Finite Group Databasebeta

← #31911855

Subgroups — #31911855

Order 18176.

196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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#22590582C211yesmaximal, characteristic, complemented
2#22590516C211yesmaximal, characteristic, complemented
3#22590416C211yesmaximal, characteristic
4#22586408C211yesmaximal, characteristic, complemented
5#22586402C211yesmaximal, characteristic, complemented
6#22586376C211yesmaximal, characteristic, complemented
7C71×#2171C211yesmaximal, characteristic, complemented
8#17986434—21no—
9#17986401C2211yescharacteristic
10#17986376C2211yescharacteristic
11#17985823—21no—
12#17985820C2211yescharacteristic
13#17985820C2211yescharacteristic
14#17985811—21no—
15#17985809—21no—
16#17985801—21no—
17#17985801—21no—
18#504×C71—21no—
19#443×C71C2211yescharacteristic
20#364×C71C2211yescharacteristic
21#350×C71—21no—
22#346×C71C2211yescharacteristic
23C2×D568—21no—
24C2×D568—21no—
25#14556125—21no—
26#14556120—21no—
27#14556071—41no—
28#14556069—41no—
29#14556068—21no—
30#14556068—21no—
31#14556062—21no—
32#14556060—21no—
33#135×C71D411yescharacteristic
34#131×C71—21no—
35M32×C71—21no—
36M32×C71—41no—
37C2×C16×C71—41no—
38C2×C16×C71D411yescharacteristic
39C2×C16×C71D411yescharacteristic
40#108×C71D411yescharacteristic
41#104×C71—21no—
42C4×C8×C71C2311yescharacteristic
43C2×D284—41no—
44C2×D284—81no—
45D568—41no—
46D568—41no—
47D568—41no—
48D568—41no—
49C4⋊Dic71—41no—
50Dic71⋊C4—81no—
51C71⋊C16—41no—
52C71⋊C16—41no—
53#41×C71—21no—
54Q16×C71SD1611yescharacteristic
55SD16×C71—21no—
56D8×C71SD1611yescharacteristic
57M16×C71—21no—
58M16×C71—41no—
59C2×C8×C71—41no—
60C2×C8×C71C2×D411yescharacteristic
61C2×C8×C71C2×D411yescharacteristic
62C42×C71#4111yescharacteristic
63C16×C71—21no—
64C16×C71—21no—
65C16×C71—41no—
66C22×D71—81no—
67Q8×C71—21no—
68D4×C71—21no—
69D4×C71—41no—
70C2×C4×C71—21no—
71C2×C4×C71—41no—
72C2×C4×C71#12111yescharacteristic
73C2×Dic71—81no—
74D284—81no—
75D284—81no—
76D284—162no—
77C8×C71—41no—
78C8×C71C2×SD1611yescharacteristic
79C8×C71#13511yescharacteristic
80C8×C71#13511yescharacteristic
81C8×C71AGL(1,ℤ/8)11yescharacteristic
82C22×C71—41no—
83C22×C71#47211yescharacteristic
84C2×D71—162no—
85C2×D71—161no—
86C4×C71—41no—
87C4×C71—41no—
88C4×C71#46311yescharacteristic
89C4×C71#46511yescharacteristic
90Dic71—161no—
91#12465—711nomaximal, Sylow
92C2×C71—21no—
93C2×C71—81no—
94C2×C71#167711yescharacteristic
95D71—321no—
96#2244—711no—
97#2231—711no—
98#2213—711no—
99#2171D7111yescharacteristic, complemented
100#1369—711no—
101#1366—711no—
102#1357—711no—
103C71#1246511yescharacteristic, complemented, Sylow
104#509—1421no—
105#504—21no—
106#499—711no—
107#493—711no—
108#443C2×D7111yescharacteristic
109#365—1421no—
110#364—711no—
111#364—711no—
112#364C2×D7111yescharacteristic
113#361—1421no—
114#360—1421no—
115#357—1421no—
116#357—1421no—
117#350—21no—
118#346C2×D7111yescharacteristic
119#135D28411yescharacteristic
120C2×D8—1421no—
121C2×D8—1421no—
122#131—21no—
123#127—1421no—
124#125—1421no—
125SD32—2841no—
126D16—2841no—
127M32—21no—
128M32—41no—
129M32—1421no—
130M32—1421no—
131C2×C16—41no—
132C2×C16C22⋊D7111yescharacteristic
133C2×C16C22⋊D7111yescharacteristic
134#108D28411yescharacteristic
135#107—1421no—
136#106—1421no—
137#104—21no—
138C4×C8C22×D7111yescharacteristic
139#41—21no—
140C2×D4—2841no—
141C2×D4—5681no—
142Q16#1176585411yescharacteristic
143SD16—21no—
144D8—2841no—
145D8—2841no—
146D8—2841no—
147D8—2841no—
148D8#1176585411yescharacteristic
149M16—21no—
150M16—41no—
151C2×C8—41no—
152C2×C8C2×C22⋊D7111yescharacteristic
153C2×C8C2×D28411yescharacteristic
154C4⋊C4—2841no—
155C4⋊C4—5681no—
156C42#1176586511yescharacteristic
157C16—21no—
158C16—21no—
159C16—41no—
160C16—2841no—
161C16—2841no—
162C23—5681no—
163Q8—21no—
164D4—21no—
165D4—41no—
166D4—5681no—
167D4—5681no—
168D4—11362no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—5681no—
172C2×C4#1455610211yescharacteristic
173C8—41no—
174C8C2×#1176585411yescharacteristic
175C8#1455614811yescharacteristic
176C8#1455615011yescharacteristic
177C8#1455615411yescharacteristic
178C22—41no—
179C22—11362no—
180C22—11361no—
181C22#1798626911yescharacteristic
182C4—41no—
183C4—41no—
184C4—11361no—
185C4#1798621411yescharacteristic
186C4#1798622811yescharacteristic
187C2—21no—
188C2—81no—
189C2—22721no—
190C2#2258774911yescharacteristic
subgroupG/H
centerC2#22587749
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2171C2
socleC2×C71#1677
prime radical#22590416C2
hypercenter#2171D71
hypocenterC71#12465
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2171

Complements

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

NH ≅ G/Ntype
1. #22590582C2semidirect
2. #22590516C2semidirect
4. #22586408C2semidirect
5. #22586402C2semidirect
6. #22586376C2semidirect
7. C71×#2171C2semidirect
99. #2171D71semidirect
103. C71#12465semidirect

Maximal subgroups

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

subgroupindex
#225905822
#225905162
#225904162
#225864082
#225864022
#225863762
C71×#21712
#1246571