Finite Group Databasebeta

← #26027221

Subgroups — #26027221

Order 11392.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667988C212yesmaximal, complemented
2#19151538C212yesmaximal, complemented
3#19151528C211yesmaximal, characteristic, complemented
4#19151525C212yesmaximal, complemented
5#19151515C211yesmaximal, characteristic
6#19151489C212yesmaximal, complemented
7#19151476C212yesmaximal
8#19151472C212yesmaximal, complemented
9#391×C89C211yesmaximal, characteristic, complemented
10C22×D356C2211yescharacteristic
11C22×C4×D89C2212yes—
12#15667988—28no—
13C2×C4⋊Dic89C2212yes—
14C2×C4⋊Dic89C2211yescharacteristic
15C2×#13083553C2212yes—
16C2×#13083553C2212yes—
17C2×#13083551C2212yes—
18C2×#13083551C2212yes—
19C2×#13083551C2212yes—
20C2×#13083551C2212yes—
21C2×#13083551C2212yes—
22#15667919C2212yes—
23#15667919C2212yes—
24#15667919C2212yes—
25#15667919C2211yescharacteristic
26#15667919C2211yescharacteristic
27C22×Q8×C89C2211yescharacteristic
28C2×C4⋊C4×C89C2211yescharacteristic
29C2×C42×C89C2211yescharacteristic
30#95×C89C2212yes—
31#95×C89C2212yes—
32C23×D89C2312yes—
33C2×D356—24no—
34C2×D356—41no—
35C22×Dic89C2312yes—
36C22×Dic89C2312yes—
37C22×Dic89C2312yes—
38C2×C4×D89—24no—
39C2×C4×D89—42no—
40C2×C4×D89—42no—
41C4⋊Dic89—24no—
42C4⋊Dic89—41no—
43#13083553—24no—
44#13083553—24no—
45#13083551—24no—
46#13083551—24no—
47#13083551—24no—
48#13083551—24no—
49#13083551—42no—
50C2×Q8×C89—41no—
51C2×Q8×C89—41no—
52C2×Q8×C89D414yescomplemented
53C22×C4×C89C2311yescharacteristic
54C22×C4×C89C2312yes—
55C22×C4×C89C2311yescharacteristic
56C22×C4×C89C2312yes—
57C22×C4×C89C2311yescharacteristic
58C4⋊C4×C89—41no—
59C42×C89—41no—
60C23×C89C2411yescharacteristic
61C22×D89—22no—
62C22×D89—22no—
63C22×D89—22no—
64C22×D89—22no—
65C22×D89—22no—
66C22×D89—22no—
67C22×D89#4114yes—
68Q8×C89—44no—
69C2×C4×C89—21no—
70C2×C4×C89—21no—
71C2×C4×C89—22no—
72C2×C4×C89—22no—
73C2×C4×C89—22no—
74C2×C4×C89—22no—
75C2×C4×C89—22no—
76C2×C4×C89—22no—
77C2×C4×C89—21no—
78C2×C4×C89—21no—
79C2×C4×C89—22no—
80C2×C4×C89C2×D412yes—
81C2×C4×C89C2×D414yes—
82C2×Dic89—22no—
83C2×Dic89—22no—
84C2×Dic89—22no—
85C2×Dic89—22no—
86C2×Dic89—22no—
87C2×Dic89—22no—
88C2×Dic89—22no—
89C2×Dic89—22no—
90C2×Dic89—22no—
91D356—44no—
92C4×D89—48no—
93C22×C89C22×D411yescharacteristic
94C22×C89C2×#4112yes—
95C22×C89H(5,𝔽2)11yescharacteristic
96C22×C89H(5,𝔽2)11yescharacteristic
97C22×C892−1+412yes—
98C2×D89—24no—
99C2×D89—24no—
100C2×D89—24no—
101C2×D89—42no—
102C2×D89—42no—
103C2×D89—42no—
104C2×D89—42no—
105C4×C89—24no—
106C4×C89—22no—
107C4×C89—42no—
108C4×C89—41no—
109C4×C89—41no—
110Dic89—42no—
111Dic89—42no—
112Dic89—42no—
113C2×C89#53411yescharacteristic
114C2×C89#53912yes—
115C2×C89#54812yes—
116C2×C89#55111yescharacteristic
117C2×C89#56311yescharacteristic
118D89—44no—
119#2728—891nomaximal, Sylow
