Finite Group Databasebeta

← #34054863

Subgroups — #34054863

Order 19072.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#23750161C211yesmaximal, characteristic
2C2×Q8×Dic149C211yesmaximal, characteristic, complemented
3#23749958C211yesmaximal, characteristic, complemented
4#23749944C211yesmaximal, characteristic
5#23749930C212yesmaximal, complemented
6#23749849C212yesmaximal, complemented
7#23749764C211yesmaximal, characteristic
8#23749732C212yesmaximal, complemented
9#23749728C212yesmaximal, complemented
10#23749679C211yesmaximal, characteristic, complemented
11#451×C149C211yesmaximal, characteristic, complemented
12C2×#14558723C2212yes—
13#18011906—21no—
14#18011902C2211yescharacteristic
15#18011900C2211yescharacteristic
16#18011895—21no—
17#18011885C2211yescharacteristic
18Q8×Dic149—21no—
19Q8×Dic149—21no—
20Q8×Dic149C2211yescharacteristic
21Q8×Dic149C2212yes—
22Q8×Dic149C2212yes—
23#18011873C2211yescharacteristic
24#18011871—21no—
25#18011869—21no—
26C2×C4⋊Dic149—21no—
27C2×C4⋊Dic149C2211yescharacteristic
28C2×C4×Dic149—21no—
29C2×C4×Dic149C2211yescharacteristic
30#18011844C2212yes—
31#18011843C2212yes—
32#18011840C2212yes—
33#18011835C2212yes—
34#18011835C2212yes—
35#18011833C2212yes—
36C22⋊#11924746C2211yescharacteristic
37#18011821C2212yes—
38C22×Q8×C149C2211yescharacteristic
39C2×Q16×C149C2212yes—
40C22⋊Q8×C149C2211yescharacteristic
41#103×C149C2212yes—
42C22⋊C8×C149C2211yescharacteristic
43C2×D596C2311yescharacteristic
44C2×C22⋊D149—21no—
45C22×Dic149—21no—
46C22×Dic149D411yescharacteristic
47C22×Dic149D411yescharacteristic
48C2×C4×D149—21no—
49#14558723—24no—
50C2×#11924746C2312yes—
51C4⋊Dic149—21no—
52C4⋊Dic149—22no—
53C4⋊Dic149—22no—
54C4⋊Dic149—21no—
55C4⋊Dic149—21no—
56C4⋊Dic149C2312yes—
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558713—21no—
60#14558713—21no—
61#14558713—21no—
62#14558713—21no—
63C4×Dic149—22no—
64C4×Dic149—22no—
65C4×Dic149—21no—
66C4×Dic149D411yescharacteristic
67C4×Dic149D411yescharacteristic
68C4×Dic149C2311yescharacteristic
69C4×Dic149C2311yescharacteristic
70C2×Q8×C149—24no—
71C2×Q8×C149—21no—
72C2×Q8×C149C2311yescharacteristic
73C2×Q8×C149C2312yes—
74C22×C4×C149—21no—
75C22×C4×C149C2311yescharacteristic
76Q16×C149—24no—
77C2×C8×C149C2312yes—
78C4⋊C4×C149—21no—
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C22×D149—21no—
82D596—22no—
83C22⋊D149—42no—
84C2×Dic149—22no—
85C2×Dic149—22no—
86C2×Dic149—21no—
87C2×Dic149—41no—
88C2×Dic149—42no—
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C2×Dic149C2×D411yescharacteristic
93C4×D149—42no—
94#11924746—22no—
95C23×C149C2×D411yescharacteristic
96Q8×C149—24no—
97Q8×C149—22no—
98Q8×C149—22no—
99Q8×C149#4114yes—
100C2×C4×C149—24no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149—21no—
104C2×C4×C149—21no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2411yescharacteristic
107C8×C149—22no—
108C22×C149—21no—
109C22×C149—21no—
110C22×C149#13712yes—
111C22×C149C22×D411yescharacteristic
112C2×D149—41no—
113C2×D149—41no—
114C2×D149—41no—
115C4×C149—24no—
116C4×C149—21no—
117C4×C149—41no—
118C4×C149—41no—
119C4×C149C2×#4112yes—
120Dic149—21no—
121Dic149—21no—
122Dic149—21no—
123Dic149—21no—
124Dic149—41no—
125Dic149—42no—
126C2×C149—22no—
