Finite Group Databasebeta

← #49168192

Subgroups — #49168192

Order 37504.

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

normal subgroups only

This group has 716 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
1#34036965C211yesmaximal, characteristic
2#34036705C212yesmaximal, complemented
3#34036700C211yesmaximal, characteristic, complemented
4#34036320C218yesmaximal, complemented
5#34036102C212yesmaximal, complemented
6#410×C293C211yesmaximal, characteristic, complemented
7#22627672—21no—
8#22627671—22no—
9#22627670—22no—
10#22627669—21no—
11#22627669C2214yes—
12C2×#17989802—22no—
13C2×#17989802C2212yes—
14#41×D293—24no—
15C22×Dic586C2211yescharacteristic, complemented
16#22627624—22no—
17#22627622—21no—
18#22627612—22no—
19#22627611—21no—
20#22627608C2218yescomplemented
21#22627605C2214yescomplemented
22#22627604—24no—
23#22627603—22no—
24#22627583C2211yescharacteristic, complemented
25#22627538C2218yescomplemented
26C2×#41×C293C2211yescharacteristic
27#117×C293C2212yescomplemented
28#99×C293C2214yescomplemented
29#17989803—24no—
30#17989802—24no—
31#17989802—24no—
32#17989802—22no—
33#17989802—22no—
34#17989802—24no—
35#17989802—22no—
36#17989802—22no—
37#17989801—24no—
38#17989801—28no—
39C2×Dic586—22no—
40C2×Dic586—21no—
41C2×Dic586—22no—
42C2×Dic586—21no—
43C2×Dic586—22no—
44C2×Dic586D412yes—
45Q8×D293—24no—
46C2×D1172—22no—
47C2×D1172D412yes—
48C2×C22⋊D293—24no—
49C2×C22⋊D293C2314yes—
50D4×D293—24no—
51D4×D293—28no—
52C22×Dic293C2312yes—
53C2×C4×D293—24no—
54C2×C4×D293D414yes—
55C4⋊Dic293—28no—
56C4⋊Dic293—21no—
57#17989781—24no—
58C22⋊Dic293—24no—
59C22⋊Dic293C2312yes—
60#17989779—28no—
61C4×Dic293—21no—
62#41×C293—22no—
63#41×C293—22no—
64C2×Q8×C293D411yescharacteristic
65C2×D4×C293D411yescharacteristic
66C2×D4×C293C2312yes—
67C22×C4×C293D412yes—
68C22×C4×C293C2311yescharacteristic
69C4⋊C4×C293—22no—
70C22⋊C4×C293C2314yes—
71C42×C293—22no—
72C22×D293—24no—
73C22×D293C2×D414yes—
74Dic586—24no—
75Dic586—22no—
76Dic586—22no—
77Dic586—41no—
78Dic586—41no—
79Dic586—42no—
80D1172—24no—
81D1172—22no—
82D1172—22no—
83C22⋊D293—28no—
84C22⋊D293—24no—
85C22⋊D293—24no—
86C2×Dic293—24no—
87C2×Dic293—22no—
88C2×Dic293—24no—
89C2×Dic293—22no—
90C2×Dic293C2×D414yes—
91C4×D293—28no—
92C4×D293—24no—
93C4×D293—24no—
94C23×C293C2×D412yes—
95C23×C293C2411yescharacteristic
96Q8×C293—22no—
97D4×C293—24no—
98D4×C293—22no—
99C2×C4×C293—22no—
100C2×C4×C293—22no—
101C2×C4×C293—24no—
102C2×C4×C293C2×D412yes—
103C2×C4×C293C2×D412yes—
104C2×C4×C293C2×D412yes—
105C2×C4×C293C2×D411yescharacteristic
106C2×C4×C293C2×D411yescharacteristic
107C22×C293—22no—
108C22×C293—41no—
109C22×C293—42no—
110C22×C293C22×D412yes—
111C22×C293C22×D411yescharacteristic
112C2×D293—24no—
113C2×D293—44no—
114C2×D293—84no—
115C4×C293—22no—
116C4×C293—44no—
117C4×C293#12011yescharacteristic
118C4×C293#12011yescharacteristic
119C4×C293#12012yes—
120Dic293—24no—
121Dic293—42no—
122Dic293—42no—
123C2×C293—21no—
124C2×C293—22no—
125C2×C293—42no—
126C2×C293C2×#12011yescharacteristic
127D293—44no—
128C293#302811yescharacteristic, complemented, Sylow
129#3028—2931nomaximal, Sylow
