Finite Group Databasebeta

← #34055917

Subgroups — #34055917

Order 19072.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D149C211yesmaximal, characteristic, complemented
2#23750154C211yesmaximal, characteristic
3#23750034C211yesmaximal, characteristic, complemented
4#23750031C211yesmaximal, characteristic, complemented
5#23749984C211yesmaximal, characteristic, complemented
6#23749959C212yesmaximal, complemented
7#23749942C212yesmaximal
8#23749908C212yesmaximal, complemented
9#23749903C212yesmaximal, complemented
10#23749680C211yesmaximal, characteristic, complemented
11#489×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149—21no—
13#18011911C2211yescharacteristic
14#18011910—21no—
15#18011910C2212yes—
16#18011909—21no—
17#18011904—21no—
18#18011903C2211yescharacteristic
19#18011893—21no—
20#18011887C2211yescharacteristic
21#18011882C2211yescharacteristic
22Q8×Dic149C2211yescharacteristic
23D4×Dic149C2211yescharacteristic
24#18011871—21no—
25#18011867—21no—
26C4⋊C4×D149—21no—
27C4⋊C4×D149C2212yes—
28C42×D149C2211yescharacteristic
29#18011845C2212yes—
30#18011844C2212yes—
31#18011843C2212yes—
32#18011842C2212yes—
33#18011840C2212yes—
34#18011835C2212yes—
35#18011832C2212yes—
36#18011823C2212yes—
37#18011822C2211yescharacteristic
38C4⋊Q8×C149C2211yescharacteristic
39#124×C149C2211yescharacteristic
40#103×C149C2212yes—
41#102×C149C2212yes—
42#97×C149C2211yescharacteristic
43C2×Dic298—22no—
44C2×Dic298—21no—
45Q8×D149—44no—
46C2×C22⋊D149—21no—
47C22×Dic149—21no—
48C2×C4×D149—22no—
49C2×C4×D149D411yescharacteristic
50C2×C4×D149D411yescharacteristic
51C2×C4×D149C2311yescharacteristic
52C2×#11924746C2312yes—
53C4⋊Dic149—21no—
54C4⋊Dic149—21no—
55C4⋊Dic149C2312yes—
56C4⋊Dic149C2311yescharacteristic
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558715—22no—
60#14558715—22no—
61#14558715—21no—
62#14558715—21no—
63C22⋊Dic149—21no—
64C22⋊Dic149—21no—
65#14558713—21no—
66C4×Dic149—21no—
67C4×Dic149D411yescharacteristic
68C4×Dic149D411yescharacteristic
69C4×Dic149C2311yescharacteristic
70C2×Q8×C149—21no—
71C2×Q8×C149C2311yescharacteristic
72C2×D4×C149C2311yescharacteristic
73C2×C8×C149C2312yes—
74C4⋊C4×C149—21no—
75C4⋊C4×C149C2312yes—
76C22⋊C4×C149—21no—
77C22⋊C4×C149—21no—
78C42×C149C2311yescharacteristic
79C22×D149C2×D411yescharacteristic
80Dic298—42no—
81Dic298—44no—
82C22⋊D149—42no—
83C2×Dic149—22no—
84C2×Dic149—21no—
85C2×Dic149—21no—
86C2×Dic149—42no—
87C2×Dic149C2×D411yescharacteristic
88C2×Dic149C2×D411yescharacteristic
89C2×Dic149C2×D411yescharacteristic
90C4×D149—22no—
91C4×D149—22no—
92C4×D149—22no—
93C4×D149—44no—
94#11924746—22no—
95C23×C149—21no—
96Q8×C149—22no—
97Q8×C149—42no—
98D4×C149—22no—
99C2×C4×C149—22no—
100C2×C4×C149—21no—
101C2×C4×C149C2×D411yescharacteristic
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2411yescharacteristic
104C8×C149—22no—
105C22×C149—41no—
106C22×C149—41no—
107C22×C149—41no—
108C22×C149C22×D411yescharacteristic
109C2×D149—21no—
110C2×D149—21no—
111C2×D149#13712yes—
112C4×C149—21no—
113C4×C149—21no—
114C4×C149—41no—
115C4×C149—42no—
116C4×C1492−1+412yes—
117Dic149—21no—
118Dic149—21no—
119Dic149—42no—
120Dic149—41no—
121Dic149—41no—
122Dic149AGL(1,ℤ/8)12yes—