120C89#272811yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—892no—
122#400—892no—
123#398—891no—
124#397—892no—
125#394—891no—
126#391D8911yescharacteristic, complemented
127#388—892no—
128#386—892no—
129#385—892no—
130C22×Q8C2×D8911yescharacteristic
131C22×D4—891no—
132C23×C4—892no—
133C22⋊Q8—1788no—
134C2×C4⋊C4—891no—
135C2×C4⋊C4—892no—
136C2×C4⋊C4—892no—
137C2×C4⋊C4—892no—
138C2×C4⋊C4C2×D8911yescharacteristic
139C2×C22⋊C4—892no—
140C2×C22⋊C4—892no—
141C2×C22⋊C4—892no—
142C2×C22⋊C4—892no—
143C2×C22⋊C4—892no—
144C2×C42C2×D8911yescharacteristic
145#95—892no—
146#95—892no—
147#95—892no—
148#95—891no—
149#95—891no—
150#95C2×D8912yes—
151#95C2×D8912yes—
152C24—892no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D35614yescomplemented
156C2×D4—3561no—
157C2×D4—1784no—
158C22×C4—892no—
159C22×C4—892no—
160C22×C4—892no—
161C22×C4—3562no—
162C22×C4—3562no—
163C22×C4—1784no—
164C22×C4C22×D8911yescharacteristic
165C22×C4C22×D8912yes—
166C22×C4C22×D8911yescharacteristic
167C22×C4C22×D8912yes—
168C22×C4C22×D8911yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—3561no—
171C4⋊C4—1784no—
172C4⋊C4—1784no—
173C4⋊C4—1784no—
174C22⋊C4—3562no—
175C22⋊C4—1784no—
176C22⋊C4—1784no—
177C22⋊C4—1784no—
178C22⋊C4—1784no—
179C42—41no—
180C23—894no—
181C23—1782no—
182C23—1782no—
183C23—1782no—
184C23—1782no—
185C23—1782no—
186C23—1782no—
187C23C23×D8911yescharacteristic
188Q8—44no—
189D4—3564no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—3568no—
202C2×C4—1782no—
203C2×C4—1782no—
204C2×C4—1782no—
205C2×C4—1782no—
206C2×C4—1782no—
207C2×C4—1782no—
208C2×C4—1782no—
209C2×C4—1782no—
210C2×C4—1782no—
211C2×C4C2×D35612yes—
212C2×C4C2×D35614yes—
213C22—3562no—
214C22—3562no—
215C22—3562no—
216C22—3562no—
217C22—1784no—
218C22—1784no—
219C22—1784no—
220C22C22×D35611yescharacteristic
221C22#41×D8912yes—
222C22#1566805711yescharacteristic
223C22#1566805711yescharacteristic
224C22#1566806012yes—
225C4—24no—
226C4—22no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—3562no—
231C4—3562no—
232C4—3562no—
233C2—3564no—
234C2#1915216011yescharacteristic
235C2#1915219412yes—
236C2#1915226012yes—
237C2#1915227811yescharacteristic
238C2#1915234711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#391×C89C2
prime radicalC22×D356C22
hypercenter#391D89
hypocenterC89#2728
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#15667988C2semidirect
2. #19151538C2semidirect
3. #19151528C2semidirect
4. #19151525C2semidirect
6. #19151489C2semidirect
8. #19151472C2semidirect
9. #391×C89C2semidirect
52. C2×Q8×C89D4semidirect
120. C89#2728semidirect
126. #391D89semidirect
155. C2×Q8D356semidirect

Maximal subgroups

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

subgroupindex
C2×#156679882
#191515382
#191515282
#191515252
#191515152
#191514892
#191514762
#191514722
#391×C892
#272889