Finite Group Databasebeta

← #34056337

Subgroups — #34056337

Order 19072.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18011942C212yesmaximal, complemented
2#23750280C211yesmaximal, characteristic
3#23750004C211yesmaximal, characteristic, complemented
4#23749881C212yesmaximal, complemented
5#23749835C211yesmaximal, characteristic, complemented
6#23749827C211yesmaximal, characteristic, complemented
7#23749785C212yesmaximal, complemented
8#23749781C212yesmaximal, complemented
9C4×#14558723C211yesmaximal, characteristic, complemented
10#23749703C211yesmaximal, characteristic, complemented
11D4×Q8×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2212yes—
13#18011942—28no—
14C2×#14558726C2212yes—
15C2×#14558726C2212yes—
16C2×#14558726C2212yes—
17C2×#14558723C2212yes—
18C2×#14558723C2211yescharacteristic
19C2×#14558720C2212yes—
20#18011911C2211yescharacteristic
21#18011893—22no—
22#18011890C2212yes—
23#18011882—21no—
24C4×D596C2211yescharacteristic
25C4⋊C4×D149—21no—
26#18011842C2211yescharacteristic
27C4⋊#11924746C2211yescharacteristic
28#18011835C2211yescharacteristic
29#18011835C2212yes—
30#18011834C2212yes—
31#18011831C2211yescharacteristic
32C22⋊#11924746C2212yes—
33C4×#11924746C2211yescharacteristic
34C22×Q8×C149C2212yes—
35C4⋊Q8×C149—21no—
36C4⋊Q8×C149C2211yescharacteristic
37C22⋊Q8×C149—22no—
38C22⋊Q8×C149C2212yes—
39C4×Q8×C149C2211yescharacteristic
40C4×D4×C149—21no—
41C4×D4×C149C2211yescharacteristic
42#14558736—24no—
43#14558736—42no—
44C2×Dic298C2312yes—
45C2×D596C2311yescharacteristic
46C2×C22⋊D149—22no—
47C2×C4×D149—22no—
48C2×C4×D149—21no—
49#14558726—24no—
50#14558726—24no—
51#14558726—42no—
52#14558723—24no—
53#14558723D414yescomplemented
54#14558720—24no—
55C2×#11924746C2312yes—
56C2×#11924746C2311yescharacteristic
57C2×#11924746C2311yescharacteristic
58C4⋊Dic149—21no—
59C4⋊Dic149—21no—
60C4⋊Dic149C2311yescharacteristic
61#14558715—22no—
62C22⋊Dic149—22no—
63#14558713—21no—
64C2×Q8×C149—22no—
65C2×Q8×C149—24no—
66C2×Q8×C149—42no—
67C2×Q8×C149C2312yes—
68C2×Q8×C149C2311yescharacteristic
69C2×D4×C149D411yescharacteristic
70C22×C4×C149—22no—
71C22×C4×C149C2312yes—
72C4⋊C4×C149—21no—
73C4⋊C4×C149—21no—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—22no—
76C4⋊C4×C149D411yescharacteristic
77C4⋊C4×C149C2311yescharacteristic
78C22⋊C4×C149—22no—
79C22⋊C4×C149D412yes—
80C42×C149—21no—
81C42×C149C2311yescharacteristic
82C22×D149—21no—
83Dic298—22no—
84Dic298—22no—
85D596—21no—
86D596C2×D412yes—
87C22⋊D149—44no—
88C2×Dic149—22no—
89C2×Dic149—21no—
90C4×D149—44no—
91C4×D149—42no—
92#11924746—22no—
93#11924746—21no—
94#11924746C2×D412yes—
95C23×C149C2×D412yes—
96Q8×C149—21no—
97Q8×C149—22no—
98Q8×C149—22no—
99Q8×C149—44no—
100Q8×C149C2×D412yes—
101D4×C149—22no—
102C2×C4×C149—21no—
103C2×C4×C149—22no—
104C2×C4×C149—22no—
105C2×C4×C149—21no—
106C2×C4×C149—22no—
107C2×C4×C149—44no—
108C2×C4×C149C2×D412yes—
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149C2×D411yescharacteristic
111C2×C4×C149C2411yescharacteristic
112C22×C149—22no—
113C22×C149—22no—
114C22×C149—22no—
115C22×C149C22×D411yescharacteristic
116C2×D149—22no—
117C2×D149—41no—
118C2×D149—41no—
119C4×C149—22no—
120C4×C149—22no—
121C4×C149—21no—
122C4×C149—42no—
123C4×C149—41no—
124C4×C149#13712yescomplemented
125C4×C149C22×D411yescharacteristic
126C4×C149H(5,𝔽2)11yescharacteristic
127Dic149—42no—
128Dic149—41no—
129C2×C149—42no—
130C2×C149D4211yescharacteristic
131C2×C149C2×#13711yescharacteristic
