Finite Group Databasebeta

← #31911593

Subgroups — #31911593

Order 18176.

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

normal subgroups only

This group has 234 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#22590484C211yesmaximal, characteristic
2#22590382C211yesmaximal, characteristic
3C8×D568C211yesmaximal, characteristic
4#22586404C211yesmaximal, characteristic, complemented
5#22586398C211yesmaximal, characteristic, complemented
6#22586389C211yesmaximal, characteristic, complemented
7C71×#1707C211yesmaximal, characteristic, complemented
8C2×#14556071—21no—
9C4×D568C2211yescharacteristic
10C8×D284—21no—
11#17985826—21no—
12C4⋊(C71⋊C16)C2211yescharacteristic
13C4⋊(C71⋊C16)C2211yescharacteristic
14#17985803—21no—
15#17985799—21no—
16#17985752C2211yescharacteristic
17#17985728—21no—
18#500×C71C2211yescharacteristic
19#498×C71—21no—
20#494×C71C2211yescharacteristic
21#487×C71—21no—
22C82×C71C2211yescharacteristic
23C2×D568D411yescharacteristic
24C2×C8×D71—41no—
25C4×D284—21no—
26#14556071—42no—
27C2×C71⋊C16—21no—
28C2×C71⋊C16—21no—
29#14556062D411yescharacteristic
30C4⋊(C71⋊C8)—21no—
31#14556050—21no—
32(C2×D71)⋊C8—41no—
33C2×Q16×C71—21no—
34C2×Q16×C71—21no—
35C4⋊Q8×C71—21no—
36C4⋊Q8×C71—21no—
37#107×C71—21no—
38#107×C71—21no—
39#106×C71—41no—
40#106×C71—41no—
41#103×C71—42no—
42C4×C8×C71D411yescharacteristic
43C4×C8×C71D411yescharacteristic
44C4×C8×C71C2311yescharacteristic
45C2×D284—21no—
46C2×C4×D71—41no—
47D568—21no—
48D568SD1612yes—
49C2×C71⋊C8—41no—
50C8×D71—82no—
51C4⋊Dic71—21no—
52(C2×D71)⋊C4—41no—
53C71⋊C16—41no—
54C71⋊C16—41no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57Q16×C71—41no—
58Q16×C71—41no—
59Q16×C71—81no—
60C2×C8×C71—21no—
61C2×C8×C71D811yescharacteristic
62C2×C8×C71D811yescharacteristic
63C2×C8×C71C2×D411yescharacteristic
64C2×C8×C71#4111yescharacteristic
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C42×C71C2×D411yescharacteristic
70C22×D71—41no—
71Q8×C71—82no—
72Q8×C71—81no—
73Q8×C71—81no—
74C2×C4×C71—81no—
75C2×C4×C71—81no—
76C2×C4×C71#12111yescharacteristic
77C2×C4×C71C2×D811yescharacteristic
78C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
79C2×Dic71—41no—
80D284—22no—
81D284—41no—
82C4×D71—81no—
83C4×D71—81no—
84C8×C71—21no—
85C8×C71—21no—
86C8×C71—41no—
87C8×C71SD3212yescomplemented
88C8×C71C2×SD1611yescharacteristic
89C8×C71#13511yescharacteristic
90C71⋊C8—81no—
91C22×C71#45911yescharacteristic
92C2×D71—42no—
93C2×D71—81no—
94C2×D71—81no—
95C4×C71—21no—
96C4×C71—161no—
97C4×C71—161no—
98C4×C71#46511yescharacteristic
99C4×C71#47111yescharacteristic
100C4×C71C2×SD3211yescharacteristic
101C4×C71#51011yescharacteristic
102Dic71—81no—
103#12420—711nomaximal, Sylow
104C2×C71#166511yescharacteristic
105C2×C71#220811yescharacteristic
106C2×C71#221211yescharacteristic
107D71—82no—
108#2226—711no—
109#2208—711no—
110#1707D7111yescharacteristic, complemented
111C8×D8—711no—
112#1367—711no—
113#1365—711no—
114#1362—711no—
115C71#1242011yescharacteristic, complemented, Sylow
116C2×SD32—1421no—
117#500C2×D7111yescharacteristic
118#498—21no—
119#494C2×D7111yescharacteristic
120#487—21no—
121C4×D8—711no—
122C8×D4—1421no—
123#367—1421no—
124C4⋊C16—711no—
125C4⋊C16—711no—
126#358—1421no—
127#357—1421no—
128#335—711no—
129D4⋊C8—1421no—
130C82C2×D7111yescharacteristic
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8—711no—
134C22×C8—2841no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4×D4—1421no—
138SD32—2842no—
139C2×C16—1421no—
140C2×C16—1421no—
141#107—21no—
142#107—21no—
143#107—711no—
144#106—41no—
145#106—41no—
146C4⋊C8—1421no—
147#103—42no—
148#102—1421no—
149C22⋊C8—2841no—
150C4×C8C22⋊D7111yescharacteristic
151C4×C8C22⋊D7111yescharacteristic
152C4×C8C22×D7111yescharacteristic
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—1421no—
156C22×C4—2841no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—712no—
161D8—1421no—
162C2×C8—21no—
163C2×C8—2841no—
164C2×C8—5682no—
165C2×C8#1176585011yescharacteristic
166C2×C8#1176585011yescharacteristic
167C2×C8D4×D7111yescharacteristic
168C2×C8#1176586411yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—1421no—
174C22⋊C4—2841no—
175C42C2×C22⋊D7111yescharacteristic
176C16—2841no—
177C16—2841no—
178C23—2841no—
179Q8—82no—
180Q8—81no—
181Q8—81no—
182D4—2841no—
183D4—1422no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—2841no—
187C2×C4—5681no—
188C2×C4—5681no—
189C2×C4#1455609911yescharacteristic
190C2×C4C2×#1176585011yescharacteristic
191C2×C4#1455615411yescharacteristic
192C8—21no—
193C8—21no—
194C8—41no—
195C8—5681no—
196C8#1455607112yescomplemented
197C8SD16×D7111yescharacteristic
198C8#1455614911yescharacteristic
199C22—2842no—
200C22—5681no—
201C22—5681no—
202C22#1798619111yescharacteristic
203C4—21no—
204C4—161no—
205C4—161no—
206C4—5681no—
207C4#1798622511yescharacteristic
208C4#1798625911yescharacteristic
209C4C2×#1455607111yescharacteristic
210C4#1798644011yescharacteristic
211C2—5682no—
212C2#2258767811yescharacteristic
213C2#2259038211yescharacteristic
214C2#2259040611yescharacteristic
subgroupG/H
centerC22#17986191
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1707C2
socleC22×C71#459
prime radicalC2×D568D4
hypercenter#1707D71
hypocenterC71#12420
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1707

Complements

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

NH ≅ G/Ntype
4. #22586404C2semidirect
5. #22586398C2semidirect
6. #22586389C2semidirect
7. C71×#1707C2semidirect
87. C8×C71SD32semidirect
110. #1707D71semidirect
115. C71#12420semidirect
196. C8#14556071semidirect

Maximal subgroups

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

subgroupindex
#225904842
#225903822
C8×D5682
#225864042
#225863982
#225863892
C71×#17072
#1242071