Finite Group Databasebeta

← #31908963

Subgroups — #31908963

Order 18176.

308 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 85 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#22598927C211yesmaximal, characteristic
2#22590094C211yesmaximal, characteristic
3#22587097C211yesmaximal, characteristic
4#22586395C211yesmaximal, characteristic, complemented
5#22586390C211yesmaximal, characteristic, complemented
6#22586388C211yesmaximal, characteristic, complemented
7C71×#1364C211yesmaximal, characteristic, complemented
8#17986554—21no—
9C2×#14556067—21no—
10#17986379C2211yescharacteristic
11#17986351—21no—
12#17986241—21no—
13#17986241C412yes—
14#17986093—21no—
15#17986093C412yes—
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×Dic284—21no—
26C2×Dic284C2×C412yes—
27C2×#11765848—41no—
28C2×C8×D71C2×C411yescharacteristic
29C2×C8×D71D411yescharacteristic
30#14556128—21no—
31#14556119—41no—
32#14556108—22no—
33#14556104—41no—
34Q8×Dic71—41no—
35#14556093—22no—
36#14556086—41no—
37C42×D71D411yescharacteristic
38#14556067—42no—
39C2×C71⋊C16—21no—
40C4⋊(C71⋊C8)—21no—
41Dic71⋊C8—41no—
42#14556051—22no—
43#14556051—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
53C2×Dic142—22no—
54C2×C4×D71C22⋊C411yescharacteristic
55C2×C4×D71D811yescharacteristic
56C2×C4×D71SD1611yescharacteristic
57Dic284—42no—
58Dic284—41no—
59Dic284—41no—
60#11765848—82no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8C22⋊C411yescharacteristic
63C2×C71⋊C8C22⋊C411yescharacteristic
64C8×D71—21no—
65C8×D71—21no—
66C8×D71—41no—
67C4⋊Dic71—41no—
68C4⋊Dic71—41no—
69C4⋊Dic71—42no—
70Dic71⋊C4—42no—
71(C2×D71)⋊C4—42no—
72C4×Dic71—42no—
73C4×Dic71C22⋊C411yescharacteristic
74C4×Dic71D811yescharacteristic
75C4×Dic71SD1611yescharacteristic
76C71⋊C16—41no—
77C2×Q8×C71—41no—
78C2×C8×C71—41no—
79C2×C8×C71C22×C411yescharacteristic
80C2×C8×C71C2×D411yescharacteristic
81C4⋊C4×C71—22no—
82C4⋊C4×C71—41no—
83C42×C71C2×D411yescharacteristic
84C16×C71—41no—
85C22×D71#10211yescharacteristic
86Q8×C71—81no—
87Q8×C71—81no—
88C2×C4×C71—42no—
89C2×C4×C71C2×C22⋊C411yescharacteristic
90C2×C4×C71C2×D811yescharacteristic
91C2×C4×C71C2×SD1611yescharacteristic
92C2×Dic71—42no—
93C2×Dic71#10211yescharacteristic
94C2×Dic71#10211yescharacteristic
95C2×Dic71#10211yescharacteristic
96C4×D71—21no—
97C4×D71—21no—
98C4×D71—21no—
99C4×D71—21no—
100C4×D71—22no—
101Dic142—42no—
102Dic142—42no—
103C8×C71—21no—
104C8×C71—81no—
105C8×C71AGL(1,ℤ/8)11yescharacteristic
106C8×C71#13711yescharacteristic
107C71⋊C8—21no—
108C71⋊C8—21no—
109C71⋊C8—81no—
110C22×C71C2×#10211yescharacteristic
111C2×D71—21no—
112C2×D71—21no—
113C2×D71—21no—
114C4×C71—21no—
115C4×C71—82no—
116C4×C71#41311yescharacteristic
117C4×C71#41711yescharacteristic
118C4×C71#50911yescharacteristic
119C4×C71#51011yescharacteristic
120Dic71—21no—
121Dic71—21no—
122Dic71—82no—
123Dic71#36212yes—
124#11988—711nomaximal, Sylow
125C2×C71#154711yescharacteristic
126C2×C71#213911yescharacteristic
127C2×C71C2×#36211yescharacteristic
128D71—41no—
