Finite Group Databasebeta

← #34060109

Subgroups — #34060109

Order 19072.

286 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 58 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#23750597C211yesmaximal, characteristic, complemented
2#23750567C211yesmaximal, characteristic, complemented
3#23750479C212yesmaximal, complemented
4#23750477C212yesmaximal, complemented
5#23750138C211yesmaximal, characteristic
6C2×#14558745—21no—
7C2×C4⋊#10597010—21no—
8C2×C22⋊#10597010—21no—
9#18011968—21no—
10#18011968C2212yes—
11(C2×#10597010)⋊C4C2212yes—
12C2×#14558737C411yescharacteristic, complemented
13C22×C4×D149C2211yescharacteristic
14#18011896—21no—
15#18011887C2211yescharacteristic
16#18011886C2211yescharacteristic
17#18011885C411yescharacteristic, complemented
18#18011881—21no—
19#18011873—21no—
20#18011871—21no—
21#18011869C411yescharacteristic, complemented
22C22⋊Q8×C149C411yescharacteristic, complemented
23C22×#10597010—22no—
24#14558745—42no—
25C2×#11924753—22no—
26C4⋊#10597010—42no—
27C22⋊#10597010—42no—
28C23×D149D411yescharacteristic
29#14558737—42no—
30C2×D596—21no—
31C2×D596—21no—
32C2×D596C2×C411yescharacteristic
33C2×C22⋊D149—21no—
34C22×Dic149D411yescharacteristic
35C2×C4×D149—21no—
36C2×C4×D149—42no—
37C2×C4×D149C2×C411yescharacteristic
38C2×C4×D149D411yescharacteristic
39C2×C4×D149D411yescharacteristic
40C2×C4×D149C2311yescharacteristic
41C4⋊Dic149C2×C411yescharacteristic
42#14558715—21no—
43#14558713—21no—
44#14558713—21no—
45#14558713—21no—
46#14558713—21no—
47C4×Dic149—21no—
48C2×Q8×C149C2×C411yescharacteristic
49C22×C4×C149C2×C411yescharacteristic
50C4⋊C4×C149—21no—
51C4⋊C4×C149C2×C411yescharacteristic
52C22⋊C4×C149—21no—
53C2×#10597010—42no—
54C2×#10597010—42no—
55C2×#10597010—42no—
56#11924753—42no—
57C22×D149—21no—
58C22×D149—21no—
59C22×D149—21no—
60C22×D149—41no—
61C22×D149—41no—
62C22×D149C22⋊C411yescharacteristic
63C22×D149C2×D411yescharacteristic
64D596—41no—
65D596—42no—
66D596—41no—
67D596—41no—
68C22⋊D149—42no—
69C2×Dic149—22no—
70C2×Dic149—21no—
71C2×Dic149C22⋊C411yescharacteristic
72C2×Dic149C2×D411yescharacteristic
73C4×D149—22no—
74C4×D149—22no—
75C4×D149—41no—
76C4×D149—41no—
77C4×D149—42no—
78C23×C149C22⋊C411yescharacteristic
79Q8×C149—41no—
80C2×C4×C149—21no—
81C2×C4×C149—21no—
82C2×C4×C149—41no—
83C2×C4×C149C22⋊C411yescharacteristic
84C2×C4×C149C22×C411yescharacteristic
85C22×C149—21no—
86C22×C149—21no—
87C22×C149—21no—
88C22×C149C2×C22⋊C411yescharacteristic
89C2×D149—21no—
90C2×D149—21no—
91C2×D149—21no—
92C2×D149—21no—
93C2×D149—21no—
94C2×D149—42no—
95C2×D149—42no—
96C2×D149—41no—
97C2×D149—41no—
98C2×D149—41no—
99C2×D149AGL(1,ℤ/8)11yescharacteristic
100C2×D149#13711yescharacteristic
101#10597010—82no—
102C4×C149—21no—
103C4×C149—21no—
104C4×C149—41no—
105C4×C149—41no—
106Dic149—21no—
107Dic149—41no—
108Dic149#9912yes—
109C2×C149—41no—
110C2×C149C2×#9911yescharacteristic
111C2×C149#41711yescharacteristic
