Finite Group Databasebeta

← #49174834

Subgroups — #49174834

Order 37504.

380 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 66 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#34037196C211yesmaximal, characteristic, complemented
2D293⋊#99C211yesmaximal, characteristic, complemented
3C2×#22627682C212yesmaximal, complemented
4#34037006C212yesmaximal, complemented
5#34036689C211yesmaximal, characteristic
6D4×#11924078—24no—
7#22627703—21no—
8C2×C22⋊#11924078—21no—
9C2×C22⋊#11924078C2212yes—
10C2×C22⋊#11924078C2212yes—
11#22627683—21no—
12#22627682—24no—
13#22627682—21no—
14C22×C22⋊D293C2211yescharacteristic
15C2×D4×D293C2211yescharacteristic
16#22627608C411yescharacteristic, complemented
17#22627606—21no—
18#22627605C411yescharacteristic, complemented
19#22627597—21no—
20#22627595—21no—
21#22627593C411yescharacteristic, complemented
22#22627592—21no—
23C22⋊C4×D293C2211yescharacteristic
24#120×C293C411yescharacteristic, complemented
25C22×#11924078—22no—
26C22×#11924078—22no—
27C22×#11924078—22no—
28C4⋊#11924078—22no—
29C22⋊#11924078—22no—
30C22⋊#11924078—22no—
31C22⋊#11924078—22no—
32C22⋊#11924078—22no—
33C22⋊#11924078D414yescomplemented
34C4×#11924078—22no—
35C23×D293D411yescharacteristic
36C23×D293C2311yescharacteristic
37C2×D1172D411yescharacteristic
38C2×C22⋊D293—21no—
39C2×C22⋊D293—21no—
40C2×C22⋊D293—22no—
41C2×C22⋊D293—22no—
42C2×C22⋊D293—21no—
43C2×C22⋊D293C2×C411yescharacteristic
44C2×C22⋊D293D411yescharacteristic
45D4×D293—22no—
46D4×D293—22no—
47C22×Dic293C2×C411yescharacteristic, complemented
48C2×C4×D293—21no—
49C2×C4×D293D411yescharacteristic
50C4⋊Dic293—21no—
51#17989781—21no—
52C22⋊Dic293—21no—
53C22⋊Dic293—21no—
54C22⋊Dic293C2×C411yescharacteristic, complemented
55#17989779—21no—
56#17989779—21no—
57C24×C293C2×C411yescharacteristic, complemented
58C2×D4×C293—21no—
59C2×D4×C293C2×C411yescharacteristic
60C22⋊C4×C293—21no—
61C22⋊C4×C293C2×C411yescharacteristic, complemented
62C2×#11924078—24no—
63C2×#11924078—42no—
64C2×#11924078—82no—
65C2×#11924078—82no—
66C2×#11924078—82no—
67C22×D293—21no—
68C22×D293—21no—
69C22×D293—81no—
70C22×D293—81no—
71C22×D293C22⋊C411yescharacteristic
72C22×D293C2×D412yes—
73C22×D293C2×D411yescharacteristic
74C22×D293C2×D411yescharacteristic
75C22×D293#4112yes—
76D1172—22no—
77C22⋊D293—22no—
78C22⋊D293—22no—
79C22⋊D293—41no—
80C22⋊D293—42no—
81C22⋊D293—41no—
82C22⋊D293—81no—
83C2×Dic293—21no—
84C2×Dic293—21no—
85C2×Dic293—22no—
86C2×Dic293C22⋊C411yescharacteristic
87C4×D293—21no—
88C4×D293—21no—
89C4×D293—81no—
90C23×C293—21no—
91C23×C293—22no—
92C23×C293—81no—
93C23×C293C22⋊C411yescharacteristic
94C23×C293C22×C411yescharacteristic
95D4×C293—22no—
96D4×C293—81no—
97C2×C4×C293—21no—
98C2×C4×C293C22⋊C411yescharacteristic
99C22×C293—21no—
100C22×C293—41no—
101C22×C293—41no—
102C22×C293—41no—
103C22×C293—41no—
104C22×C293—41no—
105C22×C293—82no—
106C22×C293C2×C22⋊C411yescharacteristic
107C22×C293C4×D412yes—
108C2×D293—21no—
109C2×D293—41no—
110C2×D293—41no—
111C2×D293—42no—
112C2×D293—42no—
113C2×D293—42no—
114C2×D293—41no—
115C2×D293—81no—
116C2×D293#12011yescharacteristic
117C2×D293#12112yes—
118C2×D293#12311yescharacteristic
119#11924078—44no—
120#11924078—82no—
121C4×C293—21no—
122C4×C293—41no—
123Dic293—21no—
124Dic293—41no—
125Dic293—41no—
126C2×C293—21no—
127C2×C293—22no—
128C2×C293—41no—
129C2×C293—81no—
130C2×C293#38611yescharacteristic
131D293—21no—
