Finite Group Databasebeta

← #27173145

Subgroups — #27173145

Order 12928.

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

normal subgroups only

This group has 226 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#20265366C212yesmaximal
2#20265354C211yesmaximal, characteristic
3#20264972C211yesmaximal, characteristic, complemented
4#20264967C212yesmaximal, complemented
5#563×C101C211yesmaximal, characteristic, complemented
6#15692300C2211yescharacteristic
7#15692286—21no—
8#15692238—22no—
9#15692183—21no—
10#15692172C2211yescharacteristic
11#15692170C2212yes—
12#15692169—22no—
13C4⋊Q8×C101—21no—
14#126×C101—22no—
15#125×C101—21no—
16#124×C101C2211yescharacteristic
17#123×C101C2212yes—
18C22⋊Q8×C101—22no—
19#13108074—21no—
20#13108073—41no—
21C2×C22⋊D101—22no—
22D4×D101—41no—
23#13108064—41no—
24#13108061—41no—
25#13108058—21no—
26#13108053—42no—
27C22⋊Dic101—22no—
28#13108051—42no—
29C2×Q8×C101—21no—
30C2×Q8×C101D411yescharacteristic
31C2×D4×C101C2311yescharacteristic
32C22×C4×C101D412yes—
33C4⋊C4×C101—21no—
34C4⋊C4×C101—22no—
35C4⋊C4×C101—22no—
36C4⋊C4×C101—21no—
37C4⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—22no—
40C22⋊C4×C101D412yes—
41C42×C101—21no—
42C42×C101D411yescharacteristic
43C23×C101C2×D412yes—
44C22×D101—22no—
45Q8×C101—41no—
46Q8×C101—81no—
47D4×C101—41no—
48C2×C4×C101—21no—
49C2×C4×C101—22no—
50C2×C4×C101—21no—
51C2×C4×C101—22no—
52C2×C4×C101—42no—
53C2×C4×C101C2×D411yescharacteristic
54C22⋊D101—42no—
55C22⋊D101—41no—
56C2×Dic101—22no—
57C2×Dic101—42no—
58D404—21no—
59C4×D101—41no—
60Dic202—21no—
61#11717579—41no—
62C22×C101—22no—
63C22×C101—42no—
64C22×C101#12011yescharacteristic
65C2×D101—41no—
66C2×D101—82no—
67C4×C101—21no—
68C4×C101—42no—
69C4×C101—41no—
70C4×C101—42no—
71C4×C101—41no—
72Dic101—41no—
73Dic101—82no—
74C2×C101—21no—
75C2×C101—42no—
76C2×C101B(4,𝔽2)11yescharacteristic
77D101—81no—
78#2201—1011nomaximal, Sylow
79C101#220111yescharacteristic, complemented, Sylow
80#563D10111yescharacteristic, complemented
81#458—1012no—
82#455—1011no—
83#354—1011no—
84#352—1012no—
85H(5,𝔽2)—1011no—
86#137—2021no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D10111yescharacteristic
91#123—2022no—
92#123C2×D10112yes—
93C22⋊Q8—22no—
94#104—2021no—
95#100—1011no—
96#99—1012no—
97#99—2022no—
98#41—4041no—
99#41—2021no—
100C2×Q8—21no—
101C2×Q8C22⋊D10111yescharacteristic
102C2×D4—4041no—
103C2×D4—2022no—
104C2×D4C22×D10111yescharacteristic
105C22×C4C22⋊D10112yes—
106Q16—4041no—
107SD16—4041no—
108M16—2021no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—4042no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—4042no—
118C22⋊C4—2022no—
119C22⋊C4C22⋊D10112yes—
120C42—21no—
121C42C22⋊D10111yescharacteristic
122C23—2022no—
123C23C2×C22⋊D10112yes—
124Q8—41no—
125Q8—81no—
126Q8—2021no—
127D4—41no—
128D4—4042no—
129D4—4041no—
130D4—2021no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—4041no—
137C2×C4—4042no—
138C2×C4—2022no—
139C2×C4C2×C22⋊D10111yescharacteristic
140C8—4041no—
141C22—22no—
142C22—42no—
143C22—8082no—
144C22—4041no—
145C22#1569222411yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—8082no—
152C4—4041no—
153C2—21no—
154C2—42no—
155C2—8081no—
156C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#563×C101C2
socleC2×C101B(4,𝔽2)
prime radical#15692300C22
hypercenter#563D101
hypocenterC101#2201
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #20264972C2semidirect
4. #20264967C2semidirect
5. #563×C101C2semidirect
79. C101#2201semidirect
80. #563D101semidirect

Maximal subgroups

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

subgroupindex
#202653662
#202653542
#202649722
#202649672
#563×C1012
#2201101