Finite Group Databasebeta

← #34060119

Subgroups — #34060119

Order 19072.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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×#18011968C211yesmaximal, characteristic, complemented
2#23750568C211yesmaximal, characteristic, complemented
3#23750478C212yesmaximal, complemented
4#23750477C212yesmaximal, complemented
5#23750137C211yesmaximal, characteristic
6C22×#11924753—21no—
7C2×C22⋊#10597010—21no—
8#18011968—21no—
9#18011968C2212yes—
10(C2×#10597010)⋊C4—21no—
11(C2×#10597010)⋊C4C2212yes—
12C2×#14558736C411yescharacteristic, complemented
13C22×C4×D149C2211yescharacteristic
14#18011895—21no—
15#18011891C411yescharacteristic, complemented
16#18011891C2211yescharacteristic
17#18011886C411yescharacteristic, complemented
18#18011886C2211yescharacteristic
19#18011881—21no—
20C4×D596—21no—
21C4×D596—21no—
22#117×C149C411yescharacteristic, complemented
23C22×#10597010—22no—
24C2×#11924753—22no—
25C2×#11924753—42no—
26C22⋊#10597010—42no—
27C23×D149D411yescharacteristic
28#14558736—42no—
29C2×Dic298C2×C411yescharacteristic
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⋊Dic149—21no—
42C4⋊Dic149—21no—
43#14558713—21no—
44#14558713—21no—
45#14558713—21no—
46#14558713—21no—
47C22×C4×C149C2×C411yescharacteristic
48C4⋊C4×C149C2×C411yescharacteristic
49C4⋊C4×C149C2×C411yescharacteristic
50C22⋊C4×C149—21no—
51C42×C149—21no—
52C2×#10597010—42no—
53C2×#10597010—42no—
54C2×#10597010—42no—
55#11924753—42no—
56C22×D149—22no—
57C22×D149—22no—
58C22×D149—21no—
59C22×D149—41no—
60C22×D149C22⋊C411yescharacteristic
61C22×D149C2×D411yescharacteristic
62Dic298—41no—
63D596—41no—
64D596—42no—
65D596—42no—
66C22⋊D149—42no—
67C2×Dic149—22no—
68C2×Dic149—21no—
69C2×Dic149C22⋊C411yescharacteristic
70C2×Dic149C2×D411yescharacteristic
71C4×D149—41no—
72C4×D149—41no—
73C4×D149—41no—
74C4×D149—41no—
75C4×D149—42no—
76C23×C149C22⋊C411yescharacteristic
77C2×C4×C149—22no—
78C2×C4×C149—21no—
79C2×C4×C149—21no—
80C2×C4×C149C22⋊C411yescharacteristic
81C2×C4×C149C22×C411yescharacteristic
82C22×C149—21no—
83C22×C149—21no—
84C22×C149#9912yes—
85C22×C149C2×C22⋊C411yescharacteristic
86C2×D149—21no—
87C2×D149—21no—
88C2×D149—21no—
89C2×D149—21no—
90C2×D149—21no—
91C2×D149—44no—
92C2×D149—41no—
93C2×D149—41no—
94C2×D149—41no—
95C2×D149#13512yes—
96#10597010—82no—
97C4×C149—21no—
98C4×C149—21no—
99C4×C149—41no—
100C4×C149—41no—
101Dic149—21no—
102Dic149—21no—
103Dic149—41no—
104C2×C149—22no—
105C2×C149C2×#9911yescharacteristic
106C2×C149#41611yescharacteristic
107C2×C149#42111yescharacteristic
108D149—22no—
109D149—41no—
110D149—81no—
111C149#151611yescharacteristic, complemented, Sylow
112#1516—1491nomaximal, Sylow
113#525—1491no—
114C2×C22⋊C8—1491no—
115#381—1491no—
116#328—1492no—
117#327—1492no—
118C2×#41—1491no—
119C23×C4—1491no—
120C22×C8—2981no—
121#123—2981no—
122C22⋊Q8—1491no—
123C22⋊Q8—1491no—
124#121—1491no—
125#121—1491no—
126#120—2981no—
127C4×D4—2981no—
128C4×D4—2981no—
129#117#1059701011yescharacteristic, complemented
130C2×C22⋊C4—2981no—
131C22⋊C8—2981no—
132C22⋊C8—1492no—
133#95—2981no—
134#95—1492no—
135C24—1491no—
136#41—5962no—
137C2×Q8—1491no—
138C2×D4—2981no—
139C2×D4—2981no—
140C2×D4—2981no—
141C2×D4—1491no—
142C22×C4—2981no—
143C22×C4—2982no—
144C22×C4—5962no—
145C22×C4—1491no—
146C22×C4—1491no—
147C22×C4—1491no—
148C22×C4—1491no—
149C22×C4—1491no—
150C22×C4C2×#1059701011yescharacteristic
151C2×C8—2982no—
152C2×C8—5962no—
153C4⋊C4—2981no—
154C4⋊C4—2981no—
155C4⋊C4C2×#1059701011yescharacteristic
156C4⋊C4C2×#1059701011yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2981no—
159C22⋊C4—2981no—
160C22⋊C4—2981no—
161C22⋊C4—2981no—
162C22⋊C4—5962no—
163C42—21no—
164C23—2982no—
165C23—2982no—
166C23—2981no—
167C23—5961no—
168C23—1491no—
169C23—1491no—
170C23C22⋊#1059701011yescharacteristic
171Q8—5961no—
172D4—5961no—
173D4—5962no—
174D4—5962no—
175D4—5962no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—2982no—
180C2×C4—2981no—
181C2×C4—5961no—
182C2×C4—5961no—
183C2×C4—5961no—
184C2×C4—5961no—
185C2×C4—5962no—
186C2×C4—5962no—
187C2×C4—5962no—
188C2×C4—5962no—
189C2×C4—1491no—
190C2×C4—1491no—
191C2×C4C22⋊#1059701011yescharacteristic
192C2×C4C22×#1059701011yescharacteristic
193C8—5962no—
194C22—21no—
195C22—21no—
196C22—2981no—
197C22—2981no—
198C22—2981no—
199C22—2981no—
200C22—2981no—
201C22—5964no—
202C22—5961no—
203C22—5961no—
204C22—5961no—
205C22—1492no—
206C22#1801197012yes—
207C22C2×C22⋊#1059701011yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—2981no—
213C4—2981no—
214C4—5961no—
215C4—11922no—
216C2—22no—
217C2—2982no—
218C2—5961no—
219C2—11921no—
220C2C2×#1801197011yescharacteristic
221C2#2375061511yescharacteristic
222C2#2375062211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
Fitting#117×C149C4
socleC22×C149C2×C22⋊C4
prime radical#23750137C2
cosocleC2×C4×D149C23
hypercenter#117#10597010
hypocenterC149#1516
LCSG ▷ C2×C4×C149 ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#18011968C2semidirect
2. #23750568C2semidirect
3. #23750478C2semidirect
4. #23750477C2semidirect
12. C2×#14558736C4semidirect
15. #18011891C4semidirect
17. #18011886C4semidirect
22. #117×C149C4semidirect
111. C149#1516semidirect
129. #117#10597010semidirect

Maximal subgroups

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

subgroupindex
C2×#180119682
#237505682
#237504782
#237504772
#237501372
#1516149