Finite Group Databasebeta

← #34056062

Subgroups — #34056062

Order 19072.

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

normal subgroups only

This group has 348 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#23750180C211yesmaximal, characteristic, complemented
2#23750162C211yesmaximal, characteristic
3#23749930C212yesmaximal, complemented
4#23749920C211yesmaximal, characteristic, complemented
5#23749916C211yesmaximal, characteristic, complemented
6#23749880C212yesmaximal, complemented
7#23749874C212yesmaximal
8#23749849C212yesmaximal, complemented
9#23749827C211yesmaximal, characteristic, complemented
10#23749647C211yesmaximal, characteristic, complemented
11#462×C149C211yesmaximal, characteristic, complemented
12C2×#14558723C2212yes—
13C2×#14558719C2212yes—
14#18011911C2211yescharacteristic
15#18011906—21no—
16#18011902—21no—
17#18011902C2211yescharacteristic
18#18011900C2211yescharacteristic
19#18011896—21no—
20#18011891—21no—
21#18011887C2211yescharacteristic
22#18011887C2212yes—
23#18011886C2211yescharacteristic
24Q8×Dic149C2212yes—
25#18011876—21no—
26C4×D596—21no—
27C4×D596C2211yescharacteristic
28#18011869—21no—
29#18011868—21no—
30C42×D149C2211yescharacteristic
31#18011844C2212yes—
32#18011843C2212yes—
33C4⋊#11924746C2211yescharacteristic
34#18011840C2212yes—
35#18011835C2212yes—
36#18011835C2212yes—
37#18011833C2212yes—
38C2×Q16×C149C2212yes—
39C4⋊Q8×C149C2211yescharacteristic
40C4×Q8×C149C2211yescharacteristic
41C4⋊C8×C149C2211yescharacteristic
42#103×C149C2212yes—
43C2×Dic298—21no—
44C2×D596—21no—
45C2×D596C2311yescharacteristic
46C2×C4×D149—21no—
47C2×C4×D149D411yescharacteristic
48C2×C4×D149D411yescharacteristic
49C2×C4×D149C2311yescharacteristic
50#14558723—42no—
51#14558719—42no—
52C2×#11924746C2312yes—
53C4⋊Dic149—21no—
54C4⋊Dic149—22no—
55C4⋊Dic149—21no—
56C4⋊Dic149C2312yes—
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558715—21no—
60#14558715—22no—
61#14558713—22no—
62#14558713—21no—
63#14558713—21no—
64#14558713—21no—
65#14558713—21no—
66#14558713—21no—
67C4×Dic149—22no—
68C4×Dic149D411yescharacteristic
69C4×Dic149D411yescharacteristic
70C4×Dic149C2311yescharacteristic
71C2×Q8×C149C2312yes—
72C2×Q8×C149C2311yescharacteristic
73Q16×C149—42no—
74C2×C8×C149C2312yes—
75C4⋊C4×C149—21no—
76C4⋊C4×C149—21no—
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2311yescharacteristic
79C42×C149—21no—
80C42×C149C2311yescharacteristic
81C22×D149—21no—
82C22×D149C2×D411yescharacteristic
83Dic298—42no—
84D596—21no—
85D596—21no—
86D596—42no—
87C2×Dic149—21no—
88C2×Dic149—22no—
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C4×D149—22no—
93C4×D149—21no—
94C4×D149—21no—
95C4×D149—41no—
96C4×D149—42no—
97#11924746—22no—
98Q8×C149—22no—
99Q8×C149—21no—
100Q8×C149—22no—
101Q8×C149—21no—
102C2×C4×C149—21no—
103C2×C4×C149—21no—
104C2×C4×C149—22no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2411yescharacteristic
108C8×C149—22no—
109C22×C149C22×D411yescharacteristic
110C2×D149—21no—
111C2×D149—21no—
112C2×D149—21no—
113C2×D149—41no—
114C2×D149—41no—
115C2×D149—41no—
116C4×C149—21no—
117C4×C149—41no—
118C4×C149—42no—
119C4×C149—41no—
120C4×C149#13712yes—
121C4×C149H(5,𝔽2)11yescharacteristic
122C4×C1492−1+411yescharacteristic
123Dic149—21no—
124Dic149—21no—
