Finite Group Databasebeta

← #31900429

Subgroups — #31900429

Order 18176.

544 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 117 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#22588424C212yesmaximal
2#22586607C211yesmaximal, characteristic
3#22586218C212yesmaximal
4#22586183C211yesmaximal, characteristic
5C71×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic71C2211yescharacteristic
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×C71⋊#95—21no—
9C2×C71⋊#95—21no—
10C2×C71⋊#95C2211yescharacteristic
11#17985761—22no—
12#17985761—22no—
13C2×C22⋊Q8×C71—21no—
14#398×C71—22no—
15#395×C71—22no—
16#380×C71—21no—
17#380×C71C2212yes—
18C2×#95×C71C2211yescharacteristic
19C23×Dic71—21no—
20C23×Dic71—21no—
21C22×C71⋊C8—22no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71Q812yescomplemented
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C71⋊#95—22no—
29C71⋊#95—42no—
30C71⋊#95—42no—
31C22×Q8×C71—21no—
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35C22⋊Q8×C71—42no—
36C22⋊Q8×C71—42no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71D412yes—
42C2×C4⋊C4×C71D412yes—
43C2×C22⋊C4×C71—22no—
44#95×C71—22no—
45#95×C71—22no—
46#95×C71—22no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C22×Dic71—41no—
50C22×Dic71—41no—
51C22×Dic71—41no—
52C22×Dic71—41no—
53C22×Dic71—41no—
54C22×Dic71—41no—
55C22×Dic71—41no—
56C22×Dic71—41no—
57C22×Dic71—41no—
58C22×Dic71—41no—
59C22×Dic71—41no—
60C22×Dic71—41no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C4⋊Dic71—24no—
65C4⋊Dic71—42no—
66C24×C71C2×D411yescharacteristic
67C2×Q8×C71—42no—
68C2×Q8×C71—81no—
69C2×Q8×C71—81no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71D812yes—
83C22×C4×C71SD1612yes—
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×Q811yescharacteristic
86C22×C4×C71#4112yes—
87C22×C4×C71#4111yescharacteristic
88C4⋊C4×C71—42no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—44no—
92C4⋊C4×C71—81no—
93C4⋊C4×C71—81no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71#12011yescharacteristic
101C23×C71C22⋊Q811yescharacteristic
102C23×C71#12411yescharacteristic
103C23×C71C2×D811yescharacteristic
104C23×C71C2×SD1611yescharacteristic
105C23×C71AGL(1,ℤ/8)11yescharacteristic
106C23×C71#13711yescharacteristic
107Q8×C71—84no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—81no—
128C2×C4×C71—81no—
129C2×C4×C71C22⋊Q812yes—
130C2×C4×C71#12412yes—
131C2×Dic71—42no—
132C2×Dic71—42no—
133C2×Dic71—42no—
134C2×Dic71—42no—
135C2×Dic71—42no—
136C2×Dic71—42no—
137C2×Dic71—81no—
138C2×Dic71—81no—
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71—81no—
142C2×Dic71—81no—
143C2×Dic71—81no—
144C2×Dic71—81no—
145C71⋊C8—82no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71#39311yescharacteristic
159C22×C71#44711yescharacteristic
160C22×C71#44811yescharacteristic
161C22×C71#45511yescharacteristic
162C22×C71#45611yescharacteristic
163C22×C71#45811yescharacteristic
164C22×C71#45811yescharacteristic
165C22×C71#47411yescharacteristic
166C22×C71#47611yescharacteristic
167C22×C71#48611yescharacteristic
168C22×C71#48911yescharacteristic
169C4×C71—22no—
170C4×C71—22no—
171C4×C71—42no—
172C4×C71—42no—
173C4×C71—82no—
174C4×C71—82no—
175Dic71—82no—
176Dic71—82no—
177#10342—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—21no—
182C2×C71#160711yescharacteristic
183C2×C71#160811yescharacteristic
184C2×C71#161011yescharacteristic
185C2×C71#161111yescharacteristic
186C2×C71#202311yescharacteristic
