Finite Group Databasebeta

← #27143687

Subgroups — #27143687

Order 12416.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261175C212yesmaximal, complemented
2#20260993C211yesmaximal, characteristic, complemented
3#20260545C212yesmaximal, complemented
4#20260517C211yesmaximal, characteristic, complemented
5#575×C97C211yesmaximal, characteristic
6C2×Dic388C2211yescharacteristic, complemented
7#15690789—21no—
8#15690756—22no—
9#15690754—22no—
10#15690750C2212yescomplemented
11#15690742C2211yescharacteristic, complemented
12#15690735—21no—
13C2×#41×C97—21no—
14#137×C97—22no—
15AGL(1,ℤ/8)×C97—22no—
16#135×C97—21no—
17#135×C97C2212yes—
18C2×M16×C97C2211yescharacteristic
19C2×Dic194—21no—
20Dic388—22no—
21C4⋊Dic97—21no—
22#13095490—41no—
23C22⋊Dic97—41no—
24#13095486—22no—
25#41×C97—22no—
26#41×C97—22no—
27#41×C97—41no—
28#41×C97—41no—
29C2×Q8×C97—21no—
30C2×D4×C97—21no—
31C2×D4×C97—41no—
32C22×C4×C97—41no—
33C22×C4×C97D411yescharacteristic
34Q16×C97—21no—
35Q16×C97D412yes—
36SD16×C97—22no—
37SD16×C97—22no—
38D8×C97—21no—
39D8×C97D412yes—
40M16×C97—22no—
41C2×C8×C97D411yescharacteristic
42C2×C8×C97C2311yescharacteristic
43C23×C97—41no—
44C23×C97D811yescharacteristic
45Q8×C97—22no—
46Q8×C97—41no—
47D4×C97—22no—
48D4×C97—41no—
49D4×C97—42no—
50D4×C97—42no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—42no—
54C2×C4×C97—41no—
55C2×C4×C97—42no—
56C2×C4×C97D811yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—41no—
59C2×Dic97—41no—
60Dic194—41no—
61Dic194—41no—
62C8×C97—21no—
63C8×C97C2×D412yes—
64C22×C97—21no—
65C22×C97—41no—
66C22×C97—42no—
67C22×C97—82no—
68C22×C97C2×D811yescharacteristic
69C4×C97—21no—
70C4×C97—42no—
71C4×C97#12011yescharacteristic
72C4×C97AGL(1,ℤ/8)11yescharacteristic
73Dic97—81no—
74Dic97—81no—
75C2×C97—21no—
76C2×C97—41no—
77C2×C97—82no—
78C2×C97#44711yescharacteristic
79#2191—971nomaximal, Sylow
80C97#219111yescharacteristic, complemented, Sylow
81#575D9711yescharacteristic
82#510—972no—
83#470—971no—
84#360—972no—
85#349—971no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D9712yes—
91C2×Q16—971no—
92C2×M16C2×D9711yescharacteristic
93C22⋊Q8—1941no—
94Q32—1942no—
95SD32—1942no—
96M32—972no—
97#106—971no—
98#103—1941no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—1941no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D9711yescharacteristic
109Q16—21no—
110Q16—1942no—
111Q16C22⋊D9712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D9712yes—
116M16—22no—
117C2×C8C22⋊D9711yescharacteristic
118C2×C8C22×D9711yescharacteristic
119C4⋊C4—3881no—
120C4⋊C4—1941no—
121C22⋊C4—3881no—
122C16—1942no—
123C23—41no—
124C23#1309549611yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—3881no—
128Q8—3881no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4#1309549611yescharacteristic
141C2×C4C2×C22⋊D9711yescharacteristic
142C8—21no—
143C8C2×C22⋊D9712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1309549611yescharacteristic
149C4—21no—
150C4—42no—
151C4—7761no—
152C4—7761no—
153C4#1569077911yescharacteristic
154C4#1569083611yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2026086911yescharacteristic
subgroupG/H
centerC2#20260869
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting, prime radical#575×C97C2
socleC2×C97#447
hypercenter#575D97
hypocenterC97#2191
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20261175C2semidirect
2. #20260993C2semidirect
3. #20260545C2semidirect
4. #20260517C2semidirect
6. C2×Dic388C22semidirect
10. #15690750C22semidirect
11. #15690742C22semidirect
80. C97#2191semidirect

Maximal subgroups

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

subgroupindex
#202611752
#202609932
#202605452
#202605172
#575×C972
#219197