Finite Group Databasebeta

← #31908970

Subgroups — #31908970

Order 18176.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590106C211yesmaximal, characteristic, complemented
3#22586400C212yesmaximal, complemented
4#22586376C212yesmaximal, complemented
5C71×#2166C211yesmaximal, characteristic, complemented
6#17986604C412yes—
7C2×#14556068—21no—
8#17986401C2211yescharacteristic
9#17986376C2211yescharacteristic
10#17986364C412yes—
11#17986252—21no—
12#17986235—21no—
13#17986112—21no—
14C4×D568—21no—
15#17985820—21no—
16#17985820C2212yes—
17#405×C71C2211yescharacteristic
18#364×C71C2212yes—
19#349×C71—21no—
20#347×C71—21no—
21C2×D568—21no—
22C2×D568—21no—
23#14556125C2×C412yes—
24#14556120C2×C412yes—
25#14556113—42no—
26#14556102—22no—
27#14556101—42no—
28C4×D284—22no—
29#14556068—22no—
30#14556068—41no—
31C2×C71⋊C16—41no—
32#14556062—21no—
33#14556060—21no—
34#14556050—22no—
35#14556050—22no—
36C2×M16×C71D411yescharacteristic
37C22×C8×C71D411yescharacteristic
38#117×C71C2×C411yescharacteristic
39M32×C71—22no—
40#97×C71C2×C411yescharacteristic
41#97×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C4×C8×C71C2311yescharacteristic
44C2×D284—22no—
45C2×D284—42no—
46C2×C22⋊D71—42no—
47C2×C4×D71—42no—
48D568—42no—
49D568—41no—
50D568—41no—
51C4⋊Dic71—22no—
52Dic71⋊C4—42no—
53(C2×D71)⋊C4—42no—
54(C2×D71)⋊C4—42no—
55C71⋊C16—42no—
56C22×C4×C71C22⋊C411yescharacteristic
57M16×C71—22no—
58C2×C8×C71—21no—
59C2×C8×C71—21no—
60C2×C8×C71C22⋊C411yescharacteristic
61C2×C8×C71C22⋊C411yescharacteristic
62C2×C8×C71C2×D411yescharacteristic
63C2×C8×C71C2×D411yescharacteristic
64C4⋊C4×C71—21no—
65C22⋊C4×C71—21no—
66C42×C71C22⋊C411yescharacteristic
67C42×C71C22×C411yescharacteristic
68C16×C71—42no—
69C23×C71#10011yescharacteristic
70C22×D71—42no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71#10011yescharacteristic
76C2×C4×C71C2×C22⋊C411yescharacteristic
77C22⋊D71—162no—
78C2×Dic71—42no—
79D284—42no—
80D284—42no—
81D284—162no—
82C4×D71—162no—
83C8×C71—21no—
84C8×C71—21no—
85C8×C71#13512yes—
86C8×C71AGL(1,ℤ/8)11yescharacteristic
87C8×C71#13711yescharacteristic
88C22×C71—21no—
89C22×C71—41no—
90C22×C71C2×#10011yescharacteristic
91C2×D71—82no—
92C2×D71—82no—
93C2×D71—82no—
94C4×C71—21no—
95C4×C71—41no—
96C4×C71—41no—
97C4×C71#41611yescharacteristic
98C4×C71#41711yescharacteristic
99Dic71—82no—
100#11989—711nomaximal, Sylow
101C2×C71—21no—
102C2×C71—41no—
103C2×C71#155111yescharacteristic
104D71—162no—
105#3176—711no—
106#2166D7111yescharacteristic, complemented
107#2153—711no—
108#1366—712no—
109#1357—712no—
110C71#1198911yescharacteristic, complemented, Sylow
111#540—712no—
112C2×M32—1421no—
113#499—711no—
114#493—711no—
115#490—712no—
116#469—1421no—
117#466—1421no—
118#442—1421no—
119C4×D8—1421no—
120#405C2×D7111yescharacteristic
121#364—712no—
122#364—1421no—
123#364C2×D7112yes—
124#349—21no—
125#347—21no—
126C2×D8—1421no—
127C2×D8—1421no—
128C2×M16C22⋊D7111yescharacteristic
129C22×C8D28411yescharacteristic
130#127—712no—
131#125—712no—
132#123—2842no—
133#121—2842no—
134#121—1422no—
135C4×D4—1422no—
136#117C4×D7111yescharacteristic
137M32—22no—
138M32—2841no—
139M32—1422no—
140C2×C16—2841no—
141#107—1421no—
142#106—1421no—
143#102—1422no—
144#102—1422no—
145#97C4×D7111yescharacteristic
146#97C22⋊D7111yescharacteristic
147C4×C8D28411yescharacteristic
148C4×C8C22×D7111yescharacteristic
149C2×D4—2842no—
150C2×D4—2842no—
151C2×D4—1422no—
152C22×C4—2842no—
153C22×C4(C2×D71)⋊C411yescharacteristic
154D8—2841no—
155D8—2842no—
156D8—2841no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8(C2×D71)⋊C411yescharacteristic
161C2×C8(C2×D71)⋊C411yescharacteristic
162C2×C8C2×C22⋊D7111yescharacteristic
163C2×C8C2×D28411yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—2842no—
166C4⋊C4—1422no—
167C22⋊C4—21no—
168C22⋊C4—2842no—
169C22⋊C4—2842no—
170C42(C2×D71)⋊C411yescharacteristic
171C42C2×C4×D7111yescharacteristic
172C16—42no—
173C16—2842no—
174C23—2842no—
175C23#1455604411yescharacteristic
176D4—2842no—
177D4—2842no—
178D4—11362no—
179D4—11362no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—2842no—
185C2×C4—11362no—
186C2×C4#1455604411yescharacteristic
187C2×C4C2×(C2×D71)⋊C411yescharacteristic
188C8—21no—
189C8—21no—
190C8#1455614812yes—
191C8#1455615611yescharacteristic
192C8#1455616111yescharacteristic
193C22—21no—
194C22—41no—
195C22—5682no—
196C22—5682no—
197C22—5682no—
198C22C2×#1455604411yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—5682no—
203C4#1798599311yescharacteristic
204C4#1798600111yescharacteristic
205C2—21no—
206C2—41no—
207C2—11362no—
208C2#2258703611yescharacteristic
subgroupG/H
centerC2#22587036
commutatorC42×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#2166C2
socleC2×C71#1551
prime radical#22598990C2
cosocleC4×C8×C71C23
hypercenter#2166D71
hypocenterC71#11989
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #22590106C2semidirect
3. #22586400C2semidirect
4. #22586376C2semidirect
5. C71×#2166C2semidirect
106. #2166D71semidirect
110. C71#11989semidirect

Maximal subgroups

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

subgroupindex
#225989902
#225901062
#225864002
#225863762
C71×#21662
#1198971