Finite Group Databasebeta

← #34060662

Subgroups — #34060662

Order 19072.

374 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 68 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#23750692C211yesmaximal, characteristic
2#23750573C211yesmaximal, characteristic
3#23750572C212yesmaximal, complemented
4#23750502C212yesmaximal, complemented
5#99×D149C211yesmaximal, characteristic
6C23×#10597010C2211yescharacteristic
7D4×#10597010—22no—
8D4×#10597010—22no—
9#18011990—21no—
10C2×C4⋊#10597010—22no—
11C2×C22⋊#10597010—21no—
12C2×C22⋊#10597010—21no—
13C2×C22⋊#10597010C2211yescharacteristic
14C2×C22⋊#10597010C2212yes—
15(C2×#10597010)⋊C4—22no—
16(C2×#10597010)⋊C4—21no—
17C2×D4×D149C2211yescharacteristic
18C22⋊C4×D149C2212yes—
19#18011826C411yescharacteristic, complemented
20#18011825C412yescomplemented
21#99×C149C411yescharacteristic, complemented
22C22×#10597010—21no—
23C22×#10597010—21no—
24C22×#10597010—21no—
25C22×#10597010—22no—
26C22×#10597010—21no—
27C22×#10597010—42no—
28C22×#10597010D412yes—
29C22×#10597010D412yes—
30C4⋊#10597010—21no—
31C4⋊#10597010—21no—
32C4⋊#10597010—44no—
33C22⋊#10597010—22no—
34C22⋊#10597010—22no—
35C22⋊#10597010—22no—
36C22⋊#10597010—22no—
37C22⋊#10597010—42no—
38C4×#10597010—22no—
39C23×D149D411yescharacteristic
40C23×D149C2311yescharacteristic
41C2×D596Q811yescharacteristic
42C2×C22⋊D149C2×C411yescharacteristic
43C2×C22⋊D149Q811yescharacteristic
44D4×D149—22no—
45D4×D149—22no—
46C2×C4×D149—22no—
47C2×C4×D149D411yescharacteristic
48C22⋊Dic149C2×C412yescomplemented
49#14558713—22no—
50C2×D4×C149C2×C411yescharacteristic
51C22⋊C4×C149C2×C412yescomplemented
52C2×#10597010—22no—
53C2×#10597010—24no—
54C2×#10597010—22no—
55C2×#10597010—22no—
56C2×#10597010—42no—
57C2×#10597010—41no—
58C2×#10597010—41no—
59C2×#10597010—42no—
60C2×#10597010—42no—
61C2×#10597010—41no—
62C2×#10597010—41no—
63C2×#10597010—41no—
64C2×#10597010—81no—
65C2×#10597010—81no—
66C22×D149—21no—
67C22×D149—21no—
68C22×D149—41no—
69C22×D149—41no—
70C22×D149—41no—
71C22×D149—41no—
72C22×D149C4⋊C411yescharacteristic, complemented
73C22×D149C2×D412yes—
74C22×D149C2×D411yescharacteristic
75C22×D149C2×Q811yescharacteristic
76C22×D149#4112yes—
77D596—22no—
78C22⋊D149—22no—
79C22⋊D149—22no—
80C2×Dic149—22no—
81C2×Dic149C4⋊C411yescharacteristic, complemented
82C4×D149—21no—
83C4×D149—21no—
84C4×D149—42no—
85C4×D149—42no—
86C23×C149C4⋊C411yescharacteristic, complemented
87C23×C149C22×C411yescharacteristic
88D4×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149C4⋊C411yescharacteristic, complemented
91C22×C149—21no—
92C22×C149—41no—
93C22×C149—41no—
94C22×C149C2×C4⋊C411yescharacteristic
95C22×C149C4×D412yes—
96C2×D149—22no—
97C2×D149—21no—
98C2×D149—22no—
99C2×D149—21no—
100C2×D149—21no—
101C2×D149—41no—
102C2×D149—42no—
103C2×D149—41no—
104C2×D149—41no—
105C2×D149—41no—
106C2×D149—81no—
107C2×D149#12011yescharacteristic
108C2×D149C22⋊Q812yes—
109C2×D149#12311yescharacteristic
110#10597010—42no—
111#10597010—42no—
112#10597010—42no—
113#10597010—81no—
114C4×C149—21no—
115C4×C149—42no—
116Dic149—21no—
117Dic149—42no—
118C2×C149—22no—
119C2×C149—21no—
120C2×C149—41no—
121C2×C149#38511yescharacteristic
122D149—21no—
123D149—22no—
124D149—41no—