132C2×C149#57811yescharacteristic
133D149—42no—
134C149#327611yescharacteristic, complemented, Sylow
135#3276—1491nomaximal, Sylow
136C2×#137—1492no—
137D4×Q8D14911yescharacteristic, complemented
138#547—1491no—
139#494—1491no—
140#470—1492no—
141#462—1491no—
142#461—1491no—
143#452—1492no—
144#451—1492no—
145C4×SD16—1491no—
146#436—1491no—
147C2×#41—1492no—
148C22×Q8C2×D14912yes—
149#137—2988no—
150C2×Q16—1492no—
151C2×Q16—1492no—
152C2×Q16—1492no—
153C2×SD16—1492no—
154C2×SD16—1491no—
155C2×M16—1492no—
156C4⋊Q8—21no—
157C4⋊Q8—1491no—
158C4⋊Q8C2×D14911yescharacteristic
159#123—2982no—
160C22⋊Q8—22no—
161C22⋊Q8—1492no—
162C22⋊Q8C2×D14912yes—
163#121—2981no—
164C4×Q8C2×D14911yescharacteristic
165C4×D4—21no—
166C4×D4—1491no—
167C4×D4C2×D14911yescharacteristic
168C2×C4⋊C4—2981no—
169#106—1491no—
170C4⋊C8—1491no—
171#103—1491no—
172#103—1492no—
173#103—1492no—
174#102—1491no—
175C22⋊C8—1492no—
176C4×C8—1491no—
177#41—2984no—
178#41—5962no—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—1492no—
183C2×Q8C22×D14912yes—
184C2×Q8C22×D14911yescharacteristic
185C2×D4—2982no—
186C2×D4—1491no—
187C2×D4C22⋊D14911yescharacteristic
188C22×C4—22no—
189C22×C4—2981no—
190C22×C4—2982no—
191C22×C4C22×D14912yes—
192Q16—2984no—
193Q16—2984no—
194Q16—5962no—
195SD16—2984no—
196SD16—1494no—
197M16—2984no—
198C2×C8—1491no—
199C2×C8—1492no—
200C2×C8—1491no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—2981no—
206C4⋊C4—2982no—
207C4⋊C4—2981no—
208C4⋊C4—1491no—
209C4⋊C4C22⋊D14911yescharacteristic
210C4⋊C4C22×D14911yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—2982no—
213C22⋊C4—2981no—
214C22⋊C4C22⋊D14912yes—
215C42—21no—
216C42C22×D14911yescharacteristic
217C23—2981no—
218C23C2×C22⋊D14912yes—
219Q8—21no—
220Q8—22no—
221Q8—22no—
222Q8—44no—
223Q8—2982no—
224Q8—2982no—
225Q8D4×D14912yes—
226D4—22no—
227D4—2981no—
228D4—5964no—
229D4—1492no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—2981no—
237C2×C4—2982no—
238C2×C4—5964no—
239C2×C4—5962no—
240C2×C4C2×C22⋊D14911yescharacteristic
241C2×C4C2×C22⋊D14912yes—
242C2×C4C2×C22⋊D14911yescharacteristic
243C2×C4C23×D14911yescharacteristic
244C8—2982no—
245C8—2981no—
246C8—1492no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—2982no—
251C22—5961no—
252C22—5961no—
253C22C22×C22⋊D14911yescharacteristic
254C4—21no—
255C4—22no—
256C4—22no—
257C4—41no—
258C4—42no—
259C4—5961no—
260C4—5962no—
261C4#1801194212yescomplemented
262C4C2×D4×D14911yescharacteristic
263C4#1801195611yescharacteristic
264C2—42no—
265C2—5962no—
266C2D4×C22⋊D14911yescharacteristic
267C2C2×#1801194211yescharacteristic
268C2#2375044011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingD4×Q8×C149C2
socleC22×C149C22×D4
prime radical#23750280C2
hypercenterD4×Q8D149
hypocenterC149#3276
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#18011942C2semidirect
3. #23750004C2semidirect
4. #23749881C2semidirect
5. #23749835C2semidirect
6. #23749827C2semidirect
7. #23749785C2semidirect
8. #23749781C2semidirect
9. C4×#14558723C2semidirect
10. #23749703C2semidirect
11. D4×Q8×C149C2semidirect
53. #14558723D4semidirect
124. C4×C149#137semidirect
134. C149#3276semidirect
137. D4×Q8D149semidirect
261. C4#18011942semidirect

Maximal subgroups

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

subgroupindex
C2×#180119422
#237502802
#237500042
#237498812
#237498352
#237498272
#237497852
#237497812
C4×#145587232
#237497032
D4×Q8×C1492
#3276149