Finite Group Databasebeta

← #28365001

Subgroups — #28365001

Order 13952.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20295831C211yesmaximal, characteristic
2#20295467C211yesmaximal, characteristic, complemented
3C2×#16812642C211yesmaximal, characteristic, complemented
4#20294838C211yesmaximal, characteristic, complemented
5#20294832C211yesmaximal, characteristic, complemented
6#20294810C211yesmaximal, characteristic, complemented
7#473×C109C211yesmaximal, characteristic, complemented
8#16812745—21no—
9#16812727—22no—
10#16812727—21no—
11#16812721—21no—
12#16812721C2211yescharacteristic
13C2×D872—21no—
14C2×D872C2211yescharacteristic
15#16812645—21no—
16#16812642—22no—
17#16812642—21no—
18#16812639C2211yescharacteristic, complemented
19C2×#13182004C2211yescharacteristic, complemented
20#137×C109—21no—
21C2×Q16×C109C2211yescharacteristic, complemented
22#131×C109C2211yescharacteristic
23#108×C109C2211yescharacteristic, complemented
24#101×C109—21no—
25#13182030—41no—
26#13182029—22no—
27#13182029—21no—
28C2×D436—22no—
29C2×D436—21no—
30D4×D109—42no—
31D4×D109—41no—
32Dic436D411yescharacteristic
33#13182018—22no—
34#13182018—21no—
35D872—22no—
36D872—21no—
37D872D411yescharacteristic
38D872D412yes—
39#13182004—21no—
40#13182004—21no—
41#41×C109D411yescharacteristic
42C2×Q8×C109—21no—
43Q16×C109—21no—
44Q16×C109—21no—
45Q16×C109—41no—
46SD16×C109—41no—
47M16×C109—21no—
48M16×C109—21no—
49M16×C109D411yescharacteristic
50C2×C8×C109—21no—
51C2×C8×C109C2311yescharacteristic
52C22×D109—42no—
53C22×D109—41no—
54Q8×C109—41no—
55Q8×C109—41no—
56Q8×C109D811yescharacteristic
57D4×C109—21no—
58D4×C109D811yescharacteristic
59C2×C4×C109—21no—
60C2×C4×C109—41no—
61C2×C4×C109C2×D411yescharacteristic
62C22⋊D109—42no—
63C22⋊D109—41no—
64D436—21no—
65D436—22no—
66D436—42no—
67D436—41no—
68C4×D109—42no—
69C4×D109—41no—
70Dic218—21no—
71C8×C109—21no—
72C8×C109—41no—
73C8×C109C2×D411yescharacteristic
74C8×C109C2×D411yescharacteristic
75C22×C109—21no—
76C22×C109AGL(1,ℤ/8)11yescharacteristic
77C2×D109—42no—
78C2×D109—41no—
79C2×D109—161no—
80C2×D109—162no—
81C4×C109—21no—
82C4×C109—81no—
83C4×C109#12011yescharacteristic
84C4×C109C2×D811yescharacteristic
85Dic109—41no—
86C2×C109—21no—
87C2×C109—41no—
88C2×C109#44711yescharacteristic
89D109—82no—
90D109—81no—
91#2194—1091nomaximal, Sylow
92C109#219411yescharacteristic, complemented, Sylow
93#576—1091no—
94#510—1091no—
95C2×SD32—1091no—
96#473D10911yescharacteristic, complemented
97#361—1091no—
98#359—1091no—
99#350—1091no—
100H(5,𝔽2)—2181no—
101#137—21no—
102AGL(1,ℤ/8)—2182no—
103AGL(1,ℤ/8)—2181no—
104#135—1091no—
105#135—2181no—
106C2×Q16C2×D10911yescharacteristic, complemented
107C2×D8—1091no—
108C2×D8—2181no—
109#131C2×D10911yescharacteristic
110Q32—2181no—
111SD32—2182no—
112SD32—2181no—
113M32—1091no—
114C2×C16—1091no—
115#108C2×D10911yescharacteristic, complemented
116#101—21no—
117#41—4361no—
118#41—2182no—
119#41—2181no—
120#41C22⋊D10911yescharacteristic
121C2×Q8—21no—
122C2×D4—4362no—
123C2×D4—4361no—
124C2×D4—2182no—
125C2×D4—2181no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16—1091no—
130SD16—41no—
131SD16—2182no—
132SD16—2181no—
133D8—1092no—
134D8—1091no—
135D8—2182no—
136D8—2181no—
137M16—21no—
138M16—21no—
139M16C22⋊D10911yescharacteristic
140C2×C8—21no—
141C2×C8C22×D10911yescharacteristic
142C16—2181no—
143C16—2181no—
144C23—4362no—
145C23—4361no—
146Q8—41no—
147Q8—41no—
148Q8—2181no—
149Q8#1318201411yescharacteristic
150D4—21no—
151D4—4362no—
152D4—4362no—
153D4—4361no—
154D4—4361no—
155D4—2182no—
156D4—2181no—
157D4#1318201411yescharacteristic
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—4362no—
161C2×C4—4361no—
162C2×C4C2×C22⋊D10911yescharacteristic
163C8—21no—
164C8—41no—
165C8D4×D10911yescharacteristic
166C8D4×D10911yescharacteristic
167C22—21no—
168C22—4362no—
169C22—4361no—
170C22—17441no—
171C22—17442no—
172C22#1681272411yescharacteristic
173C4—21no—
174C4—81no—
175C4—4361no—
176C4#1681266711yescharacteristic
177C4C2×#1318201411yescharacteristic
178C2—21no—
179C2—41no—
180C2—8722no—
181C2—8721no—
182C2#2029515911yescharacteristic
subgroupG/H
centerC2#20295159
commutator, cosocleC2×C8×C109C23
FrattiniC2×C8C22×D109
Fitting#473×C109C2
socleC2×C109#447
prime radical#20295831C2
hypercenter#473D109
hypocenterC109#2194
LCSG ▷ C2×C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #20295467C2semidirect
3. C2×#16812642C2semidirect
4. #20294838C2semidirect
5. #20294832C2semidirect
6. #20294810C2semidirect
7. #473×C109C2semidirect
18. #16812639C22semidirect
19. C2×#13182004C22semidirect
21. C2×Q16×C109C22semidirect
23. #108×C109C22semidirect
92. C109#2194semidirect
96. #473D109semidirect
106. C2×Q16C2×D109semidirect
115. #108C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958312
#202954672
C2×#168126422
#202948382
#202948322
#202948102
#473×C1092
#2194109