Finite Group Databasebeta

← #34055896

Subgroups — #34055896

Order 19072.

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

normal subgroups only

This group has 324 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
1#23750177C211yesmaximal, characteristic
2C2×#18011902C211yesmaximal, characteristic
3#23750033C212yesmaximal, complemented
4#23749992C212yesmaximal, complemented
5#23749960C211yesmaximal, characteristic
6#23749956C211yesmaximal, characteristic
7#23749953C211yesmaximal, characteristic
8#23749931C212yesmaximal, complemented
9#23749930C212yesmaximal, complemented
10#23749679C211yesmaximal, characteristic
11#484×C149C211yesmaximal, characteristic
12#18011910C2211yescharacteristic
13#18011909C2211yescharacteristic
14#18011902—21no—
15#18011902—21no—
16#18011902C2212yes—
17#18011902C2212yes—
18#18011901—21no—
19#18011890C2211yescharacteristic
20#18011888—21no—
21Q8×Dic149C2211yescharacteristic
22#18011876—21no—
23#18011876C2211yescharacteristic
24#18011867—21no—
25C2×C4⋊Dic149—21no—
26C2×C4⋊Dic149C2211yescharacteristic
27C2×#14558715—21no—
28C2×C4×Dic149C2211yescharacteristic
29#18011844C2212yes—
30#18011843C2212yes—
31#18011842C2212yes—
32#18011840C2212yes—
33#18011836C2212yes—
34#18011835C2212yes—
35#18011834C2212yes—
36C22⋊#11924746C2211yescharacteristic
37#18011821C2212yes—
38C22⋊Q8×C149C2211yescharacteristic
39C2×C4⋊C4×C149C2211yescharacteristic
40#107×C149C2212yes—
41#103×C149C2212yes—
42C22⋊C8×C149C2211yescharacteristic
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45C22×Dic149—21no—
46C22×Dic149D411yescharacteristic
47C22×Dic149D411yescharacteristic
48C2×#11924746C2312yes—
49C4⋊Dic149—22no—
50C4⋊Dic149—22no—
51C4⋊Dic149—21no—
52C4⋊Dic149—21no—
53C4⋊Dic149C2311yescharacteristic
54C4⋊Dic149C2312yes—
55#14558715—21no—
56#14558715—21no—
57#14558715—21no—
58#14558715—22no—
59#14558715—22no—
60#14558715—21no—
61#14558715—21no—
62C22⋊Dic149—21no—
63C4×Dic149—21no—
64C4×Dic149—21no—
65C4×Dic149D411yescharacteristic
66C4×Dic149D411yescharacteristic
67C4×Dic149C2311yescharacteristic
68C4×Dic149C2311yescharacteristic
69C2×Q8×C149C2311yescharacteristic
70C22×C4×C149—21no—
71C22×C4×C149C2311yescharacteristic
72C2×C8×C149C2312yes—
73C4⋊C4×C149—21no—
74C4⋊C4×C149—21no—
75C4⋊C4×C149C2311yescharacteristic
76C4⋊C4×C149C2312yes—
77C22⋊C4×C149—21no—
78Dic298—22no—
79Dic298—42no—
80C2×Dic149—22no—
81C2×Dic149—22no—
82C2×Dic149—21no—
83C2×Dic149—21no—
84C2×Dic149—41no—
85C2×Dic149—42no—
86C2×Dic149C2×D411yescharacteristic
87C2×Dic149C2×D411yescharacteristic
88C2×Dic149C2×D411yescharacteristic
89C2×Dic149C2×D411yescharacteristic
90#11924746—22no—
91C23×C149C2×D411yescharacteristic
92Q8×C149—22no—
93C2×C4×C149—22no—
94C2×C4×C149—21no—
95C2×C4×C149—21no—
96C2×C4×C149—22no—
97C2×C4×C149—42no—
98C2×C4×C149C2×D411yescharacteristic
99C2×C4×C149C2411yescharacteristic
100C8×C149—22no—
101C22×C149—21no—
102C22×C149—21no—
103C22×C149#13712yes—
104C22×C149C22×D411yescharacteristic
105C4×C149—21no—
106C4×C149—41no—
107C4×C149—42no—
108C4×C149—41no—
109C4×C1492−1+412yes—
110Dic149—21no—
