Finite Group Databasebeta

← #27170057

Subgroups — #27170057

Order 12928.

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265803C211yesmaximal, characteristic, complemented
2#20265330C212yesmaximal, complemented
3#20265086C211yesmaximal, characteristic
4#20264915C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7C22×C22⋊D101—21no—
8C2×#13108063—22no—
9#15692234—21no—
10#15692234C2212yes—
11#15692232—22no—
12#15692232—22no—
13C2×C22⋊Dic101—21no—
14C2×#13108051—21no—
15#15692177—22no—
16#15692162C2211yescharacteristic
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101—21no—
20C2×Dic202—21no—
21C2×Dic202D412yescomplemented
22C2×D404—22no—
23C2×D404—42no—
24C2×C22⋊D101—42no—
25C2×C22⋊D101—42no—
26C2×C22⋊D101—42no—
27C22×Dic101—21no—
28C22×Dic101—21no—
29C22×Dic101—21no—
30#13108063—44no—
31C4⋊Dic101—21no—
32C4⋊Dic101D412yescomplemented
33#13108053—22no—
34#13108053—22no—
35C22⋊Dic101—22no—
36C22⋊Dic101—22no—
37#13108051—42no—
38C24×C101D411yescharacteristic
39C22×C4×C101D411yescharacteristic
40C22×C4×C101C2311yescharacteristic
41C2×C8×C101—22no—
42C22⋊C4×C101—22no—
43C23×C101—21no—
44C23×C101—21no—
45C23×C101—41no—
46C23×C101—41no—
47C23×C101SD1612yescomplemented
48C23×C101C2×D411yescharacteristic
49C22×D101—22no—
50C22×D101—41no—
51C22×D101—41no—
52C22×D101—41no—
53C22×D101—41no—
54C22×D101—41no—
55C22×D101—41no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×D412yes—
60C22⋊D101—48no—
61C2×Dic101—22no—
62C2×Dic101—22no—
63C2×Dic101—22no—
64C2×Dic101—41no—
65C2×Dic101—41no—
66C2×Dic101—81no—
67C2×Dic101—81no—
68D404—24no—
69D404—42no—
70Dic202—42no—
71Dic202—81no—
72C8×C101—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—22no—
76C22×C101—42no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—81no—
80C22×C101#12011yescharacteristic
81C22×C101C2×SD1611yescharacteristic
82C22×C101#13711yescharacteristic
83C2×D101—42no—
84C2×D101—42no—
85C2×D101—42no—
86C2×D101—81no—
87C2×D101—81no—
88C2×D101—81no—
89C2×D101—81no—
90C4×C101—22no—
91C4×C101—41no—
92Dic101—42no—
93Dic101—42no—
94Dic101—42no—
95C2×C101—21no—
96C2×C101—21no—
97C2×C101—42no—
98C2×C101#44811yescharacteristic
99C2×C101#45311yescharacteristic
100C2×C101B(4,𝔽2)11yescharacteristic
101D101—82no—
102#1596—1011nomaximal, Sylow
103C101#159611yescharacteristic, complemented, Sylow
104#543—1011no—
105#450—1012no—
106#392—1011no—
107#328—1012no—
108#323D10111yescharacteristic, complemented
109C22×D4—1011no—
110C22×D4—2021no—
111C2×SD16—2022no—
112C22⋊Q8—1012no—
113C22⋊Q8—2022no—
114C22⋊Q8—2022no—
115C22⋊Q8—2021no—
116C2×C22⋊C4—2021no—
117C2×C22⋊C4—2021no—
118C2×C22⋊C4C2×D10111yescharacteristic
119#102—2022no—
120C22⋊C8C2×D10112yes—
121#95—1011no—
122C24—2021no—
123C24D40411yescharacteristic
124C2×Q8—1012no—
125C2×Q8—2021no—
126C2×D4—4042no—
127C2×D4—4042no—
128C2×D4—4042no—
129C2×D4—4042no—
130C2×D4—2022no—
131C22×C4—2021no—
132C22×C4—2021no—
133C22×C4—2021no—
134C22×C4D40411yescharacteristic
135C22×C4C22×D10111yescharacteristic
136SD16—4044no—
137C2×C8—22no—
138C4⋊C4—1012no—
139C4⋊C4—2022no—
140C4⋊C4—2021no—
141C4⋊C4—2022no—
142C22⋊C4—22no—
143C22⋊C4—4042no—
144C22⋊C4—2022no—
145C22⋊C4—2022no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—4041no—
151C23—4041no—
152C23—4041no—
153C23—4041no—
154C23—4041no—
155C23—4041no—
156C23—2022no—
157C23#1310806312yescomplemented
158C23C2×D40411yescharacteristic
159Q8—8081no—
160Q8—4042no—
161D4—4042no—
162D4—4048no—
163D4—2024no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8081no—
168C2×C4—8081no—
169C2×C4—4041no—
170C2×C4—4041no—
171C2×C4—2022no—
172C2×C4—2022no—
173C2×C4—2022no—
174C2×C4D4×D10112yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—8081no—
184C22—8081no—
185C22—8081no—
186C22—8081no—
187C22—4042no—
188C22—4042no—
189C22—4042no—
190C22#1569222511yescharacteristic
191C22C2×#1310806311yescharacteristic
192C22#1569228811yescharacteristic
193C4—22no—
194C4—41no—
195C4—4042no—
196C4—4042no—
197C4—4042no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—8082no—
202C2#2026531811yescharacteristic
203C2#2026534711yescharacteristic
204C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radical#20265086C2
hypercenter#323D101
hypocenterC101#1596
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #20265803C2semidirect
2. #20265330C2semidirect
4. #20264915C2semidirect
5. #323×C101C2semidirect
21. C2×Dic202D4semidirect
32. C4⋊Dic101D4semidirect
47. C23×C101SD16semidirect
103. C101#1596semidirect
108. #323D101semidirect
157. C23#13108063semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202653302
#202650862
#202649152
#323×C1012
#1596101