Finite Group Databasebeta

← #27186572

Subgroups — #27186572

Order 12928.

288 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 70 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#20266031C214yesmaximal, complemented
4C2×#15692231C211yesmaximal, characteristic
5C2×D101⋊C8—21no—
6C2×C4⋊#87938—21no—
7C2×C22⋊#87938—21no—
8#15692312C2212yes—
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4C2212yes—
11C22×C4×D101C2211yescharacteristic
12#15692231—21no—
13#15692231—21no—
14#15692231C412yescomplemented
15#15692231C2212yes—
16C2×C4⋊Dic101C411yescharacteristic, complemented
17C2×#13108053—21no—
18C2×#13108051—21no—
19C22×Q8×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21D101⋊C8—42no—
22C2×#11717581—22no—
23C4⋊#87938—42no—
24C22⋊#87938—42no—
25C23×D101D411yescharacteristic
26C22×Dic101—21no—
27C22×Dic101D411yescharacteristic
28C2×C4×D101—42no—
29C2×C4×D101C2×C411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101D411yescharacteristic
32C2×C4×D101C2311yescharacteristic
33C4⋊Dic101—21no—
34C4⋊Dic101C2×C412yes—
35#13108053—22no—
36#13108053—22no—
37#13108051—22no—
38#13108051—22no—
39C2×Q8×C101—21no—
40C2×Q8×C101—42no—
41C2×Q8×C101C2×C412yes—
42C22×C4×C101—21no—
43C22×C4×C101C2×C411yescharacteristic
44C23×C101C22⋊C411yescharacteristic
45C22×D101—41no—
46C22×D101—41no—
47C22×D101C22⋊C411yescharacteristic
48C22×D101SD1612yes—
49C22×D101Q1612yes—
50C22×D101C2×D411yescharacteristic
51C2×#87938—42no—
52C2×#87938—42no—
53C2×#87938—42no—
54Q8×C101—42no—
55Q8×C101—42no—
56C2×C4×C101—22no—
57C2×C4×C101—22no—
58C2×C4×C101—42no—
59C2×C4×C101C22⋊C411yescharacteristic
60C2×C4×C101C22×C411yescharacteristic
61C2×Dic101—22no—
62C2×Dic101—41no—
63C2×Dic101—42no—
64C2×Dic101C22⋊C411yescharacteristic
65C2×Dic101C2×D411yescharacteristic
66C4×D101—22no—
67C4×D101—22no—
68C4×D101—41no—
69C4×D101—41no—
70#11717581—42no—
71C22×C101—21no—
72C22×C101—21no—
73C22×C101#10312yes—
74C22×C101C2×C22⋊C411yescharacteristic
75C2×D101—21no—
76C2×D101—21no—
77C2×D101—21no—
78C2×D101—21no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101#10312yes—
82C2×D101C2×SD1611yescharacteristic
83C2×D101C2×Q1611yescharacteristic
84#87938—82no—
85C4×C101—21no—
86C4×C101—42no—
87C4×C101#9912yes—
88Dic101—21no—
89Dic101—21no—
90Dic101—42no—
91C2×C101—22no—
92C2×C101C2×#9911yescharacteristic
93C2×C101C2×#10311yescharacteristic
94C2×C101#42111yescharacteristic
95D101—21no—
96D101—22no—
97D101—21no—
98#1520—1011nomaximal, Sylow
99C101#152011yescharacteristic, complemented, Sylow
100C2×C22⋊Q8—1011no—
101C2×C22⋊C8—1011no—
102#380—1011no—
103#328—1014no—
104C22×Q8#8793811yescharacteristic, complemented
105C23×C4—1011no—
106C22×C8—2021no—
107C22⋊Q8—1012no—
108C22⋊Q8—1012no—
109C22⋊Q8—2021no—
110C22⋊Q8—2021no—
111C2×C4⋊C4—1011no—
112C2×C4⋊C4—2021no—
113C2×C4⋊C4—2021no—
114C2×C22⋊C4—2021no—
115C2×C22⋊C4—2021no—
116C22⋊C8—1012no—
117C22⋊C8—2021no—
118#95—1012no—
119C24—1011no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×#8793812yes—
123C22×C4—21no—
124C22×C4—1011no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—1011no—
128C22×C4—1011no—
129C22×C4—4042no—
130C22×C4—2022no—
131C22×C4—2021no—
132C22×C4C2×#8793811yescharacteristic
133C2×C8—4042no—
134C2×C8—2022no—
135C4⋊C4—1012no—
136C4⋊C4—4042no—
137C4⋊C4—2022no—
138C4⋊C4—2022no—
139C4⋊C4—2021no—
140C22⋊C4—4042no—
141C22⋊C4—2022no—
142C22⋊C4—2022no—
143C23—1012no—
144C23—1012no—
145C23—1011no—
146C23—1011no—
147C23—4041no—
148C23—4041no—
149C23C22⋊#8793811yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—1011no—
156C2×C4—1011no—
157C2×C4—4041no—
158C2×C4—4041no—
159C2×C4—4042no—
160C2×C4—4042no—
161C2×C4—4042no—
162C2×C4—4042no—
163C2×C4—4041no—
164C2×C4—2022no—
165C2×C4—2022no—
166C2×C4—2022no—
167C2×C4C22⋊#8793811yescharacteristic
168C2×C4C22×#8793811yescharacteristic
169C8—4042no—
170C22—21no—
171C22—21no—
172C22—1011no—
173C22—1012no—
174C22—1011no—
175C22—4042no—
176C22—4042no—
177C22—2021no—
178C22—2021no—
179C22—2021no—
180C22—2021no—
181C22#1569231812yes—
182C22C2×C22⋊#8793811yescharacteristic
183C4—21no—
184C4—42no—
185C4—8082no—
186C4—4042no—
187C4—2021no—
188C4—2021no—
189C4#1569231312yes—
190C2—22no—
191C2—2021no—
192C2—2022no—
193C2—2021no—
194C2C2×#1569231311yescharacteristic
195C2C2×#1569231811yescharacteristic
196C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22×Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22×Q8#87938
hypocenterC101#1520
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20266031C2semidirect
14. #15692231C4semidirect
16. C2×C4⋊Dic101C4semidirect
19. C22×Q8×C101C4semidirect
99. C101#1520semidirect
104. C22×Q8#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
#202660312
C2×#156922312
#1520101