Finite Group Databasebeta

← #49159232

Subgroups — #49159232

Order 37504.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#34036641C211yesmaximal, characteristic
2C2×#22627581C211yesmaximal, characteristic, complemented
3C2×#22627552C211yesmaximal, characteristic, complemented
4C2×#22627550C211yesmaximal, characteristic, complemented
5#34036092C211yesmaximal, characteristic, complemented
6#34035888C211yesmaximal, characteristic, complemented
7#339×C293C211yesmaximal, characteristic, complemented
8C2×#17989802C2211yescharacteristic
9C22×C4×D293C2211yescharacteristic
10C2×#17989786C2211yescharacteristic
11#22627608—21no—
12#22627604C412yescomplemented
13#22627599C412yescomplemented
14#22627594—21no—
15C4×D1172—22no—
16#22627581—22no—
17#22627581—22no—
18C2×#17989781—21no—
19C2×#17989779—21no—
20C2×C4×Dic293C2211yescharacteristic
21#22627552—22no—
22#22627550—22no—
23#22627536—22no—
24C2×M16×C293C2211yescharacteristic
25#117×C293C2211yescharacteristic
26C2×C42×C293C2211yescharacteristic
27C23×D293D411yescharacteristic
28#17989802—21no—
29#17989802—21no—
30#17989802—21no—
31#17989802—21no—
32C2×Dic586C2×C411yescharacteristic
33C2×D1172—22no—
34C2×D1172C2×C411yescharacteristic
35C2×C22⋊D293—21no—
36C22×Dic293—21no—
37C22×Dic293D411yescharacteristic
38C2×C4×D293—21no—
39C2×C4×D293—21no—
40C2×C4×D293—21no—
41C2×C4×D293—21no—
42C2×C4×D293—21no—
43C2×C4×D293C2×C412yes—
44C2×C4×D293D412yes—
45#17989786—21no—
46#17989786—21no—
47C2×#14556873—21no—
48C4⋊Dic293—22no—
49#17989781—24no—
50#17989779—24no—
51#17989779—22no—
52#17989779—22no—
53C4×Dic293—21no—
54C4×Dic293D412yescomplemented
55C22×C4×C293—21no—
56C22×C4×C293C2311yescharacteristic
57M16×C293—21no—
58M16×C293—21no—
59C2×C8×C293—21no—
60C4⋊C4×C293C2×C412yes—
61C22⋊C4×C293—21no—
62C42×C293—21no—
63C42×C293—21no—
64C42×C293D412yescomplemented
65C22×D293—21no—
66C22×D293—21no—
67C22×D293—21no—
68C22×D293—42no—
69C22×D293C22⋊C412yes—
70Dic586—21no—
71Dic586—21no—
72D1172—21no—
73D1172—21no—
74D1172—44no—
75C22⋊D293—41no—
76C22⋊D293—41no—
77C2×Dic293—21no—
78C2×Dic293—21no—
79C2×Dic293—21no—
80C2×Dic293—22no—
81C2×Dic293—42no—
82C2×Dic293C22⋊C412yes—
83C4×D293—22no—
84C4×D293—22no—
85C4×D293—22no—
86C4×D293—22no—
87C4×D293—41no—
88C4×D293—41no—
89#14556873—41no—
90C23×C293C2×D411yescharacteristic
91C2×C4×C293—22no—
92C2×C4×C293—22no—
93C2×C4×C293—42no—
94C2×C4×C293C22×C411yescharacteristic
95C2×C4×C293C2×D411yescharacteristic
96C2×C4×C293C2×D411yescharacteristic
97C2×C4×C293C2×D411yescharacteristic
98C2×C4×C293#4111yescharacteristic
99C2×C4×C293#4111yescharacteristic
100C8×C293—41no—
101C22×C293—22no—
102C22×C293C2×C22⋊C411yescharacteristic
103C22×C293#12011yescharacteristic
104C22×C293#12311yescharacteristic
105C2×D293—22no—
106C2×D293—22no—
107C2×D293—22no—
108C2×D293—42no—
109C2×D293—42no—
110C2×D293—41no—
111C2×D293—42no—
112C4×C293—42no—
113C4×C293—42no—
114C4×C293C4×D411yescharacteristic
115C4×C293C4×D411yescharacteristic
116C4×C293#12111yescharacteristic
117C4×C293#12111yescharacteristic
118Dic293—22no—
119Dic293—42no—
120Dic293—41no—
121C2×C293—21no—
122C2×C293—21no—
123C2×C293#38611yescharacteristic
124C2×C293#42112yes—
125D293—42no—
126D293—81no—
127C293#189711yescharacteristic, complemented, Sylow
