Finite Group Databasebeta

← #28379315

Subgroups — #28379315

Order 13952.

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

normal subgroups only

This group has 238 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#20296007C212yesmaximal
2#20295933C212yesmaximal, complemented
3#20295932C212yesmaximal, complemented
4#20295601C211yesmaximal, characteristic
5#16812778—22no—
6#16812758—22no—
7#16812758C2214yes—
8C2×#13182030C2211yescharacteristic
9C2×Q8×D109C411yescharacteristic, complemented
10#16812697C411yescharacteristic, complemented
11#16812688C2211yescharacteristic
12#16812687C2211yescharacteristic
13#16812684—22no—
14#16812677—22no—
15#16812656C411yescharacteristic, complemented
16C4⋊Q8×C109C411yescharacteristic, complemented
17C4⋊#89137—42no—
18C22⋊#89137—24no—
19C4×#89137—42no—
20#13182030—42no—
21C2×Dic218—21no—
22C2×Dic218C2×C411yescharacteristic
23Q8×D109—42no—
24C2×D436D412yes—
25C2×D436C2311yescharacteristic
26C2×C4×D109—21no—
27C2×C4×D109C2×C411yescharacteristic
28C2×C4×D109D412yes—
29#13182008—22no—
30#13182008—21no—
31#13182006—21no—
32#13182006—22no—
33#13182006—22no—
34C4×Dic109C2×C411yescharacteristic
35C2×Q8×C109C2×C411yescharacteristic
36C2×Q8×C109C2×C411yescharacteristic
37C4⋊C4×C109—22no—
38C42×C109C2×C411yescharacteristic
39C22×D109C22⋊C411yescharacteristic
40C22×D109C2×D412yes—
41C2×#89137—44no—
42Q8×C109—41no—
43Q8×C109—41no—
44C2×C4×C109—21no—
45C2×C4×C109—21no—
46C2×C4×C109C22⋊C412yes—
47C2×C4×C109C22×C411yescharacteristic
48C2×Dic109—21no—
49C2×Dic109—21no—
50C2×Dic109C22⋊C411yescharacteristic
51D436—42no—
52D436—41no—
53C4×D109—22no—
54C4×D109—22no—
55C4×D109—22no—
56C4×D109—42no—
57Dic218—42no—
58Dic218—41no—
59C22×C109C2×C22⋊C411yescharacteristic
60C2×D109—21no—
61C2×D109—22no—
62C2×D109—41no—
63C2×D109—42no—
64#89137—84no—
65C4×C109—21no—
66C4×C109—22no—
67C4×C109—41no—
68C4×C109—41no—
69Dic109—41no—
70Dic109—41no—
71Dic109#9912yes—
72C2×C109—21no—
73C2×C109C2×#9911yescharacteristic
74D109—42no—
75D109—41no—
76#2129—1091nomaximal, Sylow
77C109#212911yescharacteristic, complemented, Sylow
78#536—1091no—
79#410—1092no—
80#354—1092no—
81#354—1092no—
82C2×#41—1091no—
83C22×Q8—1091no—
84C4⋊Q8—1091no—
85C4⋊Q8#8913711yescharacteristic, complemented
86#124—1091no—
87#124—1091no—
88#123—2182no—
89C22⋊Q8—2182no—
90#117—1091no—
91#117—2182no—
92#99—1094no—
93#99—2182no—
94#41—4362no—
95C2×Q8—1091no—
96C2×Q8—4362no—
97C2×Q8—2181no—
98C2×Q8C2×#8913711yescharacteristic
99C2×Q8C2×#8913711yescharacteristic
100C2×D4—1092no—
101C2×D4—1091no—
102C22×C4—1092no—
103C22×C4—1091no—
104C22×C4—2181no—
105C4⋊C4—22no—
106C4⋊C4—4362no—
107C4⋊C4—2181no—
108C4⋊C4—2182no—
109C22⋊C4—2181no—
110C22⋊C4—2182no—
111C22⋊C4—2182no—
112C22⋊C4—2184no—
113C42—1091no—
114C42—4362no—
115C42C2×#8913711yescharacteristic
116C23—1092no—
117C23—1091no—
118Q8—41no—
119Q8—41no—
120Q8—4361no—
121Q8—4362no—
122D4—4362no—
123D4—4361no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—1091no—
127C2×C4—4362no—
128C2×C4—4364no—
129C2×C4—2182no—
130C2×C4—2182no—
131C2×C4—2182no—
132C2×C4—2181no—
133C2×C4—2181no—
134C2×C4C22⋊#8913712yes—
135C2×C4C22×#8913711yescharacteristic
136C22—4362no—
137C22—4361no—
138C22—2182no—
139C22—2181no—
140C22C2×C22⋊#8913711yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—8724no—
146C4—1092no—
147C4—4361no—
148C4—4361no—
149C2—21no—
150C2—4362no—
151C2—4361no—
152C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC4⋊Q8×C109C4
socleC2×C109C2×#99
prime radicalC2×#13182030C22
cosocleC2×D436C23
hypercenterC4⋊Q8#89137
hypocenterC109#2129
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20295933C2semidirect
3. #20295932C2semidirect
9. C2×Q8×D109C4semidirect
10. #16812697C4semidirect
15. #16812656C4semidirect
16. C4⋊Q8×C109C4semidirect
77. C109#2129semidirect
85. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202960072
#202959332
#202959322
#202956012
#2129109