125D149B(4,𝔽2)11yescharacteristic
126D149#45811yescharacteristic
127C149#186711yescharacteristic, complemented, Sylow
128#1867—1491nomaximal, Sylow
129#518—1491no—
130C2×#99—1491no—
131#386—1491no—
132#385—1492no—
133#342—1492no—
134C22×D4—1491no—
135C23×C4—1491no—
136C4×D4—2982no—
137C4×D4—2982no—
138#117—2981no—
139C2×C4⋊C4—2982no—
140C2×C22⋊C4—2981no—
141C2×C22⋊C4—2981no—
142C2×C22⋊C4—1491no—
143C2×C22⋊C4—1492no—
144C2×C22⋊C4—1492no—
145#99—1491no—
146#99—1492no—
147#99#1059701011yescharacteristic, complemented
148#95—2982no—
149#95—2981no—
150C24—1491no—
151C24—1491no—
152C2×D4—2982no—
153C2×D4—2982no—
154C2×D4—1491no—
155C2×D4—1491no—
156C2×D4—1491no—
157C2×D4C2×#1059701011yescharacteristic
158C22×C4—2982no—
159C22×C4—2981no—
160C22×C4—2982no—
161C22×C4—2981no—
162C22×C4—2981no—
163C22×C4—2981no—
164C22×C4—5962no—
165C22×C4—1491no—
166C22×C4—1492no—
167C22×C4—1492no—
168C4⋊C4—2981no—
169C4⋊C4—2981no—
170C4⋊C4—5964no—
171C22⋊C4—2982no—
172C22⋊C4—2982no—
173C22⋊C4—2982no—
174C22⋊C4—2982no—
175C22⋊C4—2982no—
176C22⋊C4—5962no—
177C22⋊C4—1492no—
178C22⋊C4C2×#1059701012yescomplemented
179C42—2982no—
180C23—2981no—
181C23—2981no—
182C23—5961no—
183C23—5961no—
184C23—5961no—
185C23—5961no—
186C23—1491no—
187C23—1492no—
188C23—1491no—
189C23—1491no—
190C23—1492no—
191C23C4⋊#1059701011yescharacteristic, complemented
192C23C22×#1059701011yescharacteristic
193D4—22no—
194D4—2982no—
195D4—2982no—
196D4—2982no—
197C2×C4—22no—
198C2×C4—2981no—
199C2×C4—2981no—
200C2×C4—2982no—
201C2×C4—2984no—
202C2×C4—2982no—
203C2×C4—2982no—
204C2×C4—2982no—
205C2×C4—5962no—
206C2×C4—5962no—
207C2×C4—5962no—
208C2×C4—5961no—
209C2×C4—5961no—
210C2×C4—5962no—
211C2×C4—5962no—
212C2×C4—5961no—
213C2×C4—5961no—
214C2×C4—5961no—
215C2×C4—1491no—
216C2×C4—11921no—
217C2×C4—11921no—
218C2×C4C4⋊#1059701011yescharacteristic, complemented
219C22—21no—
220C22—41no—
221C22—41no—
222C22—2982no—
223C22—2981no—
224C22—2982no—
225C22—2981no—
226C22—2981no—
227C22—5962no—
228C22—5961no—
229C22—5961no—
230C22—5961no—
231C22—5961no—
232C22—1491no—
233C22—1492no—
234C22—1491no—
235C22—11921no—
236C22C2×C4⋊#1059701011yescharacteristic
237C22D4×#1059701012yes—
238C4—21no—
239C4—42no—
240C4—2981no—
241C4—5962no—
242C4—5962no—
243C4—5962no—
244C4—5962no—
245C4—11921no—
246C2—22no—
247C2—21no—
248C2—41no—
249C2—2981no—
250C2—2982no—
251C2—5961no—
252C2—1491no—
253C2—1491no—
254C2#2375057211yescharacteristic
subgroupG/H
centerC2#23750572
commutatorC23×C149C22×C4
FrattiniC23C22×#10597010
Fitting#99×C149C4
socleC2×C149#385
prime radicalC2×D4×D149C22
cosocleC23×D149C23
hypercenter#99#10597010
hypocenterC149#1867
LCSG ▷ C23×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
3. #23750572C2semidirect
4. #23750502C2semidirect
19. #18011826C4semidirect
20. #18011825C4semidirect
21. #99×C149C4semidirect
48. C22⋊Dic149C2×C4semidirect
51. C22⋊C4×C149C2×C4semidirect
72. C22×D149C4⋊C4semidirect
81. C2×Dic149C4⋊C4semidirect
86. C23×C149C4⋊C4semidirect
90. C2×C4×C149C4⋊C4semidirect
127. C149#1867semidirect
147. #99#10597010semidirect
178. C22⋊C4C2×#10597010semidirect
191. C23C4⋊#10597010semidirect
218. C2×C4C4⋊#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506922
#237505732
#237505722
#237505022
#99×D1492
#1867149