Finite Group Databasebeta

← #28362303

Subgroups — #28362303

Order 13952.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#20295750C211yesmaximal, characteristic
2#20295422C211yesmaximal, characteristic, complemented
3#20295307C211yesmaximal, characteristic
4#20295237C211yesmaximal, characteristic, complemented
5#20294756C211yesmaximal, characteristic, complemented
6#20294753C211yesmaximal, characteristic
7#332×C109C211yesmaximal, characteristic, complemented
8C2×Dic436—21no—
9#16812699C2211yescharacteristic
10#16812694—22no—
11#16812688—21no—
12#16812679—21no—
13Q8×Dic109—21no—
14C4×Dic218C2211yescharacteristic
15C4×D436C2211yescharacteristic
16#16812657—21no—
17#16812633—21no—
18#16812630—22no—
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—
29Dic436—42no—
30C2×#11724784—21no—
31C4⋊Dic109—22no—
32C4⋊Dic109—21no—
33C4⋊Dic109—21no—
34C4⋊Dic109D411yescharacteristic
35C4⋊Dic109D411yescharacteristic
36#13182008—21no—
37#13182006—22no—
38#13182006—22no—
39#13182006—21no—
40C4×Dic109—22no—
41C4×Dic109—21no—
42C2×Q8×C109—21no—
43C2×C8×C109—21no—
44C2×C8×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109D412yes—
47C42×C109C2311yescharacteristic
48C22×D109—21no—
49Q8×C109—41no—
50Q8×C109—41no—
51C2×C4×C109—22no—
52C2×C4×C109C2×D411yescharacteristic
53C2×C4×C109C2×D411yescharacteristic
54C2×C4×C109C2×D411yescharacteristic
55C2×Dic109—21no—
56C2×Dic109—21no—
57C2×Dic109—21no—
58C2×Dic109—41no—
59D436—21no—
60D436—21no—
61C4×D109—41no—
62C4×D109—41no—
63Dic218—21no—
64Dic218—21no—
65Dic218—41no—
66Dic218—41no—
67C8×C109—41no—
68C8×C109—41no—
69#11724784—22no—
70C22×C109#12011yescharacteristic
71C2×D109—41no—
72C2×D109—41no—
73C2×D109—41no—
74C4×C109—21no—
75C4×C109—42no—
76C4×C109#13511yescharacteristic
77C4×C109#13511yescharacteristic
78C4×C109AGL(1,ℤ/8)11yescharacteristic
79C4×C109#13711yescharacteristic
80Dic109—41no—
81Dic109—41no—
82Dic109—41no—
83Dic109—81no—
84C2×C109#44911yescharacteristic
85C2×C109#45211yescharacteristic
86C2×C109#45611yescharacteristic
87D109—81no—
88#1652—1091nomaximal, Sylow
89C109#165211yescharacteristic, complemented, Sylow
90#556—1091no—
91#498—1091no—
92#474—1091no—
93#462—1091no—
94#332D10911yescharacteristic, complemented
95Q8⋊C8—1091no—
96D4⋊C8—1091no—
97C2×Q16—2181no—
98C4⋊Q8—1091no—
99C4⋊Q8C2×D10911yescharacteristic
100#126—2182no—
101#124—2181no—
102C22⋊Q8—2181no—
103C4×Q8—1091no—
104C4×Q8—2181no—
105C4×D4—1091no—
106#117—2181no—
107#107—2181no—
108#106—2182no—
109C4⋊C8C2×D10911yescharacteristic
110C4⋊C8C2×D10911yescharacteristic
111#103—2181no—
112#102—2181no—
113C4×C8—1091no—
114C2×Q8—21no—
115C2×Q8—1091no—
116C2×Q8—2181no—
117C2×D4—1091no—
118C22×C4—2181no—
119Q16—4362no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2181no—
123C4⋊C4—21no—
124C4⋊C4—1091no—
125C4⋊C4—1091no—
126C4⋊C4—2182no—
127C4⋊C4—2181no—
128C4⋊C4—2181no—
129C4⋊C4—2181no—
130C4⋊C4D43612yes—
131C22⋊C4—2182no—
132C22⋊C4—2182no—
133C22⋊C4—2181no—
134C42—2182no—
135C42—2181no—
136C42C22×D10911yescharacteristic
137C23—2181no—
138Q8—41no—
139Q8—41no—
140Q8—4361no—
141Q8—4361no—
142Q8—2181no—
143Q8—2181no—
144D4—2181no—
145D4—2181no—
146C2×C4—22no—
147C2×C4—4361no—
148C2×C4—4361no—
149C2×C4—4361no—
150C2×C4—2181no—
151C2×C4—2181no—
152C2×C4—2181no—
153C2×C4D4×D10911yescharacteristic
154C2×C4D4×D10911yescharacteristic
155C2×C4C2×D43611yescharacteristic
156C8—41no—
157C8—41no—
158C8—2182no—
159C22—4361no—
160C22—4361no—
161C22—4361no—
162C22#1681266911yescharacteristic
163C4—21no—
164C4—42no—
165C4—8721no—
166C4—4361no—
167C4—4361no—
168C4—4361no—
169C4#1681271811yescharacteristic
170C4#1681272211yescharacteristic
171C4#1681272611yescharacteristic
172C4#1681272811yescharacteristic
173C2—8721no—
174C2#2029517411yescharacteristic
175C2#2029518911yescharacteristic
176C2#2029521111yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#332×C109C2
socleC22×C109#120
prime radicalC2×D436D4
hypercenter#332D109
hypocenterC109#1652
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20295422C2semidirect
4. #20295237C2semidirect
5. #20294756C2semidirect
7. #332×C109C2semidirect
89. C109#1652semidirect
94. #332D109semidirect

Maximal subgroups

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

subgroupindex
#202957502
#202954222
#202953072
#202952372
#202947562
#202947532
#332×C1092
#1652109