Finite Group Databasebeta

← #28361028

Subgroups — #28361028

Order 13952.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#20294935C211yesmaximal, characteristic
2#20294750C211yesmaximal, characteristic, complemented
3#342×C109C211yesmaximal, characteristic, complemented
4C22×D436C411yescharacteristic, complemented
5C22×C22⋊D109—21no—
6C2×#13182006—21no—
7C22⋊#11724784—21no—
8#16812606C411yescharacteristic, complemented
9C2×C22⋊C4×C109—21no—
10C2×C22⋊C4×C109C2211yescharacteristic
11C23×D109—21no—
12C2×D436—41no—
13C2×D436—42no—
14C2×C22⋊D109—42no—
15C2×C22⋊D109—42no—
16C2×C22⋊D109—41no—
17C2×C22⋊D109—41no—
18C22×Dic109—21no—
19C2×#11724784—41no—
20#13182006—42no—
21C24×C109D411yescharacteristic
22C22×C4×C109—41no—
23C22×C4×C109C2×C411yescharacteristic
24C22×C4×C109D411yescharacteristic
25C22⋊C4×C109—22no—
26C22⋊C4×C109—41no—
27C22⋊C4×C109—41no—
28C23×C109—21no—
29C23×C109—21no—
30C23×C109—41no—
31C23×C109—41no—
32C23×C109C22⋊C411yescharacteristic
33C23×C109D811yescharacteristic
34C23×C109SD1611yescharacteristic
35C22×D109—22no—
36C22×D109—41no—
37C22×D109—41no—
38C22×D109—41no—
39C22×D109—41no—
40C22×D109—41no—
41C22×D109—41no—
42C2×C4×C109—21no—
43C2×C4×C109—21no—
44C2×C4×C109—41no—
45C2×C4×C109—41no—
46C2×C4×C109—42no—
47C2×C4×C109—81no—
48C2×C4×C109—81no—
49C22⋊D109—44no—
50C22⋊D109—44no—
51C2×Dic109—22no—
52C2×Dic109—41no—
53C2×Dic109—41no—
54D436—42no—
55D436—42no—
56#11724784—81no—
57C22×C109—21no—
58C22×C109—21no—
59C22×C109—21no—
60C22×C109—21no—
61C22×C109—41no—
62C22×C109—41no—
63C22×C109—41no—
64C22×C109—41no—
65C22×C109—81no—
66C22×C109#9911yescharacteristic
67C22×C109#10211yescharacteristic
68C22×C109#10411yescharacteristic
69C2×D109—42no—
70C2×D109—42no—
71C2×D109—42no—
72C2×D109—81no—
73C2×D109—81no—
74C2×D109—81no—
75C2×D109—81no—
76C4×C109—41no—
77C4×C109—41no—
78C4×C109—82no—
79Dic109—42no—
80C2×C109—21no—
81C2×C109—21no—
82C2×C109—41no—
83C2×C109—41no—
84C2×C109#32711yescharacteristic
85C2×C109AΣL(1,𝔽16)11yescharacteristic
86C2×C109#35311yescharacteristic
87D109—82no—
88#1343—1091nomaximal, Sylow
89C109#134311yescharacteristic, complemented, Sylow
90#392—1091no—
91#342D10911yescharacteristic, complemented
92#323—1091no—
93C22×D4—1091no—
94C22×D4—2181no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—2181no—
97C2×C22⋊C4C2×D10911yescharacteristic
98C22⋊C8—2181no—
99#95—1091no—
100C24—2181no—
101C24C22⋊D10911yescharacteristic
102C2×D4—4361no—
103C2×D4—4362no—
104C2×D4—4362no—
105C2×D4—4361no—
106C2×D4—4361no—
107C2×D4—4362no—
108C22×C4—41no—
109C22×C4—2181no—
110C22×C4C4×D10911yescharacteristic
111C22×C4D43611yescharacteristic
112C2×C8—4361no—
113C22⋊C4—22no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—4362no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—4361no—
122C23—4361no—
123C23—4361no—
124C23—4361no—
125C23—4361no—
126C23—4361no—
127C23—2182no—
128C23#1318200611yescharacteristic
129C23#1318201411yescharacteristic
130C23#1318201611yescharacteristic
131D4—4362no—
132D4—4362no—
133D4—4364no—
134D4—4364no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—4361no—
143C2×C4—4361no—
144C2×C4—2182no—
145C8—8721no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—8721no—
156C22—8721no—
157C22—8721no—
158C22—8721no—
159C22—4362no—
160C22—4362no—
161C22—4362no—
162C22#1681261311yescharacteristic
163C22#1681261911yescharacteristic
164C22#1681262711yescharacteristic
165C4—41no—
166C4—41no—
167C4—82no—
168C4—4362no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—8722no—
174C2#2029475911yescharacteristic
175C2#2029481211yescharacteristic
176C2#2029481811yescharacteristic
subgroupG/H
centerC22#16812613
commutatorC22×C4×C109C2×C4
FrattiniC2×C22⋊C4C2×D109
Fitting#342×C109C2
socleC22×C109#99
prime radical#20294935C2
cosocleC2×C22⋊C4×C109C22
hypercenter#342D109
hypocenterC109#1343
LCSG ▷ C22×C4×C109 ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #20294750C2semidirect
3. #342×C109C2semidirect
4. C22×D436C4semidirect
8. #16812606C4semidirect
89. C109#1343semidirect
91. #342D109semidirect

Maximal subgroups

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

subgroupindex
#202949352
#202947502
#342×C1092
#1343109