112C2×C149#42111yescharacteristic
113D149—21no—
114D149—21no—
115D149—41no—
116D149—81no—
117C149#151311yescharacteristic, complemented, Sylow
118#1513—1491nomaximal, Sylow
119#525—1491no—
120#407—1491no—
121#380—1491no—
122#328—1492no—
123#327—1492no—
124C2×#41—1491no—
125C23×C4—1491no—
126C2×M16—2981no—
127#123—2981no—
128C22⋊Q8—1491no—
129C22⋊Q8#1059701011yescharacteristic, complemented
130#121—1491no—
131#121—1491no—
132#120—2981no—
133C4×D4—2981no—
134C4×D4—2981no—
135#117—1491no—
136C2×C4⋊C4—2981no—
137C2×C22⋊C4—2981no—
138C22⋊C8—2981no—
139C22⋊C8—1492no—
140#95—1492no—
141C24—1491no—
142#41—5962no—
143C2×Q8C2×#1059701011yescharacteristic
144C2×D4—2981no—
145C2×D4—2981no—
146C2×D4—2981no—
147C2×D4—1491no—
148C22×C4—2981no—
149C22×C4—2982no—
150C22×C4—5962no—
151C22×C4—1491no—
152C22×C4—1491no—
153C22×C4—1491no—
154C22×C4—1491no—
155C22×C4—1491no—
156C22×C4C2×#1059701011yescharacteristic
157M16—5962no—
158C2×C8—2982no—
159C4⋊C4—21no—
160C4⋊C4—2981no—
161C4⋊C4—5962no—
162C4⋊C4—1491no—
163C4⋊C4C2×#1059701011yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2981no—
166C22⋊C4—2981no—
167C22⋊C4—2981no—
168C22⋊C4—2981no—
169C22⋊C4—5962no—
170C42—2981no—
171C23—2981no—
172C23—2981no—
173C23—2981no—
174C23—5961no—
175C23—5961no—
176C23—1491no—
177C23—1491no—
178C23C22⋊#1059701011yescharacteristic
179Q8—41no—
180D4—5961no—
181D4—5962no—
182D4—5961no—
183D4—5961no—
184D4—5962no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—2982no—
189C2×C4—2982no—
190C2×C4—2982no—
191C2×C4—2981no—
192C2×C4—5961no—
193C2×C4—5961no—
194C2×C4—5962no—
195C2×C4—5962no—
196C2×C4—5962no—
197C2×C4—5962no—
198C2×C4—1491no—
199C2×C4—1491no—
200C2×C4C22⋊#1059701011yescharacteristic
201C2×C4C22×#1059701011yescharacteristic
202C8—5962no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—2981no—
207C22—2981no—
208C22—2981no—
209C22—2981no—
210C22—2981no—
211C22—5962no—
212C22—5962no—
213C22—5961no—
214C22—5961no—
215C22—5961no—
216C22—1491no—
217C22—1491no—
218C22C2×C22⋊#1059701011yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—2981no—
224C4—5961no—
225C4—1492no—
226C4—11922no—
227C2—41no—
228C2—2981no—
229C2—2981no—
230C2—5961no—
231C2—11921no—
232C2D149⋊#9911yescharacteristic
233C2#2375061611yescharacteristic
234C2#2375062411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC22⋊Q8×C149C4
socleC22×C149C2×C22⋊C4
prime radical#23750138C2
cosocleC2×C4×D149C23
hypercenterC22⋊Q8#10597010
hypocenterC149#1513
LCSG ▷ C2×C4×C149 ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23750597C2semidirect
2. #23750567C2semidirect
3. #23750479C2semidirect
4. #23750477C2semidirect
12. C2×#14558737C4semidirect
17. #18011885C4semidirect
21. #18011869C4semidirect
22. C22⋊Q8×C149C4semidirect
117. C149#1513semidirect
129. C22⋊Q8#10597010semidirect

Maximal subgroups

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

subgroupindex
#237505972
#237505672
#237504792
#237504772
#237501382
#1513149