Finite Group Databasebeta

← #49168148

Subgroups — #49168148

Order 37504.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22627672C211yesmaximal, characteristic, complemented
2#34036927C212yesmaximal, complemented
3#34036311C214yesmaximal, complemented
4#34036301C214yesmaximal, complemented
5#34036147C212yesmaximal, complemented
6C2×#22627543C211yesmaximal, characteristic, complemented
7#536×C293C211yesmaximal, characteristic, complemented
8#22627672—24no—
9#22627672—22no—
10#22627672C2214yes—
11C2×#17989803—22no—
12C2×#17989802—22no—
13C2×#17989802C2212yes—
14C2×Q8×D293—22no—
15#22627655C2214yescomplemented
16#22627654—22no—
17#22627650C2214yescomplemented
18#22627649—22no—
19#22627644—24no—
20C2×#17989786C2212yes—
21#22627552C2218yescomplemented
22#22627543C2214yescomplemented
23C2×#41×C293C2211yescharacteristic
24C22×Q8×C293C2211yescharacteristic
25C4⋊Q8×C293C2212yescomplemented
26#124×C293C2212yescomplemented
27#123×C293—22no—
28C22⋊Q8×C293—22no—
29#117×C293C2211yescharacteristic
30#17989803—28no—
31#17989803—28no—
32#17989802—28no—
33#17989802—28no—
34#17989802—22no—
35#17989802—22no—
36#17989802D414yes—
37#17989802C2314yes—
38C2×Dic586—22no—
39C2×Dic586D412yes—
40Q8×D293—28no—
41Q8×D293—28no—
42C2×D1172—22no—
43C2×D1172D412yes—
44C2×C22⋊D293—22no—
45C2×C4×D293—24no—
46C2×C4×D293—22no—
47#17989792—24no—
48#17989790—24no—
49#17989789—24no—
50#17989787—24no—
51#17989786—22no—
52#17989786C2314yes—
53C2×#14556873—22no—
54#41×C293—22no—
55#41×C293—22no—
56C2×Q8×C293—22no—
57C2×Q8×C293—22no—
58C2×Q8×C293D412yes—
59C2×Q8×C293C2311yescharacteristic
60C2×Q8×C293C2312yes—
61C2×D4×C293—21no—
62C2×D4×C293C2311yescharacteristic
63C22×C4×C293—21no—
64C22×C4×C293D412yes—
65C22×C4×C293C2311yescharacteristic
66C4⋊C4×C293—21no—
67C4⋊C4×C293—24no—
68C22⋊C4×C293—21no—
69C22⋊C4×C293—24no—
70C42×C293C2312yes—
71C22×D293—22no—
72Dic586—28no—
73Dic586—22no—
74Dic586C2×D414yes—
75D1172—28no—
76D1172—22no—
77D1172C2×D414yes—
78C22⋊D293—24no—
79C22⋊D293—24no—
80C2×Dic293—22no—
81C4×D293—216no—
82C4×D293—24no—
83C4×D293—24no—
84#14556873—24no—
85C23×C293—21no—
86C23×C293C2×D411yescharacteristic
87Q8×C293—24no—
88Q8×C293—24no—
89Q8×C293—22no—
90D4×C293—22no—
91D4×C293—41no—
92D4×C293—41no—
93C2×C4×C293—22no—
94C2×C4×C293—22no—
95C2×C4×C293—22no—
96C2×C4×C293—21no—
97C2×C4×C293—42no—
98C2×C4×C293C2×D412yes—
99C2×C4×C293C2×D412yes—
100C2×C4×C293C2×D414yes—
101C2×C4×C293C2×D411yescharacteristic
102C2×C4×C293C2411yescharacteristic
103C22×C293—41no—
104C22×C293—42no—
105C22×C293—41no—
106C22×C293#12011yescharacteristic
107C22×C293#12011yescharacteristic
108C22×C293C22×D411yescharacteristic
109C2×D293—24no—
110C2×D293—44no—
111C4×C293—24no—
112C4×C293—42no—
113C4×C293—41no—
114C4×C293#12012yes—
115C4×C293C22×D412yes—
116Dic293—24no—
117C2×C293—21no—
118C2×C293—21no—
119C2×C293—21no—
120C2×C293—81no—
121C2×C293C2×#12011yescharacteristic
122D293—44no—
123C293#302011yescharacteristic, complemented, Sylow
124#3020—2931nomaximal, Sylow
125C2×2−1+4—2931no—
