Finite Group Databasebeta

← #31886509

Subgroups — #31886509

Order 18176.

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

normal subgroups only

This group has 462 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
1#22589898C211yesmaximal, characteristic, complemented
2#22589816C211yesmaximal, characteristic, complemented
3#22589743C211yesmaximal, characteristic
4#22586163C211yesmaximal, characteristic, complemented
5#22586163C211yesmaximal, characteristic, complemented
6#22586161C211yesmaximal, characteristic
7C71×#2683C211yesmaximal, characteristic
8C2×#14556059—21no—
9C2×#14556049—21no—
10#17985869C2211yescharacteristic
11#17985761—21no—
12#17985761—21no—
13#17985761—21no—
14#17985761—21no—
15#17985754C2211yescharacteristic
16#17985754C2211yescharacteristic
17#17985754C2211yescharacteristic
18C2×C22⋊Q8×C71—21no—
19C2×C22⋊Q8×C71—21no—
20#396×C71—21no—
21#396×C71—21no—
22#395×C71—21no—
23#390×C71C2211yescharacteristic
24#389×C71C2211yescharacteristic
25#389×C71C2211yescharacteristic
26#386×C71—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33#14556059—42no—
34#14556049—42no—
35C22×Q8×C71—21no—
36C22×D4×C71D411yescharacteristic
37C23×C4×C71—21no—
38C22⋊Q8×C71—42no—
39C22⋊Q8×C71—42no—
40C22⋊Q8×C71—42no—
41C22⋊Q8×C71—42no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—21no—
56C2×C42×C71C2311yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60C22×Dic71—41no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8—41no—
70C4⋊Dic71—42no—
71C4⋊Dic71—41no—
72C4⋊Dic71—41no—
73C4⋊Dic71—41no—
74C4⋊Dic71—41no—
75C24×C71—21no—
76C2×Q8×C71—42no—
77C2×Q8×C71—42no—
78C2×Q8×C71—81no—
79C2×D4×C71—21no—
80C2×D4×C71—41no—
81C2×D4×C71—41no—
82C2×D4×C71SD1612yes—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—42no—
89C22×C4×C71—42no—
90C22×C4×C71—81no—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—42no—
100C4⋊C4×C71—42no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C4⋊C4×C71—41no—
105C4⋊C4×C71D812yes—
106C4⋊C4×C71SD1612yes—
107C22⋊C4×C71—42no—
108C22⋊C4×C71—42no—
109C22⋊C4×C71—41no—
110C22⋊C4×C71—41no—
111C22⋊C4×C71—41no—
112C22⋊C4×C71—41no—
113C22⋊C4×C71—81no—
114C42×C71—41no—
115C23×C71—22no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71#12011yescharacteristic
123Q8×C71—84no—
124D4×C71—42no—
125D4×C71—81no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—84no—
148C2×C4×C71C2×D811yescharacteristic
149C2×C4×C71C2×SD1611yescharacteristic
150C2×C4×C71C2×SD1611yescharacteristic
151C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C71#13711yescharacteristic
154C2×Dic71—81no—
155C2×Dic71—81no—
156C2×Dic71—81no—
157C71⋊C8—42no—
158C71⋊C8—81no—
159C71⋊C8—81no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—42no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71—81no—
166C22×C71—81no—
167C22×C71#44711yescharacteristic
168C22×C71#44811yescharacteristic
169C22×C71#45011yescharacteristic
170C22×C71#45411yescharacteristic
171C22×C71#45511yescharacteristic
172C22×C71#45611yescharacteristic
173C22×C71#45811yescharacteristic
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—42no—
181C4×C71—42no—
182C4×C71—81no—
183C4×C71—81no—
184C4×C71—81no—
185Dic71—161no—
186#7791—711nomaximal, Sylow
187C2×C71—82no—
188C2×C71#160811yescharacteristic
189C2×C71#161011yescharacteristic
190C2×C71#161611yescharacteristic
191C2×C71#162711yescharacteristic
192C2×C71#163911yescharacteristic
193C2×C71#165111yescharacteristic
194C2×C71#166411yescharacteristic
195#2683D7111yescharacteristic
196#2099—711no—
197#2084—711no—
198#2071—711no—
199#1271—711no—
200#1271—711no—
201#1270—711no—
202C71#779111yescharacteristic, complemented, Sylow
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8—21no—
205C2×#106—1421no—
206C2×#102—1421no—
207#396—21no—
208#396—21no—
209#395—21no—
210#390C2×D7111yescharacteristic
211#389C2×D7111yescharacteristic
212#389C2×D7111yescharacteristic
213#386—21no—
214#384—711no—
215#340—1421no—
216#340—1421no—
217#340—1421no—
218#340—1421no—
219#336—711no—
220#336—711no—
221#336—711no—
222C22×Q8—21no—
223C22×D4C22⋊D7111yescharacteristic
224C23×C4—21no—
225C22×C8—1421no—
226C22×C8—1421no—
227C22×C8—1421no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—1421no—
237C2×C4⋊C4—1421no—
238C2×C4⋊C4—1421no—
239C2×C4⋊C4C22⋊D7111yescharacteristic
240C2×C4⋊C4C22⋊D7111yescharacteristic
241C2×C4⋊C4C22⋊D7111yescharacteristic
242C2×C4⋊C4C22⋊D7111yescharacteristic
243C2×C4⋊C4C22⋊D7111yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42C22×D7111yescharacteristic
250#106—2842no—
251#102—2842no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—81no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4#1176585212yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—2841no—
272C22×C4C2×C22⋊D7111yescharacteristic
273C22×C4C2×C22⋊D7111yescharacteristic
274C22×C4C2×C22⋊D7111yescharacteristic
275C2×C8—2841no—
276C2×C8—2841no—
277C2×C8—2841no—
278C2×C8—2841no—
279C2×C8—2841no—
280C2×C8—2841no—
281C2×C8—2841no—
282C2×C8—2841no—
283C2×C8—1422no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—2842no—
296C4⋊C4—2841no—
297C4⋊C4—2841no—
298C4⋊C4—2841no—
299C4⋊C4—2841no—
300C4⋊C4#1176585012yes—
301C4⋊C4#1176585312yes—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—81no—
309C42—41no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1455609711yescharacteristic
318Q8—84no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—5681no—
344C2×C4—5681no—
345C2×C4—5681no—
346C2×C4C2×#1176585011yescharacteristic
347C2×C4C2×#1176585211yescharacteristic
348C2×C4C2×#1176585311yescharacteristic
349C2×C4#1455615411yescharacteristic
350C2×C4#1455615411yescharacteristic
351C2×C4#1455615911yescharacteristic
352C8—2842no—
353C8—5681no—
354C8—5681no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#1798613311yescharacteristic
363C22#1798613811yescharacteristic
364C22#1798615011yescharacteristic
365C22#1798617211yescharacteristic
366C22#1798617711yescharacteristic
367C22#1798618211yescharacteristic
368C22#1798618911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—11361no—
381C2—82no—
382C2#2258733111yescharacteristic
383C2#2258734311yescharacteristic
384C2#2258737711yescharacteristic
385C2#2258744311yescharacteristic
386C2#2258751911yescharacteristic
387C2#2258760311yescharacteristic
388C2#2258767411yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2683C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2683D71
hypocenterC71#7791
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #22589898C2semidirect
2. #22589816C2semidirect
4. #22586163C2semidirect
5. #22586163C2semidirect
202. C71#7791semidirect

Maximal subgroups

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

subgroupindex
#225898982
#225898162
#225897432
#225861632
#225861612
C71×#26832
#779171