Finite Group Databasebeta

← #28378442

Subgroups — #28378442

Order 13952.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#20295998C211yesmaximal, characteristic
2#20295969C211yesmaximal, characteristic
3#20295880C214yesmaximal, complemented
4C2×#16812675C211yesmaximal, characteristic
5C2×#13182038—21no—
6C2×C22⋊#89137—21no—
7#16812756—21no—
8#16812756C2212yes—
9(C2×#89137)⋊C4—21no—
10(C2×#89137)⋊C4C2212yes—
11C22×C4×D109C2211yescharacteristic
12#16812675—21no—
13#16812675—21no—
14#16812675C412yescomplemented
15#16812675C2212yes—
16C2×C4⋊Dic109C411yescharacteristic, complemented
17C2×#13182008—21no—
18C2×#13182006—21no—
19C22×Q8×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21#13182038—42no—
22C2×#11724786—22no—
23C22⋊#89137—42no—
24C23×D109D411yescharacteristic
25C22×Dic109—21no—
26C22×Dic109D411yescharacteristic
27C2×C4×D109—42no—
28C2×C4×D109C2×C411yescharacteristic
29C2×C4×D109D411yescharacteristic
30C2×C4×D109D411yescharacteristic
31C2×C4×D109C2311yescharacteristic
32C4⋊Dic109—21no—
33C4⋊Dic109C2×C412yes—
34#13182008—22no—
35#13182008—22no—
36#13182006—22no—
37#13182006—22no—
38C2×Q8×C109—21no—
39C2×Q8×C109—42no—
40C2×Q8×C109C2×C412yes—
41C22×C4×C109—21no—
42C22×C4×C109C2×C411yescharacteristic
43C23×C109C22⋊C411yescharacteristic
44C22×D109—22no—
45C22×D109—22no—
46C22×D109—41no—
47C22×D109C22⋊C411yescharacteristic
48C22×D109C2×D411yescharacteristic
49C2×#89137—42no—
50C2×#89137—42no—
51C2×#89137—42no—
52Q8×C109—42no—
53Q8×C109—42no—
54C2×C4×C109—22no—
55C2×C4×C109—22no—
56C2×C4×C109—42no—
57C2×C4×C109C22⋊C411yescharacteristic
58C2×C4×C109C22×C411yescharacteristic
59C2×Dic109—22no—
60C2×Dic109—22no—
61C2×Dic109—42no—
62C2×Dic109C22⋊C411yescharacteristic
63C2×Dic109C2×D411yescharacteristic
64C4×D109—41no—
65C4×D109—41no—
66C4×D109—41no—
67C4×D109—41no—
68#11724786—42no—
69C22×C109—21no—
70C22×C109—21no—
71C22×C109#9912yes—
72C22×C109C2×C22⋊C411yescharacteristic
73C2×D109—21no—
74C2×D109—21no—
75C2×D109—21no—
76C2×D109—21no—
77C2×D109—21no—
78C2×D109—44no—
79C2×D109#13712yes—
80#89137—82no—
81C4×C109—21no—
82C4×C109—21no—
83C4×C109—42no—
84Dic109—21no—
85Dic109—21no—
86Dic109—42no—
87C2×C109—22no—
88C2×C109C2×#9911yescharacteristic
89C2×C109#41911yescharacteristic
90C2×C109#42111yescharacteristic
91D109—22no—
92D109—41no—
93#1511—1091nomaximal, Sylow
94C109#151111yescharacteristic, complemented, Sylow
95C2×C22⋊Q8—1091no—
96#407—1091no—
97#381—1091no—
98#328—1094no—
99C22×Q8#8913711yescharacteristic, complemented
100C23×C4—1091no—
101C2×M16—2181no—
102C22⋊Q8—1092no—
103C22⋊Q8—1092no—
104C22⋊Q8—2181no—
105C22⋊Q8—2181no—
106C2×C4⋊C4—1091no—
107C2×C4⋊C4—2181no—
108C2×C22⋊C4—2181no—
109C2×C22⋊C4—2181no—
110C22⋊C8—1092no—
111C22⋊C8—2181no—
112#95—1092no—
113#95—2181no—
114C24—1091no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×#8913712yes—
118C22×C4—21no—
119C22×C4—1091no—
120C22×C4—1091no—
121C22×C4—1091no—
122C22×C4—1091no—
123C22×C4—1091no—
124C22×C4—4362no—
125C22×C4—2182no—
126C22×C4—2181no—
127C22×C4C2×#8913711yescharacteristic
128M16—4362no—
129C2×C8—2182no—
130C4⋊C4—1092no—
131C4⋊C4—2182no—
132C4⋊C4—2182no—
133C4⋊C4—2181no—
134C22⋊C4—4362no—
135C22⋊C4—2182no—
136C22⋊C4—2182no—
137C23—1091no—
138C23—1091no—
139C23—4361no—
140C23—2182no—
141C23—2182no—
142C23C22⋊#8913711yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—1091no—
149C2×C4—1091no—
150C2×C4—4361no—
151C2×C4—4361no—
152C2×C4—4361no—
153C2×C4—4361no—
154C2×C4—4362no—
155C2×C4—4362no—
156C2×C4—4362no—
157C2×C4—4362no—
158C2×C4—2182no—
159C2×C4—2182no—
160C2×C4C22⋊#8913711yescharacteristic
161C2×C4C22×#8913711yescharacteristic
162C8—4362no—
163C22—21no—
164C22—21no—
165C22—1092no—
166C22—4364no—
167C22—2181no—
168C22—2181no—
169C22—2181no—
170C22—2181no—
171C22—2181no—
172C22#1681275712yes—
173C22C2×C22⋊#8913711yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—8722no—
178C4—4362no—
179C4—2181no—
180C4—2181no—
181C2—22no—
182C2—4361no—
183C2—2182no—
184C2C2×#1681275711yescharacteristic
185C2#2029601911yescharacteristic
186C2#2029602511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22×Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC2×C4×D109C23
hypercenterC22×Q8#89137
hypocenterC109#1511
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20295880C2semidirect
14. #16812675C4semidirect
16. C2×C4⋊Dic109C4semidirect
19. C22×Q8×C109C4semidirect
94. C109#1511semidirect
99. C22×Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202959982
#202959692
#202958802
C2×#168126752
#1511109