127C2×C149#52511yescharacteristic
128C2×C149C2×#13711yescharacteristic
129C2×C149#57511yescharacteristic
130D149—81no—
131C149#311711yescharacteristic, complemented, Sylow
132#3117—1491nomaximal, Sylow
133#529—1491no—
134C2×C4×Q8—1491no—
135#484—1491no—
136#481—1491no—
137#479—1492no—
138#464—1492no—
139#451D14911yescharacteristic, complemented
140#448—1491no—
141#441—1492no—
142#440—1492no—
143#432—1491no—
144C22×Q8C2×D14911yescharacteristic
145C2×Q16C2×D14912yes—
146C2×SD16—1492no—
147#126—2981no—
148#125—1491no—
149#124—1491no—
150#123—2981no—
151C22⋊Q8C2×D14911yescharacteristic
152#121—1491no—
153C4×Q8—2981no—
154C4×Q8—2981no—
155C4×Q8—1492no—
156C4×Q8—1492no—
157C4×Q8—1491no—
158C4×D4—2981no—
159C4×D4—1491no—
160#117—2981no—
161C2×C4⋊C4—2981no—
162C2×C4⋊C4—1491no—
163C2×C42—2981no—
164C2×C42—1491no—
165#107—1492no—
166#106—1492no—
167C4⋊C8—1492no—
168#103—1492no—
169#103—1492no—
170#103C2×D14912yes—
171#102—1492no—
172C22⋊C8—1491no—
173C22⋊C8C2×D14911yescharacteristic
174#97—1492no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D14911yescharacteristic
178C2×Q8C22×D14912yes—
179C2×D4—2981no—
180C2×D4—1491no—
181C22×C4—21no—
182C22×C4—2981no—
183C22×C4—2981no—
184C22×C4—1491no—
185C22×C4—1491no—
186C22×C4C22×D14911yescharacteristic
187Q16—24no—
188SD16—2984no—
189C2×C8—1492no—
190C2×C8C22×D14912yes—
191C4⋊C4—21no—
192C4⋊C4—2982no—
193C4⋊C4—2982no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—2981no—
197C4⋊C4—2981no—
198C4⋊C4—1492no—
199C4⋊C4—1491no—
200C4⋊C4C22×D14911yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2981no—
203C22⋊C4—2981no—
204C22⋊C4—2981no—
205C22⋊C4—2981no—
206C42—2982no—
207C42—2982no—
208C42—2981no—
209C42—1491no—
210C42—1491no—
211C42—1491no—
212C42—1491no—
213C23—2981no—
214C23D4×D14911yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1455873514yes—
219D4—2982no—
220D4—5962no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—2982no—
227C2×C4—2982no—
228C2×C4—2981no—
229C2×C4—5962no—
230C2×C4—5961no—
231C2×C4—5962no—
232C2×C4—1491no—
233C2×C4—1491no—
234C2×C4—1491no—
235C2×C4—1491no—
236C2×C4D4×D14911yescharacteristic
237C2×C4C23×D14911yescharacteristic
238C8—22no—
239C8—2982no—
240C22—21no—
241C22—21no—
242C22—5961no—
243C22—5961no—
244C22—5961no—
245C22#1801194312yes—
246C22C2×D4×D14911yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—2981no—
252C4—2981no—
253C4—2981no—
254C4—2981no—
255C4—5961no—
256C4—5962no—
257C4C2×#1455873512yes—
258C2—22no—
259C2—11921no—
260C2#2375013111yescharacteristic
261C2C2×#1801194311yescharacteristic
262C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#451×C149C2
socleC22×C149C22×D4
prime radical#18011885C22
hypercenter#451D149
hypocenterC149#3117
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic149C2semidirect
3. #23749958C2semidirect
5. #23749930C2semidirect
6. #23749849C2semidirect
8. #23749732C2semidirect
9. #23749728C2semidirect
10. #23749679C2semidirect
11. #451×C149C2semidirect
131. C149#3117semidirect
139. #451D149semidirect

Maximal subgroups

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

subgroupindex
#237501612
C2×Q8×Dic1492
#237499582
#237499442
#237499302
#237498492
#237497642
#237497322
#237497282
#237496792
#451×C1492
#3117149