Finite Group Databasebeta

← #27170371

Subgroups — #27170371

Order 12928.

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

normal subgroups only

This group has 194 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#20265902C211yesmaximal, characteristic
2#20265459C211yesmaximal, characteristic
3#20265384C211yesmaximal, characteristic
4#20264923C211yesmaximal, characteristic
5#20264909C211yesmaximal, characteristic
6#20264909C211yesmaximal, characteristic
7#500×C101C211yesmaximal, characteristic
8C2×#13108064—21no—
9#15692253—21no—
10#15692246—21no—
11#15692245—22no—
12Q8×Dic101—21no—
13C4×Dic202C2211yescharacteristic
14C4×Dic202C2211yescharacteristic
15#15692220—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692179—21no—
20#15692178—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#107×C101—22no—
24C4×C8×C101C2211yescharacteristic
25C2×Dic202D411yescharacteristic
26C2×Dic202D411yescharacteristic
27#13108064—42no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101D411yescharacteristic
32C4⋊Dic101D411yescharacteristic
33#13108053—22no—
34#13108053—22no—
35#13108053—21no—
36#13108053—21no—
37C4×Dic101—22no—
38C4×Dic101—21no—
39C4×Dic101—21no—
40C2×Q8×C101—21no—
41C2×Q8×C101—21no—
42C2×C8×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101D412yes—
47C42×C101C2311yescharacteristic
48Q8×C101—41no—
49Q8×C101—41no—
50Q8×C101—41no—
51Q8×C101—41no—
52C2×C4×C101—22no—
53C2×C4×C101—41no—
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—21no—
61Dic202—21no—
62Dic202—21no—
63Dic202—21no—
64Dic202Q1612yescomplemented
65C8×C101—22no—
66#11717579—41no—
67#11717579—41no—
68C22×C101#12011yescharacteristic
69C4×C101—21no—
70C4×C101—42no—
71C4×C101—81no—
72C4×C101C2×Q1611yescharacteristic
73C4×C101#13511yescharacteristic
74C4×C101AGL(1,ℤ/8)11yescharacteristic
75C4×C101#13711yescharacteristic
76Dic101—22no—
77Dic101—41no—
78Dic101—41no—
79Dic101—41no—
80C2×C101#44911yescharacteristic
81C2×C101#45111yescharacteristic
82C2×C101#45611yescharacteristic
83#1648—1011nomaximal, Sylow
84C101#164811yescharacteristic, complemented, Sylow
85#557—1011no—
86#500D10111yescharacteristic
87#475—1011no—
88#462—1011no—
89#332—1011no—
90Q8⋊C8—1011no—
91Q8⋊C8—1011no—
92C2×Q16—2021no—
93C4⋊Q8—2021no—
94C4⋊Q8C2×D10111yescharacteristic
95C4⋊Q8C2×D10111yescharacteristic
96#125—2021no—
97#125—2022no—
98C4×Q8—1011no—
99C4×Q8—1011no—
100C4×Q8—2021no—
101C4×Q8—2021no—
102#107—22no—
103#106—2021no—
104C4⋊C8—1011no—
105C4⋊C8—1011no—
106#103—2021no—
107#103—2021no—
108C4×C8C2×D10111yescharacteristic
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—1011no—
112C2×Q8—1011no—
113Q16—4042no—
114C2×C8—21no—
115C2×C8—2021no—
116C2×C8—2021no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—1011no—
121C4⋊C4—1011no—
122C4⋊C4—2022no—
123C4⋊C4—2022no—
124C4⋊C4—2022no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4C22⋊D10112yes—
128C42—2022no—
129C42—2021no—
130C42—2021no—
131C42C22×D10111yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—41no—
135Q8—41no—
136Q8—1012no—
137Q8—2021no—
138Q8—2021no—
139Q8—2021no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—2021no—
143C2×C4—2021no—
144C2×C4—2021no—
145C2×C4—2021no—
146C2×C4D4×D10111yescharacteristic
147C2×C4D4×D10111yescharacteristic
148C2×C4C2×C22⋊D10111yescharacteristic
149C8—22no—
150C8—4041no—
151C8—4041no—
152C22#1569222311yescharacteristic
153C4—21no—
154C4—42no—
155C4—81no—
156C4—4041no—
157C4—4041no—
158C4—4041no—
159C4—2022no—
160C4Q16×D10111yescharacteristic
161C4#1569227411yescharacteristic
162C4#1569227911yescharacteristic
163C4#1569228711yescharacteristic
164C2#2026531911yescharacteristic
165C2#2026533111yescharacteristic
166C2#2026535911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#500×C101C2
socle, prime radicalC22×C101#120
hypercenter#500D101
hypocenterC101#1648
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
64. Dic202Q16semidirect
84. C101#1648semidirect

Maximal subgroups

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

subgroupindex
#202659022
#202654592
#202653842
#202649232
#202649092
#500×C1012
#1648101