Finite Group Databasebeta

← #34060951

Subgroups — #34060951

Order 19072.

266 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 52 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#23750612C211yesmaximal, characteristic, complemented
2#23750605C211yesmaximal, characteristic, complemented
3#23750526C212yesmaximal, complemented
4#23750525C212yesmaximal, complemented
5#23750190C211yesmaximal, characteristic
6C2×#14558745—21no—
7#18011990—21no—
8#18011972—21no—
9#18011971C2212yes—
10#18011969—21no—
11#18011969C2212yes—
12C2×#14558735C2211yescharacteristic
13C22×C22⋊D149C411yescharacteristic, complemented
14#18011907—21no—
15#18011897—21no—
16#18011884—21no—
17#18011883—21no—
18#18011881C2211yescharacteristic
19#18011880C2211yescharacteristic
20#18011879C411yescharacteristic, complemented
21#18011867C411yescharacteristic, complemented
22#120×C149C411yescharacteristic, complemented
23#14558745—22no—
24#14558745—41no—
25C2×#11924753—41no—
26C4⋊#10597010—41no—
27C22⋊#10597010—22no—
28C4×#10597010—41no—
29C23×D149C2×C411yescharacteristic
30#14558735—41no—
31#14558735—41no—
32C2×Dic298D411yescharacteristic
33C2×D596—21no—
34C2×C22⋊D149—21no—
35C2×C22⋊D149—21no—
36C2×C22⋊D149—21no—
37C2×C22⋊D149—21no—
38C2×C22⋊D149—41no—
39C2×C22⋊D149—41no—
40C2×C22⋊D149D411yescharacteristic
41C2×C22⋊D149C2311yescharacteristic
42C22×Dic149C2×C411yescharacteristic
43C22×Dic149D411yescharacteristic
44C2×C4×D149D411yescharacteristic
45#14558715—21no—
46C22⋊Dic149—21no—
47C22⋊Dic149C2×C411yescharacteristic
48#14558713—21no—
49#14558713—21no—
50C4×Dic149—21no—
51C24×C149C2×C411yescharacteristic
52C2×D4×C149—21no—
53C2×D4×C149C2×C411yescharacteristic
54C22⋊C4×C149—21no—
55C22⋊C4×C149C2×C411yescharacteristic
56C2×#10597010—42no—
57#11924753—42no—
58C22×D149—21no—
59C22×D149—41no—
60C22×D149—81no—
61C22×D149C2×D411yescharacteristic
62Dic298—41no—
63D596—81no—
64C22⋊D149—41no—
65C22⋊D149—41no—
66C22⋊D149—42no—
67C22⋊D149—41no—
68C22⋊D149—41no—
69C22⋊D149—81no—
70C22⋊D149—81no—
71C2×Dic149—21no—
72C2×Dic149—21no—
73C2×Dic149—21no—
74C2×Dic149—21no—
75C2×Dic149—21no—
76C2×Dic149C22⋊C411yescharacteristic
77C2×Dic149C22⋊C411yescharacteristic
78C2×Dic149C2×D411yescharacteristic
79C4×D149—21no—
80C4×D149—21no—
81C23×C149—21no—
82C23×C149—41no—
83C23×C149—81no—
84C23×C149C22⋊C411yescharacteristic
85C23×C149C22×C411yescharacteristic
86D4×C149—41no—
87D4×C149—81no—
88C2×C4×C149—21no—
89C2×C4×C149C22⋊C411yescharacteristic
90C22×C149—21no—
91C22×C149—21no—
92C22×C149—41no—
93C22×C149—41no—
94C22×C149—41no—
95C22×C149—41no—
96C22×C149—41no—
97C22×C149—82no—
98C22×C149C2×C22⋊C411yescharacteristic
99C2×D149—21no—
100C2×D149—41no—
101C2×D149—41no—
102C2×D149—41no—
103C2×D149—41no—
104C2×D149—82no—
105#10597010—82no—
106C4×C149—21no—
107C4×C149—41no—
108Dic149—21no—
109Dic149—21no—
110Dic149—41no—
111Dic149#9911yescharacteristic
112Dic149#9911yescharacteristic
