Finite Group Databasebeta

← #15674210

Subgroups — #15674210

Order 2944.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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×#10625118C211yesmaximal, characteristic
2C2×#10625017C211yesmaximal, characteristic, complemented
3C2×#10625003C211yesmaximal, characteristic
4C2×#10624991C211yesmaximal, characteristic
5C2×#10624997C211yesmaximal, characteristic, complemented
6#13092286C211yesmaximal, characteristic, complemented
7C2×C23×#103C211yesmaximal, characteristic, complemented
8C22×Q8×C23C2211yescharacteristic
9C22×D92C2211yescharacteristic
10C22×C8×C23C2211yescharacteristic
11C2×C4⋊C4×C23C2211yescharacteristic
12#10625118—22no—
13#10625118C414yescomplemented
14C2×(C2×D23)⋊C4—22no—
15C2×C4⋊Dic23C2211yescharacteristic
16C2×Dic23⋊C4—22no—
17C2×C4×Dic23C2211yescharacteristic
18C22×C23⋊C8C2211yescharacteristic
19C23×#103—22no—
20#10625017—22no—
21#10625003—22no—
22#10624997—22no—
23#10624991—22no—
24C23×D23—21no—
25C2×Q8×C23—21no—
26C2×Q8×C23—21no—
27C2×Q8×C23—21no—
28C2×Q8×C23—21no—
29C2×Q8×C23—21no—
30C2×Q8×C23C2×C412yes—
31C22×C4×C23—21no—
32C22×C4×C23—21no—
33C22×C4×C23C2311yescharacteristic
34C22×Dic23—21no—
35C22×Dic23D412yes—
36C2×D92—21no—
37C2×D92—21no—
38C2×D92—21no—
39C2×D92—21no—
40C2×D92—21no—
41C2×D92C2×C412yes—
42C2×C8×C23—21no—
43C2×C8×C23—21no—
44C2×C8×C23D412yes—
45C4⋊C4×C23—21no—
46C4⋊C4×C23—21no—
47(C2×D23)⋊C4—28no—
48C4⋊Dic23—21no—
49C4⋊Dic23—21no—
50Dic23⋊C4—44no—
51C4×Dic23C2×C412yes—
52C4×Dic23D412yes—
53C2×C23⋊C8—21no—
54C2×C23⋊C8—21no—
55C2×C23⋊C8D412yes—
56C23×C23C2×D411yescharacteristic
57C22×D23—22no—
58C22×D23—41no—
59C22×D23—41no—
60C22×D23—42no—
61C22×D23—42no—
62Q8×C23—22no—
63Q8×C23—22no—
64Q8×C23—41no—
65Q8×C23—41no—
66C2×C4×C23—22no—
67C2×C4×C23—41no—
68C2×C4×C23—41no—
69C2×C4×C23—41no—
70C2×C4×C23—42no—
71C2×C4×C23C22×C411yescharacteristic
72C2×C4×C23C2×D411yescharacteristic
73C2×C4×C23C2×D411yescharacteristic
74C2×C4×C23C2×D411yescharacteristic
75C2×C4×C23#4111yescharacteristic
76C2×C4×C23#4111yescharacteristic
77C2×Dic23—22no—
78C2×Dic23—22no—
79C2×Dic23—41no—
80C2×Dic23—42no—
81C2×Dic23C22⋊C414yes—
82D92—22no—
83D92—22no—
84D92—41no—
85D92—41no—
86C8×C23—22no—
87C23⋊C8—22no—
88#1959—231nomaximal, Sylow
89C22×C23C2×C22⋊C411yescharacteristic
90C22×C23#12411yescharacteristic
91C22×C23#12711yescharacteristic
92C22×C23AGL(1,ℤ/8)12yes—
93C22×C23#13712yes—
94C2×D23—42no—
95C2×D23—44no—
96C2×D23—82no—
97C2×D23—81no—
98C2×D23—81no—
99C4×C23—42no—
100C4×C23—81no—
101C4×C23C4×D411yescharacteristic
102C4×C23C4×D411yescharacteristic
103C4×C23#12111yescharacteristic
104C4×C23#12111yescharacteristic
105Dic23—24no—
106Dic23—81no—
107C2×#124—231no—
108C2×#103—231no—
109C2×#103D2311yescharacteristic, complemented
110C2×#102—231no—
111C2×#102—231no—
112C2×#97—231no—
113#384—231no—
