Finite Group Databasebeta

← #31907538

Subgroups — #31907538

Order 18176.

348 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 83 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#22597251C211yesmaximal, characteristic
2#22590152C211yesmaximal, characteristic, complemented
3#22586317C212yesmaximal, complemented
4#22586314C212yesmaximal, complemented
5C71×#1453C211yesmaximal, characteristic, complemented
6#17986799C411yescharacteristic
7#17986765—21no—
8#17986122—21no—
9#17986122C412yes—
10#17986117—21no—
11#17986117C2212yes—
12#17986048C411yescharacteristic
13C2×#14556056—21no—
14#17985820C2212yescomplemented
15#17985781—21no—
16#17985777—21no—
17#431×C71—21no—
18C4×M16×C71C2211yescharacteristic
19#322×C71—21no—
20#322×C71C2211yescharacteristic, complemented
21C82×C71C2211yescharacteristic, complemented
22C2×#11765865—21no—
23#14556161—22no—
24#14556156—22no—
25#14556150—21no—
26#14556150C2×C412yes—
27#14556136—42no—
28#14556134—22no—
29#14556134—22no—
30C2×#11765849—21no—
31M16×D71—42no—
32#14556068—22no—
33#14556064—21no—
34#14556064C2×C412yes—
35#14556056—22no—
36#14556056—22no—
37C2×M16×C71C2×C411yescharacteristic
38C2×M16×C71D411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C4⋊C8×C71—22no—
41C22⋊C8×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71—21no—
45C4×C8×C71—21no—
46C4×C8×C71D411yescharacteristic
47C4×C8×C71C2311yescharacteristic
48#11765865—22no—
49#11765865—22no—
50#11765865—41no—
51#11765865—41no—
52C2×Dic142—21no—
53C2×D284—21no—
54C2×C22⋊D71—41no—
55C2×C4×D71—41no—
56Dic284—21no—
57Dic284M1612yes—
58#11765854—22no—
59#11765854—22no—
60D568—21no—
61D568M1612yes—
62#11765849—22no—
63#11765849—41no—
64#11765848—42no—
65#11765848—42no—
66C2×C71⋊C8—41no—
67C8×D71—42no—
68C8×D71—42no—
69C71⋊C16—42no—
70C22×C4×C71—21no—
71C22×C4×C71C22⋊C411yescharacteristic
72M16×C71—22no—
73M16×C71—22no—
74C2×C8×C71—22no—
75C2×C8×C71—22no—
76C2×C8×C71C22⋊C411yescharacteristic
77C2×C8×C71C22⋊C411yescharacteristic
78C2×C8×C71C22×C411yescharacteristic
79C2×C8×C71C2×D411yescharacteristic
80C42×C71—21no—
81C42×C71C22⋊C411yescharacteristic
82C42×C71C2×D411yescharacteristic
83C23×C71#10411yescharacteristic
84C22×D71—41no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—42no—
90C2×C4×C71#10411yescharacteristic
91C2×C4×C71C2×C22⋊C411yescharacteristic
92C22⋊D71—42no—
93C22⋊D71—42no—
94C2×Dic71—41no—
95D284—22no—
96D284—41no—
97C4×D71—42no—
98C4×D71—42no—
99Dic142—22no—
100Dic142—41no—
101C8×C71—21no—
102C8×C71—21no—
103C8×C71—42no—
104C8×C71—42no—
105C8×C71C2×M1612yes—
106C8×C71AGL(1,ℤ/8)12yes—
107C71⋊C8—42no—
108C22×C71—21no—
109C22×C71—41no—
110C22×C71C2×#10411yescharacteristic
111C2×D71—42no—
112C2×D71—161no—
113C4×C71—22no—
114C4×C71—21no—
115C4×C71—41no—
116C4×C71#40711yescharacteristic
117C4×C71#41811yescharacteristic
118Dic71—42no—
119#11670—711nomaximal, Sylow
120C2×C71—21no—
121C2×C71—41no—
122C2×C71#149011yescharacteristic
123D71—82no—
124#2956—711no—
125#2165—711no—
126#1453D7111yescharacteristic, complemented
127#1336—712no—
