Finite Group Databasebeta

← #28364073

Subgroups — #28364073

Order 13952.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812725C212yesmaximal, complemented
2#20295607C211yesmaximal, characteristic, complemented
3C2×#16812627C212yesmaximal, complemented
4#20294795C211yesmaximal, characteristic, complemented
5#390×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2212yes—
7#16812725—24no—
8#16812725—24no—
9C2×#13182017—22no—
10C2×#13182014—22no—
11C2×#13182013C2212yes—
12#16812679—22no—
13#16812674—22no—
14#16812673—22no—
15#16812627—24no—
16C22×D4×C109C2211yescharacteristic
17C2×C4⋊C4×C109C2211yescharacteristic
18C2×C22⋊C4×C109—22no—
19C2×C42×C109C2211yescharacteristic
20#13182029—22no—
21#13182029—22no—
22#13182029—22no—
23#13182029—22no—
24C2×Dic218D412yes—
25C2×D436—22no—
26C2×D436D412yes—
27C2×C22⋊D109—22no—
28C2×C4×D109—22no—
29#13182017—28no—
30#13182014—28no—
31#13182013—22no—
32#13182013D414yescomplemented
33C2×#11724784—22no—
34C4⋊Dic109—22no—
35#13182006—24no—
36C24×C109—21no—
37C2×D4×C109—22no—
38C2×D4×C109—22no—
39C2×D4×C109—21no—
40C2×D4×C109D412yescomplemented
41C22×C4×C109—21no—
42C22×C4×C109D412yes—
43C22×C4×C109C2311yescharacteristic
44C4⋊C4×C109—22no—
45C22⋊C4×C109—44no—
46C42×C109—22no—
47C23×C109—22no—
48C23×C109—42no—
49C23×C109—42no—
50C23×C109—41no—
51C23×C109—41no—
52C23×C109C2×D411yescharacteristic
53C22×D109—22no—
54D4×C109—24no—
55D4×C109—42no—
56C2×C4×C109—22no—
57C2×C4×C109—22no—
58C2×C4×C109—41no—
59C2×C4×C109—41no—
60C2×C4×C109—42no—
61C2×C4×C109C2×D412yes—
62C2×C4×C109C2×D412yes—
63C2×C4×C109C2×D411yescharacteristic
64C2×C4×C109#4111yescharacteristic
65C22⋊D109—42no—
66C22⋊D109—42no—
67C2×Dic109—22no—
68D436—22no—
69D436—22no—
70D436—82no—
71C4×D109—42no—
72C4×D109—42no—
73Dic218—22no—
74Dic218—22no—
75#11724784—24no—
76C22×C109—21no—
77C22×C109—21no—
78C22×C109—44no—
79C22×C109—42no—
80C22×C109—82no—
81C22×C109—81no—
82C22×C109—81no—
83C22×C109#12011yescharacteristic
84C22×C109#12111yescharacteristic
85C22×C109#12711yescharacteristic
86C2×D109—42no—
87C2×D109—42no—
88C2×D109—42no—
89C4×C109—42no—
90C4×C109—42no—
91C4×C109#12012yes—
92C4×C109#12112yes—
93Dic109—42no—
94C2×C109—21no—
95C2×C109—21no—
96C2×C109—82no—
97C2×C109#39211yescharacteristic
98C2×C109#45311yescharacteristic
99C2×C109#45411yescharacteristic
100D109—82no—
101#2006—1091nomaximal, Sylow
102C109#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1092no—
104#537—1091no—
105C2×#104—1092no—
106#390D10911yescharacteristic, complemented
107#343—1091no—
108C2×#41—1092no—
109C22×D4C2×D10911yescharacteristic
110AGL(1,ℤ/8)—2184no—
111AGL(1,ℤ/8)—2184no—
112C2×SD16—2182no—
113C2×D8—2182no—
114C2×M16—1092no—
115C22⋊Q8—2182no—
116#121—2182no—
117#121—2182no—
118C2×C4⋊C4C2×D10911yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D10911yescharacteristic
121#104—2184no—
122C24—21no—
123#41—2182no—
124#41—2182no—
125#41—2182no—
126#41—2182no—
127C2×Q8—1092no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—1092no—
132C2×D4—2182no—
133C2×D4—2182no—
134C2×D4D43612yescomplemented
135C22×C4—21no—
136C22×C4—2182no—
137C22×C4C22⋊D10912yes—
138C22×C4C22×D10911yescharacteristic
139SD16—2188no—
140D8—2188no—
141M16—1094no—
142M16—2182no—
143C2×C8—2182no—
144C4⋊C4—22no—
145C4⋊C4—2182no—
146C22⋊C4—44no—
147C22⋊C4—2184no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—2182no—
155C23C2×C22⋊D10911yescharacteristic
156Q8—2182no—
157Q8—2182no—
158D4—24no—
159D4—42no—
160D4—8722no—
161D4—4362no—
162D4—4362no—
163D4—2182no—
164D4—2182no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—4362no—
171C2×C4—4362no—
172C2×C4—2182no—
173C2×C4D4×D10912yes—
174C2×C4D4×D10912yes—
175C2×C4C2×D43611yescharacteristic
176C2×C4#1318202911yescharacteristic
177C8—2184no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—4362no—
186C22—4362no—
187C22—4362no—
188C22#1681266711yescharacteristic
189C22#1681267311yescharacteristic
190C22#1681269511yescharacteristic
191C4—42no—
192C4—42no—
193C4—4362no—
194C4#1681266912yes—
195C4#1681267212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8722no—
200C2#2029493511yescharacteristic
201C2#2029519311yescharacteristic
202C2#2029519811yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#390×C109C2
socleC22×C109#120
hypercenter#390D109
hypocenterC109#2006
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812725C2semidirect
2. #20295607C2semidirect
3. C2×#16812627C2semidirect
4. #20294795C2semidirect
5. #390×C109C2semidirect
32. #13182013D4semidirect
40. C2×D4×C109D4semidirect
102. C109#2006semidirect
106. #390D109semidirect
134. C2×D4D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
#202956072
C2×#168126272
#202947952
#390×C1092
#2006109