Finite Group Databasebeta

← #31908928

Subgroups — #31908928

Order 18176.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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#22598920C211yesmaximal, characteristic
2C4⋊C16×D71C211yesmaximal, characteristic, complemented
3#22587097C211yesmaximal, characteristic, complemented
4#22586395C211yesmaximal, characteristic, complemented
5#22586387C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic, complemented
7C71×#1364C211yesmaximal, characteristic, complemented
8#17986550—21no—
9C2×C16×D71—21no—
10#17986376C2211yescharacteristic
11#17986347—21no—
12#17986236—21no—
13#17986236C412yes—
14#17986081—21no—
15#17986081C412yes—
16#17986021—21no—
17C4×C8×D71C2211yescharacteristic
18C4⋊(C71⋊C16)C2211yescharacteristic
19Dic71⋊C16—21no—
20#17985752C2211yescharacteristic
21#17985751—21no—
22#500×C71C2211yescharacteristic
23C4⋊C16×C71C2211yescharacteristic
24#335×C71C2211yescharacteristic
25C2×#11765866—41no—
26C2×D568—21no—
27C2×D568C2×C412yes—
28C2×#11765848—41no—
29C2×C8×D71C2×C411yescharacteristic
30C2×C8×D71D411yescharacteristic
31#14556125—21no—
32#14556117—41no—
33#14556103—22no—
34#14556103—41no—
35#14556090—22no—
36#14556090—41no—
37C42×D71D411yescharacteristic
38C2×C71⋊C16—21no—
39C16×D71—42no—
40C4⋊(C71⋊C8)—21no—
41Dic71⋊C8—41no—
42#14556048—22no—
43#14556048—22no—
44C4×C71⋊C8D411yescharacteristic
45C8×Dic71C2×C411yescharacteristic
46C8×Dic71D411yescharacteristic
47C4⋊Q8×C71—21no—
48C2×C16×C71—21no—
49#107×C71—21no—
50#107×C71C2×C412yes—
51C4⋊C8×C71—21no—
52C4×C8×C71C2311yescharacteristic
53#11765866—82no—
54#11765866—82no—
55C2×D284—22no—
56C2×D284—42no—
57C2×D284—41no—
58C2×D284—41no—
59C2×C4×D71—42no—
60C2×C4×D71C22⋊C411yescharacteristic
61C2×C4×D71D811yescharacteristic
62C2×C4×D71SD1611yescharacteristic
63D568—42no—
64D568—41no—
65D568—41no—
66#11765848—82no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8C22⋊C411yescharacteristic
69C2×C71⋊C8C22⋊C411yescharacteristic
70C8×D71—21no—
71C8×D71—21no—
72C8×D71—41no—
73(C2×D71)⋊C4—42no—
74(C2×D71)⋊C4—42no—
75C4×Dic71C22⋊C411yescharacteristic
76C4×Dic71D811yescharacteristic
77C4×Dic71SD1611yescharacteristic
78C71⋊C16—41no—
79C2×Q8×C71—41no—
80C2×C8×C71—41no—
81C2×C8×C71C22×C411yescharacteristic
82C2×C8×C71C2×D411yescharacteristic
83C4⋊C4×C71—22no—
84C4⋊C4×C71—41no—
85C42×C71C2×D411yescharacteristic
86C16×C71—41no—
87C22×D71—42no—
88C22×D71#10211yescharacteristic
89Q8×C71—81no—
90Q8×C71—81no—
91C2×C4×C71—42no—
92C2×C4×C71C2×C22⋊C411yescharacteristic
93C2×C4×C71C2×D811yescharacteristic
94C2×C4×C71C2×SD1611yescharacteristic
95C2×Dic71#10211yescharacteristic
96C2×Dic71#10211yescharacteristic
97C2×Dic71#10211yescharacteristic
98D284—42no—
99D284—42no—
100D284—82no—
101D284—84no—
102D284—81no—
103D284—81no—
104C4×D71—21no—
105C4×D71—21no—
106C4×D71—41no—
107C4×D71—84no—
108C4×D71D1612yes—
109C4×D71SD3212yes—
110C8×C71—21no—
111C8×C71—81no—
112C8×C71AGL(1,ℤ/8)11yescharacteristic
113C8×C71#13711yescharacteristic
114C71⋊C8—21no—
115C71⋊C8—21no—
116C71⋊C8—81no—
117C22×C71C2×#10211yescharacteristic
118C2×D71—21no—
119C2×D71—21no—
120C2×D71—82no—
121C2×D71—82no—
122C2×D71—82no—
123C2×D71#35712yes—
124C4×C71—21no—
125C4×C71—82no—
126C4×C71#41311yescharacteristic
127C4×C71#41711yescharacteristic
128C4×C71C2×D1611yescharacteristic
129C4×C71C2×SD3211yescharacteristic
130Dic71—21no—
131Dic71—21no—
