Finite Group Databasebeta

← #28364700

Subgroups — #28364700

Order 13952.

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

normal subgroups only

This group has 286 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#20295594C211yesmaximal, characteristic
2#99×D109C212yesmaximal, complemented
3#20294821C211yesmaximal, characteristic, complemented
4#20294818C212yesmaximal, complemented
5#354×C109C211yesmaximal, characteristic, complemented
6C2×#13182030C411yescharacteristic, complemented
7C2×D4×D109C2211yescharacteristic
8#16812696C411yescharacteristic, complemented
9#16812695C2211yescharacteristic
10#16812688C411yescharacteristic, complemented
11#16812672—22no—
12#16812669—22no—
13#16812656C411yescharacteristic, complemented
14C22⋊C4×D109—22no—
15#16812614C2212yes—
16#16812613—22no—
17#124×C109C2211yescharacteristic
18#99×C109C2212yes—
19C23×D109D412yes—
20#13182030—42no—
21C2×D436—21no—
22C2×D436—22no—
23C2×D436C2×C411yescharacteristic
24C2×D436C2×C411yescharacteristic, complemented
25C2×C22⋊D109—22no—
26C2×C22⋊D109D412yes—
27D4×D109—42no—
28C2×C4×D109—21no—
29C2×C4×D109—42no—
30C2×C4×D109C2×C411yescharacteristic
31#13182008—21no—
32C22⋊Dic109—22no—
33#13182006—21no—
34#13182006—22no—
35#13182006—42no—
36C4×Dic109C2×C411yescharacteristic, complemented
37C2×Q8×C109C2×C411yescharacteristic, complemented
38C2×D4×C109C2311yescharacteristic
39C22⋊C4×C109—22no—
40C22⋊C4×C109—22no—
41C42×C109C2×C411yescharacteristic, complemented
42C23×C109C2×D412yes—
43C22×D109—22no—
44C22×D109—22no—
45C22×D109—21no—
46C22×D109—82no—
47C22×D109C22⋊C412yes—
48C22×D109C22⋊C411yescharacteristic
49Q8×C109—41no—
50D4×C109—41no—
51C2×C4×C109—21no—
52C2×C4×C109—21no—
53C2×C4×C109—42no—
54C2×C4×C109C22×C411yescharacteristic
55C22⋊D109—42no—
56C22⋊D109—82no—
57C2×Dic109—21no—
58C2×Dic109—42no—
59C2×Dic109C22⋊C411yescharacteristic
60D436—41no—
61D436—41no—
62D436—42no—
63D436—82no—
64C4×D109—22no—
65C4×D109—42no—
66C4×D109—84no—
67C22×C109—22no—
68C22×C109—42no—
69C22×C109C2×C22⋊C411yescharacteristic
70C2×D109—22no—
71C2×D109—41no—
72C2×D109—44no—
73C2×D109—42no—
74C2×D109—42no—
75C2×D109—41no—
76C2×D109—82no—
77C2×D109#9912yes—
78C4×C109—21no—
79C4×C109—41no—
80C4×C109—41no—
81C4×C109—82no—
82Dic109—21no—
83Dic109—41no—
84Dic109—82no—
85C2×C109—21no—
86C2×C109—42no—
87C2×C109C2×#9911yescharacteristic
88D109—22no—
89D109—42no—
90D109—81no—
91#2127—1091nomaximal, Sylow
92C109#212711yescharacteristic, complemented, Sylow
93#535—1091no—
94C2×#99—1092no—
95#354—1091no—
96#354D10911yescharacteristic, complemented
97#353—1092no—
98C2×#41—1091no—
99C22×D4—1091no—
100#127—1091no—
101#127—1091no—
102#124—1091no—
103#124C2×D10911yescharacteristic
104#121—2182no—
105#120—2182no—
106#117—1091no—
107C2×C22⋊C4—2182no—
108#99—1092no—
109#99—2182no—
110#99C2×D10912yes—
111C24—1092no—
112#41—4362no—
113C2×Q8C4×D10911yescharacteristic, complemented
114C2×D4—1092no—
115C2×D4—1091no—
116C2×D4—1091no—
117C2×D4—4362no—
118C2×D4—2182no—
119C2×D4—2181no—
120C2×D4—2182no—
121C2×D4C22×D10911yescharacteristic
122C22×C4—1091no—
123C22×C4—4362no—
124C22×C4—2181no—
125C4⋊C4—2181no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—4362no—
129C22⋊C4—2181no—
130C22⋊C4—2182no—
131C22⋊C4—2182no—
132C42—1091no—
133C42C4×D10911yescharacteristic, complemented
134C23—8722no—
135C23—1091no—
136C23—1092no—
137C23—2182no—
138C23—2182no—
139C23—2181no—
140C23D4×D10912yes—
141Q8—41no—
142D4—41no—
143D4—8722no—
144D4—8722no—
145D4—4361no—
146D4—4361no—
147D4—4362no—
148D4—4362no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—8724no—
153C2×C4—1091no—
154C2×C4—4362no—
155C2×C4—4362no—
156C2×C4—2182no—
157C2×C4—2181no—
158C2×C4C2×C4×D10911yescharacteristic
159C22—22no—
160C22—42no—
161C22—8722no—
162C22—1092no—
163C22—4362no—
164C22—4362no—
165C22—4361no—
166C22—4364no—
167C22—4361no—
168C22—2182no—
169C22C22⋊C4×D10911yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—8722no—
175C4—4361no—
176C4—2181no—
177C2—21no—
178C2—42no—
179C2—8721no—
180C2—4362no—
181C2—2182no—
182C2#99×D10911yescharacteristic
subgroupG/H
centerC2#99×D109
commutatorC2×C4×C109C22×C4
FrattiniC2×D4C22×D109
Fitting#354×C109C2
socleC2×C109C2×#99
prime radical#20295594C2
cosocleC2×D4×C109C23
hypercenter#354D109
hypocenterC109#2127
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D109C2semidirect
3. #20294821C2semidirect
4. #20294818C2semidirect
5. #354×C109C2semidirect
6. C2×#13182030C4semidirect
8. #16812696C4semidirect
10. #16812688C4semidirect
13. #16812656C4semidirect
24. C2×D436C2×C4semidirect
36. C4×Dic109C2×C4semidirect
37. C2×Q8×C109C2×C4semidirect
41. C42×C109C2×C4semidirect
92. C109#2127semidirect
96. #354D109semidirect
113. C2×Q8C4×D109semidirect
133. C42C4×D109semidirect

Maximal subgroups

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

subgroupindex
#202955942
#99×D1092
#202948212
#202948182
#354×C1092
#2127109