Finite Group Databasebeta

← #34061568

Subgroups — #34061568

Order 19072.

402 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 144 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#23750660C212yesmaximal, complemented
2#23750658C212yesmaximal, complemented
3#23750656C212yesmaximal, complemented
4#23750654C212yesmaximal, complemented
5#23750650C212yesmaximal, complemented
6#23750598C212yesmaximal, complemented
7#23750597C211yesmaximal, characteristic, complemented
8#23750596C211yesmaximal, characteristic, complemented
9#23750135C211yesmaximal, characteristic
10C2×#14558746C2212yes—
11C2×#14558745C2212yes—
12C2×#14558745C2212yescomplemented
13C22×#11924753C2212yescomplemented
14#18011978C2212yescomplemented
15C4⋊#11924753C2212yescomplemented
16#18011968C2211yescharacteristic
17#18011968C2212yes—
18#18011968C2212yes—
19#18011968C2211yescharacteristic
20C22⋊#11924753C2211yescharacteristic
21C22⋊#11924753C2211yescharacteristic
22C22⋊#11924753C2212yescomplemented
23C22⋊#11924753C2212yescomplemented
24#18011966C2212yescomplemented
25C4×#11924753C2212yescomplemented
26C2×#14558735C2211yescharacteristic
27C22×C4×D149C2211yescharacteristic
28#18011893C412yes—
29#18011891C411yescharacteristic
30#18011890C411yescharacteristic
31#18011882C411yescharacteristic
32#18011879C412yes—
33D4×Dic149C2212yes—
34#18011871C2212yes—
35#18011869C2211yescharacteristic
36#121×C149C411yescharacteristic
37#14558746—42no—
38#14558745—42no—
39#14558745—42no—
40C2×#11924753—42no—
41C2×#11924753C2312yes—
42C2×#11924753C2312yes—
43C2×#11924753C2312yes—
44C2×#11924753C2312yes—
45C23×D149C2×C411yescharacteristic
46#14558735—42no—
47C2×Dic298C2×C411yescharacteristic
48C2×C22⋊D149C2×C412yes—
49C2×C22⋊D149C2×C412yes—
50C22×Dic149C2311yescharacteristic
51C22×Dic149C2312yes—
52C2×C4×D149—42no—
53C2×C4×D149C2×C411yescharacteristic
54C2×C4×D149C2×C411yescharacteristic
55C2×C4×D149C2311yescharacteristic
56C2×C4×D149C2311yescharacteristic
57C4⋊Dic149C2×C411yescharacteristic
58C4⋊Dic149C2×C412yes—
59#14558715C2×C412yes—
60C22⋊Dic149C2×C412yes—
61C22⋊Dic149C2×C412yes—
62#14558713C2×C412yes—
63#14558713C2×C412yes—
64C4×Dic149C2312yes—
65C2×D4×C149C2×C411yescharacteristic
66C2×D4×C149C2×C412yes—
67C22×C4×C149C2×C411yescharacteristic
68C4⋊C4×C149C2×C411yescharacteristic
69C22⋊C4×C149C2×C412yes—
70#11924753—22no—
71#11924753—22no—
72#11924753—22no—
73#11924753—22no—
74C22×D149—41no—
75C22×D149—41no—
76C22×D149—41no—
77C22×D149C22×C411yescharacteristic
78C22×D149C22×C411yescharacteristic
79Dic298—41no—
80C22⋊D149—42no—
81C22⋊D149—42no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149C22×C411yescharacteristic
86C2×Dic149C22×C412yes—
87C2×Dic149C22×C412yes—
88C2×Dic149C2411yescharacteristic
89C4×D149—22no—
90C4×D149—22no—
91C4×D149—41no—
92C4×D149—41no—
93C23×C149C22×C412yes—
94C23×C149C22×C411yescharacteristic
95D4×C149—41no—
96D4×C149—42no—
97C2×C4×C149—41no—
98C2×C4×C149C22×C411yescharacteristic
99C2×C4×C149C22×C411yescharacteristic
100C2×C4×C149C22×C412yes—
101C22×C149—21no—
102C22×C149—22no—
103C22×C149—21no—
104C22×C149—22no—
105C22×C149—22no—
106C22×C149—21no—
107C22×C149C23×C411yescharacteristic
108C2×D149—21no—
109C2×D149—21no—
110C2×D149—21no—
111C2×D149—21no—
112C2×D149—21no—
113C2×D149—21no—
114C2×D149—44no—
115C4×C149—21no—
116C4×C149—21no—
117C4×C149—22no—
118Dic149—21no—
119Dic149—22no—
120Dic149—22no—
121Dic149H(5,𝔽2)12yes—
122C2×C149—41no—
123C2×C149—42no—