123C2×C149—81no—
124C2×C149#53611yescharacteristic
125C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C149C2×#13711yescharacteristic
127D149—22no—
128C149#323911yescharacteristic, complemented, Sylow
129#3239—1491nomaximal, Sylow
130C2×C4⋊Q8—1491no—
131#529—1491no—
132#501—1491no—
133#501—1491no—
134#489—1491no—
135#489D14911yescharacteristic, complemented
136#484—1492no—
137#481—1492no—
138#475—1492no—
139#474—1492no—
140#432—1491no—
141C22×Q8—2981no—
142C4⋊Q8—2981no—
143C4⋊Q8—2981no—
144C4⋊Q8—1492no—
145C4⋊Q8—1491no—
146C4⋊Q8C2×D14911yescharacteristic
147#126—2981no—
148#125—1491no—
149#124C2×D14911yescharacteristic
150#123—2981no—
151C22⋊Q8—1491no—
152#121—1491no—
153C4×Q8—1491no—
154C4×D4—2981no—
155C4×D4—1491no—
156#117—2981no—
157C2×C4⋊C4—2981no—
158C2×C4⋊C4—1492no—
159C2×C42—1491no—
160#107—1492no—
161#107—1492no—
162#106—1492no—
163#106—1492no—
164C4⋊C8—1492no—
165#103—1492no—
166#103C2×D14912yes—
167#102—1492no—
168#102C2×D14912yes—
169C22⋊C8—1492no—
170#97—1491no—
171#97C2×D14911yescharacteristic
172C2×Q8—21no—
173C2×Q8—2982no—
174C2×Q8—2981no—
175C2×Q8—5964no—
176C2×Q8C22×D14911yescharacteristic
177C2×D4—2981no—
178C2×D4C22×D14911yescharacteristic
179C22×C4—2982no—
180C22×C4—2981no—
181C22×C4—1491no—
182C22×C4—1491no—
183C22×C4—1491no—
184C2×C8—1492no—
185C2×C8C22×D14912yes—
186C4⋊C4—21no—
187C4⋊C4—2982no—
188C4⋊C4—2982no—
189C4⋊C4—2981no—
190C4⋊C4—2981no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2981no—
194C4⋊C4—1492no—
195C4⋊C4—1491no—
196C4⋊C4—1491no—
197C4⋊C4C22×D14912yes—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—2981no—
201C22⋊C4—2981no—
202C22⋊C4—2981no—
203C42—2981no—
204C42—1491no—
205C42—1491no—
206C42—1491no—
207C42C22×D14911yescharacteristic
208C23—21no—
209C23—1491no—
210Q8—22no—
211Q8—42no—
212Q8—5962no—
213Q8—5964no—
214D4—22no—
215D4—5962no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—2982no—
219C2×C4—2982no—
220C2×C4—2982no—
221C2×C4—2981no—
222C2×C4—2981no—
223C2×C4—2982no—
224C2×C4—5964no—
225C2×C4—5962no—
226C2×C4—1491no—
227C2×C4—1491no—
228C2×C4—1491no—
229C2×C4D4×D14911yescharacteristic
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—22no—
233C8—2982no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—2981no—
238C22—2981no—
239C22—1492no—
240C22C2×D4×D14911yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—2981no—
246C4—2981no—
247C4—5962no—
248C4—5961no—
249C4—5961no—
250C4—1492no—
251C4#1801195812yes—
252C2—81no—
253C2—2982no—
254C2#2375019511yescharacteristic
255C2AGL(1,ℤ/8)×D14911yescharacteristic
256C2#137×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#489×C149C2
socleC22×C149C22×D4
prime radical#18011882C22
hypercenter#489D149
hypocenterC149#3239
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D149C2semidirect
3. #23750034C2semidirect
4. #23750031C2semidirect
5. #23749984C2semidirect
6. #23749959C2semidirect
8. #23749908C2semidirect
9. #23749903C2semidirect
10. #23749680C2semidirect
11. #489×C149C2semidirect
128. C149#3239semidirect
135. #489D149semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1492
#237501542
#237500342
#237500312
#237499842
#237499592
#237499422
#237499082
#237499032
#237496802
#489×C1492
#3239149