Finite Group Databasebeta

← #35216222

Subgroups — #35216222

Order 20864.

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

normal subgroups only

This group has 230 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#23782235C211yesmaximal, characteristic
2#101×D163C212yesmaximal, complemented
3#23781463C212yesmaximal, complemented
4#23781462C211yesmaximal, characteristic, complemented
5#355×C163C211yesmaximal, characteristic, complemented
6C2×#14579698C411yescharacteristic
7C2×Q8×D163C2211yescharacteristic
8M16×D163—22no—
9#19104735C411yescharacteristic
10#19104733C2211yescharacteristic
11#19104722C411yescharacteristic
12#19104717—22no—
13#19104712—22no—
14#19104692C411yescharacteristic
15#19104654C2212yes—
16#19104653—22no—
17#124×C163C2211yescharacteristic
18#101×C163C2212yes—
19#14579698—42no—
20C2×Dic326C2×C411yescharacteristic
21C2×Dic326C2×C411yescharacteristic
22C2×Dic326D412yes—
23Q8×D163—42no—
24C2×C22⋊D163—21no—
25C22×Dic163—21no—
26C2×C4×D163C2×C411yescharacteristic
27C2×C4×D163D412yes—
28#14579683—22no—
29#14579682—42no—
30C8×D163—42no—
31C4⋊Dic163—22no—
32Dic163⋊C4—21no—
33Dic163⋊C4—22no—
34C22⋊Dic163—22no—
35(C2×D163)⋊C4—21no—
36C4×Dic163C2×C411yescharacteristic
37C2×Q8×C163C2311yescharacteristic
38C2×D4×C163C2×C411yescharacteristic
39M16×C163—22no—
40C22⋊C4×C163—22no—
41C42×C163C2×C411yescharacteristic
42C22×D163C22⋊C411yescharacteristic
43Dic326—41no—
44Dic326—42no—
45Dic326—41no—
46C22⋊D163—42no—
47C2×Dic163—21no—
48C2×Dic163—21no—
49C2×Dic163—42no—
50C2×Dic163C22⋊C412yes—
51C2×Dic163C22⋊C411yescharacteristic
52C4×D163—22no—
53C4×D163—22no—
54C4×D163—22no—
55C163⋊C8—42no—
56C23×C163—21no—
57Q8×C163—41no—
58D4×C163—41no—
59C2×C4×C163—21no—
60C2×C4×C163C22×C411yescharacteristic
61C2×C4×C163C2×D412yes—
62C8×C163—42no—
63C22×C163—42no—
64C22×C163—41no—
65C22×C163C2×C22⋊C411yescharacteristic
66C2×D163—41no—
67C2×D163#9912yes—
68C4×C163—22no—
69C4×C163—21no—
70C4×C163—41no—
71Dic163—21no—
72Dic163—22no—
73Dic163—41no—
74Dic163—41no—
75C2×C163—21no—
76C2×C163—81no—
77C2×C163C2×#9911yescharacteristic
78D163—22no—
79C163#213311yescharacteristic, complemented, Sylow
80#2133—1631nomaximal, Sylow
81#536—1631no—
82C2×#101—1632no—
83#356—1632no—
84#355—1631no—
85#355D16311yescharacteristic, complemented
86C2×#41—1631no—
87C22×Q8—1631no—
88C2×M16—3262no—
89C4⋊Q8—1631no—
90C4⋊Q8—1631no—
91#124—1631no—
92#124C2×D16311yescharacteristic
93#123—3262no—
94C22⋊Q8—3262no—
95#117—1631no—
96#101—3262no—
97#101—1632no—
98#101C2×D16312yes—
99#41—6522no—
100C2×Q8—6522no—
101C2×Q8—1631no—
102C2×Q8—1631no—
103C2×Q8—1632no—
104C2×Q8C22×D16311yescharacteristic
105C2×D4—3261no—
106C2×D4C4×D16311yescharacteristic
107C22×C4—3261no—
108C22×C4—1631no—
109C22×C4—1632no—
110M16—22no—
111M16—3262no—
112M16—6522no—
113C2×C8—6522no—
114C4⋊C4—3261no—
115C4⋊C4—3262no—
116C4⋊C4—3262no—
117C22⋊C4—22no—
118C22⋊C4—3261no—
119C22⋊C4—3262no—
120C42—1631no—
121C42C4×D16311yescharacteristic
122C23—21no—
123C23—1631no—
124Q8—41no—
125Q8—6522no—
126Q8—6521no—
127Q8—6521no—
128D4—41no—
129D4—6522no—
130C2×C4—21no—
131C2×C4—3262no—
132C2×C4—3262no—
133C2×C4—3262no—
134C2×C4—3261no—
135C2×C4—3261no—
136C2×C4—6522no—
137C2×C4—1632no—
138C2×C4—1631no—
139C2×C4C2×C4×D16311yescharacteristic
140C2×C4D4×D16312yes—
141C8—42no—
142C8—6522no—
143C22—42no—
144C22—41no—
145C22—6521no—
146C22—1632no—
147C22C22⋊C4×D16311yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—3261no—
152C4—3262no—
153C4—6521no—
154C4—6521no—
155C2—21no—
156C2—81no—
157C2—3262no—
158C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC2×C4×C163C22×C4
FrattiniC2×Q8C22×D163
Fitting#355×C163C2
socleC2×C163C2×#99
prime radicalC2×#14579698C4
cosocleC2×Q8×C163C23
hypercenter#355D163
hypocenterC163#2133
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. #101×D163C2semidirect
3. #23781463C2semidirect
4. #23781462C2semidirect
5. #355×C163C2semidirect
79. C163#2133semidirect
85. #355D163semidirect

Maximal subgroups

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

subgroupindex
#237822352
#101×D1632
#237814632
#237814622
#355×C1632
#2133163