124C2×C149#51811yescharacteristic
125C2×C149#56811yescharacteristic
126C2×C149#56811yescharacteristic
127D149—41no—
128D149—41no—
129C149#297811yescharacteristic, complemented, Sylow
130#2978—1491nomaximal, Sylow
131#525—1491no—
132#436—1492no—
133#436—1492no—
134#435—1492no—
135#435—1492no—
136#433—1492no—
137#408—1492no—
138#407—1491no—
139#407—1491no—
140C2×#41—1491no—
141C23×C4—1491no—
142C2×M16—1492no—
143C2×M16—1492no—
144C2×M16—1492no—
145C22×C8—1492no—
146#123—1492no—
147C22⋊Q8—1491no—
148C22⋊Q8—1491no—
149#121—1491no—
150#121#1059701011yescharacteristic
151#120—1492no—
152C4×D4—1492no—
153C4×D4—1492no—
154#117—1491no—
155C4⋊C8—1492no—
156C4⋊C8—1492no—
157C22⋊C8—1491no—
158C22⋊C8—1491no—
159C22⋊C8—1492no—
160C22⋊C8—1492no—
161C22⋊C8—1492no—
162C22⋊C8—1492no—
163C22⋊C8—1491no—
164C22⋊C8—1491no—
165#97—1492no—
166C4×C8—1492no—
167C24—1491no—
168#41—5962no—
169C2×Q8—1491no—
170C2×D4—1492no—
171C2×D4—1492no—
172C2×D4C2×#1059701012yes—
173C2×D4C2×#1059701011yescharacteristic
174C22×C4—5962no—
175C22×C4—1492no—
176C22×C4—1491no—
177C22×C4—1491no—
178C22×C4—1491no—
179C22×C4—1491no—
180C22×C4—1491no—
181C22×C4C2×#1059701011yescharacteristic
182M16—5962no—
183M16—5962no—
184M16—5962no—
185C2×C8—5962no—
186C2×C8—1492no—
187C2×C8—1492no—
188C2×C8—1492no—
189C2×C8—1492no—
190C4⋊C4—1491no—
191C4⋊C4—1492no—
192C4⋊C4—1492no—
193C4⋊C4C2×#1059701011yescharacteristic
194C22⋊C4—1492no—
195C22⋊C4—1492no—
196C22⋊C4—1492no—
197C22⋊C4—1492no—
198C22⋊C4C2×#1059701012yes—
199C42—1492no—
200C23—5961no—
201C23—5961no—
202C23—5961no—
203C23—1491no—
204C23—1491no—
205C23C22×#1059701011yescharacteristic
206C23C22×#1059701012yes—
207Q8—5961no—
208D4—42no—
209D4—41no—
210D4—5962no—
211D4—5962no—
212C2×C4—41no—
213C2×C4—2982no—
214C2×C4—2982no—
215C2×C4—2982no—
216C2×C4—2982no—
217C2×C4—2982no—
218C2×C4—5961no—
219C2×C4—5961no—
220C2×C4—1492no—
221C2×C4—1491no—
222C2×C4—1492no—
223C2×C4—1491no—
224C2×C4C22×#1059701012yes—
225C2×C4C22×#1059701011yescharacteristic
226C2×C4C22×#1059701011yescharacteristic
227C8—2982no—
228C8—2982no—
229C8—2982no—
230C8—2982no—
231C22—21no—
232C22—22no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—21no—
237C22—2981no—
238C22—2981no—
239C22—2981no—
240C22—2981no—
241C22—2981no—
242C22—2981no—
243C22—5964no—
244C22C23×#1059701011yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—2981no—
249C4—2982no—
250C4—2982no—
251C4—1492no—
252C2—41no—
253C2—42no—
254C2—5961no—
255C2—5961no—
256C2#2375069211yescharacteristic
257C2#2375070311yescharacteristic
258C2#2375070411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10597010
commutator, socleC22×C149C23×C4
Fitting#121×C149C4
prime radical#23750135C2
cosocleC2×Dic149C24
hypercenter#121#10597010
hypocenterC149#2978
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #23750660C2semidirect
2. #23750658C2semidirect
3. #23750656C2semidirect
4. #23750654C2semidirect
5. #23750650C2semidirect
6. #23750598C2semidirect
7. #23750597C2semidirect
8. #23750596C2semidirect
12. C2×#14558745C22semidirect
13. C22×#11924753C22semidirect
14. #18011978C22semidirect
15. C4⋊#11924753C22semidirect
22. C22⋊#11924753C22semidirect
23. C22⋊#11924753C22semidirect
24. #18011966C22semidirect
25. C4×#11924753C22semidirect
129. C149#2978semidirect

Maximal subgroups

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

subgroupindex
#237506602
#237506582
#237506562
#237506542
#237506502
#237505982
#237505972
#237505962
#237501352
#2978149