Finite Group Databasebeta

← #28364076

Subgroups — #28364076

Order 13952.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16812727C212yesmaximal, complemented
2#20295607C211yesmaximal, characteristic, complemented
3C2×#16812625C212yesmaximal, complemented
4#20294927C211yesmaximal, characteristic, complemented
5#343×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2212yes—
7C22×D436C2211yescharacteristic
8#16812727—24no—
9#16812727—24no—
10C2×#13182018—22no—
11C2×D872—22no—
12#16812679—22no—
13#16812674—22no—
14#16812673—22no—
15C2×#13182006—22no—
16C2×C4×Dic109C2211yescharacteristic
17#16812625—24no—
18C2×M16×C109C2212yes—
19C2×C4⋊C4×C109C2211yescharacteristic
20C23×D109—21no—
21#13182029—22no—
22#13182029—22no—
23#13182029—22no—
24#13182029—22no—
25C2×Dic218D412yes—
26C2×D436—22no—
27C2×D436—22no—
28C2×D436—21no—
29C2×D436—22no—
30C2×D436D412yes—
31C2×D436D412yescomplemented
32C2×C22⋊D109—22no—
33C22×Dic109—21no—
34C2×C4×D109—22no—
35#13182018—28no—
36D872—28no—
37C4⋊Dic109—22no—
38#13182006—24no—
39#13182006—44no—
40C4×Dic109—22no—
41C22×C4×C109D412yes—
42C22×C4×C109C2311yescharacteristic
43M16×C109—22no—
44M16×C109D414yescomplemented
45C2×C8×C109—22no—
46C4⋊C4×C109—22no—
47C23×C109C2×D411yescharacteristic
48C22×D109—22no—
49C22×D109—22no—
50C22×D109—42no—
51C22×D109—42no—
52C22×D109—41no—
53C22×D109—41no—
54C2×C4×C109—22no—
55C2×C4×C109—42no—
56C2×C4×C109C2×D411yescharacteristic
57C2×C4×C109C2×D412yes—
58C2×C4×C109C2×D412yes—
59C2×C4×C109#4111yescharacteristic
60C22⋊D109—42no—
61C22⋊D109—42no—
62C2×Dic109—22no—
63C2×Dic109—22no—
64C2×Dic109—41no—
65C2×Dic109—41no—
66D436—22no—
67D436—24no—
68D436—22no—
69D436—42no—
70D436—82no—
71C4×D109—42no—
72C4×D109—42no—
73Dic218—22no—
74Dic218—22no—
75C8×C109—24no—
76C22×C109—21no—
77C22×C109—21no—
78C22×C109#12011yescharacteristic
79C22×C109#12111yescharacteristic
80C22×C109#12711yescharacteristic
81C2×D109—44no—
82C2×D109—42no—
83C2×D109—42no—
84C2×D109—42no—
85C2×D109—42no—
86C2×D109—82no—
87C2×D109—81no—
88C2×D109—81no—
89C4×C109—42no—
90C4×C109#12012yes—
91C4×C109#12112yes—
92Dic109—42no—
93Dic109—42no—
94C2×C109—21no—
95C2×C109—21no—
96C2×C109#39211yescharacteristic
97C2×C109#45311yescharacteristic
98C2×C109#45411yescharacteristic
99D109—82no—
100D109—82no—
101#2006—1091nomaximal, Sylow
102C109#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1092no—
104#537—1091no—
105C2×#104—1092no—
106#390—1091no—
107#343D10911yescharacteristic, complemented
108C2×#41—1092no—
109C22×D4—1091no—
110AGL(1,ℤ/8)—2184no—
111AGL(1,ℤ/8)—2184no—
112C2×SD16—2182no—
113C2×D8—2182no—
114C2×M16C2×D10912yes—
115C22⋊Q8—2182no—
116#121—2182no—
117#121—2182no—
118C2×C4⋊C4C2×D10911yescharacteristic
119C2×C22⋊C4—2182no—
120C2×C42—1091no—
121#104—2184no—
122C24—2181no—
123#41—2182no—
124#41—2182no—
125#41—2182no—
126#41—2182no—
127C2×Q8—1092no—
128C2×D4—1092no—
129C2×D4—1092no—
130C2×D4—2182no—
131C2×D4—2182no—
132C2×D4—2182no—
133C2×D4—2182no—
134C2×D4—2181no—
135C22×C4—2181no—
136C22×C4—2182no—
137C22×C4D43612yes—
138C22×C4C22×D10911yescharacteristic
139SD16—2188no—
140D8—2188no—
141M16—22no—
142M16D43614yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—2182no—
146C22⋊C4—4364no—
147C22⋊C4—2184no—
148C42—2182no—
149C23—4362no—
150C23—4362no—
151C23—4361no—
152C23—4361no—
153C23—2182no—
154C23—2182no—
155C23C2×D43611yescharacteristic
156Q8—2182no—
157Q8—2182no—
158D4—8722no—
159D4—4362no—
160D4—4362no—
161D4—4362no—
162D4—2182no—
163D4—2184no—
164D4—2182no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—4361no—
168C2×C4—4361no—
169C2×C4—4362no—
170C2×C4—4362no—
171C2×C4—2182no—
172C2×C4—2182no—
173C2×C4D4×D10911yescharacteristic
174C2×C4D4×D10912yes—
175C2×C4C2×D43612yes—
176C2×C4#1318203011yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—8722no—
181C22—8721no—
182C22—8721no—
183C22—4364no—
184C22—4362no—
185C22—4362no—
186C22—4362no—
187C22—4362no—
188C22#1681266911yescharacteristic
189C22#1681267411yescharacteristic
190C22#1681269611yescharacteristic
191C4—42no—
192C4—4362no—
193C4—4362no—
194C4#1681266912yes—
195C4#1681267412yes—
196C2—21no—
197C2—21no—
198C2—8722no—
199C2—8722no—
200C2#2029493611yescharacteristic
201C2#2029519611yescharacteristic
202C2#2029520111yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#343×C109C2
socleC22×C109#120
hypercenter#343D109
hypocenterC109#2006
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#16812727C2semidirect
2. #20295607C2semidirect
3. C2×#16812625C2semidirect
4. #20294927C2semidirect
5. #343×C109C2semidirect
31. C2×D436D4semidirect
44. M16×C109D4semidirect
102. C109#2006semidirect
107. #343D109semidirect
142. M16D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
#202956072
C2×#168126252
#202949272
#343×C1092
#2006109