Finite Group Databasebeta

← #34055296

Subgroups — #34055296

Order 19072.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#23750160C211yesmaximal, characteristic
3#23750035C211yesmaximal, characteristic, complemented
4#23750017C211yesmaximal, characteristic, complemented
5#23749985C211yesmaximal, characteristic, complemented
6#23749882C212yesmaximal, complemented
7#23749874C212yesmaximal
8#23749717C212yesmaximal, complemented
9#23749710C212yesmaximal, complemented
10#23749577C211yesmaximal, characteristic, complemented
11#491×C149C211yesmaximal, characteristic, complemented
12C2×#14558726C2212yes—
13C2×#14558723C2212yes—
14C2×#14558719C2212yes—
15#18011911C2211yescharacteristic
16#18011906—21no—
17#18011902—21no—
18#18011899C2211yescharacteristic
19#18011896—21no—
20#18011891—21no—
21#18011891C2211yescharacteristic
22#18011887C2212yes—
23#18011886C2211yescharacteristic
24Q8×Dic149C2212yes—
25#18011876—21no—
26#18011876C2211yescharacteristic
27#18011873—21no—
28#18011873C2211yescharacteristic
29#18011869—21no—
30#18011869—21no—
31C42×D149C2211yescharacteristic
32#18011844C2212yes—
33#18011842C2212yes—
34#18011836C2212yes—
35#18011833C2212yes—
36#18011822C2211yescharacteristic
37C8×Dic149C2212yes—
38C4⋊Q8×C149C2211yescharacteristic
39#125×C149C2211yescharacteristic
40#103×C149C2212yes—
41#103×C149C2212yes—
42#97×C149C2211yescharacteristic
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45C2×D596—21no—
46C2×D596C2311yescharacteristic
47C2×C4×D149—21no—
48C2×C4×D149D411yescharacteristic
49C2×C4×D149D411yescharacteristic
50C2×C4×D149C2311yescharacteristic
51#14558726—24no—
52#14558723—24no—
53#14558719—24no—
54C2×#11924746C2312yes—
55C4⋊Dic149—21no—
56C4⋊Dic149—22no—
57C4⋊Dic149—21no—
58C4⋊Dic149C2312yes—
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—21no—
68C4×Dic149—22no—
69C4×Dic149D411yescharacteristic
70C4×Dic149D411yescharacteristic
71C4×Dic149C2311yescharacteristic
72C2×Q8×C149C2312yes—
73C2×C8×C149C2312yes—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—21no—
76C4⋊C4×C149—21no—
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2311yescharacteristic
79C4⋊C4×C149C2311yescharacteristic
80C42×C149C2311yescharacteristic
81C22×D149—21no—
82C22×D149C2×D411yescharacteristic
83Dic298—22no—
84Dic298—42no—
85D596—22no—
86D596—42no—
87C2×Dic149—22no—
88C2×Dic149—21no—
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C4×D149—22no—
93C4×D149—22no—
94C4×D149—41no—
95C4×D149—42no—
96#11924746#4114yes—
97Q8×C149—22no—
98Q8×C149—22no—
99C2×C4×C149—21no—
100C2×C4×C149—21no—
101C2×C4×C149—22no—
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2411yescharacteristic
105C8×C149—22no—
106C22×C149C22×D411yescharacteristic
107C2×D149—21no—
108C2×D149—21no—
109C2×D149—21no—
110C2×D149—41no—
111C2×D149—41no—
112C2×D149—41no—
113C4×C149—21no—
114C4×C149—21no—
115C4×C149—42no—
116C4×C149—41no—
117C4×C149—41no—
118C4×C149C2×#4112yes—
119Dic149—21no—
120Dic149—21no—
