Finite Group Databasebeta

← #27170466

Subgroups — #27170466

Order 12928.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265587C211yesmaximal, characteristic, complemented
2#20265547C211yesmaximal, characteristic, complemented
3#20265536C211yesmaximal, characteristic
4#20264919C211yesmaximal, characteristic, complemented
5#20264919C211yesmaximal, characteristic, complemented
6#20264917C211yesmaximal, characteristic
7#563×C101C211yesmaximal, characteristic
8#15692255C2211yescharacteristic
9#15692188—21no—
10#15692186—21no—
11#15692179—21no—
12#15692178—21no—
13#15692178—21no—
14#15692176—21no—
15#15692166C2211yescharacteristic
16#15692166C2211yescharacteristic
17#15692166C2211yescharacteristic
18C4⋊Q8×C101—21no—
19#126×C101—21no—
20#126×C101—21no—
21#125×C101C2211yescharacteristic
22#125×C101C2211yescharacteristic
23#124×C101C2211yescharacteristic
24#123×C101—21no—
25C22⋊Q8×C101—21no—
26C22⋊Q8×C101—21no—
27C2×Dic202—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C2×#11717579—21no—
31C4⋊Dic101—21no—
32C4⋊Dic101—21no—
33C2×Q8×C101—21no—
34C2×Q8×C101D411yescharacteristic
35C2×D4×C101D411yescharacteristic
36C22×C4×C101—21no—
37C4⋊C4×C101—21no—
38C4⋊C4×C101—21no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101—21no—
41C4⋊C4×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101D411yescharacteristic
44C4⋊C4×C101D411yescharacteristic
45C4⋊C4×C101D411yescharacteristic
46C4⋊C4×C101D411yescharacteristic
47C22⋊C4×C101—21no—
48C22⋊C4×C101—21no—
49C22⋊C4×C101—21no—
50C22⋊C4×C101—21no—
51C22⋊C4×C101—21no—
52C42×C101—21no—
53C42×C101C2311yescharacteristic
54C23×C101—21no—
55Q8×C101—41no—
56Q8×C101—81no—
57D4×C101—41no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—21no—
63C2×C4×C101—81no—
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C2×Dic101—41no—
68Dic202—42no—
69#11717579—22no—
70#11717579—41no—
71#11717579—41no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101#12011yescharacteristic
76C4×C101—21no—
77C4×C101—21no—
78C4×C101—21no—
79C4×C101—41no—
80C4×C101—41no—
81C4×C101—41no—
82C4×C101—41no—
83C4×C101—41no—
84Dic101—81no—
85C2×C101—81no—
86C2×C101#45411yescharacteristic
87C2×C101#45511yescharacteristic
88C2×C101#45611yescharacteristic
89#1664—1011nomaximal, Sylow
90C101#166411yescharacteristic, complemented, Sylow
91#563D10111yescharacteristic
92#501—1011no—
93#491—1011no—
94#489—1011no—
95#330—1011no—
96#330—1011no—
97#329—1011no—
98C4⋊Q8—21no—
99C4⋊Q8—1011no—
100#126—21no—
101#126—21no—
102#125C2×D10111yescharacteristic
103#125C2×D10111yescharacteristic
104#124C2×D10111yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—2021no—
109#106—2021no—
110#103—2021no—
111#103—2021no—
112#103—2021no—
113#102—2021no—
114#97—1011no—
115#97—1011no—
116#97—1011no—
117C2×Q8—21no—
118C2×Q8—2021no—
119C2×Q8C22⋊D10111yescharacteristic
120C2×D4C22⋊D10111yescharacteristic
121C22×C4—21no—
122C2×C8—2021no—
123C2×C8—2021no—
124C2×C8—2021no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2021no—
132C4⋊C4—2021no—
133C4⋊C4C22⋊D10111yescharacteristic
134C4⋊C4C22⋊D10111yescharacteristic
135C4⋊C4C22⋊D10111yescharacteristic
136C4⋊C4C22⋊D10111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D10111yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—4042no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—4041no—
156C2×C4C2×C22⋊D10111yescharacteristic
157C2×C4C2×C22⋊D10111yescharacteristic
158C2×C4C2×C22⋊D10111yescharacteristic
159C8—4041no—
160C8—4041no—
161C8—2022no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1569222411yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—8081no—
175C2—81no—
176C2#2026535011yescharacteristic
177C2#2026535511yescharacteristic
178C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#563×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#563D101
hypocenterC101#1664
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20265587C2semidirect
2. #20265547C2semidirect
4. #20264919C2semidirect
5. #20264919C2semidirect
90. C101#1664semidirect

Maximal subgroups

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

subgroupindex
#202655872
#202655472
#202655362
#202649192
#202649172
#563×C1012
#1664101