Finite Group Databasebeta

← #28365006

Subgroups — #28365006

Order 13952.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#20295827C211yesmaximal, characteristic
2#20295469C211yesmaximal, characteristic, complemented
3C2×#16812643C211yesmaximal, characteristic, complemented
4#20295302C211yesmaximal, characteristic, complemented
5#20294832C211yesmaximal, characteristic, complemented
6#20294811C211yesmaximal, characteristic, complemented
7#361×C109C211yesmaximal, characteristic, complemented
8#16812744—21no—
9#16812732—21no—
10#16812725—21no—
11#16812723—22no—
12#16812721C2211yescharacteristic
13#16812718—21no—
14C2×Dic436C2211yescharacteristic, complemented
15D8×D109—21no—
16#16812705C2211yescharacteristic
17Dic872—21no—
18#16812644—21no—
19#16812643—22no—
20C2×#13182004C2211yescharacteristic, complemented
21#16812617—21no—
22C2×D8×C109C2211yescharacteristic
23M32×C109C2211yescharacteristic, complemented
24#108×C109C2211yescharacteristic, complemented
25#13182029—21no—
26#13182029D411yescharacteristic
27#13182028—22no—
28#13182028—41no—
29C2×Dic218—21no—
30C2×C22⋊D109—42no—
31D4×D109—22no—
32D4×D109—41no—
33Dic436—21no—
34Dic436—21no—
35Dic436—41no—
36Dic436D411yescharacteristic
37#13182018—21no—
38#13182018—41no—
39#13182017—22no—
40D872D411yescharacteristic
41#13182014—22no—
42#13182013D411yescharacteristic
43#13182012—21no—
44C8×D109—21no—
45#13182004—21no—
46C2×D4×C109—21no—
47D8×C109—21no—
48D8×C109D412yes—
49M16×C109—21no—
50C2×C8×C109C2311yescharacteristic
51C16×C109—21no—
52C23×C109—41no—
53C22×D109—42no—
54D4×C109—22no—
55D4×C109—41no—
56C2×C4×C109C2×D411yescharacteristic
57C22⋊D109—21no—
58C22⋊D109—42no—
59C22⋊D109—42no—
60C22⋊D109—41no—
61C2×Dic109—42no—
62C2×Dic109—41no—
63D436—21no—
64D436D811yescharacteristic
65C4×D109—21no—
66C4×D109—41no—
67Dic218—21no—
68Dic218—41no—
69Dic218—41no—
70Dic218D811yescharacteristic
71C8×C109—41no—
72C8×C109C2×D411yescharacteristic
73C8×C109C2×D411yescharacteristic
74#11724784—21no—
75C22×C109—42no—
76C22×C109—161no—
77C22×C109AGL(1,ℤ/8)11yescharacteristic
78C2×D109—21no—
79C2×D109—41no—
80C2×D109—162no—
81C4×C109#12011yescharacteristic
82C4×C109C2×D811yescharacteristic
83Dic109—21no—
84Dic109—41no—
85Dic109—81no—
86C2×C109—21no—
87C2×C109—82no—
88C2×C109#44711yescharacteristic
89D109—41no—
90D109—81no—
91#2194—1091nomaximal, Sylow
92C109#219411yescharacteristic, complemented, Sylow
93#576—1091no—
94#510—1091no—
95C2×SD32—1091no—
96#473—1091no—
97#361D10911yescharacteristic, complemented
98#359—1091no—
99#350—1091no—
100H(5,𝔽2)—2181no—
101#137—2181no—
102AGL(1,ℤ/8)—2181no—
103AGL(1,ℤ/8)—2182no—
104#135—1091no—
105#135—2181no—
106C2×Q16—1091no—
107C2×D8—2181no—
108C2×D8C2×D10911yescharacteristic
109#131—1091no—
110Q32—2181no—
111SD32—2181no—
112SD32—2182no—
113M32C2×D10911yescharacteristic, complemented
114C2×C16—1091no—
115#108C2×D10911yescharacteristic, complemented
116#101—2181no—
117#41—1091no—
118#41—4361no—
119#41—2182no—
120#41—2181no—
121C2×Q8—2181no—
122C2×D4—21no—
123C2×D4—4361no—
124C2×D4—4362no—
125C2×D4—2182no—
126Q16—1091no—
127Q16—4361no—
128Q16—2181no—
129Q16—2181no—
130SD16—4361no—
131SD16—2182no—
132SD16—2181no—
133D8—21no—
134D8—1091no—
135D8—2182no—
136D8D43612yes—
137M16—21no—
138M16—1091no—
139M16—2181no—
140C2×C8—2181no—
141C2×C8C22×D10911yescharacteristic
142C16—21no—
143C16—2181no—
144C23—41no—
145C23—4362no—
146Q8—1091no—
147Q8—4361no—
148Q8—4361no—
149Q8—2181no—
150D4—22no—
151D4—41no—
152D4—1091no—
153D4—4362no—
154D4—4361no—
155D4—4362no—
156D4—2181no—
157D4—2181no—
158C2×C4—4361no—
159C2×C4—4362no—
160C2×C4—4361no—
161C2×C4—2181no—
162C2×C4D4×D10911yescharacteristic
163C8—41no—
164C8—2181no—
165C8D4×D10911yescharacteristic
166C8C2×D43611yescharacteristic
167C22—42no—
168C22—161no—
169C22—4361no—
170C22—17442no—
171C22—2181no—
172C22#1681272611yescharacteristic
173C4—8721no—
174C4—4361no—
175C4—2181no—
176C4#1681266911yescharacteristic
177C4D8×D10911yescharacteristic
178C2—21no—
179C2—82no—
180C2—8721no—
181C2—4361no—
182C2#2029516111yescharacteristic
subgroupG/H
centerC2#20295161
commutator, cosocleC2×C8×C109C23
FrattiniC2×C8C22×D109
Fitting#361×C109C2
socleC2×C109#447
prime radical#20295827C2
hypercenter#361D109
hypocenterC109#2194
LCSG ▷ C2×C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #20295469C2semidirect
3. C2×#16812643C2semidirect
4. #20295302C2semidirect
5. #20294832C2semidirect
6. #20294811C2semidirect
7. #361×C109C2semidirect
14. C2×Dic436C22semidirect
20. C2×#13182004C22semidirect
23. M32×C109C22semidirect
24. #108×C109C22semidirect
92. C109#2194semidirect
97. #361D109semidirect
113. M32C2×D109semidirect
115. #108C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958272
#202954692
C2×#168126432
#202953022
#202948322
#202948112
#361×C1092
#2194109