Finite Group Databasebeta

← #27173083

Subgroups — #27173083

Order 12928.

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#20265617C212yesmaximal, complemented
2#20265435C211yesmaximal, characteristic, complemented
3#20264987C212yesmaximal, complemented
4#20264959C211yesmaximal, characteristic, complemented
5#575×C101C211yesmaximal, characteristic
6C2×Dic404C2211yescharacteristic, complemented
7#15692234—21no—
8#15692201—22no—
9#15692199—22no—
10#15692195C2212yescomplemented
11#15692187C2211yescharacteristic, complemented
12#15692180—21no—
13C2×#41×C101—21no—
14#137×C101—22no—
15AGL(1,ℤ/8)×C101—22no—
16#135×C101—21no—
17#135×C101C2212yes—
18C2×M16×C101C2211yescharacteristic
19C2×Dic202—21no—
20Dic404—22no—
21C4⋊Dic101—21no—
22#13108053—41no—
23C22⋊Dic101—41no—
24#13108049—22no—
25#41×C101—22no—
26#41×C101—22no—
27#41×C101—41no—
28#41×C101—41no—
29C2×Q8×C101—21no—
30C2×D4×C101—21no—
31C2×D4×C101—41no—
32C22×C4×C101—41no—
33C22×C4×C101D411yescharacteristic
34Q16×C101—21no—
35Q16×C101D412yes—
36SD16×C101—22no—
37SD16×C101—22no—
38D8×C101—21no—
39D8×C101D412yes—
40M16×C101—22no—
41C2×C8×C101D411yescharacteristic
42C2×C8×C101C2311yescharacteristic
43C23×C101—41no—
44C23×C101D811yescharacteristic
45Q8×C101—22no—
46Q8×C101—41no—
47D4×C101—22no—
48D4×C101—41no—
49D4×C101—42no—
50D4×C101—42no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—42no—
54C2×C4×C101—41no—
55C2×C4×C101—42no—
56C2×C4×C101D811yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—41no—
59C2×Dic101—41no—
60Dic202—41no—
61Dic202—41no—
62C8×C101—21no—
63C8×C101C2×D412yes—
64C22×C101—21no—
65C22×C101—41no—
66C22×C101—42no—
67C22×C101—82no—
68C22×C101C2×D811yescharacteristic
69C4×C101—21no—
70C4×C101—42no—
71C4×C101#12011yescharacteristic
72C4×C101AGL(1,ℤ/8)11yescharacteristic
73Dic101—81no—
74Dic101—81no—
75C2×C101—21no—
76C2×C101—41no—
77C2×C101—82no—
78C2×C101#44711yescharacteristic
79#2191—1011nomaximal, Sylow
80C101#219111yescharacteristic, complemented, Sylow
81#575D10111yescharacteristic
82#510—1012no—
83#470—1011no—
84#360—1012no—
85#349—1011no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D10112yes—
91C2×Q16—1011no—
92C2×M16C2×D10111yescharacteristic
93C22⋊Q8—2021no—
94Q32—2022no—
95SD32—2022no—
96M32—1012no—
97#106—1011no—
98#103—2021no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—2021no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D10111yescharacteristic
109Q16—21no—
110Q16—2022no—
111Q16C22⋊D10112yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D10112yes—
116M16—22no—
117C2×C8C22⋊D10111yescharacteristic
118C2×C8C22×D10111yescharacteristic
119C4⋊C4—4041no—
120C4⋊C4—2021no—
121C22⋊C4—4041no—
122C16—2022no—
123C23—41no—
124C23#1310805911yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—4041no—
128Q8—4041no—
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—4041no—
139C2×C4—4041no—
140C2×C4#1310805911yescharacteristic
141C2×C4C2×C22⋊D10111yescharacteristic
142C8—21no—
143C8C2×C22⋊D10112yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1310805911yescharacteristic
149C4—21no—
150C4—42no—
151C4—8081no—
152C4—8081no—
153C4#1569222411yescharacteristic
154C4#1569228111yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2026531111yescharacteristic
subgroupG/H
centerC2#20265311
commutator, cosocleC2×C8×C101C23
FrattiniC2×C8C22×D101
Fitting, prime radical#575×C101C2
socleC2×C101#447
hypercenter#575D101
hypocenterC101#2191
LCSG ▷ C2×C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20265617C2semidirect
2. #20265435C2semidirect
3. #20264987C2semidirect
4. #20264959C2semidirect
6. C2×Dic404C22semidirect
10. #15692195C22semidirect
11. #15692187C22semidirect
80. C101#2191semidirect

Maximal subgroups

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

subgroupindex
#202656172
#202654352
#202649872
#202649592
#575×C1012
#2191101