Finite Group Databasebeta

← #27172794

Subgroups — #27172794

Order 12928.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265725C211yesmaximal, characteristic
2#20265149C212yesmaximal, complemented
3#20264971C212yesmaximal, complemented
4#20264970C211yesmaximal, characteristic, complemented
5#353×C101C211yesmaximal, characteristic, complemented
6C2×#13108073C411yescharacteristic, complemented
7C2×#13108073C2211yescharacteristic
8#15692255C411yescharacteristic, complemented
9#15692242C411yescharacteristic, complemented
10#15692242C2211yescharacteristic
11#15692226—22no—
12D4×Dic101—22no—
13#15692211—22no—
14C42×D101C411yescharacteristic, complemented
15#15692170C2212yes—
16#15692169—22no—
17#127×C101C2211yescharacteristic
18#99×C101C2212yes—
19#13108073—41no—
20#13108073—41no—
21#13108073—41no—
22#13108073—41no—
23C2×Dic202C2×C411yescharacteristic
24C2×Dic202C2×C411yescharacteristic
25C2×C22⋊D101—21no—
26C2×C22⋊D101D412yes—
27C22×Dic101—21no—
28C22×Dic101D412yes—
29C2×C4×D101—21no—
30C2×C4×D101C2×C411yescharacteristic
31C4⋊Dic101—22no—
32#13108053—42no—
33C22⋊Dic101—22no—
34C22⋊Dic101—22no—
35C22⋊Dic101—22no—
36C4×Dic101—21no—
37C4×Dic101—42no—
38C4×Dic101C2×C411yescharacteristic
39C2×D4×C101—22no—
40C2×D4×C101C2×C411yescharacteristic
41C2×D4×C101C2311yescharacteristic
42C22⋊C4×C101—22no—
43C42×C101C2×C411yescharacteristic
44C23×C101—21no—
45C23×C101C2×D412yes—
46C22×D101C22⋊C411yescharacteristic
47D4×C101—41no—
48D4×C101—41no—
49D4×C101—44no—
50C2×C4×C101—21no—
51C2×C4×C101—42no—
52C2×C4×C101C22×C411yescharacteristic
53C22⋊D101—42no—
54C22⋊D101—42no—
55C2×Dic101—22no—
56C2×Dic101—22no—
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—41no—
60C2×Dic101—41no—
61C2×Dic101—42no—
62C2×Dic101C22⋊C412yes—
63C2×Dic101C22⋊C411yescharacteristic
64C4×D101—21no—
65C4×D101—21no—
66C4×D101—42no—
67Dic202—41no—
68Dic202—41no—
69C22×C101—22no—
70C22×C101—42no—
71C22×C101—42no—
72C22×C101—41no—
73C22×C101C2×C22⋊C411yescharacteristic
74C2×D101—21no—
75C2×D101—41no—
76C4×C101—22no—
77C4×C101—21no—
78C4×C101—82no—
79Dic101—22no—
80Dic101—41no—
81Dic101—41no—
82Dic101—82no—
83Dic101#9911yescharacteristic
84Dic101#9911yescharacteristic
85C2×C101—21no—
86C2×C101—42no—
87C2×C101—81no—
88C2×C101C2×#9911yescharacteristic
89D101—41no—
90#2126—1011nomaximal, Sylow
91C101#212611yescharacteristic, complemented, Sylow
92#532—1011no—
93#410—1012no—
94#354—1012no—
95#353—1011no—
96#353D10111yescharacteristic, complemented
97C2×#41—1011no—
98C2×#41—1011no—
99C4⋊Q8—1011no—
100#127C2×D10111yescharacteristic
101#124—1011no—
102#124—1011no—
103#121—2022no—
104C4×D4—2022no—
105#117—2022no—
106C2×C42—1011no—
107#99—1012no—
108#99—2022no—
109#99C2×D10112yes—
110#41—4041no—
111#41—4041no—
112#41—4041no—
113#41—4041no—
114C2×Q8—1011no—
115C2×Q8—1011no—
116C2×D4—22no—
117C2×D4—1012no—
118C2×D4—2021no—
119C2×D4C4×D10111yescharacteristic
120C2×D4C22×D10111yescharacteristic
121C22×C4—1012no—
122C22×C4—1011no—
123C22×C4—2021no—
124C22×C4—2021no—
125C4⋊C4—4042no—
126C4⋊C4—2022no—
127C22⋊C4—22no—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C22⋊C4—2022no—
131C42—1011no—
132C42—4042no—
133C42—2021no—
134C42C4×D10111yescharacteristic
135C23—21no—
136C23—1011no—
137C23D4×D10112yes—
138Q8—4041no—
139Q8—4041no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—4042no—
144D4—4042no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—1011no—
148C2×C4—1012no—
149C2×C4—4042no—
150C2×C4—4042no—
151C2×C4—4041no—
152C2×C4—4041no—
153C2×C4—2021no—
154C2×C4—2021no—
155C2×C4—2022no—
156C2×C4—2022no—
157C2×C4—2021no—
158C2×C4—2021no—
159C2×C4C2×C4×D10111yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—4041no—
165C22—2021no—
166C22C22⋊C4×D10111yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—8082no—
171C4—1011no—
172C4—1011no—
173C4—4041no—
174C4—4041no—
175C4—2022no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—4041no—
180C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#353×C101C2
socleC2×C101C2×#99
prime radical#20265725C2
cosocleC2×D4×C101C23
hypercenter#353D101
hypocenterC101#2126
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20265149C2semidirect
3. #20264971C2semidirect
4. #20264970C2semidirect
5. #353×C101C2semidirect
6. C2×#13108073C4semidirect
8. #15692255C4semidirect
9. #15692242C4semidirect
14. C42×D101C4semidirect
91. C101#2126semidirect
96. #353D101semidirect

Maximal subgroups

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

subgroupindex
#202657252
#202651492
#202649712
#202649702
#353×C1012
#2126101