Finite Group Databasebeta

← #31911857

Subgroups — #31911857

Order 18176.

196 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 57 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#22590583C211yesmaximal, characteristic, complemented
2#22590416C211yesmaximal, characteristic
3#22590182C211yesmaximal, characteristic, complemented
4#22586407C211yesmaximal, characteristic, complemented
5#22586401C211yesmaximal, characteristic, complemented
6#22586376C211yesmaximal, characteristic, complemented
7C71×#2231C211yesmaximal, characteristic, complemented
8#17986434—21no—
9#17986416—21no—
10#17986376C2211yescharacteristic
11#17985820C2211yescharacteristic
12#17985820C2211yescharacteristic
13#17985811—21no—
14#17985799—21no—
15#17985799—21no—
16#17985783—21no—
17#17985775C2211yescharacteristic
18#499×C71C2211yescharacteristic
19#443×C71C2211yescharacteristic
20#365×C71—21no—
21#364×C71C2211yescharacteristic
22#360×C71—21no—
23C2×D568—21no—
24C2×D568—21no—
25#14556125—21no—
26#14556071—41no—
27#14556069—41no—
28#14556068—21no—
29#14556068—21no—
30#14556068—41no—
31C2×C71⋊C16—41no—
32C2×C71⋊C16D411yescharacteristic
33C2×C71⋊C16D411yescharacteristic
34#135×C71D411yescharacteristic
35#131×C71—21no—
36#125×C71—21no—
37M32×C71—21no—
38#108×C71D411yescharacteristic
39#107×C71—21no—
40#106×C71—21no—
41#104×C71—21no—
42C4×C8×C71C2311yescharacteristic
43C2×D284—41no—
44C2×D284—81no—
45D568—41no—
46D568—41no—
47D568—41no—
48D568—41no—
49C71⋊C16—21no—
50C71⋊C16—21no—
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
62C4⋊C4×C71—41no—
63C4⋊C4×C71—81no—
64C42×C71#4111yescharacteristic
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—81no—
73C2×C4×C71#12111yescharacteristic
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—161no—
89C4×C71#46311yescharacteristic
90C4×C71#46511yescharacteristic
91#12465—711nomaximal, Sylow
92C2×C71—21no—
93C2×C71—81no—
94C2×C71#167711yescharacteristic
95D71—321no—
96#2244—711no—
97#2231D7111yescharacteristic, complemented
98#2213—711no—
99#2171—711no—
100#1369—711no—
101#1366—711no—
102#1357—711no—
103C71#1246511yescharacteristic, complemented, Sylow
104#509—1421no—
105#504—1421no—
106#499C2×D7111yescharacteristic
107#493—711no—
108#443C2×D7111yescharacteristic
109#365—21no—
110#364—711no—
111#364—711no—
112#364C2×D7111yescharacteristic
113#361—1421no—
114#360—21no—
115#357—1421no—
116#357—1421no—
117#350—1421no—
118#346—711no—
119#135C22⋊D7111yescharacteristic
120C2×D8—1421no—
121C2×D8—1421no—
122#131—21no—
123#127—1421no—
124#125—21no—
125SD32—2841no—
126D16—2841no—
127M32—21no—
128M32—2841no—
129M32—1421no—
130M32—1421no—
131C2×C16—2841no—
132C2×C16—711no—
133C2×C16—711no—
134#108C22⋊D7111yescharacteristic
135#107—21no—
136#106—21no—
137#104—21no—
138C4×C8C22×D7111yescharacteristic
139#41—21no—
140C2×D4—2841no—
141C2×D4—5681no—
142Q16#1176585211yescharacteristic
143SD16—21no—
144D8—2841no—
145D8—2841no—
146D8—2841no—
147D8—2841no—
148D8#1176585211yescharacteristic
149M16—21no—
150M16—41no—
151C2×C8—41no—
152C2×C8D4×D7111yescharacteristic
153C2×C8C2×C22⋊D7111yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1176586411yescharacteristic
157C16—41no—
158C16—2841no—
159C16—2841no—
160C16—1421no—
161C16—1421no—
162C23—5681no—
163Q8—21no—
164D4—21no—
165D4—41no—
166D4—5681no—
167D4—5681no—
168D4—11362no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—81no—
172C2×C4#1455610011yescharacteristic
173C8—41no—
174C8C2×#1176585211yescharacteristic
175C8#1455614811yescharacteristic
176C8#1455614911yescharacteristic
177C8#1455615511yescharacteristic
178C22—41no—
179C22—11362no—
180C22—11361no—
181C22#1798626711yescharacteristic
182C4—41no—
183C4—41no—
184C4—161no—
185C4#1798621611yescharacteristic
186C4#1798623011yescharacteristic
187C2—21no—
188C2—81no—
189C2—22721no—
190C2#2258775111yescharacteristic
subgroupG/H
centerC2#22587751
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2231C2
socleC2×C71#1677
prime radical#22590416C2
hypercenter#2231D71
hypocenterC71#12465
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2231

Complements

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

NH ≅ G/Ntype
1. #22590583C2semidirect
3. #22590182C2semidirect
4. #22586407C2semidirect
5. #22586401C2semidirect
6. #22586376C2semidirect
7. C71×#2231C2semidirect
97. #2231D71semidirect
103. C71#12465semidirect

Maximal subgroups

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

subgroupindex
#225905832
#225904162
#225901822
#225864072
#225864012
#225863762
C71×#22312
#1246571