Finite Group Databasebeta

← #34043607

Subgroups — #34043607

Order 19072.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#23750183C211yesmaximal, characteristic, complemented
2#23749759C211yesmaximal, characteristic, complemented
3#23749533C211yesmaximal, characteristic, complemented
4#23749360C211yesmaximal, characteristic, complemented
5#23749359C211yesmaximal, characteristic, complemented
6#23749348C211yesmaximal, characteristic, complemented
7#447×C149C211yesmaximal, characteristic, complemented
8C22×C22⋊D149—21no—
9C2×D4×D149—21no—
10C2×#14558721—21no—
11#18011896—21no—
12#18011893—21no—
13#18011885C2211yescharacteristic
14#18011883—21no—
15#18011881—21no—
16#18011879—21no—
17C2×C22⋊Dic149—21no—
18C2×#14558713C2211yescharacteristic
19#18011832—21no—
20C22⋊#11924746C2211yescharacteristic, complemented
21#18011818C2211yescharacteristic, complemented
22C22×D4×C149C2211yescharacteristic, complemented
23C2×D8×C149—21no—
24#121×C149C2211yescharacteristic
25#102×C149—21no—
26C22⋊C8×C149C2211yescharacteristic, complemented
27C23×D149—21no—
28C23×D149D411yescharacteristic
29C2×D596—21no—
30C2×D596D411yescharacteristic, complemented
31C2×C22⋊D149—21no—
32C2×C22⋊D149—21no—
33C2×C22⋊D149—21no—
34C2×C22⋊D149—42no—
35C2×C22⋊D149—42no—
36C2×C22⋊D149—42no—
37D4×D149—42no—
38D4×D149—42no—
39C22×Dic149—21no—
40C22×Dic149D411yescharacteristic
41C2×C4×D149—21no—
42#14558721—42no—
43C2×#11924746—21no—
44C4⋊Dic149D411yescharacteristic, complemented
45#14558715—21no—
46C22⋊Dic149—21no—
47C22⋊Dic149—42no—
48#14558713—21no—
49#14558713—21no—
50#14558713—21no—
51#14558713—21no—
52C24×C149—21no—
53C2×D4×C149—21no—
54C2×D4×C149—21no—
55C2×D4×C149—21no—
56C2×D4×C149—42no—
57C2×D4×C149D411yescharacteristic, complemented
58C22×C4×C149C2311yescharacteristic
59D8×C149—42no—
60C2×C8×C149—21no—
61C4⋊C4×C149D411yescharacteristic, complemented
62C22⋊C4×C149—21no—
63C22×D149—21no—
64C22×D149—21no—
65C22×D149—21no—
66C22×D149—42no—
67C22×D149—41no—
68C22×D149—41no—
69C22×D149—81no—
70C22×D149—81no—
71C22×D149D812yescomplemented
72D596—41no—
73D596—42no—
74C22⋊D149—42no—
75C22⋊D149—48no—
76C22⋊D149—41no—
77C22⋊D149—41no—
78C22⋊D149—81no—
79C2×Dic149—22no—
80C2×Dic149—21no—
81C2×Dic149—21no—
82C2×Dic149—41no—
83C2×Dic149—41no—
84C2×Dic149—41no—
85C4×D149—42no—
86#11924746—41no—
87C23×C149—22no—
88C23×C149—21no—
89C23×C149—41no—
90C23×C149—41no—
91C23×C149—41no—
92C23×C149—41no—
93C23×C149—41no—
94C23×C149—41no—
95C23×C149C2×D411yescharacteristic
96D4×C149—22no—
97D4×C149—22no—
98D4×C149—41no—
99D4×C149—41no—
100D4×C149—41no—
101D4×C149—41no—
102D4×C149—41no—
103C2×C4×C149—21no—
104C2×C4×C149—41no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C8×C149—41no—
108C22×C149—21no—
109C22×C149—21no—
110C22×C149—42no—
111C22×C149—42no—
112C22×C149—41no—
113C22×C149—41no—
114C22×C149—42no—
115C22×C149—41no—
116C22×C149—81no—
117C22×C149—81no—
118C22×C149—81no—
119C22×C149—81no—
120C22×C149#12011yescharacteristic
121C22×C149C2×D811yescharacteristic
122C22×C149AGL(1,ℤ/8)11yescharacteristic
123C2×D149—22no—
124C2×D149—41no—
125C2×D149—41no—
126C2×D149—42no—
127C2×D149—41no—
128C2×D149—41no—
129C2×D149—41no—
130C2×D149—81no—
131C2×D149—84no—
132C4×C149—21no—
133C4×C149—21no—
134C4×C149—41no—
135Dic149—42no—
136Dic149—41no—
137Dic149—41no—
138C2×C149—21no—
139C2×C149—21no—
140C2×C149—81no—
141C2×C149—82no—
142C2×C149#44711yescharacteristic
143C2×C149#45311yescharacteristic
144C2×C149B(4,𝔽2)11yescharacteristic
