Finite Group Databasebeta

← #28378471

Subgroups — #28378471

Order 13952.

288 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊(C22⋊C8)C211yesmaximal, characteristic
2#20295968C211yesmaximal, characteristic
3#20295879C214yesmaximal, complemented
4C2×#16812679C211yesmaximal, characteristic
5C2×D109⋊C8—21no—
6C2×C4⋊#89137—21no—
7C2×C22⋊#89137—21no—
8#16812756C2212yes—
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4C2212yes—
11C22×Dic218C411yescharacteristic, complemented
12C22×C4×D109C2211yescharacteristic
13#16812679—21no—
14#16812679—21no—
15#16812679C412yescomplemented
16#16812679C2212yes—
17C2×C4⋊Dic109—21no—
18C2×#13182006—21no—
19C2×C4⋊C4×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21D109⋊C8—42no—
22C2×#11724786—22no—
23C4⋊#89137—42no—
24C22⋊#89137—42no—
25C23×D109D411yescharacteristic
26C2×Dic218—21no—
27C2×Dic218—42no—
28C2×Dic218C2×C412yes—
29C22×Dic109—21no—
30C22×Dic109D411yescharacteristic
31C2×C4×D109—42no—
32C2×C4×D109C2×C411yescharacteristic
33C2×C4×D109D411yescharacteristic
34C2×C4×D109D411yescharacteristic
35C2×C4×D109C2311yescharacteristic
36C4⋊Dic109—22no—
37C4⋊Dic109—22no—
38#13182006—22no—
39#13182006—22no—
40C22×C4×C109—21no—
41C22×C4×C109C2×C411yescharacteristic
42C4⋊C4×C109—21no—
43C4⋊C4×C109C2×C412yes—
44C23×C109C22⋊C411yescharacteristic
45C22×D109—41no—
46C22×D109—41no—
47C22×D109C22⋊C411yescharacteristic
48C22×D109SD1612yes—
49C22×D109Q1612yes—
50C22×D109C2×D411yescharacteristic
51C2×#89137—42no—
52C2×#89137—42no—
53C2×#89137—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—41no—
61C2×Dic109—42no—
62C2×Dic109C22⋊C411yescharacteristic
63C2×Dic109C2×D411yescharacteristic
64C4×D109—22no—
65C4×D109—22no—
66C4×D109—41no—
67C4×D109—41no—
68Dic218—42no—
69Dic218—42no—
70#11724786—42no—
71C22×C109—21no—
72C22×C109—21no—
73C22×C109#10312yes—
74C22×C109C2×C22⋊C411yescharacteristic
75C2×D109—21no—
76C2×D109—21no—
77C2×D109—21no—
78C2×D109—21no—
79C2×D109—42no—
80C2×D109—42no—
81C2×D109#10312yes—
82C2×D109C2×SD1611yescharacteristic
83C2×D109C2×Q1611yescharacteristic
84#89137—82no—
85C4×C109—21no—
86C4×C109—42no—
87C4×C109#9912yes—
88Dic109—21no—
89Dic109—21no—
90Dic109—42no—
91C2×C109—22no—
92C2×C109C2×#9911yescharacteristic
93C2×C109C2×#10311yescharacteristic
94C2×C109#42111yescharacteristic
95D109—21no—
96D109—22no—
97D109—21no—
98#1520—1091nomaximal, Sylow
99C109#152011yescharacteristic, complemented, Sylow
100C2×C22⋊Q8—1091no—
101C2×C22⋊C8—1091no—
102#380—1091no—
103#328—1094no—
104C22×Q8—1091no—
105C23×C4—1091no—
106C22×C8—2181no—
107C22⋊Q8—1092no—
108C22⋊Q8—1092no—
109C22⋊Q8—2181no—
110C22⋊Q8—2181no—
111C2×C4⋊C4—2181no—
112C2×C4⋊C4—2181no—
113C2×C4⋊C4#8913711yescharacteristic, complemented
114C2×C22⋊C4—2181no—
115C2×C22⋊C4—2181no—
116C22⋊C8—1092no—
117C22⋊C8—2181no—
118#95—1092no—
119C24—1091no—
120C2×Q8—1092no—
121C2×Q8—4362no—
122C2×Q8—2181no—
123C22×C4—21no—
124C22×C4—1091no—
125C22×C4—1091no—
126C22×C4—1091no—
127C22×C4—1091no—
128C22×C4—1091no—
129C22×C4—4362no—
130C22×C4—2182no—
131C22×C4—2181no—
132C22×C4C2×#8913711yescharacteristic
133C2×C8—4362no—
134C2×C8—2182no—
135C4⋊C4—21no—
136C4⋊C4—4362no—
137C4⋊C4—2182no—
138C4⋊C4—2182no—
139C4⋊C4C2×#8913712yes—
140C22⋊C4—4362no—
141C22⋊C4—2182no—
142C22⋊C4—2182no—
143C23—1092no—
144C23—1092no—
145C23—1091no—
146C23—1091no—
147C23—4361no—
148C23—4361no—
149C23C22⋊#8913711yescharacteristic
150Q8—4362no—
151Q8—4362no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—1091no—
156C2×C4—1091no—
157C2×C4—4361no—
158C2×C4—4361no—
159C2×C4—4362no—
160C2×C4—4362no—
161C2×C4—4362no—
162C2×C4—4362no—
163C2×C4—4361no—
164C2×C4—2182no—
165C2×C4—2182no—
166C2×C4—2182no—
167C2×C4C22⋊#8913711yescharacteristic
168C2×C4C22×#8913711yescharacteristic
169C8—4362no—
170C22—21no—
171C22—21no—
172C22—1091no—
173C22—1092no—
174C22—1091no—
175C22—4362no—
176C22—4362no—
177C22—2181no—
178C22—2181no—
179C22—2181no—
180C22—2181no—
181C22#1681276212yes—
182C22C2×C22⋊#8913711yescharacteristic
183C4—21no—
184C4—42no—
185C4—8722no—
186C4—4362no—
187C4—2181no—
188C4—2181no—
189C4#1681275812yes—
190C2—22no—
191C2—2181no—
192C2—2182no—
193C2—2181no—
194C2C2×#1681275811yescharacteristic
195C2C2×#1681276211yescharacteristic
196C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC2×C4⋊C4×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC2×C4×D109C23
hypercenterC2×C4⋊C4#89137
hypocenterC109#1520
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20295879C2semidirect
11. C22×Dic218C4semidirect
15. #16812679C4semidirect
19. C2×C4⋊C4×C109C4semidirect
99. C109#1520semidirect
113. C2×C4⋊C4#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
#202959682
#202958792
C2×#168126792
#1520109