132D293—22no—
133D293—21no—
134D293—41no—
135C293#191311yescharacteristic, complemented, Sylow
136#1913—2931nomaximal, Sylow
137#534—2931no—
138#518—2931no—
139C2×#99—2932no—
140C2×#99—2931no—
141#342—2932no—
142C22×D4—2931no—
143C22×D4—2931no—
144#123—2931no—
145#123—2931no—
146#123—5861no—
147#121—5861no—
148#121—5861no—
149#120—2931no—
150#120—5861no—
151#120#1192407811yescharacteristic, complemented
152C4×D4—5864no—
153#117—5861no—
154C2×C22⋊C4—2932no—
155C2×C22⋊C4—2931no—
156C2×C22⋊C4—2932no—
157C2×C22⋊C4—5861no—
158#99—5864no—
159#99—5861no—
160#99—5861no—
161C24—2931no—
162C24—2931no—
163C24C2×#1192407811yescharacteristic, complemented
164C2×D4—21no—
165C2×D4—2931no—
166C2×D4—2931no—
167C2×D4—2931no—
168C2×D4—5862no—
169C2×D4—5862no—
170C2×D4—5861no—
171C2×D4—5862no—
172C2×D4—5861no—
173C2×D4—5861no—
174C2×D4—5862no—
175C2×D4C2×#1192407811yescharacteristic
176C22×C4—2931no—
177C22×C4—2931no—
178C22×C4—5861no—
179C22×C4—5862no—
180C22×C4—5862no—
181C22×C4—5862no—
182C4⋊C4—5861no—
183C4⋊C4—5862no—
184C4⋊C4—5861no—
185C22⋊C4—21no—
186C22⋊C4—2934no—
187C22⋊C4—2931no—
188C22⋊C4—5861no—
189C22⋊C4—5861no—
190C22⋊C4—5861no—
191C22⋊C4—5862no—
192C22⋊C4—5862no—
193C22⋊C4—5861no—
194C22⋊C4—5862no—
195C22⋊C4—5862no—
196C22⋊C4C2×#1192407811yescharacteristic, complemented
197C42—5862no—
198C23—21no—
199C23—22no—
200C23—81no—
201C23—2931no—
202C23—2931no—
203C23—2932no—
204C23—2932no—
205C23—2931no—
206C23—23441no—
207C23—23441no—
208C23—5861no—
209C23—5861no—
210C23C22⋊#1192407811yescharacteristic
211C23C22×#1192407811yescharacteristic
212D4—22no—
213D4—81no—
214D4—23441no—
215D4—5862no—
216D4—5862no—
217D4—5862no—
218D4—11721no—
219D4—11721no—
220D4—11722no—
221C2×C4—21no—
222C2×C4—2931no—
223C2×C4—23441no—
224C2×C4—23442no—
225C2×C4—23442no—
226C2×C4—23442no—
227C2×C4—5861no—
228C2×C4—5861no—
229C2×C4—5864no—
230C2×C4—5861no—
231C2×C4—5861no—
232C2×C4—5862no—
233C2×C4—11722no—
234C2×C4C22⋊#1192407811yescharacteristic
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—82no—
242C22—2931no—
243C22—2931no—
244C22—2932no—
245C22—23441no—
246C22—5861no—
247C22—11721no—
248C22—11722no—
249C22—11722no—
250C22—11721no—
251C22—11721no—
252C22—11722no—
253C22C2×C22⋊#1192407811yescharacteristic
254C22D4×#1192407812yes—
255C4—21no—
256C4—41no—
257C4—23442no—
258C4—5861no—
259C4—11721no—
260C4—11724no—
261C4—11721no—
262C2—21no—
263C2—22no—
264C2—41no—
265C2—81no—
266C2—5862no—
267C2—5861no—
268C2—5861no—
269C2—11721no—
270C2#3403707711yescharacteristic
subgroupG/H
centerC2#34037077
commutatorC23×C293C22×C4
FrattiniC23C22×#11924078
Fitting#120×C293C4
socleC2×C293#386
prime radical#34036689C2
cosocleC23×D293C23
hypercenter#120#11924078
hypocenterC293#1913
LCSG ▷ C23×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #34037196C2semidirect
2. D293⋊#99C2semidirect
3. C2×#22627682C2semidirect
4. #34037006C2semidirect
16. #22627608C4semidirect
18. #22627605C4semidirect
21. #22627593C4semidirect
24. #120×C293C4semidirect
33. C22⋊#11924078D4semidirect
47. C22×Dic293C2×C4semidirect
54. C22⋊Dic293C2×C4semidirect
57. C24×C293C2×C4semidirect
61. C22⋊C4×C293C2×C4semidirect
135. C293#1913semidirect
151. #120#11924078semidirect
163. C24C2×#11924078semidirect
196. C22⋊C4C2×#11924078semidirect

Maximal subgroups

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

subgroupindex
#340371962
D293⋊#992
C2×#226276822
#340370062
#340366892
#1913293