187C2×C71#202811yescharacteristic
188C2×C71#203011yescharacteristic
189#2659D7111yescharacteristic
190#1823—712no—
191#1445—711no—
192#1294—712no—
193#1280—711no—
194C71#1034211yescharacteristic, complemented, Sylow
195C2×C22⋊Q8—21no—
196C22×C4⋊C4—711no—
197C2×C22⋊C8—712no—
198#398—22no—
199#395—22no—
200#380—21no—
201#380C2×D7112yes—
202C2×#95—711no—
203C2×#95—1421no—
204C2×#95—1421no—
205C2×#95C2×D7111yescharacteristic
206#340—1422no—
207#340—1422no—
208C22×Q8—21no—
209C23×C4—1421no—
210C23×C4—1421no—
211C23×C4C22⋊D7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22×D7111yescharacteristic
214C22×C8—1422no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—712no—
222C2×C4⋊C4—1421no—
223C2×C4⋊C4—1422no—
224C2×C4⋊C4—1422no—
225C2×C4⋊C4C22⋊D7112yes—
226C2×C4⋊C4C22⋊D7112yes—
227C2×C22⋊C4—22no—
228C22⋊C8—1422no—
229C22⋊C8—1422no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—2842no—
234#95—2842no—
235#95—1422no—
236C24C2×C22⋊D7111yescharacteristic
237C2×Q8—42no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—2841no—
253C22×C4—2841no—
254C22×C4—2841no—
255C22×C4—2841no—
256C22×C4—2841no—
257C22×C4—2841no—
258C22×C4—2841no—
259C22×C4—2841no—
260C22×C4—2841no—
261C22×C4—2841no—
262C22×C4—2841no—
263C22×C4—2841no—
264C22×C4—1422no—
265C22×C4—1422no—
266C22×C4#1176585012yes—
267C22×C4#1176585212yes—
268C22×C4C2×C22⋊D7112yes—
269C22×C4Q8×D7111yescharacteristic
270C22×C4#1176586612yes—
271C22×C4#1176586611yescharacteristic
272C2×C8—2842no—
273C2×C8—2842no—
274C2×C8—2842no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—2842no—
282C4⋊C4—1424no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1455609711yescharacteristic
290C23#1455610411yescharacteristic
291C23#1455611711yescharacteristic
292C23C2×#1176585011yescharacteristic
293C23C2×#1176585211yescharacteristic
294C23#1455615411yescharacteristic
295C23#1455615911yescharacteristic
296Q8—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—2842no—
319C2×C4—2842no—
320C2×C4—2842no—
321C2×C4—2842no—
322C2×C4—2842no—
323C2×C4—2842no—
324C2×C4—5681no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4—5681no—
328C2×C4—5681no—
329C2×C4—5681no—
330C2×C4—5681no—
331C2×C4—5681no—
332C2×C4#1455610412yes—
333C2×C4#1455611712yes—
334C8—5682no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1798591011yescharacteristic
348C22#1798613311yescharacteristic
349C22#1798613811yescharacteristic
350C22#1798617711yescharacteristic
351C22#1798618211yescharacteristic
352C22#1798618911yescharacteristic
353C22#1798618911yescharacteristic
354C22#1798627611yescharacteristic
355C22#1798628611yescharacteristic
356C22#1798634711yescharacteristic
357C22#1798636111yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—5682no—
365C4—5682no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2258732611yescharacteristic
371C2#2258733111yescharacteristic
372C2#2258734311yescharacteristic
373C2#2258734811yescharacteristic
374C2#2258947311yescharacteristic
375C2#2258949811yescharacteristic
376C2#2258950611yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC23×C4×C71C23
FrattiniC23×C4C22×D71
FittingC71×#2659C2
socleC23×C71#120
prime radicalC24×C71C2×D4
hypercenter#2659D71
hypocenterC71#10342
LCSG ▷ C23×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
25. C2×C4⋊Dic71Q8semidirect
194. C71#10342semidirect

Maximal subgroups

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

subgroupindex
#225884242
#225866072
#225862182
#225861832
C71×#26592
#1034271