Finite Group Databasebeta

← #28362985

Subgroups — #28362985

Order 13952.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#16812744C211yesmaximal, characteristic
2C2×#16812627C212yesmaximal, complemented
3#20295025C212yesmaximal, complemented
4C2×#16812616C211yesmaximal, characteristic, complemented
5#390×C109C211yesmaximal, characteristic, complemented
6#16812744—24no—
7#16812744—22no—
8#16812744C414yescomplemented
9C2×#13182029C2212yes—
10C2×#13182028—22no—
11C22×C22⋊D109—22no—
12C2×D4×D109—22no—
13C2×#13182013C2212yescomplemented
14#16812627—24no—
15#16812622—22no—
16#16812619—22no—
17#16812616—22no—
18C22×D4×C109C2211yescharacteristic
19C2×C4⋊C4×C109C2211yescharacteristic, complemented
20C2×C22⋊C4×C109—22no—
21C2×C42×C109C2211yescharacteristic, complemented
22C23×D109—22no—
23#13182029—22no—
24#13182029—22no—
25#13182029C2×C414yes—
26#13182029D414yes—
27#13182028—28no—
28#13182028—28no—
29C2×Dic218D412yes—
30C2×D436D412yes—
31C2×C22⋊D109—24no—
32C2×C22⋊D109—28no—
33C2×C22⋊D109—28no—
34C2×C22⋊D109—22no—
35D4×D109—28no—
36D4×D109—28no—
37C22×Dic109—22no—
38C2×C4×D109—22no—
39#13182013—22no—
40#13182013—22no—
41C2×#11724784—22no—
42C24×C109D412yes—
43C2×D4×C109—22no—
44C2×D4×C109—22no—
45C2×D4×C109C2×C412yes—
46C2×D4×C109D412yes—
47C22×C4×C109—21no—
48C22×C4×C109—21no—
49C22×C4×C109C2311yescharacteristic
50C4⋊C4×C109—22no—
51C22⋊C4×C109—44no—
52C42×C109—22no—
53C23×C109—22no—
54C23×C109—22no—
55C23×C109—42no—
56C23×C109—42no—
57C23×C109C22⋊C414yes—
58C23×C109C2×D411yescharacteristic
59C22×D109—22no—
60C22×D109—28no—
61C22×D109—82no—
62C22×D109—82no—
63D4×C109—24no—
64D4×C109—24no—
65C2×C4×C109—22no—
66C2×C4×C109—41no—
67C2×C4×C109—41no—
68C2×C4×C109—42no—
69C2×C4×C109—41no—
70C2×C4×C109C22×C411yescharacteristic
71C2×C4×C109C2×D412yes—
72C2×C4×C109C2×D412yes—
73C2×C4×C109C2×D411yescharacteristic
74C22⋊D109—216no—
75C22⋊D109—24no—
76C22⋊D109—24no—
77C2×Dic109—28no—
78C2×Dic109—22no—
79D436—22no—
80D436C22⋊C414yescomplemented
81C4×D109—24no—
82C4×D109—24no—
83Dic218—22no—
84Dic218C22⋊C414yescomplemented
85#11724784—42no—
86C22×C109—21no—
87C22×C109—21no—
88C22×C109—24no—
89C22×C109—44no—
90C22×C109—42no—
91C22×C109—42no—
92C22×C109—82no—
93C22×C109C2×C22⋊C411yescharacteristic
94C22×C109#12011yescharacteristic
95C22×C109#12011yescharacteristic
96C2×D109—24no—
97C2×D109—42no—
98C2×D109—42no—
99C2×D109—88no—
100C4×C109—42no—
101C4×C109—81no—
102C4×C109C2×C22⋊C412yes—
103C4×C109#12012yes—
104Dic109—24no—
105C2×C109—21no—
106C2×C109—21no—
107C2×C109—44no—
108C2×C109#37911yescharacteristic
109C2×C109#45311yescharacteristic
110C2×C109#45511yescharacteristic
111D109—44no—
112#1792—1091nomaximal, Sylow
113C109#179211yescharacteristic, complemented, Sylow
114C2×H(5,𝔽2)—1091no—
115C2×#104—1092no—
116#417—1092no—
117C2×#100—1091no—
118#390D10911yescharacteristic, complemented
119H(5,𝔽2)—1094no—
120H(5,𝔽2)—2184no—
121H(5,𝔽2)—2182no—
122C2×#41—1092no—
123C2×#41—2182no—
124C22×D4—2182no—
125C22×D4—2182no—
126C22×D4C2×D10911yescharacteristic
127C2×M16—1092no—
128C2×C4⋊C4C2×D10911yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D10911yescharacteristic, complemented
131#104—2184no—
132#103—2182no—
133#102—2182no—
134#100—2182no—
135C24—2182no—
136C24C22⋊D10912yes—
137#41—1094no—
138#41—1094no—
139#41—2188no—
140#41—2188no—
141#41—2182no—
142#41—2182no—
143C2×Q8—1092no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—1092no—
147C2×D4—2188no—
148C2×D4—2188no—
149C2×D4—2184no—
150C2×D4—2188no—
151C2×D4—2188no—
152C2×D4—2182no—
153C2×D4C4×D10912yes—
154C2×D4D43612yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—2182no—
158C22×C4—2182no—
159C22×C4C22×D10911yescharacteristic
160M16—2182no—
161M16—2182no—
162C2×C8—2182no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—8722no—
171C23—8722no—
172C23—2188no—
173C23—2182no—
174C23#1318200614yes—
175C23C2×C22⋊D10911yescharacteristic
176Q8—1094no—
177Q8—2182no—
178D4—24no—
179D4—24no—
180D4—1094no—
181D4—21816no—
182D4—2184no—
183D4—2184no—
184D4—2182no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—2188no—
191C2×C4—2182no—
192C2×C4—2184no—
193C2×C4—2184no—
194C2×C4C2×C4×D10911yescharacteristic
195C2×C4D4×D10912yes—
196C2×C4D4×D10912yes—
197C2×C4C2×D43611yescharacteristic
198C8—4362no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—8728no—
207C22—4362no—
208C22—4362no—
209C22—2184no—
210C22C2×#1318200611yescharacteristic
211C22#1681266711yescharacteristic
212C22#1681266711yescharacteristic
213C4—42no—
214C4—81no—
215C4—2184no—
216C4C22⋊C4×D10912yes—
217C4#1681266912yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—4364no—
222C2#2029488111yescharacteristic
223C2#2029519311yescharacteristic
224C2#2029520311yescharacteristic
subgroupG/H
centerC22#16812667
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#390×C109C2
socleC22×C109#120
prime radicalC2×#16812744C2
cosocleC22×C4×C109C23
hypercenter#390D109
hypocenterC109#1792
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16812627C2semidirect
3. #20295025C2semidirect
4. C2×#16812616C2semidirect
5. #390×C109C2semidirect
8. #16812744C4semidirect
13. C2×#13182013C22semidirect
19. C2×C4⋊C4×C109C22semidirect
21. C2×C42×C109C22semidirect
80. D436C22⋊C4semidirect
84. Dic218C22⋊C4semidirect
113. C109#1792semidirect
118. #390D109semidirect
128. C2×C4⋊C4C2×D109semidirect
130. C2×C42C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127442
C2×#168126272
#202950252
C2×#168126162
#390×C1092
#1792109