Finite Group Databasebeta

← #28364685

Subgroups — #28364685

Order 13952.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#20295597C211yesmaximal, characteristic
2#20295010C211yesmaximal, characteristic
3#20294815C214yesmaximal, complemented
4#410×C109C211yesmaximal, characteristic
5C22×Dic218C411yescharacteristic, complemented
6C2×#13182013—21no—
7#16812697—21no—
8#16812686—21no—
9#16812680C412yescomplemented
10#16812680C2212yes—
11#16812678—21no—
12#16812678—21no—
13#16812658C411yescharacteristic, complemented
14#16812618—21no—
15#16812616C2212yes—
16C2×#41×C109C2211yescharacteristic
17#117×C109—21no—
18#99×C109—21no—
19#99×C109C2212yes—
20C2×Dic218—21no—
21C2×Dic218—21no—
22C2×Dic218—41no—
23C2×Dic218—41no—
24C22×Dic109C2×C412yes—
25#13182013—22no—
26#13182013—41no—
27C2×#11724784—41no—
28C4⋊Dic109—21no—
29#13182008—22no—
30#13182008—22no—
31C22⋊Dic109—22no—
32C22⋊Dic109—22no—
33C22⋊Dic109C2×C412yes—
34C4×Dic109—21no—
35#41×C109—41no—
36#41×C109—41no—
37C2×Q8×C109D411yescharacteristic
38C2×D4×C109C2×C411yescharacteristic
39C2×D4×C109D411yescharacteristic
40C2×D4×C109C2311yescharacteristic
41C22×C4×C109C2×C411yescharacteristic
42C22×C4×C109D411yescharacteristic
43C22×C4×C109D411yescharacteristic
44C4⋊C4×C109—41no—
45C22⋊C4×C109—22no—
46C42×C109—41no—
47C23×C109C22⋊C411yescharacteristic
48C23×C109C22×C411yescharacteristic
49C23×C109C2×D411yescharacteristic
50Q8×C109—41no—
51D4×C109—41no—
52D4×C109—41no—
53D4×C109—41no—
54C2×C4×C109—21no—
55C2×C4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109—21no—
58C2×C4×C109—21no—
59C2×C4×C109—21no—
60C2×C4×C109—42no—
61C2×C4×C109C22⋊C411yescharacteristic
62C2×C4×C109C22⋊C411yescharacteristic
63C2×C4×C109C22⋊C411yescharacteristic
64C2×C4×C109C2×D411yescharacteristic
65C2×Dic109—22no—
66C2×Dic109—22no—
67C2×Dic109—42no—
68Dic218—42no—
69Dic218—41no—
70Dic218—41no—
71#11724784—42no—
72C22×C109—21no—
73C22×C109—21no—
74C22×C109—21no—
75C22×C109—41no—
76C22×C109—41no—
77C22×C109—41no—
78C22×C109C2×C22⋊C411yescharacteristic
79C4×C109—21no—
80C4×C109—21no—
81C4×C109—21no—
82C4×C109—82no—
83C4×C109#9911yescharacteristic
84C4×C109#9911yescharacteristic
85Dic109—42no—
86Dic109—42no—
87C2×C109—21no—
88C2×C109—41no—
89C2×C109—41no—
90C2×C109—41no—
91C2×C109C2×#9911yescharacteristic
92#2123—1091nomaximal, Sylow
93C109#212311yescharacteristic, complemented, Sylow
94#536—1091no—
95#413—1091no—
96#410D10911yescharacteristic
97#352—1094no—
98C2×#41C2×D10911yescharacteristic
99C22×Q8—1091no—
100C2×M16—2181no—
101C4⋊Q8—2181no—
102#124—2181no—
103#123—1092no—
104#123—1092no—
105C22⋊Q8—2181no—
106C22⋊Q8—2181no—
107#117—21no—
108#117—1091no—
109#101—2181no—
110#100—1092no—
111#99—21no—
112#99C2×D10912yes—
113#41—41no—
114#41—41no—
115C2×Q8—4361no—
116C2×Q8—4361no—
117C2×Q8—2181no—
118C2×Q8—2181no—
119C2×Q8D43611yescharacteristic
120C2×D4C4×D10911yescharacteristic
121C2×D4C22⋊D10911yescharacteristic
122C2×D4C22×D10911yescharacteristic
123C22×C4—1092no—
124C22×C4C4×D10911yescharacteristic
125C22×C4D43611yescharacteristic
126C22×C4C22⋊D10911yescharacteristic
127M16—4361no—
128M16—2182no—
129C2×C8—4361no—
130C4⋊C4—41no—
131C4⋊C4—2182no—
132C4⋊C4—2181no—
133C4⋊C4—2182no—
134C22⋊C4—22no—
135C22⋊C4—1092no—
136C22⋊C4—2182no—
137C22⋊C4—2182no—
138C42—41no—
139C42—2181no—
140C23#1318200611yescharacteristic
141C23C2×C4×D10911yescharacteristic
142C23C2×C22⋊D10911yescharacteristic
143Q8—41no—
144Q8—4362no—
145Q8—4361no—
146Q8—4361no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—4362no—
158C2×C4—2182no—
159C2×C4—2182no—
160C2×C4#1318200611yescharacteristic
161C2×C4#1318200611yescharacteristic
162C2×C4#1318200611yescharacteristic
163C2×C4C2×D43611yescharacteristic
164C8—4362no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1318200611yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—4362no—
177C4—4362no—
178C4#1681261311yescharacteristic
179C4#1681261311yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1681261311yescharacteristic
subgroupG/H
centerC2C2×#16812613
commutatorC23×C109C22×C4
FrattiniC2×D4C22×D109
Fitting#410×C109C2
socleC2×C109C2×#99
prime radicalC2×#41×C109C22
cosocleC2×D4×C109C23
hypercenter#410D109
hypocenterC109#2123
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #20294815C2semidirect
5. C22×Dic218C4semidirect
9. #16812680C4semidirect
13. #16812658C4semidirect
93. C109#2123semidirect

Maximal subgroups

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

subgroupindex
#202955972
#202950102
#202948152
#410×C1092
#2123109