Finite Group Databasebeta

← #16818720

Subgroups — #16818720

Order 3712.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#14271513C211yesmaximal, characteristic, complemented
2#14271443C211yesmaximal, characteristic, complemented
3#14271368C211yesmaximal, characteristic, complemented
4#14270912C211yesmaximal, characteristic, complemented
5D4⋊#6597C211yesmaximal, characteristic, complemented
6D4⋊#6597C211yesmaximal, characteristic, complemented
7C29×#555C211yesmaximal, characteristic
8C29×C4⋊Q8C2211yescharacteristic
9C29×#124—22no—
10C29×#121—21no—
11C29×#121—21no—
12C4×D4×C29C2211yescharacteristic
13C4×D4×C29C2211yescharacteristic
14C29×#117—21no—
15C29×#117—21no—
16C2×#90783—21no—
17#11745231C2211yescharacteristic
18#11745183—22no—
19#11745180—21no—
20#11745158—21no—
21#11745155—21no—
22C4⋊#6597C2211yescharacteristic
23C4⋊#6597C2211yescharacteristic
24C4×#6597C2211yescharacteristic
25C2×Q8×C29D412yes—
26C2×D4×C29—22no—
27C2×D4×C29D411yescharacteristic
28C2×D4×C29D411yescharacteristic
29C22×C4×C29—21no—
30C22×C4×C29—21no—
31C2×Dic58—21no—
32C4⋊C4×C29—21no—
33C4⋊C4×C29—21no—
34C4⋊C4×C29D411yescharacteristic
35C4⋊C4×C29D411yescharacteristic
36C22⋊C4×C29—22no—
37C22⋊C4×C29—22no—
38C22⋊C4×C29—21no—
39C22⋊C4×C29—21no—
40C42×C29—22no—
41C42×C29C2311yescharacteristic
42#90783—42no—
43C4⋊Dic29—21no—
44C4⋊Dic29—21no—
45C2×#6597—21no—
46C2×#6597—21no—
47C2×#6597—21no—
48C23×C29—21no—
49C23×C29—21no—
50Q8×C29—42no—
51D4×C29—21no—
52D4×C29—21no—
53D4×C29—21no—
54D4×C29—21no—
55D4×C29—82no—
56C2×C4×C29—21no—
57C2×C4×C29—21no—
58C2×C4×C29—22no—
59C2×C4×C29—41no—
60C2×C4×C29—41no—
61C2×C4×C29—41no—
62C2×C4×C29—41no—
63C2×C4×C29C2×D411yescharacteristic
64C2×C4×C29C2×D411yescharacteristic
65C2×C4×C29C2×D411yescharacteristic
66C2×Dic29—41no—
67Dic58—41no—
68Dic58—41no—
69#6597—41no—
70#6597—41no—
71#6597—41no—
72#1646—291nomaximal, Sylow
73C22×C29—41no—
74C22×C29—41no—
75C22×C29—41no—
76C22×C29—41no—
77C22×C29—41no—
78C22×C29—41no—
79C22×C29#12011yescharacteristic
80C4×C29—21no—
81C4×C29—41no—
82C4×C29—41no—
83C4×C29—42no—
84C4×C29#13511yescharacteristic
85C4×C29#13511yescharacteristic
86C4×C29AGL(1,ℤ/8)11yescharacteristic
87C4×C29#13711yescharacteristic
88Dic29—81no—
89#555D2911yescharacteristic
90#487—291no—
91#474—291no—
92#461—291no—
93#333—291no—
94D4⋊C8—291no—
95D4⋊C8—291no—
96C2×C29—81no—
97C2×C29—81no—
98C2×C29#44911yescharacteristic
99C2×C29#45211yescharacteristic
100C2×C29#45511yescharacteristic
101C2×SD16—581no—
102C4⋊Q8—291no—
103C4⋊Q8C2×D2911yescharacteristic
104#124—22no—
105#121—21no—
106#121—21no—
107C4×D4C2×D2911yescharacteristic
108C4×D4C2×D2911yescharacteristic
109#117—21no—
110#117—21no—
111#107—581no—
112C4⋊C8—291no—
113C4⋊C8—291no—
114#103—582no—
115#103—581no—
116#102—581no—
117C4×C8—291no—
118C29#164611yescharacteristic, complemented, Sylow
119C2×Q8—581no—
120C2×Q8C22⋊D2912yes—
121C2×D4—22no—
122C2×D4C22⋊D2911yescharacteristic
123C2×D4C22⋊D2911yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—1162no—
127C2×C8—581no—
128C2×C8—581no—
129C2×C8—581no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—581no—
133C4⋊C4—581no—
134C4⋊C4C22⋊D2911yescharacteristic
135C4⋊C4C22⋊D2911yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D2911yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—1161no—
146Q8—1161no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—1161no—
160C2×C4C2×C22⋊D2911yescharacteristic
161C2×C4C2×C22⋊D2911yescharacteristic
162C2×C4C2×C22⋊D2911yescharacteristic
163C8—1161no—
164C8—1161no—
165C8—1161no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1174531511yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—2321no—
178C4#1174531111yescharacteristic
179C4#1174531211yescharacteristic
180C4#1174531211yescharacteristic
181C4#1174531311yescharacteristic
182C2—81no—
183C2—81no—
184C2#1427130811yescharacteristic
185C2#1427132511yescharacteristic
186C2#1427134111yescharacteristic
subgroupG/H
centerC22#11745315
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
Fitting, prime radicalC29×#555C2
socleC22×C29#120
hypercenter#555D29
hypocenterC29#1646
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #14271513C2semidirect
2. #14271443C2semidirect
3. #14271368C2semidirect
4. #14270912C2semidirect
5. D4⋊#6597C2semidirect
6. D4⋊#6597C2semidirect
118. C29#1646semidirect

Maximal subgroups

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

subgroupindex
#142715132
#142714432
#142713682
#142709122
D4⋊#65972
C29×#5552
#164629