Finite Group Databasebeta

← #27171629

Subgroups — #27171629

Order 12928.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15692285C211yesmaximal, characteristic
2#20265654C211yesmaximal, characteristic, complemented
3#20265194C211yesmaximal, characteristic
4#20265185C211yesmaximal, characteristic, complemented
5#20264947C211yesmaximal, characteristic, complemented
6#20264933C211yesmaximal, characteristic, complemented
7#410×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9#15692285—22no—
10#15692285—22no—
11C2×#13108064C412yescomplemented
12C2×#13108062C412yescomplemented
13C2×#13108058C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15C4×Dic202—22no—
16#15692220—22no—
17#15692214C2211yescharacteristic
18#15692211—22no—
19#15692182—22no—
20#15692178—22no—
21C2×#41×C101C2211yescharacteristic
22#117×C101C2211yescharacteristic
23#117×C101C2211yescharacteristic
24#99×C101—22no—
25C2×Dic202—21no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202C2×C412yes—
29C2×Dic202D412yes—
30C2×Dic202D412yes—
31C22×Dic101—21no—
32#13108064—24no—
33#13108062—24no—
34#13108058—21no—
35#13108058—21no—
36#13108058—21no—
37#13108058—21no—
38C2×#11717579—21no—
39C2×#11717579C2×C412yes—
40C4⋊Dic101—21no—
41#13108053—24no—
42C22⋊Dic101—21no—
43C4×Dic101—24no—
44C4×Dic101D412yes—
45#41×C101—21no—
46#41×C101—21no—
47#41×C101—21no—
48#41×C101—21no—
49C2×Q8×C101C2×C411yescharacteristic
50C2×D4×C101—21no—
51C2×D4×C101C2×C411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C4⋊C4×C101—21no—
55C4⋊C4×C101—21no—
56C22⋊C4×C101—21no—
57C22⋊C4×C101D412yes—
58C42×C101—21no—
59C42×C101—21no—
60C23×C101—21no—
61C23×C101C2×D411yescharacteristic
62Q8×C101—21no—
63Q8×C101—21no—
64D4×C101—21no—
65D4×C101—21no—
66D4×C101—41no—
67D4×C101—41no—
68C2×C4×C101—21no—
69C2×C4×C101—22no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101C22×C411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101#4111yescharacteristic
78C2×C4×C101#4111yescharacteristic
79C2×Dic101—22no—
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—22no—
83Dic202—22no—
84Dic202—22no—
85Dic202—22no—
86Dic202C22⋊C414yes—
87#11717579—22no—
88#11717579—41no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—81no—
92C22×C101C4×D411yescharacteristic
93C22×C101#12011yescharacteristic
94C22×C101#12111yescharacteristic
95C4×C101—42no—
96C4×C101—41no—
97C4×C101—81no—
98C4×C101C2×C22⋊C411yescharacteristic
99C4×C101C4×D411yescharacteristic
100C4×C101#12111yescharacteristic
101C4×C101#12311yescharacteristic
102Dic101—24no—
103Dic101—42no—
104C2×C101—21no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—81no—
108C2×C101#38611yescharacteristic
109#1905—1011nomaximal, Sylow
110C101#190511yescharacteristic, complemented, Sylow
111C2×#137—1011no—
112#519—1011no—
113#421—1011no—
114#419—1011no—
115#410D10111yescharacteristic
116#344—1011no—
117H(3,ℤ/4)—1011no—
118C2×#41C2×D10111yescharacteristic
119C22×Q8—1011no—
120#137—2022no—
121#137—2022no—
122C2×Q16—1012no—
123C2×SD16—1012no—
124C2×M16—1011no—
125C2×M16—1011no—
126C4×Q8—2022no—
127C4×Q8—2022no—
128#117—1011no—
129#117—2022no—
130#117C2×D10111yescharacteristic
131#117C2×D10111yescharacteristic
132#104—2022no—
133#103—2022no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—1012no—
140C2×Q8—1012no—
141C2×Q8—1012no—
142C2×Q8—2021no—
143C2×Q8—2021no—
144C2×Q8—2021no—
145C2×Q8C4×D10111yescharacteristic
146C2×D4—21no—
147C2×D4C4×D10111yescharacteristic
148C22×C4—21no—
149C22×C4—2021no—
150C22×C4C22×D10111yescharacteristic
151Q16—2024no—
152SD16—2024no—
153M16—2021no—
154M16—2021no—
155M16—2021no—
156M16—2021no—
157C2×C8—1012no—
158C2×C8—2021no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2024no—
162C4⋊C4—2021no—
163C22⋊C4—21no—
164C22⋊C4—2021no—
165C22⋊C4C22⋊D10112yes—
166C42—21no—
167C42—21no—
168C42—1012no—
169C42—2024no—
170C23—21no—
171C23C2×C22⋊D10111yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1014no—
175Q8—2022no—
176Q8—2022no—
177Q8—2022no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2022no—
188C2×C4—2022no—
189C2×C4—2022no—
190C2×C4—2022no—
191C2×C4C2×C4×D10111yescharacteristic
192C2×C4D4×D10111yescharacteristic
193C2×C4D4×D10111yescharacteristic
194C2×C4D4×D10111yescharacteristic
195C2×C4#1310807311yescharacteristic
196C2×C4#1310807411yescharacteristic
197C8—4041no—
198C8—2022no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D10111yescharacteristic
203C22#1569222311yescharacteristic
204C22#1569222611yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—4042no—
209C4—2024no—
210C4C22⋊C4×D10111yescharacteristic
211C4#1569221511yescharacteristic
212C4#1569222811yescharacteristic
213C4#1569223711yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2026505311yescharacteristic
subgroupG/H
centerC2#20265053
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#410×C101C2
socleC2×C101#386
prime radicalC2×#41×C101C22
cosocleC22×C4×C101C23
hypercenter#410D101
hypocenterC101#1905
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20265654C2semidirect
4. #20265185C2semidirect
5. #20264947C2semidirect
6. #20264933C2semidirect
11. C2×#13108064C4semidirect
12. C2×#13108062C4semidirect
110. C101#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#156922852
#202656542
#202651942
#202651852
#202649472
#202649332
#410×C1012
#1905101