Finite Group Databasebeta

← #28363557

Subgroups — #28363557

Order 13952.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#20295793C211yesmaximal, characteristic, complemented
2#20295590C211yesmaximal, characteristic
3#20295030C212yesmaximal, complemented
4#20294782C212yesmaximal, complemented
5#410×C109C211yesmaximal, characteristic, complemented
6C22×D436C2211yescharacteristic
7#16812706—24no—
8C2×#13182013—22no—
9C2×#13182013C2212yes—
10#16812695C412yescomplemented
11#16812688C412yescomplemented
12#16812673—22no—
13#16812667—22no—
14#16812658C2211yescharacteristic
15#16812619—24no—
16C2×#41×C109C2211yescharacteristic
17#117×C109C2212yes—
18#99×C109—22no—
19C23×D109—21no—
20C2×D436—22no—
21C2×D436—22no—
22C2×D436—21no—
23C2×D436C2×C412yes—
24C2×C22⋊D109—24no—
25#13182013—24no—
26#13182013—22no—
27#13182013—22no—
28C2×#11724784—24no—
29C2×#11724784D414yescomplemented
30C4⋊Dic109—21no—
31C22⋊Dic109—21no—
32#13182006—24no—
33C4×Dic109C2×C412yes—
34#41×C109—22no—
35#41×C109—22no—
36C2×Q8×C109C2×C411yescharacteristic
37C2×D4×C109C2×C411yescharacteristic
38C2×D4×C109D412yes—
39C22×C4×C109D412yes—
40C22×C4×C109C2311yescharacteristic
41C4⋊C4×C109—22no—
42C22⋊C4×C109—22no—
43C42×C109—22no—
44C23×C109C22⋊C412yes—
45C23×C109C2×D411yescharacteristic
46C22×D109—22no—
47C22×D109—42no—
48C22×D109—82no—
49Q8×C109—22no—
50D4×C109—24no—
51D4×C109—22no—
52C2×C4×C109—22no—
53C2×C4×C109—22no—
54C2×C4×C109—42no—
55C2×C4×C109C22⋊C412yes—
56C2×C4×C109C22×C411yescharacteristic
57C2×C4×C109C2×D412yes—
58C2×C4×C109C2×D411yescharacteristic
59C2×C4×C109#4112yes—
60C22⋊D109—84no—
61C2×Dic109—22no—
62D436—24no—
63D436—41no—
64D436—41no—
65#11724784—24no—
66C22×C109—22no—
67C22×C109—41no—
68C22×C109—42no—
69C22×C109C2×C22⋊C411yescharacteristic
70C22×C109#12112yes—
71C2×D109—44no—
72C2×D109—42no—
73C2×D109—84no—
74C4×C109—22no—
75C4×C109—82no—
76C4×C109C4×D412yes—
77C4×C109#12011yescharacteristic
78C4×C109#12311yescharacteristic
79Dic109—42no—
80C2×C109—21no—
81C2×C109—22no—
82C2×C109—42no—
83C2×C109#38611yescharacteristic
84D109—82no—
85#1910—1091nomaximal, Sylow
86C109#191011yescharacteristic, complemented, Sylow
87#568—1091no—
88#535—1091no—
89#418—1092no—
90#410D10911yescharacteristic, complemented
91H(3,ℤ/4)—1092no—
92C2×#41C2×D10911yescharacteristic
93C22×D4—1091no—
94#131—2184no—
95C2×M16—1092no—
96C2×M16—2182no—
97#127—1092no—
98#124—1092no—
99#121—2182no—
100#120—2182no—
101#117—1091no—
102#117C2×D10912yes—
103#102—2184no—
104#99—22no—
105C24—2181no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D10911yescharacteristic
109C2×D4—1092no—
110C2×D4—2184no—
111C2×D4—2182no—
112C2×D4—2181no—
113C2×D4—2182no—
114C2×D4C4×D10911yescharacteristic
115C2×D4C22⋊D10912yes—
116C22×C4D43612yes—
117C22×C4C22×D10911yescharacteristic
118M16—2184no—
119M16—2182no—
120M16—2182no—
121C2×C8—1094no—
122C2×C8—2184no—
123C4⋊C4—22no—
124C4⋊C4—2181no—
125C22⋊C4—22no—
126C22⋊C4—2181no—
127C22⋊C4—2184no—
128C42—22no—
129C42—1092no—
130C23—8722no—
131C23—4362no—
132C23—2182no—
133C23#1318200612yes—
134C23C2×C22⋊D10911yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—8724no—
139D4—4361no—
140D4—4361no—
141D4—2184no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—2182no—
146C2×C4#1318200612yes—
147C2×C4C2×C4×D10911yescharacteristic
148C2×C4D4×D10912yes—
149C2×C4C2×D43611yescharacteristic
150C2×C4#1318202812yes—
151C8—2184no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—8724no—
156C22—4364no—
157C22—4362no—
158C22C2×#1318200611yescharacteristic
159C22#1681267112yes—
160C4—22no—
161C4—82no—
162C4—4362no—
163C4#1681265912yes—
164C4#1681266911yescharacteristic
165C4#1681268311yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—8722no—
170C2#2029490311yescharacteristic
subgroupG/H
centerC2#20294903
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#410×C109C2
socleC2×C109#386
prime radical#20295590C2
cosocleC22×C4×C109C23
hypercenter#410D109
hypocenterC109#1910
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #20295793C2semidirect
3. #20295030C2semidirect
4. #20294782C2semidirect
5. #410×C109C2semidirect
10. #16812695C4semidirect
11. #16812688C4semidirect
29. C2×#11724784D4semidirect
86. C109#1910semidirect
90. #410D109semidirect

Maximal subgroups

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

subgroupindex
#202957932
#202955902
#202950302
#202947822
#410×C1092
#1910109