126#575—2932no—
127#536D29311yescharacteristic, complemented
128#456—2934no—
129#454—2934no—
130#421—2932no—
131C2×#101—2931no—
1322−1+4—2934no—
1332−1+4—5864no—
1342−1+4—5862no—
135C2×#41—2932no—
136C2×#41—5862no—
137C2×#41—5862no—
138C2×#41C2×D29311yescharacteristic
139C22×Q8—5862no—
140C22×Q8C2×D29311yescharacteristic
141#137—2934no—
142#137—5862no—
143AGL(1,ℤ/8)—2934no—
144AGL(1,ℤ/8)—5862no—
145#135—5864no—
146C2×M16—2932no—
147C4⋊Q8C2×D29312yescomplemented
148#124C2×D29312yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D29311yescharacteristic
152#104—2938no—
153#101—2934no—
154#41—22no—
155#41—22no—
156#41—2934no—
157#41—2934no—
158#41—5868no—
159#41—5868no—
160#41—5868no—
161#41—5862no—
162#41—5862no—
163#41—5868no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—2932no—
167C2×Q8—5862no—
168C2×Q8—5868no—
169C2×Q8—5868no—
170C2×Q8C22⋊D29312yes—
171C2×Q8C22×D29311yescharacteristic
172C2×Q8C22×D29312yes—
173C2×D4—21no—
174C2×D4—2932no—
175C2×D4—5862no—
176C2×D4—5862no—
177C2×D4C22×D29311yescharacteristic
178C22×C4—21no—
179C22×C4—5864no—
180C22×C4—5862no—
181C22×C4C22⋊D29312yes—
182C22×C4C22×D29311yescharacteristic
183Q16—5864no—
184SD16—5864no—
185SD16—5864no—
186D8—5864no—
187M16—2934no—
188M16—5862no—
189C2×C8—5862no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D29312yes—
195C23—21no—
196C23—5862no—
197C23C2×C22⋊D29311yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—2934no—
202Q8—5868no—
203Q8—5862no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—2934no—
208D4—5868no—
209D4—5864no—
210D4—5864no—
211D4—5862no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—58616no—
218C2×C4—5862no—
219C2×C4—5864no—
220C2×C4—5864no—
221C2×C4D4×D29312yes—
222C2×C4D4×D29312yes—
223C2×C4C2×C22⋊D29314yes—
224C2×C4C2×C22⋊D29311yescharacteristic
225C2×C4C23×D29311yescharacteristic
226C8—5864no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—5864no—
231C22—11724no—
232C22#2262759211yescharacteristic
233C22#2262759211yescharacteristic
234C22C22×C22⋊D29311yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—5864no—
239C4#2262759212yes—
240C4C2×D4×D29312yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—11724no—
246C2C2×#2262759211yescharacteristic
subgroupG/H
centerC2C2×#22627592
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#536×C293C2
socleC2×C293C2×#120
hypercenter#536D293
hypocenterC293#3020
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#22627672C2semidirect
2. #34036927C2semidirect
3. #34036311C2semidirect
4. #34036301C2semidirect
5. #34036147C2semidirect
6. C2×#22627543C2semidirect
7. #536×C293C2semidirect
15. #22627655C22semidirect
17. #22627650C22semidirect
21. #22627552C22semidirect
22. #22627543C22semidirect
25. C4⋊Q8×C293C22semidirect
26. #124×C293C22semidirect
123. C293#3020semidirect
127. #536D293semidirect
147. C4⋊Q8C2×D293semidirect
148. #124C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276722
#340369272
#340363112
#340363012
#340361472
C2×#226275432
#536×C2932
#3020293