Finite Group Databasebeta

← #35215585

Subgroups — #35215585

Order 20864.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19104768C212yesmaximal, complemented
2#23782246C211yesmaximal, characteristic
3C2×#19104661C212yesmaximal, complemented
4#23781570C211yesmaximal, characteristic, complemented
5#343×C163C211yesmaximal, characteristic, complemented
6C2×#14579699C2212yes—
7C22×Dic326C2211yescharacteristic
8#19104768—24no—
9#19104768—24no—
10C2×Dic652—22no—
11C2×#14579688—22no—
12#19104715—22no—
13#19104710—22no—
14#19104709—22no—
15C2×C4×Dic163C2211yescharacteristic
16#19104661—24no—
17C163⋊#95—22no—
18C2×M16×C163C2212yes—
19C2×C4⋊C4×C163C2211yescharacteristic
20#14579699—22no—
21#14579699—22no—
22#14579699—22no—
23#14579699—22no—
24C2×Dic326—22no—
25C2×Dic326—22no—
26C2×Dic326—21no—
27C2×Dic326D412yes—
28C2×Dic326D412yescomplemented
29C2×D652—22no—
30C2×D652D412yes—
31C2×C22⋊D163—22no—
32C22×Dic163—21no—
33C22×Dic163—21no—
34C2×C4×D163—22no—
35Dic652—28no—
36#14579688—28no—
37C4⋊Dic163—22no—
38(C2×D163)⋊C4—24no—
39C4×Dic163—22no—
40C22×C4×C163D412yes—
41C22×C4×C163C2311yescharacteristic
42M16×C163—22no—
43M16×C163D414yescomplemented
44C2×C8×C163—22no—
45C4⋊C4×C163—22no—
46C22×D163—22no—
47Dic326—24no—
48Dic326—22no—
49Dic326—22no—
50Dic326—42no—
51D652—22no—
52D652—22no—
53D652—82no—
54C22⋊D163—42no—
55C22⋊D163—42no—
56C2×Dic163—22no—
57C2×Dic163—22no—
58C2×Dic163—22no—
59C2×Dic163—41no—
60C2×Dic163—41no—
61C2×Dic163—41no—
62C2×Dic163—41no—
63C4×D163—42no—
64C4×D163—42no—
65C23×C163C2×D411yescharacteristic
66C2×C4×C163—22no—
67C2×C4×C163—42no—
68C2×C4×C163C2×D411yescharacteristic
69C2×C4×C163C2×D412yes—
70C2×C4×C163C2×D412yes—
71C2×C4×C163#4111yescharacteristic
72C8×C163—24no—
73C22×C163—21no—
74C22×C163—21no—
75C22×C163#12011yescharacteristic
76C22×C163#12111yescharacteristic
77C22×C163#12711yescharacteristic
78C2×D163—42no—
79C2×D163—42no—
80C2×D163—42no—
81C4×C163—42no—
82C4×C163#12012yes—
83C4×C163#12112yes—
84Dic163—42no—
85Dic163—42no—
86Dic163—42no—
87C2×C163—21no—
88C2×C163—21no—
89C2×C163#39211yescharacteristic
90C2×C163#45511yescharacteristic
91C2×C163#45611yescharacteristic
92D163—82no—
93C163#200711yescharacteristic, complemented, Sylow
94#2007—1631nomaximal, Sylow
95C2×#137—1632no—
96#537—1631no—
97C2×#104—1632no—
98#391—1631no—
99#343D16311yescharacteristic, complemented
100C2×#41—1632no—
101C22×Q8—1631no—
102#137—3264no—
103#137—3264no—
104C2×Q16—3262no—
105C2×SD16—3262no—
106C2×M16C2×D16312yes—
107C22⋊Q8—3262no—
108#121—3262no—
109#121—3262no—
110C2×C4⋊C4C2×D16311yescharacteristic
111C2×C42—1631no—
112#104—3264no—
113#95—3262no—
114#41—3262no—
115#41—3262no—
116#41—3262no—
117#41—3262no—
118C2×Q8—3261no—
119C2×Q8—3262no—
120C2×Q8—3262no—
121C2×Q8—1632no—
122C2×Q8—1632no—
123C2×D4—3262no—
124C2×D4—3262no—
125C2×D4—1632no—
126C22×C4—3261no—
127C22×C4—3261no—
128C22×C4—3262no—
129C22×C4D65212yes—
130C22×C4C22×D16311yescharacteristic
131Q16—3268no—
132SD16—3268no—
133M16—22no—
134M16D65214yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—3262no—
138C22⋊C4—3264no—
139C42—3262no—
140C23—3262no—
141C23C2×D65211yescharacteristic
142Q8—3264no—
143Q8—3262no—
144Q8—3262no—
145Q8—6522no—
146D4—13042no—
147D4—3262no—
148D4—3262no—
149D4—6522no—
150D4—6522no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—3262no—
154C2×C4—3262no—
155C2×C4—3262no—
156C2×C4—6521no—
157C2×C4—6521no—
158C2×C4—6521no—
159C2×C4—6521no—
160C2×C4—6522no—
161C2×C4—6522no—
162C2×C4D4×D16311yescharacteristic
163C2×C4D4×D16312yes—
164C2×C4C2×D65212yes—
165C2×C4#1457970011yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—6522no—
170C22—6522no—
171C22—6522no—
172C22#1910470511yescharacteristic
173C22#1910471011yescharacteristic
174C22#1910473211yescharacteristic
175C4—42no—
176C4—6522no—
177C4—6522no—
178C4—6522no—
179C4#1910470512yes—
180C4#1910471012yes—
181C2—21no—
182C2—21no—
183C2—13042no—
184C2#2378157511yescharacteristic
185C2#2378184511yescharacteristic
186C2#2378185011yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#343×C163C2
socleC22×C163#120
prime radical#23782246C2
hypercenter#343D163
hypocenterC163#2007
LCSG ▷ C22×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#19104768C2semidirect
3. C2×#19104661C2semidirect
4. #23781570C2semidirect
5. #343×C163C2semidirect
28. C2×Dic326D4semidirect
43. M16×C163D4semidirect
93. C163#2007semidirect
99. #343D163semidirect
134. M16D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
#237822462
C2×#191046612
#237815702
#343×C1632
#2007163