125Dic149—21no—
126Dic149—41no—
127Dic149—42no—
128C2×C149#53711yescharacteristic
129C2×C149C2×#13711yescharacteristic
130C2×C149#57511yescharacteristic
131D149—41no—
132D149—81no—
133C149#325211yescharacteristic, complemented, Sylow
134#3252—1491nomaximal, Sylow
135#533—1491no—
136#529—1491no—
137#479—1492no—
138#477—1491no—
139#476—1491no—
140#470—1492no—
141#468—1492no—
142#464—1492no—
143#462D14911yescharacteristic, complemented
144#461—1491no—
145#423—1491no—
146C2×Q16C2×D14912yes—
147C2×SD16—1492no—
148C2×M16—1492no—
149C4⋊Q8—1491no—
150C4⋊Q8C2×D14911yescharacteristic
151#126—2981no—
152#125—2981no—
153#125—1491no—
154#124—1491no—
155#123—2981no—
156C22⋊Q8—2981no—
157C22⋊Q8—1491no—
158C22⋊Q8—1492no—
159#121—1491no—
160C4×Q8—2981no—
161C4×Q8—1492no—
162C4×Q8C2×D14911yescharacteristic
163C4×D4—2981no—
164C4×D4—1491no—
165#117—2981no—
166#117—2981no—
167C2×C42—1491no—
168#107—1492no—
169#106—1492no—
170C4⋊C8—1492no—
171C4⋊C8—1491no—
172C4⋊C8C2×D14911yescharacteristic
173#103—1492no—
174#103—1492no—
175#103C2×D14912yes—
176#102—1492no—
177C2×Q8—2981no—
178C2×Q8C22×D14911yescharacteristic
179C2×Q8C22×D14912yes—
180C2×D4—2981no—
181C2×D4—1491no—
182C22×C4—2981no—
183C22×C4—1491no—
184C22×C4—1491no—
185C22×C4—1491no—
186Q16—42no—
187SD16—5962no—
188M16—5962no—
189C2×C8—1492no—
190C2×C8C22×D14912yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—2981no—
197C4⋊C4—2982no—
198C4⋊C4—2982no—
199C4⋊C4—2981no—
200C4⋊C4—1492no—
201C4⋊C4—1491no—
202C4⋊C4C22×D14911yescharacteristic
203C22⋊C4—2982no—
204C22⋊C4—2981no—
205C22⋊C4—2981no—
206C22⋊C4—2981no—
207C22⋊C4—2981no—
208C22⋊C4—2981no—
209C42—21no—
210C42—2982no—
211C42—1491no—
212C42—1491no—
213C42—1491no—
214C42C22×D14911yescharacteristic
215C23—2981no—
216C23—1491no—
217Q8—22no—
218Q8—21no—
219Q8—21no—
220Q8—22no—
221Q8—5962no—
222D4—2981no—
223D4—2981no—
224D4—5962no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—2982no—
229C2×C4—2981no—
230C2×C4—2981no—
231C2×C4—2982no—
232C2×C4—2981no—
233C2×C4—5961no—
234C2×C4—5962no—
235C2×C4—1491no—
236C2×C4—1491no—
237C2×C4—1491no—
238C2×C4D4×D14911yescharacteristic
239C2×C4D4×D14911yescharacteristic
240C2×C4C23×D14911yescharacteristic
241C8—22no—
242C8—2982no—
243C22—2981no—
244C22—2981no—
245C22—2981no—
246C22—5961no—
247C22—5961no—
248C22—5961no—
249C22C2×D4×D14911yescharacteristic
250C4—21no—
251C4—41no—
252C4—41no—
253C4—42no—
254C4—2981no—
255C4—2981no—
256C4—2981no—
257C4—5961no—
258C4—5962no—
259C4#1801194312yes—
260C4#1801195611yescharacteristic
261C4#1801195811yescharacteristic
262C2—5961no—
263C2—11921no—
264C2#2375020411yescharacteristic
265C2C2×#1801194311yescharacteristic
266C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#462×C149C2
socleC22×C149C22×D4
prime radical#18011886C22
hypercenter#462D149
hypocenterC149#3252
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23749930C2semidirect
4. #23749920C2semidirect
5. #23749916C2semidirect
6. #23749880C2semidirect
8. #23749849C2semidirect
9. #23749827C2semidirect
10. #23749647C2semidirect
11. #462×C149C2semidirect
133. C149#3252semidirect
143. #462D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237501622
#237499302
#237499202
#237499162
#237498802
#237498742
#237498492
#237498272
#237496472
#462×C1492
#3252149