128#1335—712no—
129C71#1167011yescharacteristic, complemented, Sylow
130#575—711no—
131#568—1421no—
132#444—712no—
133#444—1421no—
134#443—712no—
135#443—1421no—
136#431—21no—
137#430—711no—
138C2×#104—1421no—
139C4×M16C2×D7111yescharacteristic
140#364—712no—
141#349—1421no—
142#347—1421no—
143#322—21no—
144#322C2×D7111yescharacteristic, complemented
145C82C2×D7111yescharacteristic, complemented
146C2×#41—1421no—
147#137—1422no—
148AGL(1,ℤ/8)—1422no—
149#135—712no—
150#135—1421no—
151#131—2842no—
152#131—1422no—
153#131—1422no—
154C2×M16—2842no—
155C2×M16—1421no—
156C2×M16C4×D7111yescharacteristic
157C2×M16D28411yescharacteristic
158C2×C42C22⋊D7111yescharacteristic
159M32—1422no—
160#108—712no—
161#108—1421no—
162C4⋊C8—22no—
163#104—1422no—
164#104—1422no—
165C22⋊C8—22no—
166#97C4×D7111yescharacteristic
167#97D28411yescharacteristic
168C4×C8—21no—
169C4×C8—21no—
170C4×C8C22⋊D7111yescharacteristic
171C4×C8C22×D7111yescharacteristic
172#41—2841no—
173#41—2841no—
174#41—1422no—
175#41—1422no—
176C2×Q8—1421no—
177C2×D4—2841no—
178C2×D4—1421no—
179C22×C4—21no—
180C22×C4—2841no—
181C22×C4(C2×D71)⋊C411yescharacteristic
182Q16—712no—
183Q16—1421no—
184SD16—1422no—
185SD16—1422no—
186D8—712no—
187D8—1421no—
188M16—22no—
189M16—22no—
190M16—2842no—
191M16—2841no—
192M16—2842no—
193M16—1422no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—2842no—
197C2×C8—2841no—
198C2×C8—2842no—
199C2×C8(C2×D71)⋊C411yescharacteristic
200C2×C8(C2×D71)⋊C411yescharacteristic
201C2×C8C2×C4×D7111yescharacteristic
202C2×C8C2×D28411yescharacteristic
203C42—21no—
204C42(C2×D71)⋊C411yescharacteristic
205C42C2×C22⋊D7111yescharacteristic
206C16—2842no—
207C23—2841no—
208C23#1455605611yescharacteristic
209Q8—2841no—
210Q8—1422no—
211D4—2841no—
212D4—2842no—
213D4—2842no—
214D4—1422no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—2842no—
221C2×C4—2841no—
222C2×C4—2842no—
223C2×C4#1455605611yescharacteristic
224C2×C4C2×(C2×D71)⋊C411yescharacteristic
225C8—21no—
226C8—21no—
227C8—42no—
228C8—42no—
229C8—2842no—
230C8M16×D7112yes—
231C8#1455615612yes—
232C22—21no—
233C22—41no—
234C22—2842no—
235C22—11361no—
236C22C2×#1455605611yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—2842no—
241C4#1798596211yescharacteristic
242C4#1798600511yescharacteristic
243C2—21no—
244C2—41no—
245C2—5682no—
246C2#2258678411yescharacteristic
subgroupG/H
centerC4#17986005
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1453C2
socleC2×C71#1490
prime radical#17986799C4
cosocleC4×C8×C71C23
hypercenter#1453D71
hypocenterC71#11670
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C8 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #22590152C2semidirect
3. #22586317C2semidirect
4. #22586314C2semidirect
5. C71×#1453C2semidirect
14. #17985820C22semidirect
20. #322×C71C22semidirect
21. C82×C71C22semidirect
126. #1453D71semidirect
129. C71#11670semidirect
144. #322C2×D71semidirect
145. C82C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225972512
#225901522
#225863172
#225863142
C71×#14532
#1167071