Finite Group Databasebeta

← #28379310

Subgroups — #28379310

Order 13952.

286 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 52 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, complemented
2#20295932C212yesmaximal, complemented
3#20295931C212yesmaximal, complemented
4#20295594C211yesmaximal, characteristic
5C2×C22⋊#89137—22no—
6#16812758C2214yes—
7#16812757—22no—
8C2×#13182030C411yescharacteristic, complemented
9C2×D4×D109C2211yescharacteristic
10#16812696C2211yescharacteristic
11#16812695C411yescharacteristic, complemented
12#16812688C2211yescharacteristic
13#16812672—22no—
14#16812669—22no—
15#16812656C411yescharacteristic, complemented
16#124×C109C411yescharacteristic, complemented
17C22×#89137—42no—
18C22⋊#89137—24no—
19C22⋊#89137—42no—
20C23×D109D412yes—
21#13182030—42no—
22C2×D436—21no—
23C2×D436—22no—
24C2×D436C2×C411yescharacteristic, complemented
25C2×D436C2311yescharacteristic
26C2×C22⋊D109—22no—
27C2×C22⋊D109D412yes—
28D4×D109—42no—
29C2×C4×D109—21no—
30C2×C4×D109C2×C411yescharacteristic
31#13182008—21no—
32#13182006—21no—
33#13182006—22no—
34C4×Dic109C2×C411yescharacteristic, complemented
35C2×Q8×C109C2×C411yescharacteristic, complemented
36C2×D4×C109C2×C411yescharacteristic
37C22⋊C4×C109—22no—
38C42×C109C2×C411yescharacteristic, complemented
39C23×C109C22⋊C412yes—
40C22×D109—22no—
41C22×D109—22no—
42C22×D109—21no—
43C22×D109—82no—
44C22×D109C22⋊C411yescharacteristic
45C22×D109C2×D412yes—
46C2×#89137—44no—
47C2×#89137—84no—
48Q8×C109—41no—
49D4×C109—41no—
50C2×C4×C109—21no—
51C2×C4×C109—21no—
52C2×C4×C109C22×C411yescharacteristic
53C22⋊D109—42no—
54C22⋊D109—82no—
55C2×Dic109—21no—
56C2×Dic109C22⋊C411yescharacteristic
57D436—41no—
58D436—41no—
59D436—42no—
60D436—82no—
61C4×D109—22no—
62C4×D109—42no—
63C22×C109—22no—
64C22×C109—42no—
65C22×C109C2×C22⋊C411yescharacteristic
66C2×D109—22no—
67C2×D109—41no—
68C2×D109—44no—
69C2×D109—42no—
70C2×D109—42no—
71C2×D109—41no—
72C2×D109—82no—
73C2×D109#9912yes—
74#89137—84no—
75C4×C109—21no—
76C4×C109—41no—
77C4×C109—41no—
78Dic109—21no—
79Dic109—41no—
80C2×C109—21no—
81C2×C109—42no—
82C2×C109C2×#9911yescharacteristic
83D109—22no—
84D109—42no—
85D109—81no—
86#2127—1091nomaximal, Sylow
87C109#212711yescharacteristic, complemented, Sylow
88#535—1091no—
89C2×#99—1092no—
90#354—1092no—
91#353—1092no—
92C2×#41—1091no—
93C22×D4—1091no—
94#127—1091no—
95#127—1091no—
96#124—1091no—
97#124#8913711yescharacteristic, complemented
98#121—2182no—
99#120—2182no—
100#117—1091no—
101C2×C22⋊C4—2182no—
102#99—1094no—
103#99—2182no—
104C24—1092no—
105#41—4362no—
106C2×Q8C2×#8913711yescharacteristic, complemented
107C2×D4—1092no—
108C2×D4—1091no—
109C2×D4—1091no—
110C2×D4—4362no—
111C2×D4—2182no—
112C2×D4—2181no—
113C2×D4—2182no—
114C2×D4C2×#8913711yescharacteristic
115C22×C4—1091no—
116C22×C4—4362no—
117C22×C4—2181no—
118C4⋊C4—2181no—
119C22⋊C4—22no—
120C22⋊C4—4362no—
121C22⋊C4—2181no—
122C22⋊C4—2182no—
123C22⋊C4—2184no—
124C42—1091no—
125C42C2×#8913711yescharacteristic, complemented
126C23—8722no—
127C23—1091no—
128C23—1092no—
129C23—2182no—
130C23—2182no—
131C23—2181no—
132C23C22⋊#8913712yes—
133Q8—41no—
134D4—41no—
135D4—8722no—
136D4—8722no—
137D4—4361no—
138D4—4361no—
139D4—4362no—
140D4—4362no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—8724no—
144C2×C4—1091no—
145C2×C4—4362no—
146C2×C4—4364no—
147C2×C4—2182no—
148C2×C4—2181no—
149C2×C4C22×#8913711yescharacteristic
150C22—22no—
151C22—42no—
152C22—8722no—
153C22—1092no—
154C22—4362no—
155C22—4362no—
156C22—4361no—
157C22—4364no—
158C22—4361no—
159C22—2182no—
160C22C2×C22⋊#8913711yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—8724no—
165C4—4361no—
166C4—2181no—
167C2—21no—
168C2—42no—
169C2—8721no—
170C2—4362no—
171C2—2182no—
172C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#124×C109C4
socleC2×C109C2×#99
prime radical#20295594C2
cosocleC2×D436C23
hypercenter#124#89137
hypocenterC109#2127
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
1. D109⋊#99C2semidirect
2. #20295932C2semidirect
3. #20295931C2semidirect
8. C2×#13182030C4semidirect
11. #16812695C4semidirect
15. #16812656C4semidirect
16. #124×C109C4semidirect
24. C2×D436C2×C4semidirect
34. C4×Dic109C2×C4semidirect
35. C2×Q8×C109C2×C4semidirect
38. C42×C109C2×C4semidirect
87. C109#2127semidirect
97. #124#89137semidirect
106. C2×Q8C2×#89137semidirect
125. C42C2×#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊#992
#202959322
#202959312
#202955942
#2127109