Finite Group Databasebeta

← #27171651

Subgroups — #27171651

Order 12928.

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

normal subgroups only

This group has 312 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
1#20265943C211yesmaximal, characteristic
2#20265751C211yesmaximal, characteristic
3#20265186C211yesmaximal, characteristic, complemented
4#20265150C211yesmaximal, characteristic
5#20264934C211yesmaximal, characteristic, complemented
6#20264933C211yesmaximal, characteristic, complemented
7#419×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9#15692263—22no—
10#15692261—22no—
11C2×#13108058C2211yescharacteristic
12M16×D101—22no—
13#15692253C412yescomplemented
14#15692244C412yescomplemented
15#15692238—22no—
16#15692231—22no—
17#15692214C2211yescharacteristic
18#15692214C2211yescharacteristic
19#15692179—22no—
20#15692169—22no—
21C22×Q8×C101C2211yescharacteristic
22C2×M16×C101C2211yescharacteristic
23#117×C101C2211yescharacteristic
24#103×C101—22no—
25#13108074—21no—
26#13108074—21no—
27#13108074—21no—
28#13108074—21no—
29C2×Dic202C2×C411yescharacteristic
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101D412yes—
32C2×C4×D101D412yes—
33#13108058—21no—
34#13108058—21no—
35#13108057—24no—
36C2×#11717579D412yes—
37C8×D101—24no—
38C4⋊Dic101—21no—
39C4⋊Dic101—21no—
40#13108053—24no—
41C22⋊Dic101—21no—
42C22⋊Dic101—21no—
43#13108051—24no—
44C4×Dic101—21no—
45C4×Dic101C2×C412yes—
46C2×Q8×C101—21no—
47C2×Q8×C101—21no—
48C2×Q8×C101—21no—
49C2×Q8×C101—21no—
50C2×Q8×C101—21no—
51C2×Q8×C101C2×C412yes—
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54M16×C101—21no—
55M16×C101—21no—
56C2×C8×C101D412yes—
57C4⋊C4×C101—21no—
58C22⋊C4×C101—21no—
59C42×C101—21no—
60C23×C101C2×D411yescharacteristic
61C22×D101C22⋊C412yes—
62Q8×C101—22no—
63Q8×C101—22no—
64Q8×C101—41no—
65Q8×C101—41no—
66C2×C4×C101—22no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C22×C411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101#4111yescharacteristic
75C2×C4×C101#4111yescharacteristic
76C22⋊D101—22no—
77C22⋊D101—22no—
78C2×Dic101—22no—
79C2×Dic101—41no—
80C2×Dic101C22⋊C412yes—
81D404—21no—
82D404—21no—
83C4×D101—22no—
84C4×D101—22no—
85Dic202—21no—
86Dic202—21no—
87C8×C101—22no—
88#11717579—22no—
89C22×C101—41no—
90C22×C101C2×C22⋊C411yescharacteristic
91C22×C101#12111yescharacteristic
92C22×C101#12111yescharacteristic
93C2×D101—22no—
94C2×D101—42no—
95C4×C101—42no—
96C4×C101—81no—
97C4×C101C4×D411yescharacteristic
98C4×C101C4×D411yescharacteristic
99C4×C101#12011yescharacteristic
100C4×C101#12311yescharacteristic
101Dic101—22no—
102Dic101—42no—
103Dic101—81no—
104C2×C101—21no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101#38611yescharacteristic
108D101—42no—
109#1909—1011nomaximal, Sylow
110C101#190911yescharacteristic, complemented, Sylow
111#568—1011no—
112#536—1011no—
113#419—1011no—
114#419D10111yescharacteristic, complemented
115#410—1011no—
116H(3,ℤ/4)—1011no—
117H(3,ℤ/4)—1011no—
118C2×#41—1011no—
119C22×Q8C2×D10111yescharacteristic
120#131—2022no—
121#131—2022no—
122C2×M16—1011no—
123C2×M16—2022no—
124C2×M16C2×D10111yescharacteristic
125C4⋊Q8—1012no—
126#124—1012no—
127#123—2022no—
128C22⋊Q8—2022no—
129#117—1011no—
130#117—1011no—
131#117C2×D10111yescharacteristic
132#103—22no—
133#103—2022no—
134#99—2022no—
135#41—2021no—
136#41—2021no—
137#41—2021no—
138#41—2021no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—1011no—
145C2×Q8C4×D10112yes—
146C2×D4—1012no—
147C2×D4—1011no—
148C22×C4—21no—
149C22×C4—1012no—
150C22×C4C22×D10111yescharacteristic
151M16—21no—
152M16—21no—
153M16—2024no—
154M16—2021no—
155M16—2021no—
156C2×C8—1012no—
157C2×C8—2024no—
158C2×C8C22⋊D10112yes—
159C4⋊C4—21no—
160C4⋊C4—2024no—
161C4⋊C4—2021no—
162C4⋊C4—2021no—
163C22⋊C4—21no—
164C22⋊C4—2024no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C42—21no—
168C42—1012no—
169C42—2021no—
170C23—1012no—
171C23D4×D10111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—2021no—
177Q8—2021no—
178D4—2022no—
179D4—2022no—
180D4—2021no—
181D4—2021no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1012no—
187C2×C4—4041no—
188C2×C4—2022no—
189C2×C4—2022no—
190C2×C4—2022no—
191C2×C4C2×C4×D10111yescharacteristic
192C2×C4D4×D10111yescharacteristic
193C2×C4D4×D10111yescharacteristic
194C2×C4C2×C22⋊D10111yescharacteristic
195C2×C4#1310807311yescharacteristic
196C2×C4#1310807411yescharacteristic
197C8—22no—
198C8—2022no—
199C22—41no—
200C22—4042no—
201C22—2022no—
202C22C22⋊C4×D10111yescharacteristic
203C22#1569222611yescharacteristic
204C22#1569222811yescharacteristic
205C4—42no—
206C4—81no—
207C4—8081no—
208C4—4042no—
209C4—2022no—
210C4#1569221511yescharacteristic
211C4C4×C22⋊D10111yescharacteristic
212C4#1569222311yescharacteristic
213C4#1569223711yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—4042no—
218C2#2026505311yescharacteristic
subgroupG/H
centerC2#20265053
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#419×C101C2
socleC2×C101#386
prime radicalC2×#13108074C22
cosocleC22×C4×C101C23
hypercenter#419D101
hypocenterC101#1909
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20265186C2semidirect
5. #20264934C2semidirect
6. #20264933C2semidirect
7. #419×C101C2semidirect
13. #15692253C4semidirect
14. #15692244C4semidirect
110. C101#1909semidirect
114. #419D101semidirect

Maximal subgroups

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

subgroupindex
#202659432
#202657512
#202651862
#202651502
#202649342
#202649332
#419×C1012
#1909101