145D149—42no—
146D149—81no—
147C149#159511yescharacteristic, complemented, Sylow
148#1595—1491nomaximal, Sylow
149#534—1491no—
150#447—1491no—
151#447D14911yescharacteristic, complemented
152#392—1491no—
153#327—1491no—
154#327—1491no—
155#323—1491no—
156C22×D4—2981no—
157C22×D4—2981no—
158C22×D4C2×D14911yescharacteristic, complemented
159C2×D8—21no—
160C2×D8—2981no—
161#123—2981no—
162#123—2981no—
163#121—2981no—
164#121—1491no—
165#121C2×D14911yescharacteristic
166#120—2981no—
167#120—2981no—
168C2×C22⋊C4—2981no—
169C2×C22⋊C4—1491no—
170#102—21no—
171#102—2981no—
172C22⋊C8—1491no—
173C22⋊C8C2×D14911yescharacteristic, complemented
174#95—1491no—
175C24—21no—
176C24—2981no—
177C24—1491no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—2981no—
183C2×D4—2981no—
184C2×D4—2981no—
185C2×D4—2981no—
186C2×D4—5962no—
187C2×D4—5962no—
188C2×D4—5962no—
189C2×D4—5962no—
190C2×D4—5962no—
191C2×D4—1491no—
192C2×D4C22⋊D14911yescharacteristic, complemented
193C22×C4—2981no—
194C22×C4—2981no—
195C22×C4—1491no—
196C22×C4C22×D14911yescharacteristic
197D8—42no—
198D8—5962no—
199C2×C8—21no—
200C2×C8—2981no—
201C4⋊C4—2981no—
202C4⋊C4—1491no—
203C4⋊C4C22⋊D14911yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—2981no—
206C22⋊C4—2981no—
207C22⋊C4—2981no—
208C22⋊C4—2981no—
209C22⋊C4—2981no—
210C22⋊C4—5962no—
211C23—22no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—2981no—
220C23—2981no—
221C23—2981no—
222C23—5961no—
223C23—5961no—
224C23—5962no—
225C23—1492no—
226C23—11921no—
227C23—11921no—
228C23D4×D14911yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—5961no—
237D4—5962no—
238D4—5962no—
239D4—5968no—
240D4—5961no—
241D4—5961no—
242D4—11921no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—2982no—
246C2×C4—2981no—
247C2×C4—2981no—
248C2×C4—5962no—
249C2×C4—5961no—
250C2×C4—5961no—
251C2×C4—5961no—
252C2×C4D4×D14911yescharacteristic
253C2×C4C2×C22⋊D14911yescharacteristic
254C8—41no—
255C8—5961no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—2982no—
269C22—5962no—
270C22—5961no—
271C22—5961no—
272C22—5961no—
273C22—5961no—
274C22—5961no—
275C22—11921no—
276C22—11924no—
277C22#1801187911yescharacteristic
278C22D8×D14911yescharacteristic
279C22#1801193511yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—5962no—
284C4—5961no—
285C4—5961no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—81no—
290C2—5962no—
291C2—11921no—
292C2#2374975711yescharacteristic
293C2#2374979111yescharacteristic
294C2#2374981111yescharacteristic
subgroupG/H
centerC22#18011879
commutator, cosocleC22×C4×C149C23
FrattiniC22×C4C22×D149
Fitting#447×C149C2
socleC22×C149#120
hypercenter#447D149
hypocenterC149#1595
LCSG ▷ C22×C4×C149 ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23750183C2semidirect
2. #23749759C2semidirect
3. #23749533C2semidirect
4. #23749360C2semidirect
5. #23749359C2semidirect
6. #23749348C2semidirect
7. #447×C149C2semidirect
20. C22⋊#11924746C22semidirect
21. #18011818C22semidirect
22. C22×D4×C149C22semidirect
26. C22⋊C8×C149C22semidirect
30. C2×D596D4semidirect
44. C4⋊Dic149D4semidirect
57. C2×D4×C149D4semidirect
61. C4⋊C4×C149D4semidirect
71. C22×D149D8semidirect
147. C149#1595semidirect
151. #447D149semidirect
158. C22×D4C2×D149semidirect
173. C22⋊C8C2×D149semidirect
192. C2×D4C22⋊D149semidirect
203. C4⋊C4C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
#237501832
#237497592
#237495332
#237493602
#237493592
#237493482
#447×C1492
#1595149