Finite Group Databasebeta

← #34045703

Subgroups — #34045703

Order 19072.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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#23750189C211yesmaximal, characteristic, complemented
2C2×#18011907C211yesmaximal, characteristic, complemented
3C2×#18011838C211yesmaximal, characteristic, complemented
4#23749630C211yesmaximal, characteristic, complemented
5C2×#18011827C211yesmaximal, characteristic, complemented
6#23749386C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
8C22×D596C2211yescharacteristic
9C22×C22⋊D149—22no—
10C2×#14558720C2211yescharacteristic
11#18011907—22no—
12#18011907—22no—
13#18011907—21no—
14#18011900—21no—
15#18011885—22no—
16#18011879—22no—
17#18011870C2211yescharacteristic
18C2×C4×Dic149C2211yescharacteristic
19#18011838—22no—
20#18011832—22no—
21#18011827—22no—
22C2×#41×C149C2211yescharacteristic
23C22×D4×C149C2211yescharacteristic
24AGL(1,ℤ/8)×C149—22no—
25AGL(1,ℤ/8)×C149—22no—
26C2×SD16×C149—21no—
27C2×D8×C149—21no—
28C2×M16×C149C2211yescharacteristic
29C23×D149D412yes—
30C2×D596—21no—
31C2×D596—21no—
32C2×D596—21no—
33C2×D596—21no—
34C2×D596—21no—
35C2×D596—21no—
36C2×C22⋊D149—28no—
37C2×C22⋊D149—24no—
38C2×C22⋊D149—48no—
39C22×Dic149—21no—
40#14558720—21no—
41#14558720—21no—
42C2×#11924746—21no—
43C4⋊Dic149D412yescomplemented
44C22⋊Dic149—21no—
45#14558713—24no—
46C4×Dic149—21no—
47C4×Dic149—21no—
48C4×Dic149D412yescomplemented
49C24×C149—21no—
50#41×C149—21no—
51#41×C149—21no—
52#41×C149—21no—
53#41×C149—21no—
54C2×Q8×C149D411yescharacteristic, complemented
55C2×D4×C149—21no—
56C2×D4×C149—21no—
57C2×D4×C149—21no—
58C2×D4×C149—21no—
59C2×D4×C149—21no—
60C2×D4×C149—21no—
61C2×D4×C149D411yescharacteristic, complemented
62C2×D4×C149D412yescomplemented
63C22×C4×C149—21no—
64C22×C4×C149C2311yescharacteristic
65SD16×C149—24no—
66D8×C149—24no—
67M16×C149—21no—
68M16×C149D412yescomplemented
69C2×C8×C149—21no—
70C22×D149—22no—
71C22×D149—22no—
72C22×D149—22no—
73C22×D149—44no—
74D596—41no—
75D596—41no—
76D596—41no—
77D596—41no—
78C22⋊D149—416no—
79C22⋊D149—84no—
80C2×Dic149—24no—
81C2×Dic149—22no—
82C2×Dic149—22no—
83#11924746—41no—
84C23×C149—21no—
85C23×C149—22no—
86C23×C149—44no—
87C23×C149—41no—
88C23×C149—41no—
89C23×C149C2×D411yescharacteristic
90Q8×C149—21no—
91Q8×C149—21no—
92D4×C149—21no—
93D4×C149—22no—
94D4×C149—22no—
95D4×C149—21no—
96D4×C149—41no—
97D4×C149—41no—
98D4×C149—41no—
99D4×C149—41no—
100C2×C4×C149—21no—
101C2×C4×C149—41no—
102C2×C4×C149—41no—
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2×D411yescharacteristic
108C2×C4×C149#4111yescharacteristic
109C8×C149—22no—
110C22×C149—22no—
111C22×C149—44no—
112C22×C149—41no—
113C22×C149—42no—
114C22×C149—42no—
115C22×C149—82no—
116C22×C149—82no—
117C22×C149#12011yescharacteristic
118C22×C149#12011yescharacteristic
119C22×C149#12111yescharacteristic
120C2×D149—44no—
121C2×D149—42no—
122C2×D149—44no—
123C2×D149—82no—
124C4×C149—41no—
125C4×C149#12011yescharacteristic
126C4×C149#12111yescharacteristic
127C4×C149#12111yescharacteristic
128C4×C149#12711yescharacteristic
129Dic149—24no—
130Dic149—42no—
131C2×C149—21no—
132C2×C149—21no—
133C2×C149—81no—
134C2×C149—82no—
135C2×C149#39211yescharacteristic
136C2×C149#45312yes—
137D149—82no—
138C149#200211yescharacteristic, complemented, Sylow
139#2002—1491nomaximal, Sylow
