Finite Group Databasebeta

← #28363424

Subgroups — #28363424

Order 13952.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#16812699C211yesmaximal, characteristic
2#20295503C211yesmaximal, characteristic
3#20295036C211yesmaximal, characteristic
4#20295034C211yesmaximal, characteristic
5C2×#16812617C211yesmaximal, characteristic
6#20294786C211yesmaximal, characteristic
7#339×C109C211yesmaximal, characteristic
8C22×Dic218C2211yescharacteristic
9C22×Dic218C2211yescharacteristic
10C2×#13182013C2211yescharacteristic
11#16812699—22no—
12#16812699C414yescomplemented
13C4×Dic218—22no—
14#16812664—22no—
15#16812658C2211yescharacteristic
16#16812655—22no—
17C2×C4⋊Dic109—22no—
18#16812624—22no—
19#16812622—22no—
20#16812617—22no—
21C2×M16×C109C2211yescharacteristic
22#117×C109C2211yescharacteristic
23C2×C42×C109C2211yescharacteristic
24C2×Dic218—21no—
25C2×Dic218—21no—
26C2×Dic218—21no—
27C2×Dic218—21no—
28C2×Dic218—21no—
29C2×Dic218—21no—
30C2×Dic218—21no—
31C2×Dic218—21no—
32C2×Dic218—21no—
33C2×Dic218C2×C412yes—
34C2×Dic218C2×C412yes—
35C2×Dic218D412yes—
36C22×Dic109—21no—
37C22×Dic109D412yes—
38#13182013—21no—
39#13182013—21no—
40C2×#11724784—21no—
41C4⋊Dic109—28no—
42C4⋊Dic109D412yes—
43#13182008—24no—
44C22⋊Dic109—21no—
45C4×Dic109—24no—
46C4×Dic109—21no—
47C22×C4×C109—21no—
48C22×C4×C109C2311yescharacteristic
49M16×C109—21no—
50M16×C109—21no—
51C2×C8×C109—21no—
52C4⋊C4×C109D412yes—
53C22⋊C4×C109—21no—
54C42×C109—21no—
55C42×C109—21no—
56C42×C109C2×C412yes—
57C23×C109C2×D411yescharacteristic
58C2×C4×C109—24no—
59C2×C4×C109—22no—
60C2×C4×C109—22no—
61C2×C4×C109C22×C411yescharacteristic
62C2×C4×C109C2×D411yescharacteristic
63C2×C4×C109C2×D411yescharacteristic
64C2×C4×C109C2×D411yescharacteristic
65C2×C4×C109#4111yescharacteristic
66C2×C4×C109#4111yescharacteristic
67C2×Dic109—22no—
68C2×Dic109—22no—
69C2×Dic109—22no—
70C2×Dic109—22no—
71C2×Dic109—41no—
72C2×Dic109—41no—
73C2×Dic109C22⋊C414yes—
74Dic218—22no—
75Dic218—22no—
76Dic218—22no—
77Dic218—22no—
78Dic218—22no—
79Dic218—22no—
80Dic218—41no—
81Dic218—41no—
82C8×C109—41no—
83#11724784—41no—
84C22×C109—22no—
85C22×C109C2×C22⋊C411yescharacteristic
86C22×C109#12011yescharacteristic
87C22×C109#12311yescharacteristic
88C4×C109—24no—
89C4×C109—42no—
90C4×C109C4×D411yescharacteristic
91C4×C109C4×D411yescharacteristic
92C4×C109#12111yescharacteristic
93C4×C109#12111yescharacteristic
94Dic109—24no—
95Dic109—42no—
96Dic109—42no—
97C2×C109—21no—
98C2×C109—21no—
99C2×C109#38611yescharacteristic
100C2×C109#45612yes—
101#1886—1091nomaximal, Sylow
102C109#188611yescharacteristic, complemented, Sylow
103C2×C4⋊Q8—1091no—
104#519—1091no—
105#419—1091no—
106#419—1091no—
107C2×#101—1091no—
108#339—1091no—
109#339D10911yescharacteristic
110C22×Q8—1091no—
111C22×Q8—1091no—
112C2×M16—1091no—
113C2×M16C2×D10911yescharacteristic
114C4⋊Q8—1094no—
115C4⋊Q8—2182no—
116C4×Q8—2182no—
117C4×Q8—2182no—
118#117—1091no—
119#117—2182no—
120#117C2×D10911yescharacteristic
121C2×C4⋊C4—2182no—
122C2×C42C2×D10911yescharacteristic
123#103—2182no—
124#103—2182no—
125#101—2182no—
126C2×Q8—1092no—
127C2×Q8—1092no—
128C2×Q8—1092no—
129C2×Q8—2181no—
130C2×Q8—2181no—
131C2×Q8—2181no—
132C2×Q8—2181no—
133C2×Q8—2181no—
134C2×Q8—2181no—
135C2×Q8—2181no—
136C2×Q8—2181no—
137C2×Q8—2181no—
138C22×C4—21no—
139C22×C4—1092no—
140C22×C4—2181no—
141C22×C4C22×D10911yescharacteristic
142M16—21no—
143M16—21no—
144M16—2181no—
145M16—2181no—
146C2×C8—21no—
147C2×C8—2181no—
148C4⋊C4—1092no—
149C4⋊C4—2188no—
150C4⋊C4—2184no—
151C4⋊C4D43612yes—
152C22⋊C4—21no—
153C22⋊C4—2181no—
154C42—21no—
155C42—21no—
156C42—2184no—
157C42—2181no—
158C42C4×D10912yes—
159C23D4×D10911yescharacteristic
160Q8—4361no—
161Q8—4361no—
162Q8—2182no—
163Q8—2182no—
164Q8—2182no—
165Q8—2182no—
166Q8—2182no—
167Q8—2182no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—1094no—
172C2×C4—4361no—
173C2×C4—4361no—
174C2×C4—2182no—
175C2×C4—2182no—
176C2×C4—2182no—
177C2×C4—2182no—
178C2×C4C2×C4×D10911yescharacteristic
179C2×C4D4×D10911yescharacteristic
180C2×C4D4×D10911yescharacteristic
181C2×C4C2×D43611yescharacteristic
182C2×C4#1318202911yescharacteristic
183C2×C4#1318203011yescharacteristic
184C8—41no—
185C8—4361no—
186C22—22no—
187C22C22⋊C4×D10911yescharacteristic
188C22#1681266911yescharacteristic
189C22#1681268411yescharacteristic
190C4—24no—
191C4—42no—
192C4—4362no—
193C4—4362no—
194C4—2184no—
195C4#1681266111yescharacteristic
196C4C4×D43611yescharacteristic
197C4#1681267211yescharacteristic
198C4#1681267411yescharacteristic
199C2—21no—
200C2—21no—
201C2#2029490411yescharacteristic
202C2#2029521012yes—
subgroupG/H
centerC22#16812669
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#339×C109C2
socleC22×C109#120
prime radicalC23×C109C2×D4
cosocleC22×C4×C109C23
hypercenter#339D109
hypocenterC109#1886
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
12. #16812699C4semidirect
102. C109#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#168126992
#202955032
#202950362
#202950342
C2×#168126172
#202947862
#339×C1092
#1886109