Finite Group Databasebeta

← #34055512

Subgroups — #34055512

Order 19072.

410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18011890C211yesmaximal, characteristic
2#23749955C212yesmaximal, complemented
3#23749953C212yesmaximal, complemented
4#23749881C214yesmaximal, complemented
5#23749781C212yesmaximal, complemented
6#23749780C212yesmaximal, complemented
7#23749589C211yesmaximal, characteristic
8C2×C22⋊Q8×C149C211yesmaximal, characteristic
9C22×Dic298C2211yescharacteristic
10C2×#14558726C2214yescomplemented
11C2×#14558720C2212yes—
12#18011890—21no—
13#18011890—21no—
14#18011890C2212yes—
15#18011890C2212yes—
16C2×C4⋊Dic149C2211yescharacteristic
17C2×#14558715—21no—
18C2×C22⋊Dic149—21no—
19#18011842C2214yescomplemented
20#18011835C2214yescomplemented
21#18011834C2214yescomplemented
22C22⋊#11924746C2212yes—
23C22⋊#11924746C2212yes—
24C22×Q8×C149C2211yescharacteristic
25C23×C4×C149C2211yescharacteristic
26C22⋊Q8×C149—21no—
27C22⋊Q8×C149—21no—
28C22⋊Q8×C149C2212yes—
29C22⋊Q8×C149C2212yes—
30C2×C4⋊C4×C149—21no—
31C2×C4⋊C4×C149C2211yescharacteristic
32C2×C22⋊C4×C149—21no—
33C2×Dic298—21no—
34C2×Dic298—42no—
35C2×Dic298C2312yes—
36C22×Dic149—21no—
37C22×Dic149—21no—
38#14558726—44no—
39#14558720—42no—
40C2×#11924746C2314yes—
41C4⋊Dic149—21no—
42C4⋊Dic149C2312yes—
43#14558715—22no—
44#14558715—22no—
45C22⋊Dic149—22no—
46C22⋊Dic149—22no—
47C24×C149D411yescharacteristic
48C2×Q8×C149—21no—
49C2×Q8×C149—42no—
50C2×Q8×C149C2312yes—
51C22×C4×C149—21no—
52C22×C4×C149—21no—
53C22×C4×C149—42no—
54C22×C4×C149D411yescharacteristic
55C22×C4×C149D411yescharacteristic
56C22×C4×C149D411yescharacteristic
57C22×C4×C149C2311yescharacteristic
58C22×C4×C149C2311yescharacteristic
59C22×C4×C149C2311yescharacteristic
60C4⋊C4×C149—22no—
61C4⋊C4×C149—22no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149C2312yes—
64C22⋊C4×C149—22no—
65C22⋊C4×C149—22no—
66Dic298—24no—
67Dic298—42no—
68C2×Dic149—22no—
69C2×Dic149—22no—
70C2×Dic149—42no—
71C2×Dic149—42no—
72#11924746—24no—
73C23×C149—22no—
74C23×C149—22no—
75C23×C149—41no—
76C23×C149C2×D411yescharacteristic
77C23×C149C2×D411yescharacteristic
78C23×C149C2×D411yescharacteristic
79Q8×C149—24no—
80Q8×C149—42no—
81C2×C4×C149—22no—
82C2×C4×C149—22no—
83C2×C4×C149—22no—
84C2×C4×C149—22no—
85C2×C4×C149—22no—
86C2×C4×C149—41no—
87C2×C4×C149—41no—
88C2×C4×C149—41no—
89C2×C4×C149—42no—
90C2×C4×C149—42no—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2×D411yescharacteristic
93C2×C4×C149C2×D411yescharacteristic
94C2×C4×C149C2411yescharacteristic
95C22×C149—21no—
96C22×C149—21no—
97C22×C149—21no—
98C22×C149—21no—
99C22×C149—21no—
100C22×C149—21no—
101C22×C149—21no—
102C22×C149—44no—
103C22×C149#13712yes—
104C22×C149#13712yes—
105C22×C149C22×D411yescharacteristic
106C4×C149—21no—
107C4×C149—21no—
108C4×C149—21no—
109C4×C149—42no—
110C4×C149—42no—
111C4×C149H(5,𝔽2)12yes—
112Dic149—42no—
113Dic149—42no—
114C2×C149—22no—
115C2×C149—22no—
116C2×C149—41no—
117C2×C149#53411yescharacteristic
118C2×C149C2×#13711yescharacteristic
