Finite Group Databasebeta

← #34060945

Subgroups — #34060945

Order 19072.

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

normal subgroups only

This group has 264 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#23750527C212yesmaximal, complemented
4#23750526C212yesmaximal, complemented
5#23750133C211yesmaximal, characteristic
6C2×#14558745—21no—
7#18011990—21no—
8#18011972—21no—
9#18011971C2212yes—
10#18011969—21no—
11#18011969C2212yes—
12C2×#14558735C2211yescharacteristic
13C22×C4×D149C411yescharacteristic, complemented
14#18011894C2211yescharacteristic
15#18011889—21no—
16#18011884—21no—
17#18011881C2211yescharacteristic
18#18011879C411yescharacteristic, complemented
19#18011873—21no—
20#18011871—21no—
21#18011867C411yescharacteristic, complemented
22#123×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⋊D149D411yescharacteristic
36C2×C22⋊D149C2311yescharacteristic
37C22×Dic149C2×C411yescharacteristic
38C22×Dic149D411yescharacteristic
39C2×C4×D149—21no—
40C2×C4×D149—21no—
41C2×C4×D149—41no—
42C2×C4×D149—41no—
43C2×C4×D149D411yescharacteristic
44#14558715—21no—
45#14558715—21no—
46C22⋊Dic149C2×C411yescharacteristic
47#14558713—21no—
48#14558713—21no—
49#14558713—21no—
50C4×Dic149—21no—
51C2×D4×C149C2×C411yescharacteristic
52C22×C4×C149C2×C411yescharacteristic
53C4⋊C4×C149—21no—
54C22⋊C4×C149—21no—
55C22⋊C4×C149C2×C411yescharacteristic
56C2×#10597010—42no—
57#11924753—42no—
58C22×D149—21no—
59C22×D149—42no—
60C22×D149—41no—
61C22×D149C2×D411yescharacteristic
62Dic298—41no—
63D596—42no—
64C22⋊D149—41no—
65C22⋊D149—41no—
66C22⋊D149—42no—
67C2×Dic149—21no—
68C2×Dic149—21no—
69C2×Dic149—21no—
70C2×Dic149—21no—
71C2×Dic149—21no—
72C2×Dic149C22⋊C411yescharacteristic
73C2×Dic149C22⋊C411yescharacteristic
74C2×Dic149C2×D411yescharacteristic
75C4×D149—21no—
76C4×D149—21no—
77C4×D149—42no—
78C4×D149—41no—
79C4×D149—41no—
80C23×C149C22⋊C411yescharacteristic
81C23×C149C22×C411yescharacteristic
82D4×C149—41no—
83C2×C4×C149—21no—
84C2×C4×C149—21no—
85C2×C4×C149—41no—
86C2×C4×C149C22⋊C411yescharacteristic
87C22×C149—21no—
88C22×C149—21no—
89C22×C149—41no—
90C22×C149—41no—
91C22×C149C2×C22⋊C411yescharacteristic
92C2×D149—22no—
93C2×D149—21no—
94C2×D149—41no—
95C2×D149—42no—
96C2×D149—41no—
97C2×D149—41no—
98C2×D149—42no—
99#10597010—82no—
100C4×C149—21no—
101C4×C149—41no—
102C4×C149—41no—
103Dic149—21no—
104Dic149—21no—
105Dic149—41no—
106Dic149#9911yescharacteristic
107Dic149#9911yescharacteristic
108C2×C149—21no—
109C2×C149—41no—
110C2×C149—41no—
111C2×C149C2×#9911yescharacteristic
112D149—41no—
113D149—42no—
114C149#212011yescharacteristic, complemented, Sylow
115#2120—1491nomaximal, Sylow
116#525—1491no—
117#413—1491no—
118#410—1491no—
119#352—1492no—
120AΣL(1,𝔽16)—1492no—
121C2×#41—1491no—
122C23×C4—1491no—
123C2×M16—2981no—
124#123—1491no—
125#123#1059701011yescharacteristic, complemented
126C22⋊Q8—2981no—
127#121—2981no—
128#120—1491no—
129#120—1491no—
130C4×D4—2981no—
131C4×D4—2981no—
132#117—2981no—
133#117—1491no—
134#101—2981no—
135#100—1492no—
136#99—2981no—
137#99—1492no—
138C24—1491no—
139#41—5961no—
140#41—5961no—
141C2×Q8—1491no—
142C2×D4—2981no—
143C2×D4—2981no—
144C2×D4—1491no—
145C2×D4—1491no—
146C2×D4C2×#1059701011yescharacteristic
147C22×C4—2981no—
148C22×C4—2981no—
149C22×C4—5961no—
150C22×C4—5961no—
151C22×C4—1491no—
152C22×C4—1491no—
153C22×C4—1491no—
154C22×C4C2×#1059701011yescharacteristic
155M16—2982no—
156M16—5961no—
157C2×C8—5961no—
158C4⋊C4—21no—
159C4⋊C4—2981no—
160C4⋊C4—2981no—
161C4⋊C4—5961no—
162C22⋊C4—21no—
163C22⋊C4—2981no—
164C22⋊C4—2981no—
165C22⋊C4—2981no—
166C22⋊C4—2982no—
167C22⋊C4—1491no—
168C22⋊C4C2×#1059701011yescharacteristic
169C42—2981no—
170C42—5961no—
171C23—2981no—
172C23—5961no—
173C23—5962no—
174C23—1491no—
175C23C22⋊#1059701011yescharacteristic
176C23C22×#1059701011yescharacteristic
177Q8—5961no—
178D4—41no—
179D4—5962no—
180D4—5961no—
181D4—5961no—
182D4—5962no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—2981no—
187C2×C4—2981no—
188C2×C4—2981no—
189C2×C4—2981no—
190C2×C4—2981no—
191C2×C4—2981no—
192C2×C4—2981no—
193C2×C4—5962no—
194C2×C4—5961no—
195C2×C4—5961no—
196C2×C4—5962no—
197C2×C4—1491no—
198C2×C4—1491no—
199C2×C4—1491no—
200C2×C4C22⋊#1059701011yescharacteristic
201C8—5962no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—2982no—
207C22—2981no—
208C22—5961no—
209C22—5961no—
210C22—5962no—
211C22—5962no—
212C22—5961no—
213C22C2×C22⋊#1059701011yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—2981no—
218C4—2981no—
219C4—5961no—
220C4—1491no—
221C4—1491no—
222C4—11922no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—5961no—
227C2—5962no—
228C2D149⋊#9911yescharacteristic
subgroupG/H
centerC2D149⋊#99
commutatorC23×C149C22×C4
FrattiniC23C22×#10597010
Fitting#123×C149C4
socleC2×C149C2×#99
prime radical#23750133C2
cosocleC2×C22⋊D149C23
hypercenter#123#10597010
hypocenterC149#2120
LCSG ▷ C23×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #23750612C2semidirect
2. #23750605C2semidirect
3. #23750527C2semidirect
4. #23750526C2semidirect
13. C22×C4×D149C4semidirect
18. #18011879C4semidirect
21. #18011867C4semidirect
22. #123×C149C4semidirect
114. C149#2120semidirect
125. #123#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506122
#237506052
#237505272
#237505262
#237501332
#2120149