Finite Group Databasebeta

← #31892457

Subgroups — #31892457

Order 18176.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590938C211yesmaximal, characteristic
2#22588275C212yesmaximal, complemented
3C2×#17985723C211yesmaximal, characteristic
4#22586219C212yesmaximal, complemented
5C2×#393×C71C211yesmaximal, characteristic
6C2×#14556052—24no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985892—24no—
9#17985859C2212yes—
10#17985855—21no—
11C4×C22⋊Dic71—21no—
12#17985723—22no—
13C23×Q8×C71C2211yescharacteristic
14C22×C22⋊C4×C71—21no—
15C22×C22⋊C4×C71C2211yescharacteristic
16#393×C71—22no—
17#393×C71C414yescomplemented
18C2×#95×C71—21no—
19C2×#95×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71D414yes—
22C2×C22⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26#14556052—48no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—22no—
30C71⋊#95—22no—
31C25×C71D411yescharacteristic
32C22×Q8×C71—21no—
33C22×Q8×C71—22no—
34C22×Q8×C71—22no—
35C22×Q8×C71—44no—
36C22×Q8×C71C2×C412yes—
37C23×C4×C71—21no—
38C23×C4×C71—21no—
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—24no—
44C2×C22⋊C4×C71—44no—
45C2×C22⋊C4×C71C2×C412yes—
46C2×C22⋊C4×C71D412yes—
47#95×C71—24no—
48#95×C71—21no—
49#95×C71C2×C412yes—
50C22×Dic71—24no—
51C2×C71⋊C8—44no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—44no—
54C22⋊Dic71—42no—
55C22⋊Dic71—42no—
56C22⋊Dic71—82no—
57C22⋊Dic71—82no—
58C4×Dic71—82no—
59C24×C71—21no—
60C24×C71—22no—
61C24×C71—21no—
62C24×C71—22no—
63C24×C71—42no—
64C24×C71—42no—
65C24×C71C22⋊C412yes—
66C24×C71C2×D411yescharacteristic
67C2×Q8×C71—28no—
68C2×Q8×C71—24no—
69C2×Q8×C71—44no—
70C2×Q8×C71—44no—
71C2×Q8×C71—44no—
72C2×Q8×C71—48no—
73C2×Q8×C71—42no—
74C22×C4×C71—24no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—22no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—42no—
84C22×C4×C71—42no—
85C22×C4×C71—42no—
86C22×C4×C71—42no—
87C22×C4×C71C22⋊C412yes—
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D412yes—
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71#4112yes—
92C22⋊C4×C71—24no—
93C22⋊C4×C71—24no—
94C22⋊C4×C71—48no—
95C23×C71—22no—
96C23×C71—22no—
97C23×C71—22no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—42no—
101C23×C71—42no—
102C23×C71—48no—
103C23×C71—42no—
104C23×C71—42no—
105C23×C71—42no—
106C23×C71—42no—
107C23×C71—81no—
108C23×C71—81no—
109C23×C71—84no—
110C23×C71C2×C22⋊C411yescharacteristic
111C23×C71#12011yescharacteristic
112C23×C71#12311yescharacteristic
113C23×C71#13512yes—
114C23×C71#13712yes—
115Q8×C71—28no—
116Q8×C71—48no—
117Q8×C71—44no—
118C2×C4×C71—24no—
119C2×C4×C71—24no—
120C2×C4×C71—22no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—44no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—44no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—81no—
134C2×C4×C71—84no—
135C2×C4×C71C4×D412yes—
136C2×C4×C71#12112yes—
137C2×Dic71—44no—
138C2×Dic71—44no—
139C2×Dic71—44no—
140C71⋊C8—82no—
141C22×C71—24no—
142C22×C71—24no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—44no—
148C22×C71—42no—
149C22×C71—42no—
150C22×C71—42no—
151C22×C71—82no—
152C22×C71—84no—
153C22×C71—84no—
154C22×C71#38611yescharacteristic
155C22×C71#41611yescharacteristic
156C22×C71#41911yescharacteristic
157C22×C71#45212yes—
158C22×C71#45512yes—
159C22×C71B(4,𝔽2)12yes—
160C22×C71#48512yes—
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—42no—
164C4×C71—44no—
165C4×C71—82no—
166Dic71—84no—
167#9016—711nomaximal, Sylow
168C2×C71—21no—
169C2×C71—21no—
170C2×C71—22no—
171C2×C71—82no—
172C2×C71#160614yes—
173C2×C71#187311yescharacteristic
174C2×C71#188511yescharacteristic
175C2×C71#188911yescharacteristic
176C2×#393D7111yescharacteristic
177#2312—711no—
178#1787—712no—
179C2×#323—711no—
180#1294—712no—
181C71#901611yescharacteristic, complemented, Sylow
182C23×Q8C2×D7111yescharacteristic
183C22×C22⋊C4—21no—
184C22×C22⋊C4C2×D7111yescharacteristic
185C2×#103—1424no—
186C2×C22⋊C8—712no—
187#393—22no—
188#393Dic7114yescomplemented
189#388—1424no—
190#380—712no—
191#379—1421no—
192C4×C22⋊C4—1421no—
193C2×#95—21no—
194C2×#95C2×D7111yescharacteristic
195#323—1422no—
196C25C22⋊D7111yescharacteristic
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—44no—
201C22×Q8C2×Dic7112yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C22⋊D7111yescharacteristic
205C23×C4C22×D7111yescharacteristic
206C22×C8—1422no—
207C2×C4⋊C4—714no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—1422no—
213C2×C22⋊C4—1422no—
214C2×C22⋊C4—1422no—
215C2×C22⋊C4C2×Dic7112yes—
216C2×C22⋊C4C22⋊D7112yes—
217C2×C42—1422no—
218#103—2848no—
219C22⋊C8—1422no—
220C22⋊C8—1422no—
221#95—24no—
222#95—21no—
223#95—1422no—
224#95—1422no—
225#95C2×Dic7112yes—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24C22⋊Dic7112yes—
233C24C2×C22⋊D7111yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—1424no—
255C22×C4C22⋊Dic7112yes—
256C22×C4C22×Dic7111yescharacteristic
257C22×C4D4×D7112yes—
258C22×C4C2×C22⋊D7111yescharacteristic
259C22×C4#1176586412yes—
260C2×C8—2844no—
261C2×C8—2842no—
262C4⋊C4—2844no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—2842no—
267C22⋊C4—2842no—
268C22⋊C4—5682no—
269C22⋊C4—5682no—
270C42—5682no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×C22⋊Dic7111yescharacteristic
287C23#1455609611yescharacteristic
288C23#1455610911yescharacteristic
289C23#1455614812yes—
290C23#1455615912yes—
291Q8—28no—
292Q8—48no—
293Q8—44no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—84no—
311C2×C4—2844no—
312C2×C4—2844no—
313C2×C4—2844no—
314C2×C4D4×Dic7112yes—
315C2×C4#1455610012yes—
316C8—5682no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1798587911yescharacteristic
331C22#1798599511yescharacteristic
332C22#1798600811yescharacteristic
333C22#1798615912yes—
334C22#1798617612yes—
335C22#1798618512yes—
336C22#1798633712yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—5684no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2258731814yes—
348C2#2258865311yescharacteristic
349C2#2258872911yescharacteristic
350C2#2258874911yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#393×C71C2
socleC23×C71#120
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenterC2×#393D71
hypocenterC71#9016
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #22588275C2semidirect
4. #22586219C2semidirect
17. #393×C71C4semidirect
181. C71#9016semidirect
188. #393Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909382
#225882752
C2×#179857232
#225862192
C2×#393×C712
#901671