Finite Group Databasebeta

← #27173153

Subgroups — #27173153

Order 12928.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265359C212yesmaximal, complemented
2#20265349C211yesmaximal, characteristic
3#20264976C212yesmaximal, complemented
4#20264974C211yesmaximal, characteristic, complemented
5#563×C101C211yesmaximal, characteristic, complemented
6#15692303C2211yescharacteristic
7#15692286—22no—
8#15692281—21no—
9#15692183—21no—
10#15692183—22no—
11#15692174C2212yes—
12#15692174C2211yescharacteristic
13C4⋊Q8×C101C2212yes—
14#126×C101—22no—
15#125×C101—21no—
16#124×C101C2211yescharacteristic
17#123×C101—21no—
18C22⋊Q8×C101—22no—
19#13108075—41no—
20#13108074—21no—
21#13108074—22no—
22Q8×D101—41no—
23#13108064—42no—
24#13108062—41no—
25#13108061—42no—
26#13108059—41no—
27#13108058—21no—
28#13108058—22no—
29C2×Q8×C101D412yes—
30C2×Q8×C101C2311yescharacteristic
31C2×D4×C101D411yescharacteristic
32C22×C4×C101—21no—
33C4⋊C4×C101—21no—
34C4⋊C4×C101—22no—
35C4⋊C4×C101—21no—
36C4⋊C4×C101—22no—
37C4⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—21no—
40C22⋊C4×C101—22no—
41C42×C101D411yescharacteristic
42C42×C101D412yes—
43C23×C101—21no—
44Q8×C101—41no—
45Q8×C101—42no—
46D4×C101—41no—
47C2×C4×C101—21no—
48C2×C4×C101—22no—
49C2×C4×C101—22no—
50C2×C4×C101—81no—
51C2×C4×C101C2×D412yes—
52C2×C4×C101C2×D411yescharacteristic
53C22⋊D101—41no—
54D404—21no—
55D404—22no—
56C4×D101—41no—
57C4×D101—42no—
58Dic202—21no—
59Dic202—22no—
60#11717579—41no—
61#11717579—42no—
62C22×C101—42no—
63C22×C101—41no—
64C22×C101#12011yescharacteristic
65C2×D101—41no—
66C4×C101—21no—
67C4×C101—22no—
68C4×C101—41no—
69C4×C101—42no—
70C4×C101—42no—
71Dic101—41no—
72C2×C101—21no—
73C2×C101—81no—
74C2×C101B(4,𝔽2)11yescharacteristic
75D101—81no—
76#2203—1011nomaximal, Sylow
77C101#220311yescharacteristic, complemented, Sylow
78#563D10111yescharacteristic, complemented
79#456—1012no—
80#454—1011no—
81#356—1012no—
82#355—1011no—
832−1+4—1011no—
84#137—2022no—
85AGL(1,ℤ/8)—2021no—
86C4⋊Q8C2×D10112yes—
87#126—22no—
88#125—21no—
89#124C2×D10111yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—2021no—
93#104—2022no—
94#101—1012no—
95#101—1011no—
96#41—4041no—
97#41—2021no—
98#41—2022no—
99C2×Q8—4041no—
100C2×Q8C22⋊D10112yes—
101C2×Q8C22×D10111yescharacteristic
102C2×D4C22⋊D10111yescharacteristic
103C22×C4—21no—
104Q16—4042no—
105SD16—4041no—
106SD16—4042no—
107D8—4041no—
108M16—2021no—
109M16—2022no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D10111yescharacteristic
119C42C22⋊D10112yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—2021no—
124Q8—2022no—
125D4—41no—
126D4—4041no—
127D4—2021no—
128D4—2022no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—4041no—
134C2×C4—4042no—
135C2×C4C2×C22⋊D10112yes—
136C2×C4C2×C22⋊D10111yescharacteristic
137C8—4041no—
138C8—4042no—
139C22—42no—
140C22—41no—
141C22—4041no—
142C22#1569222411yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—4041no—
149C2—21no—
150C2—81no—
151C2—8081no—
152C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×Q8×C101C23
FrattiniC2×Q8C22×D101
Fitting#563×C101C2
socleC2×C101B(4,𝔽2)
prime radical#20265349C2
hypercenter#563D101
hypocenterC101#2203
LCSG ▷ C2×Q8×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20265359C2semidirect
3. #20264976C2semidirect
4. #20264974C2semidirect
5. #563×C101C2semidirect
77. C101#2203semidirect
78. #563D101semidirect

Maximal subgroups

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

subgroupindex
#202653592
#202653492
#202649762
#202649742
#563×C1012
#2203101