Finite Group Databasebeta

← #49170603

Subgroups — #49170603

Order 37504.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D1172C211yesmaximal, characteristic
2#34036477C211yesmaximal, characteristic, complemented
3#34036447C212yesmaximal, complemented
4#34036416C211yesmaximal, characteristic, complemented
5#34036360C212yesmaximal, complemented
6#34036329C211yesmaximal, characteristic, complemented
7#34036279C212yesmaximal, complemented
8C4×#17989789C211yesmaximal, characteristic, complemented
9D4×#14556873C211yesmaximal, characteristic, complemented
10C2×#22627544C212yesmaximal, complemented
11#554×C293C211yesmaximal, characteristic, complemented
12C22×D1172C2212yes—
13C2×D4×D293—21no—
14C2×#17989789C2211yescharacteristic
15C2×#17989789C2212yescomplemented
16C22×#14556873C2212yescomplemented
17#22627621C2211yescharacteristic
18#22627599—21no—
19#22627598C2212yes—
20#22627594—22no—
21C4×D1172C2211yescharacteristic
22#22627555C2212yescomplemented
23#22627555C2211yescharacteristic, complemented
24C4⋊#14556873C2211yescharacteristic, complemented
25#22627548C2211yescharacteristic, complemented
26#22627544C2212yes—
27#22627544C2211yescharacteristic
28#22627544C2212yescomplemented
29#22627544C2212yescomplemented
30#22627544C2212yescomplemented
31C22⋊#14556873C2212yescomplemented
32C4×#14556873C2211yescharacteristic, complemented
33C4⋊Q8×C293C2211yescharacteristic, complemented
34#125×C293—21no—
35C22⋊Q8×C293—22no—
36C22⋊Q8×C293C2212yescomplemented
37C4×Q8×C293C2211yescharacteristic, complemented
38C4×D4×C293—21no—
39C4×D4×C293C2211yescharacteristic
40C2×C4⋊C4×C293C2212yescomplemented
41C23×D293—22no—
42C2×D1172—24no—
43C2×D1172—22no—
44C2×D1172—21no—
45C2×D1172—21no—
46C2×D1172—42no—
47C2×D1172C2312yes—
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—22no—
50D4×D293—44no—
51C2×C4×D293—21no—
52#17989789—22no—
53#17989789—42no—
54C2×#14556873—42no—
55C2×#14556873C2311yescharacteristic
56C2×#14556873C2312yes—
57C2×#14556873C2311yescharacteristic
58C4⋊Dic293C2311yescharacteristic
59#17989779—21no—
60#17989779—22no—
61C2×Q8×C293—21no—
62C2×Q8×C293C2311yescharacteristic
63C2×D4×C293D411yescharacteristic
64C22×C4×C293—22no—
65C22×C4×C293C2312yes—
66C4⋊C4×C293—22no—
67C4⋊C4×C293—21no—
68C4⋊C4×C293—21no—
69C4⋊C4×C293—22no—
70C4⋊C4×C293D411yescharacteristic
71C4⋊C4×C293C2311yescharacteristic
72C4⋊C4×C293C2312yes—
73C22⋊C4×C293—22no—
74C22⋊C4×C293D412yes—
75C42×C293—21no—
76C42×C293C2311yescharacteristic
77C22×D293—21no—
78C22×D293—22no—
79C22×D293—44no—
80C22×D293—44no—
81C22×D293—82no—
82D1172—22no—
83D1172—21no—
84D1172—22no—
85D1172—44no—
86D1172—42no—
87D1172—81no—
88D1172#4112yes—
89C22⋊D293—44no—
90C2×Dic293—21no—
91C4×D293—42no—
92#14556873—21no—
93#14556873—21no—
94#14556873—22no—
95C23×C293C2×D412yes—
96Q8×C293—21no—
97Q8×C293—21no—
98Q8×C293—81no—
99D4×C293SD1614yescomplemented
100C2×C4×C293—22no—
101C2×C4×C293—22no—
102C2×C4×C293—21no—
103C2×C4×C293—21no—
104C2×C4×C293—22no—
105C2×C4×C293—82no—
106C2×C4×C293C2×D411yescharacteristic
107C2×C4×C293C2×D412yes—
108C2×C4×C293C2×D411yescharacteristic
109C2×C4×C293C2411yescharacteristic
110C22×C293—22no—
111C22×C293—22no—
112C22×C293C2×SD1614yes—
113C22×C293C22×D411yescharacteristic
114C2×D293—22no—
115C2×D293—41no—
116C2×D293—41no—
117C2×D293—42no—
118C2×D293—42no—
119C2×D293—42no—
120C2×D293—88no—
121C4×C293—22no—
122C4×C293—21no—
123C4×C293—41no—
124C4×C293—41no—
125C4×C293—42no—
126C4×C293C2×SD1612yes—
127C4×C293C2×#4111yescharacteristic
128C4×C293H(5,𝔽2)11yescharacteristic
129Dic293—41no—
130C2×C293—24no—
131C2×C293#54611yescharacteristic
132C2×C293C22×SD1611yescharacteristic
133C2×C293#57611yescharacteristic
134D293—42no—
135D293—82no—
136C293#329511yescharacteristic, complemented, Sylow
137#3295—2931nomaximal, Sylow
138#554D29311yescharacteristic, complemented
139D42—2931no—
