Finite Group Databasebeta

← #28364683

Subgroups — #28364683

Order 13952.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#20295590C211yesmaximal, characteristic
2#20295000C211yesmaximal, characteristic, complemented
3#20294814C214yesmaximal, complemented
4#413×C109C211yesmaximal, characteristic, complemented
5C22×D436C411yescharacteristic, complemented
6#16812695—21no—
7#16812688—21no—
8#16812673—21no—
9#16812673—21no—
10#16812667C412yescomplemented
11#16812667C2212yes—
12#16812658—21no—
13#16812658C411yescharacteristic, complemented
14#16812614—21no—
15#16812614C2212yes—
16C2×#41×C109C2211yescharacteristic
17C2×M16×C109—21no—
18#101×C109—21no—
19#100×C109C2212yes—
20C23×D109C2×C412yes—
21C2×D436—21no—
22C2×D436—21no—
23C2×D436—41no—
24C2×D436—41no—
25C2×C22⋊D109—22no—
26C2×C22⋊D109—22no—
27C4⋊Dic109—21no—
28C4⋊Dic109—41no—
29C22⋊Dic109—22no—
30C22⋊Dic109C2×C412yes—
31#13182006—22no—
32#13182006—22no—
33C4×Dic109—21no—
34C4×Dic109—41no—
35#41×C109—41no—
36#41×C109—41no—
37C2×Q8×C109D411yescharacteristic
38C2×D4×C109C2×C411yescharacteristic
39C2×D4×C109D411yescharacteristic
40C2×D4×C109C2311yescharacteristic
41C22×C4×C109C2×C411yescharacteristic
42C22×C4×C109D411yescharacteristic
43C22×C4×C109D411yescharacteristic
44M16×C109—22no—
45M16×C109—41no—
46C2×C8×C109—41no—
47C23×C109C22⋊C411yescharacteristic
48C23×C109C22×C411yescharacteristic
49C23×C109C2×D411yescharacteristic
50C22×D109—22no—
51C22×D109—42no—
52C22×D109—82no—
53Q8×C109—41no—
54D4×C109—41no—
55D4×C109—41no—
56D4×C109—41no—
57C2×C4×C109—21no—
58C2×C4×C109—21no—
59C2×C4×C109—21no—
60C2×C4×C109—21no—
61C2×C4×C109—21no—
62C2×C4×C109—21no—
63C2×C4×C109C22⋊C411yescharacteristic
64C2×C4×C109C22⋊C411yescharacteristic
65C2×C4×C109C22⋊C411yescharacteristic
66C2×C4×C109C2×D411yescharacteristic
67C22⋊D109—82no—
68C22⋊D109—82no—
69C2×Dic109—22no—
70C2×Dic109—42no—
71D436—42no—
72D436—41no—
73D436—41no—
74C8×C109—42no—
75C22×C109—21no—
76C22×C109—21no—
77C22×C109—21no—
78C22×C109—41no—
79C22×C109—41no—
80C22×C109—41no—
81C22×C109C2×C22⋊C411yescharacteristic
82C2×D109—42no—
83C2×D109—42no—
84C2×D109—42no—
85C2×D109—84no—
86C4×C109—21no—
87C4×C109—21no—
88C4×C109—21no—
89C4×C109#9911yescharacteristic
90C4×C109#9911yescharacteristic
91Dic109—42no—
92Dic109—82no—
93C2×C109—21no—
94C2×C109—41no—
95C2×C109—41no—
96C2×C109—41no—
97C2×C109C2×#9911yescharacteristic
98D109—82no—
99#2122—1091nomaximal, Sylow
100C109#212211yescharacteristic, complemented, Sylow
101#535—1091no—
102#413D10911yescharacteristic, complemented
103#410—1091no—
104AΣL(1,𝔽16)—1094no—
105C2×#41C2×D10911yescharacteristic
106C22×D4—1091no—
107C2×M16—21no—
108#127—2181no—
109#124—2181no—
110#121—2181no—
111#121—2181no—
112#120—1092no—
113#120—1092no—
114#117—1091no—
115#117—2181no—
116#101—21no—
117#100C2×D10912yes—
118#99—1092no—
119#99—2181no—
120C24—1092no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D10911yescharacteristic
124C2×D4—4361no—
125C2×D4—4361no—
126C2×D4—2182no—
127C2×D4—2182no—
128C2×D4—2181no—
129C2×D4—2181no—
130C2×D4C4×D10911yescharacteristic
131C2×D4D43611yescharacteristic
132C2×D4C22×D10911yescharacteristic
133C22×C4C4×D10911yescharacteristic
134C22×C4D43611yescharacteristic
135C22×C4C22⋊D10911yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—4361no—
140C4⋊C4—2181no—
141C22⋊C4—1092no—
142C22⋊C4—2182no—
143C22⋊C4—2182no—
144C22⋊C4—2182no—
145C42—4361no—
146C42—2181no—
147C23—8722no—
148C23—4362no—
149C23—2182no—
150C23#1318200611yescharacteristic
151C23C2×C4×D10911yescharacteristic
152C23C2×D43611yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—8722no—
158D4—8722no—
159D4—4362no—
160D4—4361no—
161D4—4361no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—4362no—
169C2×C4—2182no—
170C2×C4#1318200611yescharacteristic
171C2×C4#1318200611yescharacteristic
172C2×C4#1318200611yescharacteristic
173C2×C4C2×C22⋊D10911yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—8724no—
182C22—4362no—
183C22—4362no—
184C22—4362no—
185C22C2×#1318200611yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—8722no—
190C4—4362no—
191C4#1681261311yescharacteristic
192C4#1681261311yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—8722no—
198C2C2×#1681261311yescharacteristic
subgroupG/H
centerC2C2×#16812613
commutatorC23×C109C22×C4
FrattiniC2×D4C22×D109
Fitting#413×C109C2
socleC2×C109C2×#99
prime radical#20295590C2
cosocleC2×D4×C109C23
hypercenter#413D109
hypocenterC109#2122
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20295000C2semidirect
3. #20294814C2semidirect
4. #413×C109C2semidirect
5. C22×D436C4semidirect
10. #16812667C4semidirect
13. #16812658C4semidirect
100. C109#2122semidirect
102. #413D109semidirect

Maximal subgroups

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

subgroupindex
#202955902
#202950002
#202948142
#413×C1092
#2122109