Finite Group Databasebeta

← #31908637

Subgroups — #31908637

Order 18176.

296 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 75 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#22598990C211yesmaximal, characteristic
2#22590156C211yesmaximal, characteristic, complemented
3#22586372C214yesmaximal, complemented
4C71×#2110C211yesmaximal, characteristic, complemented
5#17986604C412yes—
6#17986401C411yescharacteristic
7#17986376C411yescharacteristic
8#17986364C2212yes—
9#17986252—21no—
10#17986235—21no—
11#17986112—21no—
12C4×D568—21no—
13#17985820—21no—
14#17985781—21no—
15#17985777C2212yes—
16C2×M32×C71—21no—
17#405×C71C2211yescharacteristic
18#347×C71—21no—
19#347×C71C2212yes—
20C2×D568—21no—
21C2×D568—21no—
22#14556125C2×C412yes—
23#14556120C2×C412yes—
24#14556113—42no—
25#14556102—22no—
26#14556101—42no—
27C4×D284—22no—
28#14556068—22no—
29#14556062—21no—
30#14556060—21no—
31#14556050—22no—
32#14556050—22no—
33C2×M16×C71D411yescharacteristic
34C22×C8×C71D411yescharacteristic
35#117×C71C2×C411yescharacteristic
36M32×C71—22no—
37M32×C71—22no—
38C2×C16×C71—22no—
39#97×C71D411yescharacteristic
40#97×C71C2311yescharacteristic
41C4×C8×C71C2×C411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C2×D284—22no—
44C2×D284—42no—
45C2×C22⋊D71—42no—
46C2×C4×D71—42no—
47D568—42no—
48D568—41no—
49D568—41no—
50C4⋊Dic71—22no—
51Dic71⋊C4—42no—
52(C2×D71)⋊C4—42no—
53(C2×D71)⋊C4—42no—
54C71⋊C16—42no—
55C22×C4×C71C22⋊C411yescharacteristic
56M16×C71—41no—
57C2×C8×C71—21no—
58C2×C8×C71—21no—
59C2×C8×C71C22⋊C411yescharacteristic
60C2×C8×C71C22⋊C411yescharacteristic
61C2×C8×C71C2×D411yescharacteristic
62C2×C8×C71C2×D411yescharacteristic
63C4⋊C4×C71—21no—
64C22⋊C4×C71—21no—
65C42×C71C22⋊C411yescharacteristic
66C42×C71C22×C411yescharacteristic
67C16×C71—24no—
68C23×C71#10011yescharacteristic
69C22×D71—42no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71#10011yescharacteristic
75C2×C4×C71C2×C22⋊C411yescharacteristic
76C22⋊D71—162no—
77C2×Dic71—42no—
78D284—42no—
79D284—42no—
80D284—162no—
81C4×D71—162no—
82C8×C71—21no—
83C8×C71—21no—
84C8×C71—21no—
85C8×C71#10411yescharacteristic
86C8×C71#10411yescharacteristic
87C22×C71—21no—
88C22×C71—41no—
89C22×C71C2×#10011yescharacteristic
90C2×D71—82no—
91C2×D71—82no—
92C2×D71—82no—
93C4×C71—21no—
94C4×C71—41no—
95C4×C71—41no—
96C4×C71C2×#10411yescharacteristic
97C4×C71#42111yescharacteristic
98Dic71—82no—
99#11913—711nomaximal, Sylow
100C2×C71—21no—
101C2×C71—41no—
102C2×C71#153011yescharacteristic
103D71—162no—
104#3176—711no—
105#2166—711no—
106#2110D7111yescharacteristic, complemented
107#1355—714no—
108C71#1191311yescharacteristic, complemented, Sylow
109#540—712no—
110C2×M32—21no—
111#499—711no—
112#493—711no—
113#490—712no—
114#469—1421no—
115#466—1421no—
116#442—1421no—
117C4×D8—1421no—
118#405C2×D7111yescharacteristic
119#364—1421no—
120#349—1421no—
121#347—21no—
122#347—712no—
123#347C2×D7112yes—
124C2×D8—1421no—
125C2×D8—1421no—
126C2×M16D28411yescharacteristic
127C22×C8C22⋊D7111yescharacteristic
128#127—712no—
129#125—712no—
130#123—2842no—
131#121—2842no—
132#121—1422no—
133C4×D4—1422no—
134#117C4×D7111yescharacteristic
135M32—22no—
136M32—22no—
137M32—1422no—
138C2×C16—22no—
139#107—1421no—
140#106—1421no—
141#102—1422no—
142#102—1422no—
143#97C22⋊D7111yescharacteristic
144#97C22×D7111yescharacteristic
145C4×C8C4×D7111yescharacteristic
146C4×C8D28411yescharacteristic
147C2×D4—2842no—
148C2×D4—2842no—
149C2×D4—1422no—
150C22×C4—2842no—
151C22×C4(C2×D71)⋊C411yescharacteristic
152D8—2842no—
153D8—2841no—
154D8—2841no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8(C2×D71)⋊C411yescharacteristic
159C2×C8(C2×D71)⋊C411yescharacteristic
160C2×C8C2×C22⋊D7111yescharacteristic
161C2×C8C2×D28411yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—2842no—
164C4⋊C4—1422no—
165C22⋊C4—21no—
166C22⋊C4—2842no—
167C22⋊C4—2842no—
168C42(C2×D71)⋊C411yescharacteristic
169C42C2×C4×D7111yescharacteristic
170C16—24no—
171C16—2842no—
172C23—2842no—
173C23#1455604411yescharacteristic
174D4—2842no—
175D4—2842no—
176D4—11362no—
177D4—11362no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—2842no—
183C2×C4—11362no—
184C2×C4#1455604411yescharacteristic
185C2×C4C2×(C2×D71)⋊C411yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8#1455605611yescharacteristic
190C8#1455605611yescharacteristic
191C22—21no—
192C22—41no—
193C22—5682no—
194C22—5682no—
195C22—5682no—
196C22C2×#1455604411yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—5682no—
201C4C2×#1455605611yescharacteristic
202C4#1798601511yescharacteristic
203C2—21no—
204C2—41no—
205C2—11362no—
206C2#2258695011yescharacteristic
subgroupG/H
centerC2#22586950
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#2110C2
socleC2×C71#1530
prime radical#22598990C2
cosocle#97×C71C23
hypercenter#2110D71
hypocenterC71#11913
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #22590156C2semidirect
3. #22586372C2semidirect
4. C71×#2110C2semidirect
106. #2110D71semidirect
108. C71#11913semidirect

Maximal subgroups

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

subgroupindex
#225989902
#225901562
#225863722
C71×#21102
#1191371