132Dic71#35712yes—
133#11981—711nomaximal, Sylow
134C2×C71#154711yescharacteristic
135C2×C71C2×#35711yescharacteristic
136C2×C71#214811yescharacteristic
137D71—22no—
138D71—162no—
139#3169—711no—
140C2×C4⋊C16—711no—
141#1567—711no—
142#1364—711no—
143#1364D7111yescharacteristic, complemented
144#1361—711no—
145#1361—711no—
146C71#1198111yescharacteristic, complemented, Sylow
147#532—1421no—
148C22×C16—1421no—
149#500C2×D7111yescharacteristic
150#493—711no—
151#486—1421no—
152#466—712no—
153#466—1421no—
154C4×D8—712no—
155C4×D8—1421no—
156#423—1421no—
157C2×C4×C8—711no—
158C4⋊C16—711no—
159C4⋊C16—1421no—
160C4⋊C16C2×D7111yescharacteristic
161#335—711no—
162#335—1421no—
163#335C2×D7111yescharacteristic
164C2×#41—2841no—
165C2×D8—712no—
166C2×D8—1421no—
167C2×M16—2841no—
168C22×C8—711no—
169C22×C8—711no—
170C4⋊Q8—21no—
171#127—1421no—
172#124—2841no—
173#121—2841no—
174#121—1422no—
175C4×D4—2841no—
176C4×D4—1422no—
177C2×C42—711no—
178C2×C16—21no—
179C2×C16—2842no—
180C2×C16—1421no—
181#107—21no—
182#107C4×D7112yes—
183C4⋊C8—21no—
184C4⋊C8—2841no—
185C4⋊C8—1421no—
186#102—1422no—
187#102—1422no—
188C4×C8—711no—
189C4×C8—711no—
190C4×C8—711no—
191C4×C8C22×D7111yescharacteristic
192#41—5682no—
193#41—5682no—
194C2×Q8—41no—
195C2×D4—2842no—
196C2×D4—2841no—
197C2×D4—2841no—
198C2×D4—1422no—
199C22×C4—2842no—
200C22×C4—711no—
201C22×C4—711no—
202C22×C4—711no—
203D8—2842no—
204D8—2841no—
205D8—2841no—
206M16—5682no—
207C2×C8—41no—
208C2×C8—2841no—
209C2×C8—2841no—
210C2×C8—711no—
211C2×C8—711no—
212C2×C8—1421no—
213C2×C8—1421no—
214C2×C8C2×C4×D7111yescharacteristic
215C2×C8D4×D7111yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—41no—
218C22⋊C4—2842no—
219C22⋊C4—2842no—
220C42—711no—
221C42—711no—
222C42—711no—
223C42D4×D7111yescharacteristic
224C16—41no—
225C16—2841no—
226C23—2842no—
227C23—711no—
228Q8—81no—
229Q8—81no—
230D4—2842no—
231D4—2842no—
232D4—5682no—
233D4—5684no—
234D4—5681no—
235D4—5681no—
236C2×C4—42no—
237C2×C4—2841no—
238C2×C4—5684no—
239C2×C4—711no—
240C2×C4—712no—
241C2×C4—712no—
242C2×C4—711no—
243C2×C4—711no—
244C2×C4—1421no—
245C2×C4—1421no—
246C2×C4C22⋊C4×D7111yescharacteristic
247C2×C4D8×D7111yescharacteristic
248C2×C4SD16×D7111yescharacteristic
249C8—21no—
250C8—81no—
251C8—5681no—
252C8—1421no—
253C8—1421no—
254C8#1455615511yescharacteristic
255C8#1455616011yescharacteristic
256C22—5682no—
257C22—5682no—
258C22—5682no—
259C22—712no—
260C22—1421no—
261C22—1421no—
262C22#102×D7111yescharacteristic
263C4—21no—
264C4—82no—
265C4—712no—
266C4—1421no—
267C4—1421no—
268C4#1798598211yescharacteristic
269C4#1798599911yescharacteristic
270C4D16×D7111yescharacteristic
271C4SD32×D7111yescharacteristic
272C2—11362no—
273C2—1422no—
274C2#2258701611yescharacteristic
275C2#357×D7111yescharacteristic
276C2#2259008811yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1364C2
socleC22×C71C2×#102
prime radical#22598920C2
cosocleC4×C8×C71C23
hypercenter#1364D71
hypocenterC71#11981
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. C4⋊C16×D71C2semidirect
3. #22587097C2semidirect
4. #22586395C2semidirect
5. #22586387C2semidirect
6. #22586385C2semidirect
7. C71×#1364C2semidirect
143. #1364D71semidirect
146. C71#11981semidirect

Maximal subgroups

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

subgroupindex
#225989202
C4⋊C16×D712
#225870972
#225863952
#225863872
#225863852
C71×#13642
#1198171