113C2×C149—21no—
114C2×C149—41no—
115C2×C149—41no—
116C2×C149—81no—
117C2×C149C2×#9911yescharacteristic
118D149—41no—
119D149—81no—
120C149#212211yescharacteristic, complemented, Sylow
121#2122—1491nomaximal, Sylow
122#535—1491no—
123#413—1491no—
124#410—1491no—
125AΣL(1,𝔽16)—1492no—
126AΣL(1,𝔽16)—1492no—
127C2×#41—1491no—
128C22×D4—1491no—
129C2×M16—2981no—
130#127—2981no—
131#124—2981no—
132#121—2981no—
133#121—2981no—
134#120—1491no—
135#120—1491no—
136#120—1491no—
137#120#1059701011yescharacteristic, complemented
138#117—2981no—
139#117—1491no—
140#101—2981no—
141#100—1492no—
142#99—2981no—
143#99—1492no—
144C24—1491no—
145C24C2×#1059701011yescharacteristic
146#41—5961no—
147#41—5961no—
148C2×Q8—1491no—
149C2×D4—21no—
150C2×D4—2981no—
151C2×D4—2981no—
152C2×D4—2981no—
153C2×D4—2981no—
154C2×D4—2981no—
155C2×D4—5961no—
156C2×D4—5961no—
157C2×D4—1491no—
158C2×D4—1491no—
159C2×D4C2×#1059701011yescharacteristic
160C22×C4—1491no—
161C22×C4—1491no—
162C22×C4—1491no—
163M16—2982no—
164M16—5961no—
165C2×C8—5961no—
166C4⋊C4—2981no—
167C4⋊C4—5961no—
168C22⋊C4—21no—
169C22⋊C4—2981no—
170C22⋊C4—2981no—
171C22⋊C4—2981no—
172C22⋊C4—2982no—
173C22⋊C4—1491no—
174C22⋊C4C2×#1059701011yescharacteristic
175C42—2981no—
176C42—5961no—
177C23—21no—
178C23—41no—
179C23—81no—
180C23—2981no—
181C23—5961no—
182C23—1491no—
183C23—11921no—
184C23C22⋊#1059701011yescharacteristic
185C23C22×#1059701011yescharacteristic
186Q8—5961no—
187D4—41no—
188D4—81no—
189D4—5961no—
190D4—5961no—
191D4—5962no—
192D4—5961no—
193D4—5961no—
194D4—11921no—
195D4—11921no—
196D4—11921no—
197C2×C4—21no—
198C2×C4—2981no—
199C2×C4—2981no—
200C2×C4—2981no—
201C2×C4—2981no—
202C2×C4—2981no—
203C2×C4—2981no—
204C2×C4—2981no—
205C2×C4—5962no—
206C2×C4—1491no—
207C2×C4—1491no—
208C2×C4—1491no—
209C2×C4C22⋊#1059701011yescharacteristic
210C8—5962no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—82no—
219C22—2981no—
220C22—5961no—
221C22—5961no—
222C22—5961no—
223C22—5961no—
224C22—11922no—
225C22C2×C22⋊#1059701011yescharacteristic
226C4—21no—
227C4—41no—
228C4—2981no—
229C4—2981no—
230C4—5961no—
231C4—1491no—
232C4—1491no—
233C4—11922no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2—81no—
238C2—5961no—
239C2—11921no—
240C2D149⋊#9911yescharacteristic
subgroupG/H
centerC2D149⋊#99
commutatorC23×C149C22×C4
FrattiniC23C22×#10597010
Fitting#120×C149C4
socleC2×C149C2×#99
prime radical#23750190C2
cosocleC2×C22⋊D149C23
hypercenter#120#10597010
hypocenterC149#2122
LCSG ▷ C23×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #23750612C2semidirect
2. #23750605C2semidirect
3. #23750526C2semidirect
4. #23750525C2semidirect
13. C22×C22⋊D149C4semidirect
20. #18011879C4semidirect
21. #18011867C4semidirect
22. #120×C149C4semidirect
120. C149#2122semidirect
137. #120#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506122
#237506052
#237505262
#237505252
#237501902
#2122149