128#1897—2931nomaximal, Sylow
129#525—2931no—
130C2×#117—2931no—
131C2×#104—2931no—
132C2×#104—2931no—
133#407—2931no—
134#339—2931no—
135#339D29311yescharacteristic, complemented
136C2×#41—2931no—
137C23×C4—2931no—
138C2×M16—2931no—
139C2×M16C2×D29311yescharacteristic
140#123—5861no—
141C22⋊Q8—2932no—
142#121—2932no—
143#120—5861no—
144C4×D4—5862no—
145#117—5862no—
146#117—5862no—
147#117C2×D29311yescharacteristic
148C2×C4⋊C4—5861no—
149C2×C22⋊C4—5861no—
150C2×C42—2931no—
151C2×C42C2×D29311yescharacteristic
152#104—5862no—
153#104—5862no—
154C22⋊C8—5862no—
155C24—2931no—
156#41—5861no—
157#41—5861no—
158#41—5861no—
159#41—5861no—
160C2×Q8—2931no—
161C2×D4—2931no—
162C2×D4—5862no—
163C2×D4—5861no—
164C22×C4—21no—
165C22×C4—2931no—
166C22×C4—2932no—
167C22×C4—2932no—
168C22×C4—5861no—
169C22×C4—5861no—
170C22×C4—5861no—
171C22×C4—5861no—
172C22×C4—5861no—
173C22×C4—5861no—
174C22×C4C22×D29311yescharacteristic
175M16—21no—
176M16—21no—
177M16—5861no—
178M16—5861no—
179C2×C8—21no—
180C2×C8—5861no—
181C4⋊C4—5862no—
182C4⋊C4—5864no—
183C4⋊C4C4×D29312yes—
184C22⋊C4—21no—
185C22⋊C4—5862no—
186C22⋊C4—5862no—
187C22⋊C4—5864no—
188C42—21no—
189C42—21no—
190C42—2932no—
191C42—5861no—
192C42D117212yescomplemented
193C23—2932no—
194C23—5861no—
195C23—5861no—
196C23—5861no—
197C23—11722no—
198C23D4×D29311yescharacteristic
199Q8—5861no—
200Q8—5861no—
201D4—5861no—
202D4—5861no—
203D4—11724no—
204D4—11721no—
205D4—11721no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2932no—
210C2×C4—5861no—
211C2×C4—5861no—
212C2×C4—5862no—
213C2×C4—5862no—
214C2×C4—5862no—
215C2×C4—5862no—
216C2×C4—5862no—
217C2×C4—5861no—
218C2×C4—11722no—
219C2×C4—11721no—
220C2×C4—11721no—
221C2×C4C2×C4×D29311yescharacteristic
222C2×C4D4×D29311yescharacteristic
223C2×C4D4×D29311yescharacteristic
224C2×C4C2×D117211yescharacteristic
225C2×C4#1798980211yescharacteristic
226C2×C4#1798980311yescharacteristic
227C8—41no—
228C8—11721no—
229C22—22no—
230C22—5862no—
231C22—5862no—
232C22—5862no—
233C22—11722no—
234C22—11722no—
235C22—11721no—
236C22—11722no—
237C22C22⋊C4×D29311yescharacteristic
238C22#2262759411yescharacteristic
239C22#2262760911yescharacteristic
240C4—42no—
241C4—42no—
242C4—5862no—
243C4—11722no—
244C4—11721no—
245C4#2262758611yescharacteristic
246C4C4×D117211yescharacteristic
247C4#2262759711yescharacteristic
248C4#2262759911yescharacteristic
249C2—21no—
250C2—21no—
251C2—23441no—
252C2—11722no—
253C2#3403600711yescharacteristic
254C2#3403614412yes—
subgroupG/H
centerC2×C4D4×D293
commutatorC2×C4×C293C22×C4
FrattiniC22×C4C22×D293
Fitting#339×C293C2
socleC22×C293C2×C22⋊C4
prime radical#34036641C2
cosocleC22×C4×C293C23
hypercenter#339D293
hypocenterC293#1897
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#22627581C2semidirect
3. C2×#22627552C2semidirect
4. C2×#22627550C2semidirect
5. #34036092C2semidirect
6. #34035888C2semidirect
7. #339×C293C2semidirect
12. #22627604C4semidirect
13. #22627599C4semidirect
54. C4×Dic293D4semidirect
64. C42×C293D4semidirect
127. C293#1897semidirect
135. #339D293semidirect
192. C42D1172semidirect

Maximal subgroups

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

subgroupindex
#340366412
C2×#226275812
C2×#226275522
C2×#226275502
#340360922
#340358882
#339×C2932
#1897293