Finite Group Databasebeta

← #27171532

Subgroups — #27171532

Order 12928.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15692242C211yesmaximal, characteristic
2#20265654C211yesmaximal, characteristic, complemented
3#20265182C211yesmaximal, characteristic
4C2×#15692173C211yesmaximal, characteristic, complemented
5#20264938C211yesmaximal, characteristic, complemented
6#20264936C211yesmaximal, characteristic, complemented
7#418×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10#15692242—22no—
11#15692242C414yescomplemented
12C4×Dic202—22no—
13#15692220—22no—
14#15692214C2211yescharacteristic
15#15692211—22no—
16C2×C22⋊Dic101—22no—
17C2×C4×Dic101C2211yescharacteristic
18#15692176—22no—
19#15692173—22no—
20C22×D4×C101C2211yescharacteristic
21C2×M16×C101C2211yescharacteristic
22#117×C101C2211yescharacteristic
23#102×C101—22no—
24C2×Dic202—21no—
25C2×Dic202—21no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202C2×C412yes—
29C2×Dic202D412yes—
30C22×Dic101—21no—
31C22×Dic101D412yes—
32#13108058—21no—
33#13108058—21no—
34C2×#11717579—21no—
35C4⋊Dic101D412yes—
36#13108053—24no—
37C22⋊Dic101—21no—
38C22⋊Dic101—28no—
39C4×Dic101—24no—
40C4×Dic101—21no—
41C4×Dic101—21no—
42C4×Dic101C2×C412yes—
43C24×C101—21no—
44C2×D4×C101—21no—
45C2×D4×C101—21no—
46C2×D4×C101—21no—
47C2×D4×C101—21no—
48C2×D4×C101—21no—
49C2×D4×C101C2×C412yes—
50C22×C4×C101C2311yescharacteristic
51M16×C101—21no—
52M16×C101—21no—
53C2×C8×C101—21no—
54C4⋊C4×C101D412yes—
55C22⋊C4×C101—21no—
56C42×C101—21no—
57C23×C101—22no—
58C23×C101—44no—
59C23×C101—41no—
60C23×C101—41no—
61C23×C101C2×D411yescharacteristic
62D4×C101—22no—
63D4×C101—22no—
64D4×C101—41no—
65D4×C101—41no—
66C2×C4×C101—22no—
67C2×C4×C101C22×C411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101#4111yescharacteristic
72C2×C4×C101#4111yescharacteristic
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101—42no—
78C2×Dic101C22⋊C414yes—
79Dic202—22no—
80Dic202—22no—
81Dic202—22no—
82Dic202—22no—
83C8×C101—41no—
84#11717579—41no—
85C22×C101—22no—
86C22×C101—44no—
87C22×C101—42no—
88C22×C101—82no—
89C22×C101—82no—
90C22×C101C2×C22⋊C411yescharacteristic
91C22×C101#12011yescharacteristic
92C22×C101#12311yescharacteristic
93C4×C101—42no—
94C4×C101C4×D411yescharacteristic
95C4×C101C4×D411yescharacteristic
96C4×C101#12111yescharacteristic
97C4×C101#12111yescharacteristic
98Dic101—24no—
99Dic101—42no—
100Dic101—42no—
101C2×C101—21no—
102C2×C101—21no—
103C2×C101—82no—
104C2×C101#38611yescharacteristic
105C2×C101#45412yes—
106#1888—1011nomaximal, Sylow
107C101#188811yescharacteristic, complemented, Sylow
108C2×#124—1011no—
109#519—1011no—
110#418—1011no—
111#418D10111yescharacteristic
112C2×#101—1011no—
113#339—1011no—
114#339—1011no—
115C22×Q8—1011no—
116C22×D4C2×D10111yescharacteristic
117C2×M16—1011no—
118C2×M16C2×D10111yescharacteristic
119#124—1014no—
120#124—2022no—
121C4×Q8—2022no—
122C4×Q8—2022no—
123#117—1011no—
124#117—2022no—
125#117C2×D10111yescharacteristic
126C2×C22⋊C4—2022no—
127C2×C42—1011no—
128#102—22no—
129#102—2022no—
130#101—2022no—
131C24—21no—
132C2×Q8—1012no—
133C2×Q8—1012no—
134C2×Q8—2021no—
135C2×Q8—2021no—
136C2×Q8—2021no—
137C2×Q8—2021no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C4×D10112yes—
144C22×C4—1012no—
145C22×C4—2021no—
146C22×C4C22×D10111yescharacteristic
147M16—21no—
148M16—21no—
149M16—2021no—
150M16—2021no—
151C2×C8—21no—
152C2×C8—2021no—
153C4⋊C4—1012no—
154C4⋊C4—2024no—
155C4⋊C4C22⋊D10112yes—
156C22⋊C4—21no—
157C22⋊C4—2028no—
158C22⋊C4—2021no—
159C42—21no—
160C42—1012no—
161C42—2024no—
162C42—2021no—
163C42—2021no—
164C23—22no—
165C23—44no—
166C23—41no—
167C23—41no—
168C23D4×D10111yescharacteristic
169Q8—2022no—
170Q8—2022no—
171Q8—2022no—
172Q8—2022no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—22no—
178C2×C4—1014no—
179C2×C4—4042no—
180C2×C4—2022no—
181C2×C4—2022no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4C2×C4×D10111yescharacteristic
185C2×C4D4×D10111yescharacteristic
186C2×C4D4×D10111yescharacteristic
187C2×C4C2×C22⋊D10111yescharacteristic
188C2×C4#1310807311yescharacteristic
189C2×C4#1310807411yescharacteristic
190C8—41no—
191C8—4041no—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22C22⋊C4×D10111yescharacteristic
198C22#1569222311yescharacteristic
199C22#1569223711yescharacteristic
200C4—42no—
201C4—4042no—
202C4—4042no—
203C4—2024no—
204C4#1569221511yescharacteristic
205C4C4×C22⋊D10111yescharacteristic
206C4#1569222611yescharacteristic
207C4#1569222811yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2026505311yescharacteristic
212C2#2026534812yes—
subgroupG/H
centerC22#15692223
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#418×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
cosocleC22×C4×C101C23
hypercenter#418D101
hypocenterC101#1888
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #20265654C2semidirect
4. C2×#15692173C2semidirect
5. #20264938C2semidirect
6. #20264936C2semidirect
11. #15692242C4semidirect
107. C101#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#156922422
#202656542
#202651822
C2×#156921732
#202649382
#202649362
#418×C1012
#1888101