Finite Group Databasebeta

← #28379335

Subgroups — #28379335

Order 13952.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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#20296013C212yesmaximal
2#20295937C212yesmaximal, complemented
3#20295936C212yesmaximal, complemented
4#20295596C211yesmaximal, characteristic
5C2×#13182038—22no—
6#16812760C2214yes—
7#16812759—22no—
8C2×#13182028C2211yescharacteristic
9C2×Q8×D109C411yescharacteristic
10#16812699C2211yescharacteristic
11#16812697C2211yescharacteristic
12#16812686C411yescharacteristic
13#16812681—22no—
14#16812676—22no—
15#16812656C411yescharacteristic
16#124×C109C411yescharacteristic
17#13182038—24no—
18#13182038—42no—
19C2×#11724786—42no—
20#13182028—42no—
21C2×Dic218—21no—
22C2×Dic218C2×C411yescharacteristic
23C2×Dic218C2311yescharacteristic
24Q8×D109—42no—
25C2×C22⋊D109D412yes—
26C22×Dic109D412yes—
27C2×C4×D109—21no—
28C2×C4×D109C2×C411yescharacteristic
29C4⋊Dic109—22no—
30#13182008—21no—
31#13182008—22no—
32C22⋊Dic109—22no—
33#13182006—21no—
34C4×Dic109C2×C411yescharacteristic
35C2×Q8×C109C2×C411yescharacteristic
36C2×D4×C109C2×C411yescharacteristic
37C22⋊C4×C109—22no—
38C42×C109C2×C411yescharacteristic
39C23×C109C22⋊C412yes—
40C22×D109C22⋊C411yescharacteristic
41Q8×C109—41no—
42D4×C109—41no—
43C2×C4×C109—21no—
44C2×C4×C109—21no—
45C2×C4×C109C22×C411yescharacteristic
46C22⋊D109—42no—
47C2×Dic109—22no—
48C2×Dic109—22no—
49C2×Dic109—21no—
50C2×Dic109—21no—
51C2×Dic109C22⋊C411yescharacteristic
52C2×Dic109C2×D412yes—
53C4×D109—22no—
54C4×D109—42no—
55Dic218—41no—
56Dic218—42no—
57Dic218—41no—
58#11724786—44no—
59C22×C109—22no—
60C22×C109—42no—
61C22×C109C2×C22⋊C411yescharacteristic
62C2×D109—21no—
63C2×D109—41no—
64C4×C109—21no—
65C4×C109—41no—
66C4×C109—41no—
67Dic109—22no—
68Dic109—41no—
69Dic109—41no—
70Dic109#9912yes—
71C2×C109—21no—
72C2×C109—42no—
73C2×C109C2×#9911yescharacteristic
74D109—41no—
75#2135—1091nomaximal, Sylow
76C109#213511yescharacteristic, complemented, Sylow
77#536—1091no—
78#413—1092no—
79#356—1092no—
80#356—1092no—
81C2×#41—1091no—
82C22×Q8—1091no—
83C2×M16—2182no—
84C4⋊Q8—1091no—
85C4⋊Q8—1091no—
86#124—1091no—
87#124#8913711yescharacteristic
88#123—2182no—
89C22⋊Q8—2182no—
90#117—1091no—
91#101—1094no—
92#100—2182no—
93#41—4362no—
94C2×Q8—1091no—
95C2×Q8—1091no—
96C2×Q8—4362no—
97C2×Q8—2181no—
98C2×Q8C2×#8913711yescharacteristic
99C2×D4—1092no—
100C2×D4C2×#8913711yescharacteristic
101C22×C4—1092no—
102C22×C4—1091no—
103C22×C4—2181no—
104M16—4362no—
105M16—2184no—
106C2×C8—4362no—
107C4⋊C4—2181no—
108C4⋊C4—2182no—
109C4⋊C4—2182no—
110C22⋊C4—22no—
111C22⋊C4—2181no—
112C22⋊C4—2182no—
113C42—1091no—
114C42C2×#8913711yescharacteristic
115C23—1091no—
116C23C22⋊#8913712yes—
117Q8—41no—
118Q8—4361no—
119Q8—4361no—
120Q8—4362no—
121D4—41no—
122D4—4362no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—1091no—
126C2×C4—1092no—
127C2×C4—4362no—
128C2×C4—2182no—
129C2×C4—2182no—
130C2×C4—2182no—
131C2×C4—2181no—
132C2×C4—2181no—
133C2×C4C22×#8913711yescharacteristic
134C8—4364no—
135C22—22no—
136C22—42no—
137C22—4361no—
138C22—2181no—
139C22C2×C22⋊#8913711yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—1092no—
144C4—4361no—
145C4—4361no—
146C4—2182no—
147C2—21no—
148C2—42no—
149C2—4361no—
150C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#124×C109C4
socleC2×C109C2×#99
prime radicalC2×#13182028C22
cosocleC2×Dic218C23
hypercenter#124#89137
hypocenterC109#2135
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20295937C2semidirect
3. #20295936C2semidirect
76. C109#2135semidirect

Maximal subgroups

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

subgroupindex
#202960132
#202959372
#202959362
#202955962
#2135109