Finite Group Databasebeta

← #28363579

Subgroups — #28363579

Order 13952.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#20295590C211yesmaximal, characteristic
2#20295010C211yesmaximal, characteristic, complemented
3C2×#16812615C212yesmaximal, complemented
4#20294798C212yesmaximal, complemented
5#568×C109C211yesmaximal, characteristic, complemented
6C22×D436C2211yescharacteristic
7C2×#13182013C2212yes—
8#16812695—21no—
9#16812688—21no—
10#16812673—22no—
11#16812667C414yes—
12#16812658C2211yescharacteristic
13#16812618—21no—
14#16812616—21no—
15#16812615—24no—
16C2×#41×C109C2211yescharacteristic
17#131×C109—24no—
18C2×M16×C109—22no—
19C2×M16×C109C2212yes—
20C23×D109C2×C412yes—
21C2×D436—22no—
22C2×D436—21no—
23C2×D436—21no—
24C2×D436—22no—
25C2×C22⋊D109—24no—
26#13182013—22no—
27#13182013—22no—
28C2×#11724784—22no—
29C4⋊Dic109—21no—
30C22⋊Dic109C2×C412yes—
31#13182006—24no—
32C4×Dic109—21no—
33#41×C109—22no—
34#41×C109—22no—
35C2×Q8×C109D411yescharacteristic
36C2×D4×C109C2×C412yes—
37C2×D4×C109D411yescharacteristic
38C22×C4×C109D412yes—
39C22×C4×C109C2311yescharacteristic
40M16×C109—24no—
41M16×C109—22no—
42M16×C109D414yescomplemented
43C2×C8×C109—24no—
44C2×C8×C109—22no—
45C23×C109C22⋊C412yes—
46C23×C109C22×C411yescharacteristic
47C22×D109—22no—
48C22×D109—24no—
49C22×D109—82no—
50Q8×C109—22no—
51D4×C109—24no—
52D4×C109—22no—
53C2×C4×C109—22no—
54C2×C4×C109—22no—
55C2×C4×C109C22⋊C412yes—
56C2×C4×C109C2×D412yes—
57C2×C4×C109C2×D411yescharacteristic
58C2×C4×C109C2×D411yescharacteristic
59C2×C4×C109#4112yes—
60C22⋊D109—84no—
61C2×Dic109—22no—
62D436—41no—
63D436—42no—
64D436—41no—
65C8×C109—24no—
66#11724784—42no—
67C22×C109—22no—
68C22×C109—41no—
69C22×C109—42no—
70C22×C109C2×C22⋊C411yescharacteristic
71C22×C109C4×D412yes—
72C2×D109—44no—
73C2×D109—42no—
74C2×D109—84no—
75C4×C109—22no—
76C4×C109#12011yescharacteristic
77C4×C109#12112yes—
78C4×C109#12311yescharacteristic
79Dic109—42no—
80C2×C109—21no—
81C2×C109—22no—
82C2×C109—42no—
83C2×C109#38611yescharacteristic
84D109—82no—
85#1914—1091nomaximal, Sylow
86C109#191411yescharacteristic, complemented, Sylow
87#568D10911yescharacteristic, complemented
88#535—1091no—
89#413—1091no—
90C2×#100—1092no—
91#344—1092no—
92C2×#41C2×D10911yescharacteristic
93C22×D4—1091no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1092no—
97C2×M16C2×D10912yes—
98#127—2181no—
99#124—2181no—
100#121—2182no—
101#120—1094no—
102#117—1091no—
103#101—2181no—
104#100—2181no—
105#100—2184no—
106C24—1092no—
107#41—22no—
108#41—22no—
109C2×Q8D43611yescharacteristic
110C2×D4—2184no—
111C2×D4—2182no—
112C2×D4—2181no—
113C2×D4—2181no—
114C2×D4—2182no—
115C2×D4C4×D10912yes—
116C2×D4D43611yescharacteristic
117C22×C4C22⋊D10912yes—
118C22×C4C22×D10911yescharacteristic
119M16—24no—
120M16—22no—
121M16—2182no—
122M16—2182no—
123M16C22⋊D10914yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—2182no—
127C4⋊C4—2181no—
128C22⋊C4—1092no—
129C22⋊C4—2184no—
130C42—2181no—
131C23—8722no—
132C23—2184no—
133C23—2182no—
134C23#1318200612yes—
135C23C2×C4×D10911yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—8724no—
140D4—4361no—
141D4—4362no—
142D4—4361no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—2182no—
146C2×C4#1318200612yes—
147C2×C4C2×C22⋊D10912yes—
148C2×C4C2×C22⋊D10911yescharacteristic
149C2×C4C2×D43611yescharacteristic
150C2×C4#1318202912yes—
151C8—24no—
152C8—4362no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—8724no—
157C22—4364no—
158C22—4362no—
159C22C2×#1318200611yescharacteristic
160C22C4×C22⋊D10912yes—
161C4—22no—
162C4—4362no—
163C4#1681266811yescharacteristic
164C4#1681267312yes—
165C4#1681268211yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—8722no—
170C2#2029490511yescharacteristic
subgroupG/H
centerC2#20294905
commutatorC23×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#568×C109C2
socleC2×C109#386
prime radical#20295590C2
cosocleC22×C4×C109C23
hypercenter#568D109
hypocenterC109#1914
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20295010C2semidirect
3. C2×#16812615C2semidirect
4. #20294798C2semidirect
5. #568×C109C2semidirect
42. M16×C109D4semidirect
86. C109#1914semidirect
87. #568D109semidirect
123. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955902
#202950102
C2×#168126152
#202947982
#568×C1092
#1914109