Finite Group Databasebeta

← #27170135

Subgroups — #27170135

Order 12928.

254 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265488C212yesmaximal, complemented
2#20265107C211yesmaximal, characteristic, complemented
3#20264911C212yesmaximal, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#537×C101C211yesmaximal, characteristic
6C2×C4⋊Dic101C2211yescharacteristic
7C2×#13108053—21no—
8#15692188—22no—
9#15692176—22no—
10C22⋊#11717579C2212yes—
11#15692162—21no—
12#15692162C2211yescharacteristic
13C2×#41×C101—21no—
14C22⋊Q8×C101—22no—
15#121×C101—22no—
16#121×C101—21no—
17#121×C101C2212yes—
18C2×C4⋊C4×C101C2211yescharacteristic
19C22×Dic101—21no—
20C22×Dic101—21no—
21C2×#11717579—22no—
22C4⋊Dic101—22no—
23#13108053—42no—
24#41×C101—42no—
25#41×C101—42no—
26C2×Q8×C101—21no—
27C2×D4×C101—21no—
28C2×D4×C101—22no—
29C2×D4×C101—21no—
30C2×D4×C101D412yes—
31C22×C4×C101—22no—
32C22×C4×C101D411yescharacteristic
33C22×C4×C101D411yescharacteristic
34C22×C4×C101C2311yescharacteristic
35C4⋊C4×C101—21no—
36C4⋊C4×C101—22no—
37C4⋊C4×C101D412yes—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—22no—
40C23×C101—22no—
41C23×C101C2×D411yescharacteristic
42Q8×C101—42no—
43D4×C101—42no—
44D4×C101—42no—
45D4×C101—42no—
46D4×C101—42no—
47D4×C101—81no—
48C2×C4×C101—21no—
49C2×C4×C101—21no—
50C2×C4×C101—21no—
51C2×C4×C101—22no—
52C2×C4×C101—41no—
53C2×C4×C101—44no—
54C2×C4×C101—41no—
55C2×C4×C101D812yes—
56C2×C4×C101C2×D412yes—
57C2×Dic101—22no—
58C2×Dic101—41no—
59C2×Dic101—41no—
60C2×Dic101—41no—
61C2×Dic101—41no—
62C2×Dic101—41no—
63#11717579—42no—
64C22×C101—21no—
65C22×C101—21no—
66C22×C101—42no—
67C22×C101—42no—
68C22×C101—42no—
69C22×C101#12011yescharacteristic
70C22×C101C2×D811yescharacteristic
71C22×C101AGL(1,ℤ/8)11yescharacteristic
72C4×C101—22no—
73C4×C101—22no—
74C4×C101—42no—
75C4×C101—41no—
76Dic101—42no—
77Dic101—81no—
78C2×C101—21no—
79C2×C101—21no—
80C2×C101—82no—
81C2×C101#44711yescharacteristic
82C2×C101#45611yescharacteristic
83C2×C101#45811yescharacteristic
84#1610—1011nomaximal, Sylow
85C101#161011yescharacteristic, complemented, Sylow
86#537D10111yescharacteristic
87#480—1012no—
88#398—1011no—
89#327—1012no—
90#324—1011no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D10112yes—
96C2×C4⋊C4—1011no—
97C2×C4⋊C4—2021no—
98C2×C4⋊C4C2×D10111yescharacteristic
99#107—2022no—
100#102—2022no—
101C22⋊C8—1012no—
102#95—1011no—
103#95—2021no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D10112yes—
111C22×C4—22no—
112C22×C4—2021no—
113C22×C4—2021no—
114C22×C4C22⋊D10111yescharacteristic
115C22×C4C22⋊D10111yescharacteristic
116C22×C4C22×D10111yescharacteristic
117C2×C8—2022no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—4042no—
121C4⋊C4—2022no—
122C4⋊C4C22⋊D10112yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D10111yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—4041no—
141C2×C4—4041no—
142C2×C4—4041no—
143C2×C4—4041no—
144C2×C4—4041no—
145C2×C4—2022no—
146C2×C4#1310805912yes—
147C2×C4C2×C22⋊D10112yes—
148C8—4042no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1569222411yescharacteristic
155C22C2×#1310805911yescharacteristic
156C22#1569228111yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—8081no—
162C4—4042no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2026531111yescharacteristic
167C2#2026536011yescharacteristic
168C2#2026536711yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#537×C101C2
socleC22×C101#120
hypercenter#537D101
hypocenterC101#1610
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20265488C2semidirect
2. #20265107C2semidirect
3. #20264911C2semidirect
4. #20264903C2semidirect
85. C101#1610semidirect

Maximal subgroups

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

subgroupindex
#202654882
#202651072
#202649112
#202649032
#537×C1012
#1610101