Finite Group Databasebeta

← #27172805

Subgroups — #27172805

Order 12928.

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

normal subgroups only

This group has 238 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#20265747C211yesmaximal, characteristic
2#20265149C212yesmaximal
3#20264972C211yesmaximal, characteristic, complemented
4#20264971C212yesmaximal, complemented
5#354×C101C211yesmaximal, characteristic, complemented
6C2×#13108073C2211yescharacteristic
7C2×Q8×D101C411yescharacteristic, complemented
8#15692255C411yescharacteristic, complemented
9#15692253C411yescharacteristic, complemented
10#15692242C2211yescharacteristic
11#15692237—22no—
12#15692232—22no—
13#15692212C411yescharacteristic, complemented
14#15692211—22no—
15#15692170C2212yes—
16#15692169—22no—
17#124×C101C2211yescharacteristic
18#99×C101C2212yes—
19#13108073—42no—
20C2×Dic202—21no—
21C2×Dic202C2×C411yescharacteristic
22C2×Dic202C2×C411yescharacteristic
23Q8×D101—42no—
24C2×C22⋊D101D412yes—
25C22×Dic101D412yes—
26C2×C4×D101—21no—
27C2×C4×D101C2×C411yescharacteristic
28C4⋊Dic101—22no—
29#13108053—21no—
30#13108053—22no—
31#13108053—42no—
32C22⋊Dic101—22no—
33C22⋊Dic101—22no—
34#13108051—21no—
35C4×Dic101—42no—
36C4×Dic101C2×C411yescharacteristic
37C2×Q8×C101C2×C411yescharacteristic
38C2×D4×C101C2311yescharacteristic
39C22⋊C4×C101—22no—
40C22⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic
42C23×C101C2×D412yes—
43C22×D101C22⋊C411yescharacteristic
44Q8×C101—41no—
45D4×C101—41no—
46C2×C4×C101—21no—
47C2×C4×C101—21no—
48C2×C4×C101—42no—
49C2×C4×C101C22×C411yescharacteristic
50C22⋊D101—42no—
51C2×Dic101—22no—
52C2×Dic101—22no—
53C2×Dic101—21no—
54C2×Dic101—21no—
55C2×Dic101—42no—
56C2×Dic101C22⋊C412yes—
57C2×Dic101C22⋊C411yescharacteristic
58C4×D101—22no—
59C4×D101—42no—
60Dic202—41no—
61Dic202—42no—
62Dic202—41no—
63C22×C101—22no—
64C22×C101—42no—
65C22×C101C2×C22⋊C411yescharacteristic
66C2×D101—21no—
67C2×D101—41no—
68C4×C101—21no—
69C4×C101—41no—
70C4×C101—41no—
71C4×C101—82no—
72Dic101—22no—
73Dic101—41no—
74Dic101—41no—
75Dic101—82no—
76Dic101#9912yes—
77C2×C101—21no—
78C2×C101—42no—
79C2×C101C2×#9911yescharacteristic
80D101—41no—
81#2129—1011nomaximal, Sylow
82C101#212911yescharacteristic, complemented, Sylow
83#536—1011no—
84#410—1012no—
85#354—1011no—
86#354—1012no—
87#354D10111yescharacteristic, complemented
88C2×#41—1011no—
89C22×Q8—1011no—
90C4⋊Q8—1011no—
91C4⋊Q8—1011no—
92#124—1011no—
93#124C2×D10111yescharacteristic
94#123—2022no—
95C22⋊Q8—2022no—
96#117—1011no—
97#117—2022no—
98#99—1012no—
99#99—2022no—
100#99C2×D10112yes—
101#41—4042no—
102C2×Q8—1011no—
103C2×Q8—1011no—
104C2×Q8—4042no—
105C2×Q8—2021no—
106C2×Q8C4×D10111yescharacteristic
107C2×D4—1012no—
108C2×D4C22×D10111yescharacteristic
109C22×C4—1012no—
110C22×C4—1011no—
111C22×C4—2021no—
112C4⋊C4—4042no—
113C4⋊C4—2021no—
114C4⋊C4—2022no—
115C4⋊C4—2022no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—2021no—
119C22⋊C4—2022no—
120C22⋊C4—2022no—
121C42—1011no—
122C42—4042no—
123C42C4×D10111yescharacteristic
124C23—1011no—
125C23D4×D10112yes—
126Q8—41no—
127Q8—4041no—
128Q8—4041no—
129Q8—4042no—
130D4—41no—
131D4—4042no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—1011no—
136C2×C4—1012no—
137C2×C4—4042no—
138C2×C4—4042no—
139C2×C4—2022no—
140C2×C4—2022no—
141C2×C4—2022no—
142C2×C4—2021no—
143C2×C4—2021no—
144C2×C4C2×C4×D10111yescharacteristic
145C22—22no—
146C22—42no—
147C22—4041no—
148C22—2021no—
149C22C22⋊C4×D10111yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—8082no—
155C4—1012no—
156C4—4041no—
157C4—4041no—
158C4—2022no—
159C2—21no—
160C2—42no—
161C2—4041no—
162C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#354×C101C2
socleC2×C101C2×#99
prime radicalC2×#13108073C22
cosocleC2×D4×C101C23
hypercenter#354D101
hypocenterC101#2129
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #20264972C2semidirect
4. #20264971C2semidirect
5. #354×C101C2semidirect
7. C2×Q8×D101C4semidirect
8. #15692255C4semidirect
9. #15692253C4semidirect
13. #15692212C4semidirect
82. C101#2129semidirect
87. #354D101semidirect

Maximal subgroups

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

subgroupindex
#202657472
#202651492
#202649722
#202649712
#354×C1012
#2129101