140C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
141#535—1491no—
142C2×#127—1491no—
143C2×#104—1491no—
144#418—1491no—
145C2×#100—1491no—
146#339—1491no—
147C2×#41C2×D14911yescharacteristic
148C22×D4—2982no—
149C22×D4—1491no—
150C22×D4C2×D14911yescharacteristic
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—22no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—1491no—
156C2×M16C2×D14911yescharacteristic
157#127—2982no—
158#127—2981no—
159#127—2982no—
160#124—2981no—
161#121—2982no—
162#120—2982no—
163#117—1491no—
164C2×C42—1491no—
165#104—2982no—
166#102—2982no—
167#100—2982no—
168C24—21no—
169C24—1492no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D14911yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—2988no—
182C2×D4—2984no—
183C2×D4—2981no—
184C2×D4—2981no—
185C2×D4—2981no—
186C2×D4—2981no—
187C2×D4—2981no—
188C2×D4—2981no—
189C2×D4—5968no—
190C2×D4C22⋊D14911yescharacteristic, complemented
191C2×D4C22⋊D14912yescomplemented
192C22×C4—21no—
193C22×C4—2981no—
194C22×C4C22×D14911yescharacteristic
195SD16—24no—
196D8—24no—
197M16—21no—
198M16—2981no—
199M16—2981no—
200M16C22⋊D14912yescomplemented
201C2×C8—21no—
202C2×C8—2981no—
203C4⋊C4—1492no—
204C22⋊C4—2984no—
205C22⋊C4—2981no—
206C42—2981no—
207C42—2981no—
208C42—1492no—
209C23—22no—
210C23—21no—
211C23—44no—
212C23—41no—
213C23—41no—
214C23—2982no—
215C23—2982no—
216C23—2982no—
217C23—5964no—
218C23D4×D14911yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—59616no—
230D4—5961no—
231D4—5961no—
232D4—5961no—
233D4—5961no—
234D4—11924no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2984no—
239C2×C4—2982no—
240C2×C4—2982no—
241C2×C4D4×D14911yescharacteristic
242C2×C4D4×D14911yescharacteristic
243C2×C4C2×C22⋊D14911yescharacteristic
244C2×C4C2×C22⋊D14911yescharacteristic
245C2×C4C2×C22⋊D14911yescharacteristic
246C2×C4#1455873511yescharacteristic
247C8—22no—
248C8—5961no—
249C22—22no—
250C22—44no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—5964no—
257C22—5962no—
258C22—5964no—
259C22—11922no—
260C22#1801187911yescharacteristic
261C22#1801187911yescharacteristic
262C22#1801188211yescharacteristic
263C4—41no—
264C4—2984no—
265C4—5962no—
266C4#1801188011yescharacteristic
267C4#1801188311yescharacteristic
268C4#1801188311yescharacteristic
269C4#1801190711yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—11922no—
275C2#2374953311yescharacteristic
276C2#2374979112yes—
subgroupG/H
centerC22#18011879
commutator, cosocleC22×C4×C149C23
FrattiniC22×C4C22×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149#120
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#2002
LCSG ▷ C22×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23750189C2semidirect
2. C2×#18011907C2semidirect
3. C2×#18011838C2semidirect
4. #23749630C2semidirect
5. C2×#18011827C2semidirect
6. #23749386C2semidirect
7. C2×AGL(1,ℤ/8)×C149C2semidirect
43. C4⋊Dic149D4semidirect
48. C4×Dic149D4semidirect
54. C2×Q8×C149D4semidirect
61. C2×D4×C149D4semidirect
62. C2×D4×C149D4semidirect
68. M16×C149D4semidirect
138. C149#2002semidirect
140. C2×AGL(1,ℤ/8)D149semidirect
174. C2×Q8C22⋊D149semidirect
190. C2×D4C22⋊D149semidirect
191. C2×D4C22⋊D149semidirect
200. M16C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
#237501892
C2×#180119072
C2×#180118382
#237496302
C2×#180118272
#237493862
C2×AGL(1,ℤ/8)×C1492
#2002149