140#486—2931no—
141#481—2932no—
142#476—2931no—
143#465—2932no—
144#460—2931no—
145#450—2932no—
146C4×SD16—2931no—
147C8×D4—2931no—
148C2×#102—2932no—
149C22×D4—2932no—
150C22×D4—5861no—
151C2×SD16—2931no—
152C2×SD16—2932no—
153C22×C8—2932no—
154C4⋊Q8C2×D29311yescharacteristic, complemented
155#127—2931no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D29312yescomplemented
159#121—2932no—
160#121—5861no—
161#120—5862no—
162C4×Q8C2×D29311yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—2931no—
165C4×D4C2×D29311yescharacteristic
166C2×C4⋊C4C2×D29312yescomplemented
167#106—2931no—
168#106—2932no—
169C4⋊C8—2931no—
170#103—2931no—
171#102—2931no—
172#102—2932no—
173#102—2932no—
174#102—2932no—
175#102—2932no—
176C22⋊C8—2932no—
177C4×C8—2931no—
178C24—5862no—
179C2×Q8—21no—
180C2×Q8C22×D29311yescharacteristic
181C2×D4—2931no—
182C2×D4—2932no—
183C2×D4—5864no—
184C2×D4—5862no—
185C2×D4—5861no—
186C2×D4—5862no—
187C2×D4—5861no—
188C2×D4—11722no—
189C2×D4—11724no—
190C2×D4C22⋊D29311yescharacteristic
191C22×C4—22no—
192C22×C4—5861no—
193C22×C4C22×D29312yes—
194SD16—5862no—
195SD16—11722no—
196C2×C8—2931no—
197C2×C8—2931no—
198C2×C8—2932no—
199C2×C8—11722no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—2931no—
205C4⋊C4C22⋊D29311yescharacteristic
206C4⋊C4C22×D29311yescharacteristic
207C4⋊C4C22×D29312yes—
208C22⋊C4—22no—
209C22⋊C4—5861no—
210C22⋊C4—5862no—
211C22⋊C4C22⋊D29312yes—
212C42—21no—
213C42C22×D29311yescharacteristic
214C23—23442no—
215C23—5861no—
216C23—5862no—
217C23—11724no—
218C23—11724no—
219C23C2×C22⋊D29312yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—2932no—
224D4—23441no—
225D4—5862no—
226D4—5861no—
227D4—5862no—
228D4—11724no—
229D4—11724no—
230D4—11722no—
231D4#1798978914yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—5861no—
239C2×C4—11722no—
240C2×C4C2×C22⋊D29311yescharacteristic
241C2×C4C2×C22⋊D29312yes—
242C2×C4C2×C22⋊D29311yescharacteristic
243C2×C4C23×D29311yescharacteristic
244C8—5861no—
245C8—5861no—
246C8—5862no—
247C22—22no—
248C22—22no—
249C22—23448no—
250C22—5862no—
251C22—11722no—
252C22—11722no—
253C22—11722no—
254C22—11721no—
255C22—11721no—
256C22C2×#1798978914yes—
257C22C22×C22⋊D29311yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—11721no—
264C4C2×#1798978912yes—
265C4#41×D29311yescharacteristic
266C4#2262767011yescharacteristic
267C2—24no—
268C2—23442no—
269C2—11722no—
270C2#3403676711yescharacteristic
271C2C22×#1798978911yescharacteristic
272C2#3403693211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#554×C293C2
socleC22×C293C22×D4
prime radicalD4×D1172C2
hypercenter#554D293
hypocenterC293#3295
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #34036477C2semidirect
3. #34036447C2semidirect
4. #34036416C2semidirect
5. #34036360C2semidirect
6. #34036329C2semidirect
7. #34036279C2semidirect
8. C4×#17989789C2semidirect
9. D4×#14556873C2semidirect
10. C2×#22627544C2semidirect
11. #554×C293C2semidirect
15. C2×#17989789C22semidirect
16. C22×#14556873C22semidirect
22. #22627555C22semidirect
23. #22627555C22semidirect
24. C4⋊#14556873C22semidirect
25. #22627548C22semidirect
28. #22627544C22semidirect
29. #22627544C22semidirect
30. #22627544C22semidirect
31. C22⋊#14556873C22semidirect
32. C4×#14556873C22semidirect
33. C4⋊Q8×C293C22semidirect
36. C22⋊Q8×C293C22semidirect
37. C4×Q8×C293C22semidirect
40. C2×C4⋊C4×C293C22semidirect
99. D4×C293SD16semidirect
136. C293#3295semidirect
138. #554D293semidirect
154. C4⋊Q8C2×D293semidirect
158. C22⋊Q8C2×D293semidirect
162. C4×Q8C2×D293semidirect
166. C2×C4⋊C4C2×D293semidirect
231. D4#17989789semidirect

Maximal subgroups

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

subgroupindex
D4×D11722
#340364772
#340364472
#340364162
#340363602
#340363292
#340362792
C4×#179897892
D4×#145568732
C2×#226275442
#554×C2932
#3295293