Finite Group Databasebeta

← #27187035

Subgroups — #27187035

Order 12928.

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D101⋊(C22⋊C8)C211yesmaximal, characteristic
2#20266119C211yesmaximal, characteristic
3#20266050C214yesmaximal, complemented
4C2×C4⋊C4×D101C211yesmaximal, characteristic
5C2×D101⋊C8C2212yes—
6C2×C4⋊#87938C2212yes—
7C2×C22⋊#87938—21no—
8#15692312—21no—
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4—21no—
11C22×C4×D101—21no—
12C22×C4×D101C2211yescharacteristic
13C2×C4⋊Dic101C411yescharacteristic, complemented
14C2×#13108053—21no—
15C4⋊C4×D101—21no—
16C4⋊C4×D101—21no—
17C4⋊C4×D101C412yescomplemented
18C4⋊C4×D101C2212yes—
19C2×C4⋊C4×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21D101⋊C8—24no—
22C2×#11717581—22no—
23C4⋊#87938—24no—
24C22⋊#87938—42no—
25C23×D101D411yescharacteristic
26C22×Dic101—21no—
27C22×Dic101D411yescharacteristic
28C2×C4×D101—24no—
29C2×C4×D101—22no—
30C2×C4×D101—22no—
31C2×C4×D101—42no—
32C2×C4×D101—42no—
33C2×C4×D101C2×C411yescharacteristic
34C2×C4×D101D411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C2×C4×D101C2311yescharacteristic
37C4⋊Dic101—21no—
38C4⋊Dic101C2×C412yes—
39#13108053—22no—
40#13108053—22no—
41C22×C4×C101—21no—
42C22×C4×C101C2×C411yescharacteristic
43C4⋊C4×C101—21no—
44C4⋊C4×C101C2×C412yes—
45C23×C101C22⋊C411yescharacteristic
46C22×D101—22no—
47C22×D101—22no—
48C22×D101C22⋊C411yescharacteristic
49C22×D101D812yes—
50C22×D101SD1612yes—
51C22×D101C2×D411yescharacteristic
52C2×#87938—42no—
53C2×#87938—42no—
54C2×#87938—42no—
55C2×C4×C101—22no—
56C2×C4×C101—22no—
57C2×C4×C101—42no—
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22×C411yescharacteristic
60C2×Dic101—22no—
61C2×Dic101—22no—
62C2×Dic101—42no—
63C2×Dic101C22⋊C411yescharacteristic
64C2×Dic101C2×D411yescharacteristic
65C4×D101—22no—
66C4×D101—22no—
67C4×D101—22no—
68C4×D101—44no—
69C4×D101—42no—
70C4×D101—42no—
71C4×D101#4114yes—
72#11717581—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101#10212yes—
76C22×C101C2×C22⋊C411yescharacteristic
77C2×D101—21no—
78C2×D101—21no—
79C2×D101—24no—
80C2×D101—24no—
81C2×D101#10212yes—
82C2×D101#12312yes—
83C2×D101C2×D811yescharacteristic
84C2×D101C2×SD1611yescharacteristic
85C2×D101#13712yes—
86#87938—82no—
87C4×C101—21no—
88C4×C101—42no—
89C4×C101#11712yes—
90Dic101—21no—
91Dic101—42no—
92Dic101#12312yes—
93C2×C101—22no—
94C2×C101#38111yescharacteristic
95C2×C101C2×#10211yescharacteristic
96C2×C101#41911yescharacteristic
97D101—22no—
98D101#48012yes—
99D101#48312yes—
100#1823—1011nomaximal, Sylow
101C101#182311yescharacteristic, complemented, Sylow
102C22×C4⋊C4—1011no—
103C2×C22⋊C8—1011no—
104#380—1011no—
105#340—1014no—
106C23×C4—1011no—
107C23×C4—2021no—
108C22×C8—1012no—
109C2×C4⋊C4—1011no—
110C2×C4⋊C4—1012no—
111C2×C4⋊C4—1012no—
112C2×C4⋊C4—1012no—
113C2×C4⋊C4—2021no—
114C2×C4⋊C4—2021no—
115C2×C4⋊C4—2021no—
116C2×C4⋊C4#8793811yescharacteristic, complemented
117C2×C22⋊C4—2021no—
118C22⋊C8—2021no—
119C22⋊C8—2021no—
120#95—2021no—
121C24—1011no—
122C22×C4—21no—
123C22×C4—1011no—
124C22×C4—1011no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—1011no—
128C22×C4—4042no—
129C22×C4—4042no—
130C22×C4—2024no—
131C22×C4—2022no—
132C22×C4—2022no—
133C22×C4—2022no—
134C22×C4—2021no—
135C22×C4C2×#8793811yescharacteristic
136C2×C8—2022no—
137C2×C8—2024no—
138C4⋊C4—21no—
139C4⋊C4—1012no—
140C4⋊C4—2022no—
141C4⋊C4—2022no—
142C4⋊C4—2024no—
143C4⋊C4—2021no—
144C4⋊C4C2×#8793812yes—
145C22⋊C4—4042no—
146C23—1011no—
147C23—1012no—
148C23—1012no—
149C23—1011no—
150C23—2022no—
151C23—2022no—
152C23C22⋊#8793811yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—1011no—
157C2×C4—1014no—
158C2×C4—1011no—
159C2×C4—4042no—
160C2×C4—4042no—
161C2×C4—4044no—
162C2×C4—4042no—
163C2×C4—4042no—
164C2×C4—4042no—
165C2×C4—4042no—
166C2×C4—2022no—
167C2×C4—2022no—
168C2×C4—2022no—
169C2×C4—2022no—
170C2×C4—2022no—
171C2×C4C22⋊#8793811yescharacteristic
172C2×C4C22×#8793811yescharacteristic
173C8—4042no—
174C22—21no—
175C22—21no—
176C22—1012no—
177C22—1011no—
178C22—1011no—
179C22—1012no—
180C22—1012no—
181C22—2024no—
182C22—2024no—
183C22—2021no—
184C22—2021no—
185C22#1569231712yes—
186C22C2×C22⋊#8793811yescharacteristic
187C4—21no—
188C4—42no—
189C4—8082no—
190C4—1012no—
191C4—4042no—
192C4—2021no—
193C4#1569233412yes—
194C2—22no—
195C2—1012no—
196C2—1012no—
197C2—2022no—
198C2#2026612011yescharacteristic
199C2C2×#1569231711yescharacteristic
200C2#2026617011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC2×C4⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1823
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20266050C2semidirect
13. C2×C4⋊Dic101C4semidirect
17. C4⋊C4×D101C4semidirect
19. C2×C4⋊C4×C101C4semidirect
101. C101#1823semidirect
116. C2×C4⋊C4#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊(C22⋊C8)2
#202661192
#202660502
C2×C4⋊C4×D1012
#1823101