130#585—2931no—
131#536—2931no—
132#536—2932no—
133#458—2938no—
134#410—2932no—
135#410D29311yescharacteristic, complemented
1362−1+4—5862no—
1372−1+4—5861no—
138H(5,𝔽2)—2934no—
139H(5,𝔽2)—5862no—
140H(5,𝔽2)—5861no—
141C2×#41—2932no—
142C2×#41—5864no—
143C2×#41—5862no—
144C2×#41C2×D29311yescharacteristic
145C22×Q8—2931no—
146C4⋊Q8—5862no—
147C4⋊Q8—5861no—
148#124—5862no—
149#124—5861no—
150#123—2934no—
151#123—2938no—
152C22⋊Q8—5864no—
153C22⋊Q8—5862no—
154#117—2931no—
155#117C2×D29312yescomplemented
156#99—2938no—
157#99C2×D29314yescomplemented
158#41—22no—
159#41—22no—
160#41—5864no—
161#41—5864no—
162#41—5868no—
163#41—5864no—
164#41—5864no—
165#41—5862no—
166#41—5862no—
167#41—5862no—
168#41—5862no—
169#41—5864no—
170C2×Q8—2932no—
171C2×Q8—5862no—
172C2×Q8—5864no—
173C2×Q8—5862no—
174C2×Q8—5862no—
175C2×Q8—5861no—
176C2×Q8—5861no—
177C2×Q8D117211yescharacteristic
178C2×D4—2934no—
179C2×D4—2932no—
180C2×D4—5862no—
181C2×D4—5868no—
182C2×D4—5864no—
183C2×D4—5864no—
184C2×D4D117211yescharacteristic
185C2×D4C22×D29312yes—
186C22×C4—2934no—
187C22×C4—2932no—
188C22×C4—5864no—
189C22×C4D117212yes—
190C22×C4C22×D29311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—5868no—
193C4⋊C4—5861no—
194C4⋊C4—5864no—
195C22⋊C4—2932no—
196C22⋊C4—5868no—
197C22⋊C4—5864no—
198C22⋊C4C22×D29314yes—
199C42—22no—
200C42—5861no—
201C23—2934no—
202C23—5864no—
203C23C2×D117212yes—
204C23C23×D29311yescharacteristic
205Q8—22no—
206Q8—5864no—
207Q8—5862no—
208Q8—5862no—
209Q8—11721no—
210Q8—11721no—
211Q8—11722no—
212D4—24no—
213D4—22no—
214D4—5864no—
215D4—5868no—
216D4—5864no—
217D4—5864no—
218D4—5862no—
219D4—5862no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—2934no—
224C2×C4—5868no—
225C2×C4—5864no—
226C2×C4—5864no—
227C2×C4—5864no—
228C2×C4—5864no—
229C2×C4—5862no—
230C2×C4—5862no—
231C2×C4D4×D29312yes—
232C2×C4D4×D29312yes—
233C2×C4C2×D117212yes—
234C2×C4C2×D117211yescharacteristic
235C2×C4C2×D117211yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—23444no—
240C22—5864no—
241C22—11724no—
242C22C2×D4×D29312yes—
243C22C22×D117211yescharacteristic
244C4—22no—
245C4—44no—
246C4—5864no—
247C4—11722no—
248C4—11722no—
249C4#2262759411yescharacteristic
250C4#2262759411yescharacteristic
251C4#2262759412yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—11724no—
256C2C2×#2262759411yescharacteristic
subgroupG/H
centerC2C2×#22627594
commutator, cosocleC23×C293C24
FrattiniC23C23×D293
Fitting#410×C293C2
socleC2×C293C2×#120
prime radical#34036965C2
hypercenter#410D293
hypocenterC293#3028
LCSG ▷ C23×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #34036705C2semidirect
3. #34036700C2semidirect
4. #34036320C2semidirect
5. #34036102C2semidirect
6. #410×C293C2semidirect
15. C22×Dic586C22semidirect
20. #22627608C22semidirect
21. #22627605C22semidirect
24. #22627583C22semidirect
25. #22627538C22semidirect
27. #117×C293C22semidirect
28. #99×C293C22semidirect
128. C293#3028semidirect
135. #410D293semidirect
155. #117C2×D293semidirect
157. #99C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369652
#340367052
#340367002
#340363202
#340361022
#410×C2932
#3028293