Finite Group Databasebeta

← #28379079

Subgroups — #28379079

Order 13952.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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#20296094C211yesmaximal, characteristic, complemented
2#20296025C212yesmaximal
3D109⋊#101C211yesmaximal, characteristic, complemented
4#20295893C212yesmaximal, complemented
5#20295601C211yesmaximal, characteristic
6D109⋊M16C2212yes—
7Q8×#89137—24no—
8#16812778—22no—
9#16812778C2212yes—
10#16812764—24no—
11#16812760—22no—
12C2×#13182030C2211yescharacteristic
13C2×Q8×D109C2211yescharacteristic
14#16812697C411yescharacteristic, complemented
15#16812688C411yescharacteristic, complemented
16#16812687C411yescharacteristic, complemented
17#16812684—22no—
18#16812677—22no—
19#16812656C2211yescharacteristic
20C4⋊Q8×C109C411yescharacteristic, complemented
21#13182039—22no—
22#13182038—22no—
23D109⋊C8—22no—
24C4⋊#89137—24no—
25C4⋊#89137—22no—
26C22⋊#89137—22no—
27C4×#89137—24no—
28C4×#89137D414yes—
29#13182030—22no—
30#13182030—22no—
31C2×Dic218C2×C411yescharacteristic
32C2×Dic218D412yes—
33Q8×D109—22no—
34Q8×D109—22no—
35C2×D436—21no—
36C2×D436C2×C411yescharacteristic
37C2×C4×D109—21no—
38C2×C4×D109D412yes—
39C2×C4×D109C2311yescharacteristic
40#13182008—22no—
41#13182008—21no—
42#13182006—21no—
43#13182006—22no—
44#13182006—22no—
45C4×Dic109C2×C411yescharacteristic
46C2×Q8×C109C2×C411yescharacteristic
47C2×Q8×C109C2×C411yescharacteristic
48C4⋊C4×C109—22no—
49C42×C109C2×C411yescharacteristic
50C22×D109—21no—
51C22×D109C2×D411yescharacteristic
52C2×#89137—24no—
53Q8×C109—22no—
54Q8×C109—22no—
55C2×C4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109C22⋊C412yes—
58C2×C4×C109C22×C411yescharacteristic
59C2×Dic109—21no—
60C2×Dic109C22⋊C412yes—
61C2×Dic109C2×D411yescharacteristic
62D436—22no—
63D436—41no—
64D436—41no—
65C4×D109—22no—
66C4×D109—22no—
67C4×D109—42no—
68C4×D109C2×D412yes—
69C4×D109#4112yes—
70Dic218—22no—
71Dic218—24no—
72#11724786—42no—
73C22×C109C2×C22⋊C411yescharacteristic
74C2×D109—41no—
75C2×D109—42no—
76C2×D109—41no—
77C2×D109#12011yescharacteristic
78C2×D109#12311yescharacteristic
79#89137—44no—
80C4×C109—22no—
81C4×C109—41no—
82C4×C109—41no—
83C4×C109C4×D412yes—
84Dic109—22no—
85Dic109—41no—
86Dic109#12112yes—
87C2×C109—21no—
88C2×C109#38611yescharacteristic
89D109—21no—
90D109—21no—
91D109—81no—
92#1912—1091nomaximal, Sylow
93C109#191211yescharacteristic, complemented, Sylow
94#536—1091no—
95#519—1091no—
96#421—1092no—
97C2×#101—1091no—
98H(3,ℤ/4)—1092no—
99C2×#41—1091no—
100C22×Q8—1091no—
101C2×M16—1092no—
102C4⋊Q8—1091no—
103C4⋊Q8#8913711yescharacteristic, complemented
104#124—1091no—
105#124—1091no—
106#123—2182no—
107C22⋊Q8—2182no—
108C4×Q8—2184no—
109#117—1092no—
110#117—1091no—
111#117—2182no—
112#104—2184no—
113#101—2182no—
114#41—2182no—
115#41—2182no—
116C2×Q8—1092no—
117C2×Q8—1091no—
118C2×Q8—2182no—
119C2×Q8—2182no—
120C2×Q8C2×#8913711yescharacteristic
121C2×Q8C2×#8913711yescharacteristic
122C2×D4—1091no—
123C2×D4—2181no—
124C22×C4—1091no—
125C22×C4—1092no—
126C22×C4—2181no—
127M16—2182no—
128M16—2182no—
129C2×C8—2182no—
130C4⋊C4—22no—
131C4⋊C4—2181no—
132C4⋊C4—2182no—
133C4⋊C4—2182no—
134C4⋊C4—2184no—
135C22⋊C4—2182no—
136C22⋊C4—2181no—
137C22⋊C4—2182no—
138C22⋊C4—2182no—
139C42—1094no—
140C42—1091no—
141C42—2184no—
142C42C2×#8913711yescharacteristic
143C23—1091no—
144C23—2181no—
145Q8—22no—
146Q8—22no—
147Q8—2184no—
148Q8—2182no—
149D4—4361no—
150D4—4361no—
151D4—2182no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—1091no—
155C2×C4—1092no—
156C2×C4—1092no—
157C2×C4—1092no—
158C2×C4—4362no—
159C2×C4—2182no—
160C2×C4—2182no—
161C2×C4—2184no—
162C2×C4—2181no—
163C2×C4C22⋊#8913712yes—
164C2×C4C22×#8913711yescharacteristic
165C8—4362no—
166C22—1091no—
167C22—1091no—
168C22—4362no—
169C22—4361no—
170C22—4361no—
171C22C2×C22⋊#8913711yescharacteristic
172C4—22no—
173C4—41no—
174C4—41no—
175C4—1092no—
176C4—4361no—
177C4—4364no—
178C4—2182no—
179C4D4×#8913712yes—
180C2—21no—
181C2—8721no—
182C2—2181no—
183C2—2181no—
184C2#2029597411yescharacteristic
subgroupG/H
centerC2#20295974
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC4⋊Q8×C109C4
socleC2×C109#386
prime radicalC2×#13182030C22
cosocleC2×C4×D109C23
hypercenterC4⋊Q8#89137
hypocenterC109#1912
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20296094C2semidirect
3. D109⋊#101C2semidirect
4. #20295893C2semidirect
14. #16812697C4semidirect
15. #16812688C4semidirect
16. #16812687C4semidirect
20. C4⋊Q8×C109C4semidirect
93. C109#1912semidirect
103. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202960942
#202960252
D109⋊#1012
#202958932
#202956012
#1912109