Finite Group Databasebeta

← D109⋊#354 #28379301

Subgroups — D109⋊#354 #28379301

Order 13952.

284 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D109⋊#99C212yesmaximal
2#20295933C212yesmaximal, complemented
3#20295932C212yesmaximal, complemented
4#124×D109C211yesmaximal, characteristic
5C2×C22⋊#89137—22no—
6#16812758C2214yes—
7#16812757—22no—
8C2×Q8×D109C411yescharacteristic, complemented
9C2×D4×D109C2211yescharacteristic
10#16812688C2211yescharacteristic
11#16812687C2211yescharacteristic
12#16812686C411yescharacteristic, complemented
13#16812685—22no—
14C22⋊C4×D109—22no—
15C42×D109C411yescharacteristic, complemented
16#124×C109C411yescharacteristic, complemented
17C22×#89137—42no—
18C22⋊#89137—24no—
19C22⋊#89137—42no—
20C23×D109D412yes—
21C2×Dic218—21no—
22C2×Dic218C2×C411yescharacteristic, complemented
23Q8×D109—41no—
24Q8×D109—41no—
25C2×D436C2311yescharacteristic
26C2×C22⋊D109D412yes—
27D4×D109—41no—
28D4×D109—41no—
29C2×C4×D109—21no—
30C2×C4×D109—21no—
31C2×C4×D109C2×C411yescharacteristic
32C22⋊Dic109—22no—
33#13182006—22no—
34#13182006—22no—
35C4×Dic109—21no—
36C4×Dic109C2×C411yescharacteristic, complemented
37C2×Q8×C109C2×C411yescharacteristic, complemented
38C2×D4×C109C2×C411yescharacteristic
39C22⋊C4×C109—22no—
40C42×C109C2×C411yescharacteristic, complemented
41C23×C109C22⋊C412yes—
42C22×D109—22no—
43C22×D109—22no—
44C22×D109—44no—
45C22×D109C22⋊C411yescharacteristic
46C22×D109C2×D412yes—
47C2×#89137—44no—
48C2×#89137—84no—
49Q8×C109—41no—
50D4×C109—41no—
51C2×C4×C109—21no—
52C2×C4×C109—21no—
53C2×C4×C109C22×C411yescharacteristic
54C22⋊D109—42no—
55C2×Dic109—21no—
56C2×Dic109—21no—
57C2×Dic109C22⋊C411yescharacteristic
58D436—41no—
59C4×D109—21no—
60C4×D109—21no—
61C4×D109—42no—
62C4×D109—41no—
63C4×D109—41no—
64Dic218—42no—
65Dic218—41no—
66C22×C109—22no—
67C22×C109—42no—
68C22×C109C2×C22⋊C411yescharacteristic
69C2×D109—22no—
70C2×D109—22no—
71C2×D109—44no—
72C2×D109—42no—
73C2×D109—82no—
74C2×D109#9911yescharacteristic
75C2×D109#9911yescharacteristic
76#89137—84no—
77C4×C109—21no—
78C4×C109—41no—
79C4×C109—41no—
80Dic109—21no—
81Dic109—41no—
82Dic109—41no—
83C2×C109—21no—
84C2×C109—42no—
85C2×C109C2×#9911yescharacteristic
86D109—21no—
87D109—42no—
88D109#35412yescomplemented
89C2×#354—1091nomaximal, Sylow
90C109C2×#35411yescharacteristic, complemented, Sylow
91C2×#124—1091no—
92C2×#99—1092no—
93#354—1092no—
94#354—1092no—
95C22×Q8—1091no—
96C22×D4—1091no—
97#124—1091no—
98#124—1091no—
99#124—1091no—
100#124—2182no—
101#124#8913711yescharacteristic, complemented
102C2×C22⋊C4—2182no—
103C2×C22⋊C4—2182no—
104C2×C42—1091no—
105#99—1094no—
106#99—2182no—
107C24—1092no—
108C2×Q8—1091no—
109C2×Q8—4361no—
110C2×Q8—4361no—
111C2×Q8—2181no—
112C2×Q8C2×#8913711yescharacteristic, complemented
113C2×D4—1092no—
114C2×D4—1091no—
115C2×D4—4361no—
116C2×D4—4361no—
117C2×D4C2×#8913711yescharacteristic
118C22×C4—1091no—
119C22×C4—4362no—
120C22×C4—2181no—
121C22×C4—2181no—
122C22⋊C4—22no—
123C22⋊C4—4362no—
124C22⋊C4—2182no—
125C22⋊C4—2182no—
126C22⋊C4—2184no—
127C22⋊C4—2182no—
128C42—1091no—
129C42—2181no—
130C42C2×#8913711yescharacteristic, complemented
131C23—1091no—
132C23—1092no—
133C23—4364no—
134C23—2182no—
135C23—2182no—
136C23C22⋊#8913712yes—
137Q8—41no—
138Q8—4361no—
139Q8—4362no—
140D4—41no—
141D4—4361no—
142D4—4362no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—8724no—
146C2×C4—1091no—
147C2×C4—4361no—
148C2×C4—4361no—
149C2×C4—4362no—
150C2×C4—4364no—
151C2×C4—2181no—
152C2×C4—2181no—
153C2×C4—2181no—
154C2×C4—2181no—
155C2×C4C22×#8913711yescharacteristic
156C22—22no—
157C22—42no—
158C22—8722no—
159C22—1091no—
160C22—1091no—
161C22—4362no—
162C22—4364no—
163C22—2182no—
164C22—2182no—
165C22C2×C22⋊#8913711yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—8724no—
170C4—4361no—
171C4—4361no—
172C4—2181no—
173C2—21no—
174C2—42no—
175C2—1092no—
176C2—4362no—
177C2—2181no—
178C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#124×C109C4
socleC2×C109C2×#99
prime radicalC2×D4×D109C22
cosocleC2×D436C23
hypercenter#124#89137
hypocenterC109C2×#354
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20295933C2semidirect
3. #20295932C2semidirect
8. C2×Q8×D109C4semidirect
12. #16812686C4semidirect
15. C42×D109C4semidirect
16. #124×C109C4semidirect
22. C2×Dic218C2×C4semidirect
36. C4×Dic109C2×C4semidirect
37. C2×Q8×C109C2×C4semidirect
40. C42×C109C2×C4semidirect
88. D109#354semidirect
90. C109C2×#354semidirect
101. #124#89137semidirect
112. C2×Q8C2×#89137semidirect
130. C42C2×#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊#992
#202959332
#202959322
#124×D1092
C2×#354109