Finite Group Databasebeta

← #28379074

Subgroups — #28379074

Order 13952.

312 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 66 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#20296106C211yesmaximal, characteristic
2#20296019C212yesmaximal, complemented
3#20296007C211yesmaximal, characteristic
4#20295893C212yesmaximal, complemented
5#20295601C211yesmaximal, characteristic
6#16812787—24no—
7D109⋊M16—22no—
8D109⋊M16C2212yes—
9#16812778C2212yes—
10#16812762—24no—
11#16812758—22no—
12C2×#13182030C2211yescharacteristic
13C2×Q8×D109C2211yescharacteristic
14#16812697C411yescharacteristic, complemented
15#16812688C411yescharacteristic, complemented
16#16812687C411yescharacteristic, complemented
17#16812684—22no—
18#16812677—22no—
19#16812656C2211yescharacteristic
20C4⋊Q8×C109C411yescharacteristic, complemented
21#13182039—24no—
22#13182039—22no—
23#13182038—22no—
24D109⋊C8—24no—
25D109⋊C8D414yes—
26C4⋊#89137—22no—
27C22⋊#89137—22no—
28C4×#89137—22no—
29#13182030—22no—
30#13182030—22no—
31C2×Dic218—21no—
32C2×Dic218C2×C411yescharacteristic
33Q8×D109—22no—
34Q8×D109—22no—
35C2×D436C2×C411yescharacteristic
36C2×D436D412yes—
37C2×C4×D109—21no—
38C2×C4×D109D412yes—
39C2×C4×D109C2311yescharacteristic
40#13182008—22no—
41#13182008—21no—
42#13182006—22no—
43#13182006—22no—
44#13182006—21no—
45C4×Dic109C2×C411yescharacteristic
46C2×Q8×C109C2×C411yescharacteristic
47C2×Q8×C109C2×C411yescharacteristic
48C4⋊C4×C109—22no—
49C42×C109C2×C411yescharacteristic
50C22×D109C22⋊C412yes—
51C22×D109C2×D411yescharacteristic
52C2×#89137—42no—
53Q8×C109—22no—
54Q8×C109—22no—
55C2×C4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109C22⋊C412yes—
58C2×C4×C109C22×C411yescharacteristic
59C2×Dic109—21no—
60C2×Dic109—21no—
61C2×Dic109C2×D411yescharacteristic
62D436—24no—
63D436—22no—
64C4×D109—22no—
65C4×D109—22no—
66C4×D109—42no—
67C4×D109C2×D412yes—
68C4×D109#4112yes—
69Dic218—22no—
70Dic218—41no—
71Dic218—41no—
72#11724786—24no—
73C22×C109C2×C22⋊C411yescharacteristic
74C2×D109—22no—
75C2×D109—41no—
76C2×D109—42no—
77C2×D109#12112yes—
78#89137—82no—
79C4×C109—22no—
80C4×C109—41no—
81C4×C109—41no—
82C4×C109C4×D412yes—
83Dic109—41no—
84Dic109—41no—
85Dic109#12011yescharacteristic
86Dic109#12311yescharacteristic
87C2×C109—21no—
88C2×C109#38611yescharacteristic
89D109—22no—
90D109—42no—
91#1909—1091nomaximal, Sylow
92C109#190911yescharacteristic, complemented, Sylow
93#568—1091no—
94#536—1091no—
95#419—1092no—
96#410—1091no—
97H(3,ℤ/4)—1092no—
98C2×#41—1091no—
99C22×Q8—1091no—
100#131—2184no—
101C2×M16—1092no—
102C2×M16—2182no—
103C4⋊Q8—1091no—
104C4⋊Q8#8913711yescharacteristic, complemented
105#124—1091no—
106#124—1091no—
107#123—2182no—
108C22⋊Q8—2182no—
109#117—1091no—
110#117—1092no—
111#103—2184no—
112#99—2182no—
113#41—2182no—
114#41—2182no—
115C2×Q8—1091no—
116C2×Q8—2181no—
117C2×Q8—2182no—
118C2×Q8—2182no—
119C2×Q8C2×#8913711yescharacteristic
120C2×Q8C2×#8913711yescharacteristic
121C2×D4—1092no—
122C2×D4—1091no—
123C22×C4—1091no—
124C22×C4—1092no—
125C22×C4—2181no—
126M16—2182no—
127M16—2182no—
128M16—2184no—
129C2×C8—1094no—
130C2×C8—2184no—
131C4⋊C4—22no—
132C4⋊C4—2181no—
133C4⋊C4—2182no—
134C4⋊C4—2182no—
135C22⋊C4—2181no—
136C22⋊C4—2182no—
137C22⋊C4—2182no—
138C22⋊C4—2182no—
139C42—1091no—
140C42—2182no—
141C42C2×#8913711yescharacteristic
142C23—1091no—
143C23—1092no—
144Q8—22no—
145Q8—22no—
146Q8—4361no—
147Q8—4361no—
148Q8—2182no—
149D4—2184no—
150D4—2182no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1091no—
154C2×C4—1092no—
155C2×C4—1092no—
156C2×C4—4362no—
157C2×C4—4362no—
158C2×C4—2182no—
159C2×C4—2182no—
160C2×C4—2181no—
161C2×C4—2181no—
162C2×C4C22⋊#8913712yes—
163C2×C4C22×#8913711yescharacteristic
164C8—2184no—
165C22—1092no—
166C22—4362no—
167C22—4361no—
168C22—2182no—
169C22C2×C22⋊#8913711yescharacteristic
170C4—22no—
171C4—41no—
172C4—41no—
173C4—8722no—
174C4—1091no—
175C4—1091no—
176C4—4361no—
177C4—4361no—
178C4D4×#8913712yes—
179C2—21no—
180C2—4362no—
181C2—2182no—
182C2#2029597411yescharacteristic
subgroupG/H
centerC2#20295974
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC4⋊Q8×C109C4
socleC2×C109#386
prime radicalC2×#13182030C22
cosocleC2×C4×D109C23
hypercenterC4⋊Q8#89137
hypocenterC109#1909
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20296019C2semidirect
4. #20295893C2semidirect
14. #16812697C4semidirect
15. #16812688C4semidirect
16. #16812687C4semidirect
20. C4⋊Q8×C109C4semidirect
92. C109#1909semidirect
104. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202961062
#202960192
#202960072
#202958932
#202956012
#1909109