Finite Group Databasebeta

← #28362995

Subgroups — #28362995

Order 13952.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#20295862C211yesmaximal, characteristic
2#20295042C212yesmaximal, complemented
3#20295023C212yesmaximal, complemented
4#20295000C211yesmaximal, characteristic, complemented
5#408×C109C211yesmaximal, characteristic, complemented
6#16812747C412yescomplemented
7#16812746—22no—
8#16812745—22no—
9#16812744—22no—
10#16812744C412yescomplemented
11C2×#13182029—22no—
12C2×#13182029C2212yes—
13#41×D109—24no—
14#16812658C2212yescomplemented
15#16812625—24no—
16#16812624—22no—
17#16812621—22no—
18#16812614—22no—
19C2×#41×C109C2211yescharacteristic
20C2×M16×C109C2211yescharacteristic, complemented
21C22×C8×C109C2211yescharacteristic, complemented
22C22⋊C8×C109—22no—
23#13182030—24no—
24#13182029—24no—
25#13182029—24no—
26#13182029—24no—
27#13182029—22no—
28#13182029—22no—
29#13182029C2×C414yes—
30#13182029D414yes—
31#13182028—28no—
32#13182028—24no—
33C2×Dic218—22no—
34C2×Dic218D412yes—
35Q8×D109—24no—
36C2×D436—22no—
37C2×D436D412yes—
38C2×C22⋊D109—24no—
39C2×C22⋊D109—22no—
40D4×D109—28no—
41D4×D109—24no—
42C2×C4×D109—24no—
43C2×C4×D109—22no—
44C4⋊Dic109—22no—
45C22⋊Dic109—22no—
46C4×Dic109—22no—
47#41×C109—22no—
48#41×C109—22no—
49C2×Q8×C109C2×C411yescharacteristic
50C2×D4×C109C2×C411yescharacteristic
51C2×D4×C109D412yes—
52C22×C4×C109D412yes—
53C22×C4×C109C2311yescharacteristic
54M16×C109—22no—
55C2×C8×C109—21no—
56C2×C8×C109—21no—
57C2×C8×C109—22no—
58C23×C109C22⋊C412yes—
59C23×C109C2×D411yescharacteristic
60C22×D109—24no—
61C22×D109—22no—
62Q8×C109—22no—
63D4×C109—24no—
64D4×C109—22no—
65C2×C4×C109—22no—
66C2×C4×C109—22no—
67C2×C4×C109C22⋊C412yes—
68C2×C4×C109C22×C411yescharacteristic
69C2×C4×C109C2×D412yes—
70C2×C4×C109C2×D412yes—
71C2×C4×C109C2×D411yescharacteristic
72C22⋊D109—28no—
73C22⋊D109—24no—
74C22⋊D109—24no—
75C2×Dic109—24no—
76C2×Dic109—22no—
77C2×Dic109—42no—
78D436—24no—
79D436—22no—
80D436C22⋊C414yes—
81C4×D109—28no—
82C4×D109—24no—
83C4×D109—24no—
84Dic218—24no—
85Dic218—22no—
86Dic218C22⋊C414yes—
87C8×C109—41no—
88C8×C109—41no—
89C22×C109—22no—
90C22×C109—41no—
91C22×C109—42no—
92C22×C109C2×C22⋊C411yescharacteristic
93C22×C109#12012yes—
94C2×D109—24no—
95C2×D109—82no—
96C2×D109—84no—
97C4×C109—22no—
98C4×C109C2×C22⋊C412yes—
99C4×C109#12011yescharacteristic
100C4×C109#12011yescharacteristic
101Dic109—24no—
102Dic109—82no—
103C2×C109—22no—
104C2×C109—21no—
105C2×C109—42no—
106C2×C109#37911yescharacteristic
107D109—44no—
108#1794—1091nomaximal, Sylow
109C109#179411yescharacteristic, complemented, Sylow
110#585—1091no—
111#421—1092no—
112#416—1092no—
113#410—1091no—
114#408D10911yescharacteristic, complemented
1152−1+4—1092no—
1162−1+4—2182no—
117H(5,𝔽2)—1092no—
118H(5,𝔽2)—2182no—
119H(5,𝔽2)—2182no—
120C2×#41—1092no—
121C2×#41—2184no—
122C2×#41—2182no—
123C2×#41C2×D10911yescharacteristic
124C2×M16C2×D10911yescharacteristic, complemented
125C22×C8C2×D10911yescharacteristic, complemented
126#117—1092no—
127#104—2184no—
128#103—2182no—
129#102—2182no—
130#99—2182no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—1094no—
135#41—1094no—
136#41—2184no—
137#41—2184no—
138#41—2184no—
139#41—2188no—
140#41—2184no—
141#41—2184no—
142#41—2182no—
143#41—2182no—
144C2×Q8—1092no—
145C2×Q8—2182no—
146C2×Q8—2184no—
147C2×Q8C4×D10911yescharacteristic
148C2×D4—1092no—
149C2×D4—2182no—
150C2×D4—2184no—
151C2×D4—2188no—
152C2×D4—2184no—
153C2×D4—2182no—
154C2×D4C4×D10911yescharacteristic
155C2×D4D43612yes—
156C22×C4—2184no—
157C22×C4—2182no—
158C22×C4C22⋊D10912yes—
159C22×C4C22×D10911yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—2182no—
165C22⋊C4—2182no—
166C42—2182no—
167C23—2184no—
168C23—2182no—
169C23#1318200612yes—
170C23C2×D43611yescharacteristic
171Q8—22no—
172Q8—1094no—
173Q8—2184no—
174Q8—2182no—
175D4—24no—
176D4—22no—
177D4—1094no—
178D4—2184no—
179D4—2188no—
180D4—2184no—
181D4—2184no—
182D4—2182no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—4362no—
186C2×C4—2188no—
187C2×C4—2184no—
188C2×C4—2182no—
189C2×C4—2184no—
190C2×C4—2184no—
191C2×C4#1318200612yes—
192C2×C4C2×C4×D10911yescharacteristic
193C2×C4D4×D10912yes—
194C2×C4D4×D10912yes—
195C2×C4C2×C22⋊D10911yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—8722no—
202C22—8724no—
203C22—2184no—
204C22C2×#1318200611yescharacteristic
205C22#1681266912yes—
206C4—22no—
207C4—8722no—
208C4—2184no—
209C4C22⋊C4×D10912yes—
210C4#1681266711yescharacteristic
211C4#1681266711yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—4364no—
216C2#2029488111yescharacteristic
subgroupG/H
centerC4#16812667
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#408×C109C2
socleC2×C109#379
prime radical#20295862C2
cosocleC22×C4×C109C23
hypercenter#408D109
hypocenterC109#1794
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20295042C2semidirect
3. #20295023C2semidirect
4. #20295000C2semidirect
5. #408×C109C2semidirect
6. #16812747C4semidirect
10. #16812744C4semidirect
14. #16812658C22semidirect
20. C2×M16×C109C22semidirect
21. C22×C8×C109C22semidirect
109. C109#1794semidirect
114. #408D109semidirect
124. C2×M16C2×D109semidirect
125. C22×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
#202950422
#202950232
#202950002
#408×C1092
#1794109