
321C22#1798597311yescharacteristic
322C4—21no—
323C4—22no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2258875911yescharacteristic
337C2#2258879911yescharacteristic
338C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2484C2
socleC22×C71#386
cosocleC23×C4×C71C23
hypercenter#2484D71
hypocenterC71#9514
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
175. C71#9514semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225884232
#225866702
#225862482
C71×#24842
#951471