Finite Group Databasebeta

← #28379293

Subgroups — #28379293

Order 13952.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#20296012C211yesmaximal, characteristic, complemented
2#20296005C211yesmaximal, characteristic, complemented
3#20295927C214yesmaximal, complemented
4#20295595C211yesmaximal, characteristic
5D109⋊M16—21no—
6#16812778—21no—
7#16812760—21no—
8#16812759C2212yes—
9#16812758—21no—
10#16812757C2212yes—
11C2×#13182029C2211yescharacteristic
12C22×D436C411yescharacteristic, complemented
13#16812696—21no—
14#16812687—21no—
15#16812674—21no—
16#16812674—21no—
17#16812669C412yescomplemented
18#16812669C2212yes—
19#117×C109C411yescharacteristic, complemented
20#13182039—41no—
21#13182038—22no—
22D109⋊C8—41no—
23C4⋊#89137—41no—
24C22⋊#89137—22no—
25C4×#89137—41no—
26C23×D109C2×C412yes—
27#13182029—41no—
28#13182029—41no—
29C2×Dic218D411yescharacteristic
30C2×D436—21no—
31C2×D436—21no—
32C2×D436—22no—
33C2×D436—22no—
34C2×D436—41no—
35C2×D436—41no—
36C2×D436D411yescharacteristic
37C2×C22⋊D109C2×C411yescharacteristic
38C2×C22⋊D109C2311yescharacteristic
39C2×C4×D109D411yescharacteristic
40C2×C4×D109D411yescharacteristic
41#13182006—22no—
42#13182006—22no—
43C22×C4×C109C2×C411yescharacteristic
44C4⋊C4×C109—21no—
45C22⋊C4×C109C2×C412yes—
46C42×C109—21no—
47C23×C109C22×C411yescharacteristic
48C22×D109—22no—
49C22×D109—42no—
50C22×D109—82no—
51C22×D109C22⋊C411yescharacteristic
52C22×D109C2×D411yescharacteristic
53C2×#89137—42no—
54C2×C4×C109—21no—
55C2×C4×C109—21no—
56C2×C4×C109—22no—
57C2×C4×C109C22⋊C411yescharacteristic
58C2×C4×C109C22⋊C411yescharacteristic
59C22⋊D109—41no—
60C22⋊D109—41no—
61C2×Dic109C22⋊C411yescharacteristic
62C2×Dic109C2×D411yescharacteristic
63D436—41no—
64D436—42no—
65D436—41no—
66D436—41no—
67D436—82no—
68D436—82no—
69C4×D109—21no—
70C4×D109—21no—
71C4×D109—21no—
72C4×D109—21no—
73Dic218—41no—
74#11724786—42no—
75C22×C109—21no—
76C22×C109—41no—
77C22×C109C2×C22⋊C411yescharacteristic
78C2×D109—21no—
79C2×D109—21no—
80C2×D109—41no—
81C2×D109—41no—
82C2×D109—42no—
83C2×D109—42no—
84C2×D109—42no—
85C2×D109—84no—
86#89137—82no—
87C4×C109—21no—
88C4×C109—42no—
89C4×C109#9911yescharacteristic
90C4×C109#9911yescharacteristic
91Dic109—21no—
92Dic109—21no—
93C2×C109—21no—
94C2×C109—41no—
95C2×C109C2×#9911yescharacteristic
96D109—41no—
97D109—41no—
98D109—82no—
99#2122—1091nomaximal, Sylow
100C109#212211yescharacteristic, complemented, Sylow
101#535—1091no—
102#413—1091no—
103#410—1091no—
104AΣL(1,𝔽16)—1094no—
105C2×#41—1091no—
106C22×D4—1091no—
107C2×M16—2181no—
108#127—2181no—
109#124—2181no—
110#121—2181no—
111#121—2181no—
112#120—1092no—
113#120—1092no—
114#117—2181no—
115#117#8913711yescharacteristic, complemented
116#101—2181no—
117#100—1092no—
118#99—1092no—
119#99—2181no—
120C24—1092no—
121#41—4361no—
122#41—4361no—
123C2×Q8—1091no—
124C2×D4—1091no—
125C2×D4—1091no—
126C2×D4—1091no—
127C2×D4—4361no—
128C2×D4—4361no—
129C2×D4—2182no—
130C2×D4—2182no—
131C2×D4—2181no—
132C2×D4—2181no—
133C22×C4—1091no—
134C22×C4—1091no—
135C22×C4C2×#8913711yescharacteristic
136M16—4361no—
137M16—2182no—
138C2×C8—4361no—
139C4⋊C4—21no—
140C4⋊C4—4361no—
141C22⋊C4—2182no—
142C22⋊C4—2182no—
143C22⋊C4—2182no—
144C22⋊C4C2×#8913712yes—
145C42—21no—
146C42—4361no—
147C23—8722no—
148C23—1091no—
149C23—1091no—
150C23—4362no—
151C23—2182no—
152C23C22×#8913711yescharacteristic
153Q8—4361no—
154D4—8722no—
155D4—8722no—
156D4—4362no—
157D4—4361no—
158D4—4361no—
159D4—4361no—
160D4—4361no—
161D4—4361no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—1091no—
166C2×C4—1091no—
167C2×C4—4362no—
168C2×C4—2181no—
169C2×C4—2181no—
170C2×C4—2181no—
171C2×C4—2181no—
172C2×C4C22⋊#8913711yescharacteristic
173C2×C4C22⋊#8913711yescharacteristic
174C8—4362no—
175C22—21no—
176C22—41no—
177C22—8724no—
178C22—4362no—
179C22—4362no—
180C22—4361no—
181C22—4361no—
182C22—4362no—
183C22—2181no—
184C22—2181no—
185C22C2×C22⋊#8913711yescharacteristic
186C4—21no—
187C4—42no—
188C4—8722no—
189C4—2181no—
190C4—2181no—
191C4#1681275811yescharacteristic
192C4#1681275811yescharacteristic
193C2—21no—
194C2—41no—
195C2—8722no—
196C2—4361no—
197C2—4361no—
198C2C2×#1681275811yescharacteristic
subgroupG/H
centerC2C2×#16812758
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#117×C109C4
socleC2×C109C2×#99
prime radical#20295595C2
cosocleC2×C22⋊D109C23
hypercenter#117#89137
hypocenterC109#2122
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20296012C2semidirect
2. #20296005C2semidirect
3. #20295927C2semidirect
12. C22×D436C4semidirect
17. #16812669C4semidirect
19. #117×C109C4semidirect
100. C109#2122semidirect
115. #117#89137semidirect

Maximal subgroups

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

subgroupindex
#202960122
#202960052
#202959272
#202955952
#2122109