Finite Group Databasebeta

← #28362224

Subgroups — #28362224

Order 13952.

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

normal subgroups only

This group has 250 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#20295697C211yesmaximal, characteristic
2#20295374C211yesmaximal, characteristic, complemented
3#20295233C211yesmaximal, characteristic, complemented
4#20294769C211yesmaximal, characteristic, complemented
5#20294756C211yesmaximal, characteristic, complemented
6D4⋊#11724784C211yesmaximal, characteristic, complemented
7#474×C109C211yesmaximal, characteristic, complemented
8C2×#13182029—21no—
9C2×#13182019—21no—
10#16812696C2211yescharacteristic
11#16812695—21no—
12#16812673—21no—
13#16812672—22no—
14C4×Dic218C2211yescharacteristic
15#16812661—21no—
16C4⋊#11724784C2211yescharacteristic
17#16812623—21no—
18#16812619—22no—
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×Dic218D411yescharacteristic
28C2×D436—21no—
29C2×D436—21no—
30C2×D436D412yes—
31C2×C22⋊D109—22no—
32C2×C22⋊D109—22no—
33C2×C4×D109—22no—
34#13182019—42no—
35C2×#11724784—21no—
36C2×#11724784—21no—
37C4⋊Dic109D411yescharacteristic
38#13182008—21no—
39#13182006—22no—
40C4×Dic109—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—22no—
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—
64C22⋊D109—82no—
65C2×Dic109—21no—
66C2×Dic109—21no—
67D436—42no—
68D436—41no—
69D436—41no—
70D436—41no—
71D436—41no—
72C4×D109—44no—
73Dic218—21no—
74Dic218D812yes—
75C8×C109—41no—
76#11724784—41no—
77#11724784—41no—
78C22×C109—41no—
79C22×C109—41no—
80C22×C109—41no—
81C22×C109#12011yescharacteristic
82C2×D109—42no—
83C2×D109—42no—
84C2×D109—42no—
85C4×C109—21no—
86C4×C109—41no—
87C4×C109—81no—
88C4×C109C2×D811yescharacteristic
89C4×C109#13511yescharacteristic
90C4×C109AGL(1,ℤ/8)11yescharacteristic
91C4×C109#13711yescharacteristic
92Dic109—22no—
93Dic109—41no—
94C2×C109—81no—
95C2×C109#44711yescharacteristic
96C2×C109#45211yescharacteristic
97C2×C109#45511yescharacteristic
98D109—82no—
99#1641—1091nomaximal, Sylow
100C109#164111yescharacteristic, complemented, Sylow
101#550—1091no—
102#486—1091no—
103#474D10911yescharacteristic, complemented
104#462—1091no—
105#331—1091no—
106Q8⋊C8—1091no—
107D4⋊C8—1091no—
108C2×#41—2181no—
109C2×Q16—2181no—
110C4⋊Q8C2×D10911yescharacteristic
111#127—1091no—
112#127—2181no—
113#121—2182no—
114#121—2181no—
115C4×Q8—1091no—
116C4×D4—2181no—
117C4×D4C2×D10911yescharacteristic
118#107—21no—
119C4⋊C8—1091no—
120C4⋊C8C2×D10911yescharacteristic
121#103—2181no—
122#102—21no—
123#102—2182no—
124C4×C8—1091no—
125#41—4362no—
126#41—4362no—
127C2×Q8—21no—
128C2×Q8—1091no—
129C2×D4—1092no—
130C2×D4—2182no—
131C2×D4—2182no—
132C2×D4—2181no—
133C2×D4—2181no—
134C2×D4C22⋊D10911yescharacteristic
135C22×C4—21no—
136C22×C4—2182no—
137Q16—4362no—
138C2×C8—21no—
139C2×C8—2181no—
140C2×C8—2181no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1091no—
144C4⋊C4—2181no—
145C4⋊C4C22⋊D10911yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—2182no—
148C42—2181no—
149C42C22×D10911yescharacteristic
150C23—21no—
151C23—2182no—
152Q8—41no—
153Q8—41no—
154Q8—1092no—
155Q8—2181no—
156D4—21no—
157D4—21no—
158D4—8722no—
159D4—4362no—
160D4—4364no—
161D4—4361no—
162D4—4361no—
163D4—4361no—
164D4—4361no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—4364no—
170C2×C4—2181no—
171C2×C4—2181no—
172C2×C4D4×D10911yescharacteristic
173C2×C4D4×D10911yescharacteristic
174C2×C4C2×C22⋊D10911yescharacteristic
175C8—41no—
176C8—4361no—
177C8—4361no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—4362no—
182C22—4362no—
183C22—4362no—
184C22#1681266711yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—4361no—
189C4—2182no—
190C4D8×D10911yescharacteristic
191C4#1681271911yescharacteristic
192C4#1681272311yescharacteristic
193C4#1681273011yescharacteristic
194C2—81no—
195C2—8722no—
196C2#2029515811yescharacteristic
197C2#2029518611yescharacteristic
198C2#2029520211yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#474×C109C2
socleC22×C109#120
prime radical#20295697C2
hypercenter#474D109
hypocenterC109#1641
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. #20295374C2semidirect
3. #20295233C2semidirect
4. #20294769C2semidirect
5. #20294756C2semidirect
6. D4⋊#11724784C2semidirect
7. #474×C109C2semidirect
100. C109#1641semidirect
103. #474D109semidirect

Maximal subgroups

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

subgroupindex
#202956972
#202953742
#202952332
#202947692
#202947562
D4⋊#117247842
#474×C1092
#1641109