119C2×C149C2×#13711yescharacteristic
120C149#319511yescharacteristic, complemented, Sylow
121#3195—1491nomaximal, Sylow
122C2×C22⋊Q8—1491no—
123C2×C22⋊Q8D14911yescharacteristic
124#483—1492no—
125#483—1492no—
126#470—1494no—
127#451—1492no—
128#451—1492no—
129#407—1491no—
130C22×Q8—1491no—
131C22×Q8C2×D14911yescharacteristic
132C23×C4C2×D14911yescharacteristic
133C2×Q16—1494no—
134C2×M16—1492no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—2981no—
138C22⋊Q8—2981no—
139C22⋊Q8—1492no—
140C22⋊Q8—1492no—
141C22⋊Q8C2×D14912yes—
142C22⋊Q8C2×D14912yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—2981no—
145C2×C4⋊C4—1491no—
146C2×C4⋊C4C2×D14911yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—2981no—
149#106—1494no—
150#103—1494no—
151#103—1494no—
152C22⋊C8—1492no—
153C22⋊C8—1492no—
154C24C22⋊D14911yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—2981no—
158C2×Q8—5962no—
159C2×Q8—1492no—
160C2×Q8C22×D14912yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—2981no—
165C22×C4—2981no—
166C22×C4C22⋊D14911yescharacteristic
167C22×C4C22⋊D14911yescharacteristic
168C22×C4C22⋊D14911yescharacteristic
169C22×C4C22×D14911yescharacteristic
170C22×C4C22×D14911yescharacteristic
171C22×C4C22×D14911yescharacteristic
172Q16—5964no—
173M16—5962no—
174C2×C8—1494no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—2982no—
179C4⋊C4—2982no—
180C4⋊C4—2981no—
181C4⋊C4—1492no—
182C4⋊C4C22×D14912yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—2982no—
186C22⋊C4—2982no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23C2×C22⋊D14911yescharacteristic
191C23C2×C22⋊D14911yescharacteristic
192C23C2×C22⋊D14911yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—2984no—
196Q8—5962no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—2982no—
208C2×C4—2982no—
209C2×C4—5962no—
210C2×C4—5962no—
211C2×C4C2×C22⋊D14911yescharacteristic
212C2×C4C2×C22⋊D14911yescharacteristic
213C2×C4C2×C22⋊D14911yescharacteristic
214C2×C4C23×D14911yescharacteristic
215C8—2984no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—44no—
224C22#1801194112yes—
225C22#1801194212yes—
226C22C22×C22⋊D14911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—5962no—
233C4—5962no—
234C4#1801195612yes—
235C2—22no—
236C2—22no—
237C2—41no—
238C2#2375018411yescharacteristic
239C2C2×#1801194111yescharacteristic
240C2C2×#1801194211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×C22⋊Q8×C149C2
socleC22×C149C22×D4
prime radicalC24×C149D4
hypercenterC2×C22⋊Q8D149
hypocenterC149#3195
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23749955C2semidirect
3. #23749953C2semidirect
4. #23749881C2semidirect
5. #23749781C2semidirect
6. #23749780C2semidirect
10. C2×#14558726C22semidirect
19. #18011842C22semidirect
20. #18011835C22semidirect
21. #18011834C22semidirect
120. C149#3195semidirect

Maximal subgroups

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

subgroupindex
C2×#180118902
#237499552
#237499532
#237498812
#237497812
#237497802
#237495892
C2×C22⋊Q8×C1492
#3195149