114C2×C23#39011yescharacteristic
115C2×C23#41712yes—
116C2×C23#48912yes—
117C2×C23#49611yescharacteristic
118C2×C23#49711yescharacteristic
119D23—82no—
120C22×Q8C2×D2311yescharacteristic
121C22×D4—231no—
122C22×C8—231no—
123C22×C8C2×D2311yescharacteristic
124#124—234no—
125#124—462no—
126C2×C4⋊C4—231no—
127C2×C4⋊C4—462no—
128C2×C4⋊C4C2×D2311yescharacteristic
129C2×C22⋊C4—462no—
130C2×C42—231no—
131#103—22no—
132#103—462no—
133#102—462no—
134#102—462no—
135#97—462no—
136C23#195911yescharacteristic, complemented, Sylow
137C24—461no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C4×D2312yes—
144C2×D4—232no—
145C2×D4—461no—
146C2×D4—461no—
147C2×D4—461no—
148C2×D4—461no—
149C2×D4—461no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—232no—
153C22×C4—461no—
154C22×C4C22×D2311yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—232no—
158C2×C8—461no—
159C2×C8—461no—
160C2×C8C22⋊D2312yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—461no—
164C4⋊C4—461no—
165C4⋊C4—924no—
166C22⋊C4—468no—
167C42—232no—
168C42—232no—
169C23—462no—
170C23—921no—
171C23—922no—
172C23—922no—
173C23—921no—
174C23D4×D2311yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—41no—
178Q8—41no—
179D4—462no—
180D4—462no—
181D4—921no—
182D4—921no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—234no—
189C2×C4—462no—
190C2×C4—462no—
191C2×C4—922no—
192C2×C4—921no—
193C2×C4C2×C4×D2311yescharacteristic
194C2×C4#6728811yescharacteristic
195C2×C4D4×D2311yescharacteristic
196C2×C4D4×D2311yescharacteristic
197C2×C4#6729011yescharacteristic
198C2×C4C2×C22⋊D2311yescharacteristic
199C8—22no—
200C8—462no—
201C22—924no—
202C22—922no—
203C22—1842no—
204C22—1841no—
205C22—1841no—
206C22C22⋊C4×D2311yescharacteristic
207C22#1062505611yescharacteristic
208C22#1062508512yes—
209C22#1062508912yes—
210C22#1062511111yescharacteristic
211C4—42no—
212C4—81no—
213C4—464no—
214C4—1841no—
215C4(C22⋊D23)⋊C411yescharacteristic
216C4#1062505511yescharacteristic
217C4C4×C22⋊D2311yescharacteristic
218C4#1062511011yescharacteristic
219C2—1842no—
220C2#1309231711yescharacteristic
221C2#1309241712yes—
222C2#1309277812yes—
223C2#1309280311yescharacteristic
224C2#1309281011yescharacteristic
subgroupG/H
centerC23D4×D23
commutatorC2×C4×C23C22×C4
FrattiniC22×C4C22×D23
FittingC2×C23×#103C2
socleC23×C23C2×D4
prime radicalC22×D92C22
cosocleC22×C4×C23C23
hypercenterC2×#103D23
hypocenterC23#1959
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#10625017C2semidirect
5. C2×#10624997C2semidirect
6. #13092286C2semidirect
7. C2×C23×#103C2semidirect
13. #10625118C4semidirect
109. C2×#103D23semidirect
136. C23#1959semidirect

Maximal subgroups

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

subgroupindex
C2×#106251182
C2×#106250172
C2×#106250032
C2×#106249912
C2×#106249972
#130922862
C2×C23×#1032
#195923