Finite Group Databasebeta

← #34060706

Subgroups — #34060706

Order 19072.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×D149⋊M16C211yesmaximal, characteristic
2C2×#18011974C212yesmaximal, complemented
3#23750567C211yesmaximal, characteristic
4#23750498C212yesmaximal, complemented
5C22⋊Q8×D149C211yesmaximal, characteristic
6C2×#14558746—21no—
7D149⋊M16—24no—
8C2×#14558745—21no—
9C2×D149⋊C8C2212yes—
10C2×C4⋊#10597010C2212yes—
11C2×C22⋊#10597010—21no—
12#18011974—24no—
13(C2×#10597010)⋊C4—21no—
14C2×Q8×D149C2211yescharacteristic
15C22×C4×D149C2211yescharacteristic
16#18011891C411yescharacteristic, complemented
17#18011890C411yescharacteristic, complemented
18#18011889—21no—
19#18011888—21no—
20#18011887C411yescharacteristic, complemented
21C4⋊C4×D149—21no—
22C4⋊C4×D149C2211yescharacteristic
23C22⋊C4×D149—21no—
24C22⋊Q8×C149C411yescharacteristic, complemented
25C22×#10597010—22no—
26#14558746—24no—
27#14558745—24no—
28D149⋊C8—22no—
29D149⋊C8D414yes—
30C2×#11924753—22no—
31C4⋊#10597010—22no—
32C4⋊#10597010—22no—
33C22⋊#10597010—42no—
34C23×D149D411yescharacteristic
35C2×Dic298—21no—
36C2×Dic298C2×C411yescharacteristic
37Q8×D149—22no—
38Q8×D149—22no—
39C22×Dic149D411yescharacteristic
40C2×C4×D149—22no—
41C2×C4×D149—22no—
42C2×C4×D149—21no—
43C2×C4×D149—21no—
44C2×C4×D149C2×C411yescharacteristic
45C2×C4×D149D411yescharacteristic
46C2×C4×D149D411yescharacteristic
47C2×C4×D149C2311yescharacteristic
48C4⋊Dic149—21no—
49C4⋊Dic149C2×C411yescharacteristic
50#14558715—21no—
51#14558715—21no—
52#14558715—21no—
53C22⋊Dic149—21no—
54#14558713—21no—
55#14558713—21no—
56C2×Q8×C149C2×C411yescharacteristic
57C22×C4×C149C2×C411yescharacteristic
58C4⋊C4×C149—21no—
59C4⋊C4×C149C2×C411yescharacteristic
60C22⋊C4×C149—21no—
61C2×#10597010—42no—
62C2×#10597010—42no—
63C2×#10597010—42no—
64#11924753—24no—
65C22×D149—22no—
66C22×D149—21no—
67C22×D149—21no—
68C22×D149—22no—
69C22×D149C22⋊C411yescharacteristic
70C22×D149C2×D411yescharacteristic
71Dic298—22no—
72Dic298—41no—
73Dic298—41no—
74C2×Dic149—21no—
75C2×Dic149—21no—
76C2×Dic149—21no—
77C2×Dic149—21no—
78C2×Dic149C22⋊C411yescharacteristic
79C2×Dic149C2×D411yescharacteristic
80C4×D149—22no—
81C4×D149—22no—
82C4×D149—21no—
83C4×D149—21no—
84C4×D149—42no—
85C4×D149—41no—
86C4×D149—41no—
87C4×D149C2×D412yes—
88C4×D149#4112yes—
89C23×C149C22⋊C411yescharacteristic
90Q8×C149—22no—
91C2×C4×C149—22no—
92C2×C4×C149—21no—
93C2×C4×C149—21no—
94C2×C4×C149C22⋊C411yescharacteristic
95C2×C4×C149C22×C411yescharacteristic
96C22×C149—21no—
97C22×C149—21no—
98C22×C149—21no—
99C22×C149C2×C22⋊C411yescharacteristic
100C2×D149—21no—
101C2×D149—21no—
102C2×D149—21no—
103C2×D149—42no—
104C2×D149—42no—
105C2×D149#12112yes—
106C2×D149AGL(1,ℤ/8)11yescharacteristic
107C2×D149#13712yes—
108C2×D149#13711yescharacteristic
109#10597010—82no—
110C4×C149—21no—
111C4×C149—41no—
112C4×C149—41no—
113C4×C149C4×D412yes—
114Dic149—21no—
115Dic149—41no—
116Dic149—41no—
117Dic149#12011yescharacteristic
118Dic149#12311yescharacteristic
119C2×C149—41no—
120C2×C149#38611yescharacteristic
121C2×C149#41711yescharacteristic
