Finite Group Databasebeta

← #27143711

Subgroups — #27143711

Order 12416.

206 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 47 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#20261549C211yesmaximal, characteristic
2C2×#15690756C211yesmaximal, characteristic, complemented
3#20261007C211yesmaximal, characteristic, complemented
4#20260552C211yesmaximal, characteristic, complemented
5#20260542C211yesmaximal, characteristic, complemented
6#20260520C211yesmaximal, characteristic, complemented
7#510×C97C211yesmaximal, characteristic, complemented
8#15690857—21no—
9#15690841—21no—
10#15690840—21no—
11#15690839—22no—
12#15690829—21no—
13C2×Dic388C2211yescharacteristic
14Q16×D97—21no—
15#15690816C2211yescharacteristic
16#15690756—22no—
17C2×#13095486C2211yescharacteristic
18#15690746C2211yescharacteristic
19#15690729—21no—
20#135×C97C2211yescharacteristic
21C2×Q16×C97C2211yescharacteristic
22Q32×C97—21no—
23SD32×C97—21no—
24M32×C97C2211yescharacteristic
25#13095511—41no—
26#13095511D411yescharacteristic
27#13095510—22no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×Dic194—41no—
31Q8×D97—22no—
32Dic388—21no—
33Dic388D412yes—
34#13095501—22no—
35#13095501—41no—
36#13095499—22no—
37#13095498—41no—
38#13095495—21no—
39#13095495D411yescharacteristic
40#13095494—21no—
41C8×D97—21no—
42#13095486—21no—
43#41×C97—21no—
44C2×Q8×C97—21no—
45Q16×C97—21no—
46Q16×C97—21no—
47Q16×C97D411yescharacteristic
48SD16×C97—21no—
49D8×C97D411yescharacteristic
50C2×C8×C97C2311yescharacteristic
51C16×C97—21no—
52Q8×C97—21no—
53Q8×C97—41no—
54Q8×C97—41no—
55D4×C97—21no—
56D4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C22⋊D97—21no—
61C22⋊D97—42no—
62C2×Dic97—42no—
63C2×Dic97—41no—
64D388D811yescharacteristic
65C4×D97—21no—
66C4×D97—42no—
67Dic194—22no—
68Dic194—42no—
69Dic194—41no—
70Dic194D811yescharacteristic
71C8×C97C2×D411yescharacteristic
72C8×C97C2×D411yescharacteristic
73#11715895—21no—
74#11715895—41no—
75C22×C97—41no—
76C22×C97AGL(1,ℤ/8)11yescharacteristic
77C2×D97—21no—
78C4×C97—41no—
79C4×C97—81no—
80C4×C97#12011yescharacteristic
81C4×C97C2×D811yescharacteristic
82Dic97—21no—
83Dic97—42no—
84C2×C97—21no—
85C2×C97—81no—
86C2×C97#44711yescharacteristic
87D97—41no—
88#2195—971nomaximal, Sylow
89C97#219511yescharacteristic, complemented, Sylow
90#578—971no—
91#510D9711yescharacteristic, complemented
92C2×Q32—971no—
93#473—971no—
94#362—971no—
95#359—971no—
96#350—971no—
972−1+4—1941no—
98#137—1941no—
99#137—1942no—
100#137—1941no—
101#135—1941no—
102#135C2×D9711yescharacteristic
103C2×Q16—971no—
104C2×Q16—1941no—
105C2×Q16C2×D9711yescharacteristic
106#131—971no—
107Q32—21no—
108Q32—1942no—
109SD32—21no—
110M32C2×D9711yescharacteristic
111C2×C16—971no—
112#108—971no—
113#101—1941no—
114#41—21no—
115#41—971no—
116#41—3881no—
117#41—3882no—
118#41—1942no—
119C2×Q8—21no—
120C2×Q8—3881no—
121C2×Q8—1942no—
122C2×Q8—1941no—
123Q16—21no—
124Q16—21no—
125Q16—972no—
126Q16—3881no—
127Q16—1942no—
128Q16—1941no—
129Q16C22⋊D9711yescharacteristic
130SD16—21no—
131SD16—3881no—
132SD16—1942no—
133D8C22⋊D9711yescharacteristic
134M16—971no—
135M16—1941no—
136M16—1941no—
137C2×C8—1941no—
138C2×C8C22×D9711yescharacteristic
139C16—21no—
140C16—1941no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—971no—
145Q8—3882no—
146Q8—3881no—
147Q8—1942no—
148D4—21no—
149D4—41no—
150D4—971no—
151D4—3882no—
152D4—1941no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—3882no—
156C2×C4—3881no—
157C2×C4—3882no—
158C2×C4—1941no—
159C2×C4D4×D9711yescharacteristic
160C8—3881no—
161C8—1941no—
162C8D4×D9711yescharacteristic
163C8C2×C22⋊D9711yescharacteristic
164C22—41no—
165C22—1941no—
166C22#1569083411yescharacteristic
167C4—41no—
168C4—81no—
169C4—3882no—
170C4—1941no—
171C4#1569077811yescharacteristic
172C4D8×D9711yescharacteristic
173C2—21no—
174C2—81no—
175C2—3881no—
176C2#2026086711yescharacteristic
subgroupG/H
centerC2#20260867
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#510×C97C2
socleC2×C97#447
prime radical#20261549C2
hypercenter#510D97
hypocenterC97#2195
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#15690756C2semidirect
3. #20261007C2semidirect
4. #20260552C2semidirect
5. #20260542C2semidirect
6. #20260520C2semidirect
7. #510×C97C2semidirect
89. C97#2195semidirect
91. #510D97semidirect

Maximal subgroups

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

subgroupindex
#202615492
C2×#156907562
#202610072
#202605522
#202605422
#202605202
#510×C972
#219597