Finite Group Databasebeta

← #31891768

Subgroups — #31891768

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596908C211yesmaximal, characteristic
2#22588360C211yesmaximal, characteristic
3#22587111C211yesmaximal, characteristic, complemented
4#22587096C211yesmaximal, characteristic, complemented
5#22586242C211yesmaximal, characteristic, complemented
6#22586180C211yesmaximal, characteristic, complemented
7C71×#1476C211yesmaximal, characteristic
8C2×#14556135—21no—
9#17986026—22no—
10#17986026—21no—
11#17986023—21no—
12#17986023C2211yescharacteristic
13#17986023C2211yescharacteristic
14C2×C4⋊(C71⋊C8)—21no—
15C2×C4⋊(C71⋊C8)C2211yescharacteristic
16C2×#14556055—21no—
17C4×C4⋊Dic71C2211yescharacteristic
18#17985758—21no—
19#17985750—22no—
20#17985750—21no—
21#17985750—21no—
22C4×#41×C71C2211yescharacteristic
23#431×C71C2211yescharacteristic
24C2×C4×C8×C71C2211yescharacteristic
25Q8⋊C8×C71C411yescharacteristic, complemented
26Q8⋊C8×C71C411yescharacteristic, complemented
27D4⋊C8×C71C411yescharacteristic, complemented
28D4⋊C8×C71C411yescharacteristic, complemented
29#14556135—42no—
30#14556135—42no—
31C2×#11765849—22no—
32C2×#11765849—21no—
33C2×#11765849—21no—
34C22×C71⋊C8—22no—
35C22×C71⋊C8—21no—
36C2×C4⋊Dic71—21no—
37C2×C4×Dic71—21no—
38C4⋊(C71⋊C8)—22no—
39C4⋊(C71⋊C8)—22no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)—21no—
42C4⋊(C71⋊C8)—21no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)D411yescharacteristic
45C4⋊(C71⋊C8)D411yescharacteristic
46C4⋊(C71⋊C8)D412yes—
47#14556055—42no—
48C71⋊#95—21no—
49C2×#41×C71D411yescharacteristic
50C22×C8×C71—21no—
51C4×Q8×C71C2×C411yescharacteristic
52C4×Q8×C71Q811yescharacteristic
53C4×D4×C71—21no—
54C4×D4×C71—21no—
55C4×D4×C71C2×C411yescharacteristic
56C4×D4×C71Q811yescharacteristic
57#117×C71—21no—
58#117×C71D411yescharacteristic
59C2×C42×C71—21no—
60C2×C42×C71C2311yescharacteristic
61C4⋊C8×C71C2×C411yescharacteristic
62C4⋊C8×C71C2×C411yescharacteristic
63C22⋊C8×C71—21no—
64C4×C8×C71—21no—
65C4×C8×C71—21no—
66C4×C8×C71C2×C411yescharacteristic
67C4×C8×C71C2×C411yescharacteristic
68C22×Dic71—41no—
69#11765849—44no—
70#11765849—41no—
71#11765849—41no—
72#11765849—41no—
73#11765849—41no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8—22no—
76C2×C71⋊C8—21no—
77C2×C71⋊C8—44no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—41no—
80C2×C71⋊C8—41no—
81C4⋊Dic71—22no—
82C4⋊Dic71—41no—
83C4×Dic71—41no—
84C4×Dic71—41no—
85#41×C71—21no—
86#41×C71—21no—
87#41×C71SD1612yes—
88#41×C71SD1612yes—
89C2×Q8×C71C4⋊C411yescharacteristic
90C2×D4×C71—21no—
91C2×D4×C71C4⋊C411yescharacteristic
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71#4111yescharacteristic
97C22×C4×C71#4111yescharacteristic
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—42no—
103C4⋊C4×C71—21no—
104C4⋊C4×C71—21no—
105C4⋊C4×C71C4⋊C411yescharacteristic
106C4⋊C4×C71C4⋊C411yescharacteristic
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C22⋊C4×C71—21no—
110C42×C71—21no—
111C42×C71—21no—
112C42×C71—21no—
113C42×C71C22×C411yescharacteristic
114C42×C71C2×D411yescharacteristic
115C42×C71C2×D411yescharacteristic
116C42×C71C2×Q811yescharacteristic
117C23×C71—21no—
118C23×C71#12311yescharacteristic
119Q8×C71—21no—
120Q8×C71#10612yes—
121D4×C71—22no—
122D4×C71—22no—
123D4×C71—21no—
124D4×C71#10612yes—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—81no—
136C2×C4×C71C2×C4⋊C411yescharacteristic
137C2×C4×C71C4×D411yescharacteristic
138C2×C4×C71C4×D411yescharacteristic
139C2×C4×C71#12011yescharacteristic
140C2×C4×C71C22⋊Q811yescharacteristic
141C2×C4×C71C22⋊Q811yescharacteristic
142C2×C4×C71C2×SD1611yescharacteristic
143C2×C4×C71C2×SD1611yescharacteristic
144C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C71#13711yescharacteristic
146C2×Dic71—42no—
147C2×Dic71—81no—
148C2×Dic71—81no—
149C8×C71—22no—
150C8×C71—41no—
151C8×C71—41no—
152C8×C71—41no—
153C71⋊C8—41no—
154C71⋊C8—42no—
155C71⋊C8—41no—
156C71⋊C8—81no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—22no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71#38511yescharacteristic
163C22×C71#48111yescharacteristic
164C22×C71#48311yescharacteristic
165C4×C71—22no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71C2×#10611yescharacteristic
172C4×C71#42811yescharacteristic
173C4×C71#44311yescharacteristic
174C4×C71#44411yescharacteristic
175C4×C71#44811yescharacteristic
176C4×C71#45011yescharacteristic
177Dic71—82no—
178#8904—711nomaximal, Sylow
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—42no—
182C2×C71#185311yescharacteristic