129#3170—711no—
130#2150—711no—
131#1567—711no—
132#1364—711no—
133#1364D7111yescharacteristic, complemented
134#1362—711no—
135#1362—711no—
136C71#1198811yescharacteristic, complemented, Sylow
137#533—1421no—
138C2×M32—1421no—
139#500C2×D7111yescharacteristic
140#494—711no—
141#487—1421no—
142#467—712no—
143#467—1421no—
144C4×Q16—712no—
145C4×Q16—1421no—
146#423—1421no—
147C2×C4×C8—711no—
148C4⋊C16—711no—
149C4⋊C16—1421no—
150C4⋊C16C2×D7111yescharacteristic
151#335—711no—
152#335—1421no—
153#335C2×D7111yescharacteristic
154C2×Q16—712no—
155C2×Q16—1421no—
156C2×M16—2841no—
157C22×C8—711no—
158C22×C8—711no—
159C4⋊Q8—21no—
160C4⋊Q8—1421no—
161#125—2841no—
162C22⋊Q8—2841no—
163C22⋊Q8—1422no—
164C4×Q8—2841no—
165C4×Q8—1422no—
166#117—2841no—
167C2×C42—711no—
168M32—2842no—
169C2×C16—21no—
170C2×C16—1421no—
171#107—21no—
172#107C4×D7112yes—
173C4⋊C8—21no—
174C4⋊C8—2841no—
175C4⋊C8—1421no—
176#103—1422no—
177#103—1422no—
178C4×C8—711no—
179C4×C8—711no—
180C4×C8—711no—
181C4×C8C22×D7111yescharacteristic
182C2×Q8—41no—
183C2×Q8—1422no—
184C22×C4—711no—
185C22×C4—711no—
186C22×C4—711no—
187Q16—2842no—
188Q16—2841no—
189Q16—2841no—
190M16—5682no—
191C2×C8—41no—
192C2×C8—2841no—
193C2×C8—2841no—
194C2×C8—711no—
195C2×C8—711no—
196C2×C8—1421no—
197C2×C8—1421no—
198C2×C8C2×C4×D7111yescharacteristic
199C2×C8D4×D7111yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—2842no—
203C4⋊C4—2842no—
204C4⋊C4—2841no—
205C4⋊C4—2841no—
206C22⋊C4—2842no—
207C42—2842no—
208C42—711no—
209C42—711no—
210C42—711no—
211C42D4×D7111yescharacteristic
212C16—41no—
213C16—2841no—
214C23—711no—
215Q8—81no—
216Q8—81no—
217Q8—2842no—
218Q8—2842no—
219C2×C4—42no—
220C2×C4—2842no—
221C2×C4—711no—
222C2×C4—711no—
223C2×C4—711no—
224C2×C4—1422no—
225C2×C4—1421no—
226C2×C4—1421no—
227C2×C4—1421no—
228C2×C4—1421no—
229C2×C4C22⋊C4×D7111yescharacteristic
230C2×C4D8×D7111yescharacteristic
231C2×C4SD16×D7111yescharacteristic
232C8—21no—
233C8—81no—
234C8—5681no—
235C8—1421no—
236C8—1421no—
237C8#1455615511yescharacteristic
238C8#1455616011yescharacteristic
239C22—1421no—
240C22—1421no—
241C22—1421no—
242C22#102×D7111yescharacteristic
243C4—21no—
244C4—82no—
245C4—5682no—
246C4—712no—
247C4—1421no—
248C4—1421no—
249C4#1798598211yescharacteristic
250C4#1798599911yescharacteristic
251C4#1798643311yescharacteristic
252C4#1798643811yescharacteristic
253C2—2841no—
254C2#2258701611yescharacteristic
255C2#2259004911yescharacteristic
256C2#362×D7111yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1364C2
socleC22×C71C2×#102
prime radicalC22×D71#102
cosocleC4×C8×C71C23
hypercenter#1364D71
hypocenterC71#11988
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
4. #22586395C2semidirect
5. #22586390C2semidirect
6. #22586388C2semidirect
7. C71×#1364C2semidirect
133. #1364D71semidirect
136. C71#11988semidirect

Maximal subgroups

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

subgroupindex
#225989272
#225900942
#225870972
#225863952
#225863902
#225863882
C71×#13642
#1198871