111Dic149—21no—
112Dic149—21no—
113Dic149—21no—
114Dic149—41no—
115Dic149—42no—
116Dic149—41no—
117C2×C149—22no—
118C2×C149#53611yescharacteristic
119C2×C149C2×#13711yescharacteristic
120C2×C149#57511yescharacteristic
121C149#323611yescharacteristic, complemented, Sylow
122#3236—1491nomaximal, Sylow
123#533—1491no—
124C2×#125—1491no—
125#501—1492no—
126#491—1492no—
127#484—1491no—
128#484D14911yescharacteristic
129#483—1491no—
130#483—1491no—
131#479—1492no—
132#479—1492no—
133#432—1491no—
134C4⋊Q8—1491no—
135C4⋊Q8—1491no—
136#125—2981no—
137#125—2981no—
138#125—2981no—
139#125—1492no—
140#125—1492no—
141C22⋊Q8—2981no—
142C22⋊Q8—1491no—
143C22⋊Q8C2×D14911yescharacteristic
144C4×Q8—2981no—
145C4×Q8—1491no—
146C4×Q8—1491no—
147#117—2981no—
148C2×C4⋊C4—2981no—
149C2×C4⋊C4—2981no—
150C2×C4⋊C4—1491no—
151C2×C4⋊C4C2×D14911yescharacteristic
152C2×C42—1491no—
153#107—1492no—
154#107C2×D14912yes—
155#106—1492no—
156#106—1492no—
157C4⋊C8—1492no—
158#103—1492no—
159#103—1492no—
160#103—1492no—
161#103C2×D14912yes—
162C22⋊C8—1491no—
163C22⋊C8C2×D14911yescharacteristic
164#97—1492no—
165C2×Q8—2981no—
166C2×Q8—1491no—
167C2×Q8C22×D14911yescharacteristic
168C22×C4—21no—
169C22×C4—2981no—
170C22×C4—1491no—
171C22×C4—1491no—
172C22×C4C22×D14911yescharacteristic
173C2×C8—1492no—
174C2×C8C22×D14912yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—2982no—
178C4⋊C4—2982no—
179C4⋊C4—2982no—
180C4⋊C4—2982no—
181C4⋊C4—2981no—
182C4⋊C4—2981no—
183C4⋊C4—2981no—
184C4⋊C4—2981no—
185C4⋊C4—2981no—
186C4⋊C4—2981no—
187C4⋊C4—2981no—
188C4⋊C4—1492no—
189C4⋊C4—1491no—
190C4⋊C4C22×D14912yes—
191C4⋊C4C22×D14911yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—2981no—
194C42—2981no—
195C42—2981no—
196C42—1491no—
197C42—1491no—
198C42—1491no—
199C42—1491no—
200C23D4×D14911yescharacteristic
201Q8—22no—
202Q8—2982no—
203Q8—5962no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2982no—
210C2×C4—2981no—
211C2×C4—2981no—
212C2×C4—2982no—
213C2×C4—5961no—
214C2×C4—5962no—
215C2×C4—1491no—
216C2×C4—1491no—
217C2×C4—1491no—
218C2×C4—1491no—
219C2×C4D4×D14911yescharacteristic
220C2×C4C23×D14911yescharacteristic
221C8—22no—
222C8—2982no—
223C22—21no—
224C22—21no—
225C22#1801194312yes—
226C22C2×D4×D14911yescharacteristic
227C4—21no—
228C4—42no—
229C4—41no—
230C4—41no—
231C4—2981no—
232C4—2981no—
233C4—2981no—
234C4—2981no—
235C4—5962no—
236C4—5961no—
237C4—5961no—
238C4#1801195812yes—
239C2—22no—
240C2#2375019711yescharacteristic
241C2C2×#1801194311yescharacteristic
242C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#484×C149C2
socleC22×C149C22×D4
prime radicalC23×C149C2×D4
hypercenter#484D149
hypocenterC149#3236
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #23750033C2semidirect
4. #23749992C2semidirect
8. #23749931C2semidirect
9. #23749930C2semidirect
121. C149#3236semidirect

Maximal subgroups

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

subgroupindex
#237501772
C2×#180119022
#237500332
#237499922
#237499602
#237499562
#237499532
#237499312
#237499302
#237496792
#484×C1492
#3236149