Finite Group Databasebeta

← #28363450

Subgroups — #28363450

Order 13952.

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

normal subgroups only

This group has 342 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#20295533C211yesmaximal, characteristic
2#20295010C211yesmaximal, characteristic, complemented
3#20294989C212yesmaximal, complemented
4#20294797C212yesmaximal, complemented
5#568×C109C211yesmaximal, characteristic, complemented
6C22×C4×D109C2211yescharacteristic
7C2×#13182013C2212yes—
8#16812680C412yes—
9#16812675—21no—
10#16812670—21no—
11#16812667C412yes—
12C4×C22⋊D109—22no—
13#16812658C2211yescharacteristic
14#16812618—21no—
15#16812616—21no—
16#16812611—24no—
17C2×#41×C109C2211yescharacteristic
18#131×C109—24no—
19C2×M16×C109—22no—
20C2×M16×C109C2212yes—
21C23×D109C2×C411yescharacteristic
22C2×C22⋊D109—22no—
23C22×Dic109C2×C411yescharacteristic
24C2×C4×D109—22no—
25C2×C4×D109—21no—
26C2×C4×D109—21no—
27C2×C4×D109—22no—
28#13182013—22no—
29#13182013—22no—
30C2×#11724784—22no—
31C4⋊Dic109—21no—
32#13182008—22no—
33C22⋊Dic109—22no—
34C22⋊Dic109C2×C412yes—
35#13182006—22no—
36C4×Dic109—21no—
37#41×C109—22no—
38#41×C109—22no—
39C2×Q8×C109D411yescharacteristic
40C2×D4×C109C2×C412yes—
41C2×D4×C109D411yescharacteristic
42C22×C4×C109D412yes—
43C22×C4×C109C2311yescharacteristic
44M16×C109—24no—
45M16×C109—22no—
46M16×C109—22no—
47C2×C8×C109—24no—
48C2×C8×C109D414yescomplemented
49C23×C109C22⋊C412yes—
50C23×C109C22×C411yescharacteristic
51C22×D109—21no—
52C22×D109—22no—
53C22×D109—42no—
54Q8×C109—22no—
55D4×C109—24no—
56D4×C109—22no—
57C2×C4×C109—22no—
58C2×C4×C109—22no—
59C2×C4×C109C22⋊C412yes—
60C2×C4×C109C2×D412yes—
61C2×C4×C109C2×D411yescharacteristic
62C2×C4×C109C2×D411yescharacteristic
63C2×C4×C109#4112yes—
64C22⋊D109—44no—
65C2×Dic109—22no—
66C2×Dic109—21no—
67C2×Dic109—22no—
68C4×D109—41no—
69C4×D109—42no—
70C4×D109—41no—
71C8×C109—24no—
72#11724784—42no—
73C22×C109—22no—
74C22×C109—41no—
75C22×C109—42no—
76C22×C109C2×C22⋊C411yescharacteristic
77C22×C109C4×D412yes—
78C2×D109—22no—
79C2×D109—24no—
80C2×D109—41no—
81C2×D109—42no—
82C2×D109—41no—
83C4×C109—22no—
84C4×C109#12011yescharacteristic
85C4×C109#12112yes—
86C4×C109#12311yescharacteristic
87Dic109—42no—
88Dic109—41no—
89C2×C109—21no—
90C2×C109—22no—
91C2×C109—42no—
92C2×C109#38611yescharacteristic
93D109—42no—
94#1891—1091nomaximal, Sylow
95C109#189111yescharacteristic, complemented, Sylow
96#568D10911yescharacteristic, complemented
97#525—1091no—
98#413—1091no—
99#407—1092no—
100#344—1092no—
101C2×#41C2×D10911yescharacteristic
102C23×C4—1091no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1092no—
106C2×M16C2×D10912yes—
107#123—1092no—
108C22⋊Q8—2181no—
109#121—2181no—
110#120—1092no—
111C4×D4—2182no—
112#117—1091no—
113#101—2181no—
114#100—2181no—
115C22⋊C8—2184no—
116C24—1091no—
117#41—22no—
118#41—22no—
119C2×Q8D43611yescharacteristic
120C2×D4—2182no—
121C2×D4C4×D10912yes—
122C2×D4D43611yescharacteristic
123C22×C4—1091no—
124C22×C4—2182no—
125C22×C4—2181no—
126C22×C4—2181no—
127C22×C4—2182no—
128C22×C4C22⋊D10912yes—
129C22×C4C22×D10911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2182no—
134M16—2182no—
135C2×C8—24no—
136C2×C8—2182no—
137C2×C8C22⋊D10914yescomplemented
138C4⋊C4—2182no—
139C4⋊C4—2181no—
140C22⋊C4—1092no—
141C22⋊C4—2182no—
142C22⋊C4—2182no—
143C42—2181no—
144C23—4362no—
145C23—2181no—
146C23—2182no—
147C23#1318200612yes—
148C23C2×C4×D10911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—4364no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—4361no—
156C2×C4—4362no—
157C2×C4—4361no—
158C2×C4—2182no—
159C2×C4—2181no—
160C2×C4—2182no—
161C2×C4#1318200612yes—
162C2×C4C2×C22⋊D10912yes—
163C2×C4C2×C22⋊D10911yescharacteristic
164C2×C4C2×D43611yescharacteristic
165C2×C4#1318202912yes—
166C8—24no—
167C8—4362no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—4361no—
172C22—4362no—
173C22—4361no—
174C22—2182no—
175C22—2184no—
176C22C2×#1318200611yescharacteristic
177C22C4×C22⋊D10912yes—
178C4—22no—
179C4—4362no—
180C4—4361no—
181C4#1681266811yescharacteristic
182C4#1681267312yes—
183C4#1681268211yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—4362no—
188C2#2029490511yescharacteristic
subgroupG/H
centerC4#16812668
commutatorC23×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#568×C109C2
socleC2×C109#386
prime radical#20295533C2
cosocleC22×C4×C109C23
hypercenter#568D109
hypocenterC109#1891
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20295010C2semidirect
3. #20294989C2semidirect
4. #20294797C2semidirect
5. #568×C109C2semidirect
48. C2×C8×C109D4semidirect
95. C109#1891semidirect
96. #568D109semidirect
137. C2×C8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955332
#202950102
#202949892
#202947972
#568×C1092
#1891109