122C2×C149#41911yescharacteristic
123D149—41no—
124D149#46812yes—
125D149#47012yes—
126C149#189411yescharacteristic, complemented, Sylow
127#1894—1491nomaximal, Sylow
128C22×M16—1491no—
129C2×C22⋊Q8—1491no—
130C2×#103—1492no—
131#380—1491no—
132#340—1492no—
133C22×Q8—1491no—
134C23×C4—1491no—
135C2×M16—2981no—
136C2×M16—2981no—
137C2×M16—2984no—
138C22×C8—1492no—
139C22⋊Q8—2981no—
140C22⋊Q8—2981no—
141C22⋊Q8—1491no—
142C22⋊Q8—1491no—
143C22⋊Q8—1491no—
144C22⋊Q8#1059701011yescharacteristic, complemented
145C2×C4⋊C4—2981no—
146C2×C4⋊C4—1492no—
147C2×C4⋊C4—1491no—
148C2×C22⋊C4—2981no—
149C2×C22⋊C4—2981no—
150#103—2984no—
151#95—2981no—
152C24—1491no—
153C2×Q8—2981no—
154C2×Q8—2982no—
155C2×Q8—2982no—
156C2×Q8—1491no—
157C2×Q8C2×#1059701011yescharacteristic
158C22×C4—2982no—
159C22×C4—2982no—
160C22×C4—2981no—
161C22×C4—2981no—
162C22×C4—2982no—
163C22×C4—1491no—
164C22×C4—1491no—
165C22×C4—1491no—
166C22×C4—1491no—
167C22×C4—1491no—
168C22×C4C2×#1059701011yescharacteristic
169M16—2984no—
170M16—2984no—
171C2×C8—2982no—
172C2×C8—2982no—
173C2×C8—1494no—
174C4⋊C4—21no—
175C4⋊C4—2981no—
176C4⋊C4—2981no—
177C4⋊C4—2981no—
178C4⋊C4—2981no—
179C4⋊C4—2982no—
180C4⋊C4—2982no—
181C4⋊C4—1491no—
182C4⋊C4C2×#1059701011yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—2981no—
185C22⋊C4—2981no—
186C22⋊C4—2981no—
187C22⋊C4—5962no—
188C23—2981no—
189C23—2982no—
190C23—2981no—
191C23—2982no—
192C23—1491no—
193C23—1491no—
194C23C22⋊#1059701011yescharacteristic
195Q8—22no—
196Q8—2982no—
197Q8—5961no—
198Q8—5961no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—2982no—
203C2×C4—2981no—
204C2×C4—2981no—
205C2×C4—2981no—
206C2×C4—2981no—
207C2×C4—2982no—
208C2×C4—2981no—
209C2×C4—2981no—
210C2×C4—5961no—
211C2×C4—5961no—
212C2×C4—5962no—
213C2×C4—5962no—
214C2×C4—5962no—
215C2×C4—5962no—
216C2×C4—1491no—
217C2×C4—1491no—
218C2×C4—1492no—
219C2×C4—1492no—
220C2×C4C22⋊#1059701011yescharacteristic
221C2×C4C22×#1059701011yescharacteristic
222C8—2984no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—2981no—
227C22—2981no—
228C22—2981no—
229C22—5962no—
230C22—5962no—
231C22—1492no—
232C22—1491no—
233C22—1491no—
234C22—1492no—
235C22C2×C22⋊#1059701011yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—2981no—
240C4—5961no—
241C4—5961no—
242C4—1491no—
243C4—1491no—
244C4—11922no—
245C4D4×#1059701012yes—
246C2—41no—
247C2—5961no—
248C2—1492no—
249C2—1492no—
250C2#2375057311yescharacteristic
251C2#2375061611yescharacteristic
252C2#2375061811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC22⋊Q8×C149C4
socleC22×C149C2×C22⋊C4
prime radicalC23×D149D4
cosocleC2×C4×D149C23
hypercenterC22⋊Q8#10597010
hypocenterC149#1894
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#18011974C2semidirect
4. #23750498C2semidirect
16. #18011891C4semidirect
17. #18011890C4semidirect
20. #18011887C4semidirect
24. C22⋊Q8×C149C4semidirect
126. C149#1894semidirect
144. C22⋊Q8#10597010semidirect

Maximal subgroups

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

subgroupindex
C2×D149⋊M162
C2×#180119742
#237505672
#237504982
C22⋊Q8×D1492
#1894149