Finite Group Databasebeta

← #28362266

Subgroups — #28362266

Order 13952.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#20295681C211yesmaximal, characteristic
2#20295429C211yesmaximal, characteristic, complemented
3#20295228C211yesmaximal, characteristic, complemented
4#20294771C211yesmaximal, characteristic, complemented
5D4⋊#11724784C211yesmaximal, characteristic, complemented
6#20294753C211yesmaximal, characteristic, complemented
7#474×C109C211yesmaximal, characteristic, complemented
8C2×#13182029—21no—
9C2×#13182016—21no—
10#16812699C2211yescharacteristic
11#16812681—22no—
12#16812678—21no—
13#16812670—21no—
14C4×D436C2211yescharacteristic
15C4⋊C4×D109—21no—
16#16812632—22no—
17C4⋊#11724784C2211yescharacteristic
18#16812623—21no—
19C4×#11724784C2211yescharacteristic
20C4⋊Q8×C109C2211yescharacteristic
21C4×D4×C109C2211yescharacteristic
22#107×C109—21no—
23C4⋊C8×C109C2211yescharacteristic
24#102×C109—21no—
25#13182029—42no—
26#13182029—42no—
27C2×Dic218—21no—
28C2×D436D411yescharacteristic
29C2×C22⋊D109—22no—
30C2×C4×D109—22no—
31C2×C4×D109—21no—
32#13182016—42no—
33C2×#11724784—21no—
34C2×#11724784—21no—
35C4⋊Dic109—21no—
36C4⋊Dic109D411yescharacteristic
37C4⋊Dic109D412yes—
38#13182008—22no—
39C22⋊Dic109—22no—
40#13182006—21no—
41C2×Q8×C109—21no—
42C2×D4×C109D411yescharacteristic
43C22×C4×C109—21no—
44C2×C8×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109—21no—
47C4⋊C4×C109D411yescharacteristic
48C22⋊C4×C109—21no—
49C42×C109C2311yescharacteristic
50C23×C109—21no—
51C22×D109—21no—
52Q8×C109—41no—
53Q8×C109—41no—
54D4×C109—21no—
55D4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109—41no—
58C2×C4×C109—41no—
59C2×C4×C109—41no—
60C2×C4×C109C2×D411yescharacteristic
61C2×C4×C109C2×D411yescharacteristic
62C2×C4×C109C2×D411yescharacteristic
63C22⋊D109—44no—
64C2×Dic109—21no—
65C2×Dic109—22no—
66D436—21no—
67D436D812yes—
68C4×D109—44no—
69C4×D109—41no—
70C4×D109—41no—
71Dic218—41no—
72Dic218—41no—
73C8×C109—41no—
74#11724784—22no—
75#11724784—41no—
76C22×C109—41no—
77C22×C109—41no—
78C22×C109—41no—
79C22×C109#12011yescharacteristic
80C2×D109—22no—
81C2×D109—41no—
82C2×D109—41no—
83C4×C109—21no—
84C4×C109—41no—
85C4×C109—81no—
86C4×C109C2×D811yescharacteristic
87C4×C109#13511yescharacteristic
88C4×C109AGL(1,ℤ/8)11yescharacteristic
89C4×C109#13711yescharacteristic
90Dic109—42no—
91Dic109—41no—
92C2×C109—81no—
93C2×C109#44711yescharacteristic
94C2×C109#45211yescharacteristic
95C2×C109#45611yescharacteristic
96D109—42no—
97#1647—1091nomaximal, Sylow
98C109#164711yescharacteristic, complemented, Sylow
99#547—1091no—
100#500—1091no—
101#474D10911yescharacteristic, complemented
102#461—1091no—
103#332—1091no—
104D4⋊C8—1091no—
105D4⋊C8—1091no—
106C2×#41—2181no—
107C2×SD16—2181no—
108C4⋊Q8—1091no—
109C4⋊Q8C2×D10911yescharacteristic
110#123—2182no—
111C22⋊Q8—2181no—
112#121—2181no—
113C4×D4—1091no—
114C4×D4C2×D10911yescharacteristic
115C2×C4⋊C4—2181no—
116#107—21no—
117#107—2182no—
118C4⋊C8—1091no—
119C4⋊C8C2×D10911yescharacteristic
120#103—2181no—
121#102—21no—
122C4×C8—1091no—
123#41—4362no—
124#41—4362no—
125C2×Q8—21no—
126C2×Q8—2181no—
127C2×D4—1091no—
128C2×D4—2182no—
129C2×D4C22⋊D10911yescharacteristic
130C22×C4—21no—
131C22×C4—2182no—
132C22×C4—2181no—
133SD16—4362no—
134C2×C8—21no—
135C2×C8—2181no—
136C2×C8—2181no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—1091no—
140C4⋊C4—1092no—
141C4⋊C4—2182no—
142C4⋊C4—2181no—
143C4⋊C4C22⋊D10911yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2182no—
146C22⋊C4—2181no—
147C42C22×D10911yescharacteristic
148C23—21no—
149C23—2181no—
150Q8—41no—
151Q8—41no—
152Q8—4361no—
153Q8—4361no—
154D4—21no—
155D4—21no—
156D4—1092no—
157D4—4364no—
158D4—2181no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—4364no—
164C2×C4—4361no—
165C2×C4—4361no—
166C2×C4—2181no—
167C2×C4—2182no—
168C2×C4D4×D10911yescharacteristic
169C2×C4D4×D10911yescharacteristic
170C2×C4C2×C22⋊D10911yescharacteristic
171C8—41no—
172C8—4361no—
173C8—2182no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—4361no—
178C22—4361no—
179C22—2182no—
180C22#1681266711yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—4361no—
185C4—4362no—
186C4D8×D10911yescharacteristic
187C4#1681271911yescharacteristic
188C4#1681272311yescharacteristic
189C4#1681273011yescharacteristic
190C2—81no—
191C2—4362no—
192C2#2029515811yescharacteristic
193C2#2029518611yescharacteristic
194C2#2029520711yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#474×C109C2
socleC22×C109#120
prime radical#20295681C2
hypercenter#474D109
hypocenterC109#1647
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20295429C2semidirect
3. #20295228C2semidirect
4. #20294771C2semidirect
5. D4⋊#11724784C2semidirect
6. #20294753C2semidirect
7. #474×C109C2semidirect
98. C109#1647semidirect
101. #474D109semidirect

Maximal subgroups

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

subgroupindex
#202956812
#202954292
#202952282
#202947712
D4⋊#117247842
#202947532
#474×C1092
#1647109