Finite Group Databasebeta

← #28365045

Subgroups — #28365045

Order 13952.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#20295215C212yesmaximal
2#20295203C211yesmaximal, characteristic
3#20294821C211yesmaximal, characteristic, complemented
4#20294816C212yesmaximal, complemented
5#563×C109C211yesmaximal, characteristic, complemented
6#16812744C2211yescharacteristic
7#16812730—21no—
8#16812682—22no—
9#16812627—21no—
10#16812616C2211yescharacteristic
11#16812614C2212yes—
12#16812613—22no—
13C4⋊Q8×C109—21no—
14#126×C109—22no—
15#125×C109—21no—
16#124×C109C2211yescharacteristic
17#123×C109C2212yes—
18C22⋊Q8×C109—22no—
19#13182029—21no—
20#13182028—41no—
21C2×C22⋊D109—22no—
22D4×D109—41no—
23#13182019—41no—
24#13182016—41no—
25#13182013—21no—
26#13182008—42no—
27C22⋊Dic109—22no—
28#13182006—42no—
29C2×Q8×C109—21no—
30C2×Q8×C109D411yescharacteristic
31C2×D4×C109C2311yescharacteristic
32C22×C4×C109D412yes—
33C4⋊C4×C109—21no—
34C4⋊C4×C109—22no—
35C4⋊C4×C109—22no—
36C4⋊C4×C109—21no—
37C4⋊C4×C109—22no—
38C22⋊C4×C109—22no—
39C22⋊C4×C109—22no—
40C22⋊C4×C109D412yes—
41C42×C109—21no—
42C42×C109D411yescharacteristic
43C23×C109C2×D412yes—
44C22×D109—22no—
45Q8×C109—41no—
46Q8×C109—81no—
47D4×C109—41no—
48C2×C4×C109—21no—
49C2×C4×C109—22no—
50C2×C4×C109—21no—
51C2×C4×C109—22no—
52C2×C4×C109—42no—
53C2×C4×C109C2×D411yescharacteristic
54C22⋊D109—42no—
55C22⋊D109—41no—
56C2×Dic109—22no—
57C2×Dic109—42no—
58D436—21no—
59C4×D109—41no—
60Dic218—21no—
61#11724784—41no—
62C22×C109—22no—
63C22×C109—42no—
64C22×C109#12011yescharacteristic
65C2×D109—41no—
66C2×D109—82no—
67C4×C109—21no—
68C4×C109—42no—
69C4×C109—41no—
70C4×C109—42no—
71C4×C109—41no—
72Dic109—41no—
73Dic109—82no—
74C2×C109—21no—
75C2×C109—42no—
76C2×C109B(4,𝔽2)11yescharacteristic
77D109—81no—
78#2201—1091nomaximal, Sylow
79C109#220111yescharacteristic, complemented, Sylow
80#563D10911yescharacteristic, complemented
81#458—1092no—
82#455—1091no—
83#354—1091no—
84#352—1092no—
85H(5,𝔽2)—1091no—
86#137—2181no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D10911yescharacteristic
91#123—2182no—
92#123C2×D10912yes—
93C22⋊Q8—22no—
94#104—2181no—
95#100—1091no—
96#99—1092no—
97#99—2182no—
98#41—4361no—
99#41—2181no—
100C2×Q8—21no—
101C2×Q8C22⋊D10911yescharacteristic
102C2×D4—4361no—
103C2×D4—2182no—
104C2×D4C22×D10911yescharacteristic
105C22×C4C22⋊D10912yes—
106Q16—4361no—
107SD16—4361no—
108M16—2181no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—4362no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—4362no—
118C22⋊C4—2182no—
119C22⋊C4C22⋊D10912yes—
120C42—21no—
121C42C22⋊D10911yescharacteristic
122C23—2182no—
123C23C2×C22⋊D10912yes—
124Q8—41no—
125Q8—81no—
126Q8—2181no—
127D4—41no—
128D4—4362no—
129D4—4361no—
130D4—2181no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—4361no—
137C2×C4—4362no—
138C2×C4—2182no—
139C2×C4C2×C22⋊D10911yescharacteristic
140C8—4361no—
141C22—22no—
142C22—42no—
143C22—8722no—
144C22—4361no—
145C22#1681266811yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—8722no—
152C4—4361no—
153C2—21no—
154C2—42no—
155C2—8721no—
156C2#2029521311yescharacteristic
subgroupG/H
centerC2#20295213
commutator, cosocleC2×D4×C109C23
FrattiniC2×D4C22×D109
Fitting#563×C109C2
socleC2×C109B(4,𝔽2)
prime radical#16812744C22
hypercenter#563D109
hypocenterC109#2201
LCSG ▷ C2×D4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #20294821C2semidirect
4. #20294816C2semidirect
5. #563×C109C2semidirect
79. C109#2201semidirect
80. #563D109semidirect

Maximal subgroups

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

subgroupindex
#202952152
#202952032
#202948212
#202948162
#563×C1092
#2201109