Finite Group Databasebeta

← #27171581

Subgroups — #27171581

Order 12928.

358 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15692214C211yesmaximal, characteristic, complemented
2C2×#15692182C212yesmaximal, complemented
3#20265141C211yesmaximal, characteristic, complemented
4#20264936C212yesmaximal, complemented
5#525×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7#15692214—24no—
8C2×C4⋊Dic101—21no—
9C2×C22⋊Dic101—21no—
10C2×C4×Dic101C2212yes—
11#15692182—24no—
12C22⋊#11717579—22no—
13C2×#41×C101C2211yescharacteristic
14C23×C4×C101C2211yescharacteristic
15#123×C101—21no—
16C22⋊Q8×C101C412yescomplemented
17#121×C101C412yescomplemented
18#120×C101—21no—
19C4×D4×C101—22no—
20#117×C101C2211yescharacteristic
21C22×Dic101—22no—
22#13108058—22no—
23#13108058—22no—
24C2×#11717579—22no—
25C4⋊Dic101—24no—
26C22⋊Dic101—24no—
27C4×Dic101—22no—
28C4×Dic101D414yescomplemented
29C24×C101D411yescharacteristic
30#41×C101—22no—
31#41×C101—22no—
32C2×Q8×C101C2×C411yescharacteristic
33C2×D4×C101—22no—
34C2×D4×C101—21no—
35C2×D4×C101C2×C411yescharacteristic
36C22×C4×C101—22no—
37C22×C4×C101—22no—
38C22×C4×C101—21no—
39C22×C4×C101C2×C412yes—
40C22×C4×C101D411yescharacteristic
41C22×C4×C101D412yes—
42C22×C4×C101C2311yescharacteristic
43C4⋊C4×C101—22no—
44C4⋊C4×C101C2×C412yes—
45C22⋊C4×C101—22no—
46C22⋊C4×C101—21no—
47C22⋊C4×C101—22no—
48C42×C101—21no—
49C23×C101—21no—
50C23×C101—21no—
51C23×C101—21no—
52C23×C101—42no—
53C23×C101C22⋊C412yes—
54C23×C101C2×D411yescharacteristic
55Q8×C101—22no—
56D4×C101—22no—
57D4×C101—44no—
58D4×C101—41no—
59D4×C101—41no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—24no—
63C2×C4×C101—24no—
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—42no—
67C2×C4×C101C22⋊C412yes—
68C2×C4×C101C22×C411yescharacteristic
69C2×C4×C101C2×D412yes—
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101#4112yes—
72C2×Dic101—24no—
73C2×Dic101—44no—
74#11717579—42no—
75C22×C101—22no—
76C22×C101—22no—
77C22×C101—22no—
78C22×C101—22no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101—41no—
82C22×C101—42no—
83C22×C101C2×C22⋊C411yescharacteristic
84C22×C101#12011yescharacteristic
85C22×C101#12311yescharacteristic
86C4×C101—22no—
87C4×C101—41no—
88C4×C101—42no—
89C4×C101C4×D412yes—
90C4×C101#12112yes—
91Dic101—44no—
92C2×C101—21no—
93C2×C101—21no—
94C2×C101—42no—
95C2×C101—81no—
96C2×C101#38611yescharacteristic
97C2×C101#42112yes—
98#1897—1011nomaximal, Sylow
99C101#189711yescharacteristic, complemented, Sylow
100#525D10111yescharacteristic
101C2×#117—1011no—
102C2×#104—1012no—
103#407—1011no—
104#339—1012no—
105C2×#41C2×D10111yescharacteristic
106C23×C4C2×D10111yescharacteristic
107C2×M16—1012no—
108#123—21no—
109C22⋊Q8Dic10112yescomplemented
110#121Dic10112yescomplemented
111#120—21no—
112C4×D4—22no—
113#117—2024no—
114#117C2×D10111yescharacteristic
115C2×C4⋊C4—2021no—
116C2×C22⋊C4—2021no—
117C2×C42—1012no—
118#104—2024no—
119C22⋊C8—2022no—
120C24C22⋊D10111yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic10111yescharacteristic
124C2×D4—22no—
125C2×D4—21no—
126C2×D4C2×Dic10111yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—2022no—
131C22×C4C2×Dic10112yes—
132C22×C4C22⋊D10111yescharacteristic
133C22×C4C22⋊D10112yes—
134C22×C4C22×D10111yescharacteristic
135M16—2022no—
136M16—2022no—
137C2×C8—2022no—
138C4⋊C4—22no—
139C4⋊C4—2024no—
140C4⋊C4C2×Dic10112yes—
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—22no—
144C22⋊C4—2024no—
145C42—21no—
146C42—1014no—
147C42—2022no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊Dic10112yes—
153C23C2×C22⋊D10111yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—4044no—
167C2×C4—2024no—
168C2×C4C22⋊Dic10112yes—
169C2×C4C22×Dic10111yescharacteristic
170C2×C4D4×D10112yes—
171C2×C4C2×C22⋊D10111yescharacteristic
172C2×C4#1310807312yes—
173C8—4042no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—42no—
182C22C2×C22⋊Dic10111yescharacteristic
183C22#1569222311yescharacteristic
184C22#1569223611yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—4044no—
189C4D4×Dic10112yes—
190C4#1569222712yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#2026505711yescharacteristic
196C2#2026519412yes—
subgroupG/H
centerC2×C4C2×C22⋊D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting, prime radical#525×C101C2
socleC22×C101C2×C22⋊C4
cosocleC22×C4×C101C23
hypercenter#525D101
hypocenterC101#1897
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#15692214C2semidirect
2. C2×#15692182C2semidirect
3. #20265141C2semidirect
4. #20264936C2semidirect
16. C22⋊Q8×C101C4semidirect
17. #121×C101C4semidirect
28. C4×Dic101D4semidirect
99. C101#1897semidirect
109. C22⋊Q8Dic101semidirect
110. #121Dic101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922142
C2×#156921822
#202651412
#202649362
#525×C1012
#1897101