Finite Group Databasebeta

← #28362167

Subgroups — #28362167

Order 13952.

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

normal subgroups only

This group has 214 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#20295737C211yesmaximal, characteristic
2#20295405C211yesmaximal, characteristic, complemented
3#20295322C211yesmaximal, characteristic, complemented
4#20295317C211yesmaximal, characteristic
5#20294756C211yesmaximal, characteristic, complemented
6#20294753C211yesmaximal, characteristic
7#333×C109C211yesmaximal, characteristic, complemented
8C2×#13182019—22no—
9#16812699C2211yescharacteristic
10#16812690—21no—
11#16812679—21no—
12#16812675—22no—
13C4×Dic218C2211yescharacteristic
14C4×D436C2211yescharacteristic
15#16812661—21no—
16C4⋊C4×D109—21no—
17#16812633—21no—
18#16812631—21no—
19#16812624—21no—
20#16812621—21no—
21C4×#11724784C2211yescharacteristic
22C4⋊Q8×C109C2211yescharacteristic
23C4⋊C8×C109C2211yescharacteristic
24C4⋊C8×C109C2211yescharacteristic
25C2×Dic218—21no—
26C2×Dic218D411yescharacteristic
27C2×D436D411yescharacteristic
28C2×C4×D109—21no—
29C2×C4×D109—22no—
30#13182019—44no—
31C2×#11724784—21no—
32C4⋊Dic109—22no—
33C4⋊Dic109—21no—
34C4⋊Dic109—21no—
35C4⋊Dic109D411yescharacteristic
36C4⋊Dic109D411yescharacteristic
37#13182008—22no—
38#13182008—21no—
39#13182006—22no—
40#13182006—21no—
41C4×Dic109—21no—
42C2×Q8×C109D412yes—
43C2×C8×C109—21no—
44C2×C8×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109—21no—
47C42×C109C2311yescharacteristic
48C22×D109—21no—
49Q8×C109—42no—
50C2×C4×C109—22no—
51C2×C4×C109C2×D411yescharacteristic
52C2×C4×C109C2×D411yescharacteristic
53C2×C4×C109C2×D411yescharacteristic
54C2×Dic109—21no—
55C2×Dic109—21no—
56C2×Dic109—21no—
57C2×Dic109—41no—
58D436—21no—
59D436Q1612yescomplemented
60C4×D109—41no—
61C4×D109—41no—
62C4×D109—82no—
63Dic218—21no—
64Dic218—21no—
65Dic218—42no—
66C8×C109—41no—
67C8×C109—41no—
68#11724784—22no—
69C22×C109#12011yescharacteristic
70C2×D109—22no—
71C2×D109—41no—
72C2×D109—41no—
73C4×C109—21no—
74C4×C109—42no—
75C4×C109C2×Q1611yescharacteristic
76C4×C109#13511yescharacteristic
77C4×C109#13711yescharacteristic
78C4×C109#13711yescharacteristic
79Dic109—41no—
80Dic109—41no—
81Dic109—41no—
82Dic109—81no—
83C2×C109#45111yescharacteristic
84C2×C109#45211yescharacteristic
85C2×C109#45311yescharacteristic
86D109—42no—
87#1632—1091nomaximal, Sylow
88C109#163211yescharacteristic, complemented, Sylow
89#554—1091no—
90#494—1091no—
91#477—1091no—
92#476—1091no—
93#333D10911yescharacteristic, complemented
94Q8⋊C8—1091no—
95D4⋊C8—1091no—
96C2×Q16—2182no—
97C4⋊Q8—1091no—
98C4⋊Q8C2×D10911yescharacteristic
99#125—2181no—
100C22⋊Q8—2181no—
101C22⋊Q8—2182no—
102C4×Q8—1091no—
103C4×D4—1091no—
104C4×D4—2181no—
105C2×C4⋊C4—2181no—
106#107—2181no—
107#106—2181no—
108C4⋊C8C2×D10911yescharacteristic
109C4⋊C8C2×D10911yescharacteristic
110#103—2181no—
111#102—2181no—
112C4×C8—1091no—
113C2×Q8—1091no—
114C2×Q8—2181no—
115C2×Q8D43612yes—
116C2×D4—1091no—
117C22×C4—2182no—
118C22×C4—2181no—
119Q16—4364no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2181no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—1091no—
126C4⋊C4—1091no—
127C4⋊C4—2182no—
128C4⋊C4—2182no—
129C4⋊C4—2181no—
130C4⋊C4—2181no—
131C4⋊C4—2181no—
132C22⋊C4—2182no—
133C22⋊C4—2181no—
134C42—2181no—
135C42C22×D10911yescharacteristic
136C23—2181no—
137Q8—42no—
138Q8—4362no—
139Q8—2181no—
140Q8—2181no—
141D4—1092no—
142D4—2181no—
143C2×C4—22no—
144C2×C4—8722no—
145C2×C4—4361no—
146C2×C4—4361no—
147C2×C4—4361no—
148C2×C4—2181no—
149C2×C4—2181no—
150C2×C4—2181no—
151C2×C4D4×D10911yescharacteristic
152C2×C4D4×D10911yescharacteristic
153C2×C4C2×D43611yescharacteristic
154C8—41no—
155C8—41no—
156C8—2182no—
157C22—4361no—
158C22—4361no—
159C22—2182no—
160C22#1681266911yescharacteristic
161C4—21no—
162C4—42no—
163C4—8721no—
164C4—4361no—
165C4—4361no—
166C4—4361no—
167C4Q16×D10911yescharacteristic
168C4#1681272011yescharacteristic
169C4#1681272811yescharacteristic
170C4#1681273111yescharacteristic
171C2—4362no—
172C2#2029518311yescharacteristic
173C2#2029519111yescharacteristic
174C2#2029519611yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#333×C109C2
socleC22×C109#120
prime radicalC2×D436D4
hypercenter#333D109
hypocenterC109#1632
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20295405C2semidirect
3. #20295322C2semidirect
5. #20294756C2semidirect
7. #333×C109C2semidirect
59. D436Q16semidirect
88. C109#1632semidirect
93. #333D109semidirect

Maximal subgroups

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

subgroupindex
#202957372
#202954052
#202953222
#202953172
#202947562
#202947532
#333×C1092
#1632109