183C2×C71#186111yescharacteristic
184C2×C71#186611yescharacteristic
185#2906—711no—
186#1806—711no—
187#1570—711no—
188#1566—711no—
189#1476D7111yescharacteristic
190#1303—711no—
191#1279—711no—
192C71#890411yescharacteristic, complemented, Sylow
193C2×#131—1421no—
194C4×#41C2×D7111yescharacteristic
195#431C2×D7111yescharacteristic
196#424—1422no—
197#424—1421no—
198#423—711no—
199#423—711no—
200#423—1421no—
201C2×C4⋊C8—711no—
202C2×C4⋊C8—1421no—
203C2×#104—1421no—
204C2×C4×C8C2×D7111yescharacteristic
205C4×C4⋊C4—711no—
206#339—1421no—
207#334—1422no—
208#334—1421no—
209#334—1421no—
210Q8⋊C8Dic7111yescharacteristic, complemented
211Q8⋊C8Dic7111yescharacteristic, complemented
212D4⋊C8Dic7111yescharacteristic, complemented
213D4⋊C8Dic7111yescharacteristic, complemented
214C2×#41D28411yescharacteristic
215#131—2842no—
216#131—2842no—
217C2×M16—1421no—
218C2×M16—1422no—
219C2×M16—1421no—
220C22×C8—21no—
221C22×C8—1422no—
222C22×C8—1421no—
223C4×Q8Dic14211yescharacteristic
224C4×Q8C2×Dic7111yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic14211yescharacteristic
228C4×D4C2×Dic7111yescharacteristic
229#117—21no—
230#117D28411yescharacteristic
231C2×C4⋊C4—1421no—
232C2×C42—21no—
233C2×C42—1421no—
234C2×C42C22×D7111yescharacteristic
235C4⋊C8—711no—
236C4⋊C8—711no—
237C4⋊C8—712no—
238C4⋊C8—1422no—
239C4⋊C8—1422no—
240C4⋊C8—1421no—
241C4⋊C8—1421no—
242C4⋊C8—1421no—
243C4⋊C8—1421no—
244C4⋊C8C2×Dic7111yescharacteristic
245C4⋊C8C2×Dic7111yescharacteristic
246#104—2842no—
247C22⋊C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C2×Dic7111yescharacteristic
251C4×C8C2×Dic7111yescharacteristic
252#95—1421no—
253#41—21no—
254#41—21no—
255#41#1176585412yes—
256#41#1176585412yes—
257C2×Q8C4⋊Dic7111yescharacteristic
258C2×D4—21no—
259C2×D4C4⋊Dic7111yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—2841no—
264C22×C4C2×D28411yescharacteristic
265C22×C4#1176586411yescharacteristic
266C22×C4#1176586411yescharacteristic
267M16—2844no—
268M16—2841no—
269M16—2841no—
270M16—2841no—
271M16—2841no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—42no—
277C2×C8—2844no—
278C2×C8—2841no—
279C2×C8—2841no—
280C2×C8—2841no—
281C2×C8—1421no—
282C2×C8—1421no—
283C2×C8—1422no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—2841no—
287C4⋊C4—1422no—
288C4⋊C4C4⋊Dic7111yescharacteristic
289C4⋊C4C4⋊Dic7111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—2841no—
297C42—2841no—
298C42C22×Dic7111yescharacteristic
299C42D4×D7111yescharacteristic
300C42D4×D7111yescharacteristic
301C42C2×Dic14211yescharacteristic
302C23—21no—
303C23#1455611211yescharacteristic
304Q8—21no—
305Q8#1455606012yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1455606012yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—2842no—
322C2×C4—5681no—
323C2×C4—5681no—
324C2×C4C2×C4⋊Dic7111yescharacteristic
325C2×C4D4×Dic7111yescharacteristic
326C2×C4D4×Dic7111yescharacteristic
327C2×C4#1455609811yescharacteristic
328C2×C4#1455610511yescharacteristic
329C2×C4#1455610511yescharacteristic
330C2×C4C2×#1176585411yescharacteristic
331C2×C4C2×#1176585411yescharacteristic
332C2×C4#1455615611yescharacteristic
333C2×C4#1455616111yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—2841no—
339C8—2842no—
340C8—2841no—
341C8—5681no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1798587411yescharacteristic
348C22#1798631811yescharacteristic
349C22#1798633011yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—5682no—
357C4C2×#1455606011yescharacteristic
358C4#1798604111yescharacteristic
359C4#1798611611yescharacteristic
360C4#1798612111yescharacteristic
361C4#1798614011yescharacteristic
362C4#1798615211yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2258853511yescharacteristic
367C2#2258858211yescharacteristic
368C2#2258861711yescharacteristic
subgroupG/H
centerC2×C4#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1476C2
socleC22×C71#385
prime radicalC2×#41×C71D4
cosocleC2×C42×C71C23
hypercenter#1476D71
hypocenterC71#8904
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
3. #22587111C2semidirect
4. #22587096C2semidirect
5. #22586242C2semidirect
6. #22586180C2semidirect
25. Q8⋊C8×C71C4semidirect
26. Q8⋊C8×C71C4semidirect
27. D4⋊C8×C71C4semidirect
28. D4⋊C8×C71C4semidirect
192. C71#8904semidirect
210. Q8⋊C8Dic71semidirect
211. Q8⋊C8Dic71semidirect
212. D4⋊C8Dic71semidirect
213. D4⋊C8Dic71semidirect

Maximal subgroups

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

subgroupindex
#225969082
#225883602
#225871112
#225870962
#225862422
#225861802
C71×#14762
#890471