Finite Group Databasebeta

← #35216218

Subgroups — #35216218

Order 20864.

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

normal subgroups only

This group has 250 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#23782214C211yesmaximal, characteristic
2#23781648C212yesmaximal, complemented
3#23781463C212yesmaximal, complemented
4#23781462C211yesmaximal, characteristic, complemented
5#355×C163C211yesmaximal, characteristic, complemented
6C2×#14579700C2211yescharacteristic
7C2×#14579698C411yescharacteristic
8M16×D163—22no—
9#19104733C2211yescharacteristic
10#19104731C411yescharacteristic
11#19104723C411yescharacteristic
12#19104708—22no—
13(C22⋊D163)⋊C4—22no—
14C42×D163C411yescharacteristic
15#19104654C2212yes—
16#19104651—22no—
17#124×C163C2211yescharacteristic
18#101×C163C2212yes—
19#14579700—41no—
20#14579700—41no—
21#14579698—41no—
22#14579698—41no—
23C2×Dic326C2×C411yescharacteristic
24C2×D652C2×C411yescharacteristic
25C2×D652D412yes—
26C2×C22⋊D163—21no—
27C2×C22⋊D163—22no—
28C22×Dic163—21no—
29C2×C4×D163—21no—
30C2×C4×D163C2×C411yescharacteristic
31C2×C4×D163D412yes—
32#14579683—22no—
33#14579682—42no—
34C8×D163—42no—
35Dic163⋊C4—22no—
36(C2×D163)⋊C4—22no—
37C4×Dic163—21no—
38C4×Dic163C2×C411yescharacteristic
39C2×Q8×C163C2311yescharacteristic
40C2×D4×C163C2×C411yescharacteristic
41M16×C163—22no—
42C22⋊C4×C163—22no—
43C42×C163C2×C411yescharacteristic
44C22×D163C22⋊C412yes—
45C22×D163C22⋊C411yescharacteristic
46Dic326—41no—
47D652—41no—
48D652—42no—
49C22⋊D163—44no—
50C22⋊D163—42no—
51C2×Dic163—21no—
52C2×Dic163—21no—
53C2×Dic163—41no—
54C2×Dic163—41no—
55C2×Dic163C22⋊C411yescharacteristic
56C4×D163—22no—
57C4×D163—22no—
58C4×D163—21no—
59C4×D163—21no—
60C4×D163—42no—
61C163⋊C8—42no—
62C23×C163—21no—
63Q8×C163—41no—
64D4×C163—41no—
65C2×C4×C163—21no—
66C2×C4×C163C22×C411yescharacteristic
67C2×C4×C163C2×D412yes—
68C8×C163—42no—
69C22×C163—42no—
70C22×C163—41no—
71C22×C163C2×C22⋊C411yescharacteristic
72C2×D163—21no—
73C2×D163—22no—
74C2×D163—41no—
75C2×D163—42no—
76C4×C163—22no—
77C4×C163—21no—
78C4×C163—41no—
79Dic163—22no—
80Dic163—41no—
81Dic163#9911yescharacteristic
82Dic163#9911yescharacteristic
83C2×C163—21no—
84C2×C163—81no—
85C2×C163C2×#9911yescharacteristic
86D163—41no—
87D163—42no—
88C163#213211yescharacteristic, complemented, Sylow
89#2132—1631nomaximal, Sylow
90#532—1631no—
91#413—1632no—
92#356—1632no—
93#355—1631no—
94#355D16311yescharacteristic, complemented
95C2×#41—1631no—
96C2×#41—1631no—
97C2×M16—3262no—
98C4⋊Q8—1631no—
99#127—1631no—
100#124—1631no—
101#124C2×D16311yescharacteristic
102#121—3262no—
103C4×D4—3262no—
104C2×C42—1631no—
105#101—1632no—
106#101C2×D16312yes—
107#100—3262no—
108#41—6521no—
109#41—6521no—
110#41—6521no—
111#41—6521no—
112C2×Q8—1631no—
113C2×Q8C22×D16311yescharacteristic
114C2×D4—3262no—
115C2×D4—3261no—
116C2×D4—1631no—
117C2×D4—1632no—
118C2×D4C4×D16311yescharacteristic
119C22×C4—3261no—
120C22×C4—3261no—
121C22×C4—1631no—
122C22×C4—1632no—
123M16—22no—
124M16—3262no—
125M16—6522no—
126C2×C8—6522no—
127C4⋊C4—3262no—
128C22⋊C4—22no—
129C22⋊C4—3262no—
130C42—3261no—
131C42—1631no—
132C42C4×D16311yescharacteristic
133C23—21no—
134C23—1632no—
135C23—1631no—
136Q8—41no—
137Q8—6521no—
138D4—41no—
139D4—6522no—
140D4—6521no—
141D4—6524no—
142D4—6522no—
143C2×C4—21no—
144C2×C4—3262no—
145C2×C4—3262no—
146C2×C4—3261no—
147C2×C4—3261no—
148C2×C4—3261no—
149C2×C4—3261no—
150C2×C4—6522no—
151C2×C4—6521no—
152C2×C4—6521no—
153C2×C4—1631no—
154C2×C4C2×C4×D16311yescharacteristic
155C2×C4D4×D16312yes—
156C8—42no—
157C8—6522no—
158C22—42no—
159C22—41no—
160C22—3261no—
161C22—3262no—
162C22—6521no—
163C22—6522no—
164C22C22⋊C4×D16311yescharacteristic
165C4—22no—
166C4—21no—
167C4—41no—
168C4—3262no—
169C4—6521no—
170C4—1631no—
171C4—1631no—
172C2—21no—
173C2—81no—
174C2—6521no—
175C2—6522no—
176C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC2×C4×C163C22×C4
FrattiniC2×Q8C22×D163
Fitting#355×C163C2
socleC2×C163C2×#99
prime radical#23782214C2
cosocleC2×Q8×C163C23
hypercenter#355D163
hypocenterC163#2132
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #23781648C2semidirect
3. #23781463C2semidirect
4. #23781462C2semidirect
5. #355×C163C2semidirect
88. C163#2132semidirect
94. #355D163semidirect

Maximal subgroups

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

subgroupindex
#237822142
#237816482
#237814632
#237814622
#355×C1632
#2132163