Finite Group Databasebeta

← #31908298

Subgroups — #31908298

Order 18176.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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#22599078C211yesmaximal, characteristic
2#22590152C211yesmaximal, characteristic, complemented
3#22586330C212yesmaximal, complemented
4#22586324C212yesmaximal, complemented
5C71×#1841C211yesmaximal, characteristic, complemented
6#17986549—21no—
7#17986397C411yescharacteristic, complemented
8#17986376C411yescharacteristic, complemented
9#17986346—21no—
10#17986252—21no—
11#17986252C2212yes—
12#17986112—21no—
13#17986112C412yescomplemented
14#17985820—21no—
15#17985781C2212yes—
16#17985777—21no—
17C4×M16×C71C2211yescharacteristic
18#344×C71—21no—
19#339×C71—21no—
20#338×C71C2211yescharacteristic
21H(3,ℤ/4)×C71C2211yescharacteristic
22C2×#11765865—41no—
23C2×D568—21no—
24C2×D568C2×C412yes—
25#14556128—21no—
26#14556125—21no—
27#14556116—41no—
28#14556102—22no—
29#14556102—41no—
30C4×D284—22no—
31C4×D284—41no—
32#14556068—22no—
33#14556060—21no—
34#14556060C2×C412yes—
35#14556050—22no—
36#14556050—22no—
37C2×M16×C71—21no—
38C2×M16×C71D411yescharacteristic
39C2×M16×C71C2311yescharacteristic
40#117×C71—21no—
41C2×C42×C71D411yescharacteristic
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46#11765865—84no—
47C2×Dic142—41no—
48C2×D284—22no—
49C2×D284—41no—
50C2×D284—41no—
51C2×C22⋊D71—42no—
52C2×C4×D71—42no—
53D568—42no—
54D568—42no—
55C4⋊Dic71—22no—
56C4⋊Dic71—41no—
57(C2×D71)⋊C4—42no—
58(C2×D71)⋊C4—42no—
59C71⋊C16—42no—
60C22×C4×C71—21no—
61C22×C4×C71C2×D411yescharacteristic
62M16×C71—41no—
63M16×C71—41no—
64M16×C71—41no—
65M16×C71—41no—
66C2×C8×C71—41no—
67C2×C8×C71C22⋊C411yescharacteristic
68C2×C8×C71C22⋊C411yescharacteristic
69C2×C8×C71C22×C411yescharacteristic
70C2×C8×C71C2×D411yescharacteristic
71C4⋊C4×C71—41no—
72C22⋊C4×C71—41no—
73C42×C71—21no—
74C42×C71—41no—
75C42×C71C22⋊C411yescharacteristic
76C42×C71C22⋊C411yescharacteristic
77C23×C71AGL(1,ℤ/8)11yescharacteristic
78C22×D71—42no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—42no—
84C2×C4×C71—81no—
85C2×C4×C71C2×C22⋊C411yescharacteristic
86C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
87C22⋊D71—84no—
88C2×Dic71—42no—
89D284—42no—
90D284—42no—
91D284—82no—
92D284—82no—
93C4×D71—84no—
94Dic142—82no—
95C8×C71—21no—
96C8×C71—21no—
97C8×C71—21no—
98C8×C71—21no—
99C8×C71—81no—
100C22×C71—21no—
101C22×C71—41no—
102C22×C71#41811yescharacteristic
103C2×D71—82no—
104C2×D71—82no—
105C2×D71—82no—
106C4×C71—22no—
107C4×C71—21no—
108C4×C71—41no—
109C4×C71—161no—
110C4×C71#41011yescharacteristic
111C4×C71#42111yescharacteristic
112Dic71—82no—
113#11837—711nomaximal, Sylow
114C2×C71—21no—
115C2×C71—41no—
116C2×C71#151411yescharacteristic
117D71—162no—
118#3184—711no—
119#2165—711no—
120#1841D7111yescharacteristic, complemented
121#1341—712no—
122#1339—712no—
123C71#1183711yescharacteristic, complemented, Sylow
124#532—1421no—
125#498—711no—
126#493—711no—
127#486—1421no—
128#469—712no—
129#469—1421no—
130#442—712no—
131#442—1421no—
132C4×M16C2×D7111yescharacteristic
133#364—1421no—
134#349—712no—
135#347—1421no—
136#344—21no—
137#339—21no—
138#338C2×D7111yescharacteristic
139H(3,ℤ/4)C2×D7111yescharacteristic
140C2×#41—2841no—
141C2×D8—712no—
142C2×D8—1421no—
143C2×M16—21no—
144C2×M16C22⋊D7111yescharacteristic
145C2×M16C22×D7111yescharacteristic
146C4⋊Q8—1421no—
147#127—1421no—
148#124—2841no—
149#121—2841no—
150#121—1422no—
151C4×D4—2841no—
152C4×D4—1422no—
153#117—21no—
154C2×C42C22⋊D7111yescharacteristic
155M32—1422no—
156#106—712no—
157#106—1421no—
158#102—1422no—
159#102—1422no—
160#97C4×D7111yescharacteristic
161#97D28411yescharacteristic
162C4×C8C4×D7111yescharacteristic
163C4×C8D28411yescharacteristic
164#41—5684no—
165C2×Q8—2841no—
166C2×D4—2841no—
167C2×D4—2841no—
168C2×D4—2842no—
169C2×D4—1422no—
170C22×C4—21no—
171C22×C4—2842no—
172C22×C4C2×C22⋊D7111yescharacteristic
173D8—2842no—
174D8—2842no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—41no—
180C2×C8(C2×D71)⋊C411yescharacteristic
181C2×C8(C2×D71)⋊C411yescharacteristic
182C2×C8C2×C4×D7111yescharacteristic
183C2×C8C2×D28411yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—2841no—
186C4⋊C4—1422no—
187C22⋊C4—41no—
188C22⋊C4—2842no—
189C22⋊C4—2842no—
190C42—21no—
191C42—41no—
192C42(C2×D71)⋊C411yescharacteristic
193C42(C2×D71)⋊C411yescharacteristic
194C16—2842no—
195C23—2842no—
196C23#1455615311yescharacteristic
197Q8—5682no—
198D4—2842no—
199D4—2842no—
200D4—5682no—
201D4—5682no—
202D4—5684no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—2842no—
210C2×C4—5684no—
211C2×C4C2×(C2×D71)⋊C411yescharacteristic
212C2×C4#1455615311yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—5682no—
221C22—5682no—
222C22—5682no—
223C22#1798600311yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—161no—
228C4—5682no—
229C4#1798597211yescharacteristic
230C4#1798601711yescharacteristic
231C2—21no—
232C2—41no—
233C2—11362no—
234C2#2258688611yescharacteristic
subgroupG/H
centerC2#22586886
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#1841C2
socleC2×C71#1514
prime radical#22599078C2
cosocleC2×M16×C71C23
hypercenter#1841D71
hypocenterC71#11837
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #22590152C2semidirect
3. #22586330C2semidirect
4. #22586324C2semidirect
5. C71×#1841C2semidirect
7. #17986397C4semidirect
8. #17986376C4semidirect
13. #17986112C4semidirect
120. #1841D71semidirect
123. C71#11837semidirect

Maximal subgroups

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

subgroupindex
#225990782
#225901522
#225863302
#225863242
C71×#18412
#1183771