121Dic149—41no—
122Dic149—42no—
123Dic149#13712yes—
124C2×C149#53211yescharacteristic
125C2×C149C2×#13711yescharacteristic
126C2×C149#57511yescharacteristic
127D149—41no—
128D149—81no—
129C149#317211yescharacteristic, complemented, Sylow
130#3172—1491nomaximal, Sylow
131#533—1491no—
132#529—1491no—
133#501—1491no—
134#497—1491no—
135#491D14911yescharacteristic, complemented
136#489—1491no—
137#470—1492no—
138#468—1492no—
139C4×Q16—1492no—
140C4×SD16—1492no—
141C4×M16—1491no—
142C2×Q16—1492no—
143C2×SD16—1492no—
144C2×M16—1492no—
145C4⋊Q8—1491no—
146C4⋊Q8C2×D14911yescharacteristic
147#126—2981no—
148#125—2981no—
149#125C2×D14911yescharacteristic
150#124—1491no—
151#123—2981no—
152C22⋊Q8—2981no—
153C22⋊Q8—1492no—
154C22⋊Q8—1491no—
155#121—1491no—
156C4×Q8—2981no—
157C4×Q8—1492no—
158C4×Q8—1491no—
159C4×D4—2981no—
160C4×D4—1491no—
161#117—2981no—
162#117—2981no—
163C2×C42—1491no—
164#107—1492no—
165#106—1492no—
166#103—1492no—
167#103C2×D14912yes—
168#103C2×D14912yes—
169#102—1492no—
170#97—1491no—
171#97C2×D14911yescharacteristic
172C4×C8—1492no—
173C2×Q8—2981no—
174C2×Q8—1491no—
175C2×Q8C22×D14912yes—
176C2×D4—2981no—
177C2×D4—1491no—
178C22×C4—2981no—
179C22×C4—1491no—
180C22×C4—1491no—
181C22×C4—1491no—
182Q16—2984no—
183SD16—2984no—
184M16—2984no—
185C2×C8—1492no—
186C2×C8C22×D14912yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—2981no—
192C4⋊C4—2982no—
193C4⋊C4—2982no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—1492no—
197C4⋊C4C22×D14911yescharacteristic
198C4⋊C4C22×D14911yescharacteristic
199C22⋊C4—2982no—
200C22⋊C4—2981no—
201C22⋊C4—2981no—
202C22⋊C4—2981no—
203C22⋊C4—2981no—
204C22⋊C4—2981no—
205C42—2981no—
206C42—2982no—
207C42—1491no—
208C42—1491no—
209C42—1491no—
210C42C22×D14911yescharacteristic
211C23—2981no—
212C23—1491no—
213Q8—22no—
214Q8—22no—
215Q8—2982no—
216Q8—5962no—
217D4—2982no—
218D4—5962no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—2982no—
223C2×C4—2982no—
224C2×C4—2981no—
225C2×C4—2982no—
226C2×C4—5961no—
227C2×C4—5962no—
228C2×C4—1491no—
229C2×C4—1491no—
230C2×C4—1491no—
231C2×C4D4×D14911yescharacteristic
232C2×C4D4×D14911yescharacteristic
233C2×C4C23×D14911yescharacteristic
234C8—22no—
235C8—1494no—
236C22—2981no—
237C22—2981no—
238C22—2981no—
239C22—5961no—
240C22—5961no—
241C22—5961no—
242C22C2×D4×D14911yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—2981no—
249C4—2981no—
250C4—5962no—
251C4—5961no—
252C4—1492no—
253C4#41×D14912yes—
254C2—5961no—
255C2—11921no—
256C2#2375017411yescharacteristic
257C2#137×D14911yescharacteristic
258C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#491×C149C2
socleC22×C149C22×D4
prime radical#18011886C22
hypercenter#491D149
hypocenterC149#3172
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23750035C2semidirect
4. #23750017C2semidirect
5. #23749985C2semidirect
6. #23749882C2semidirect
8. #23749717C2semidirect
9. #23749710C2semidirect
10. #23749577C2semidirect
11. #491×C149C2semidirect
129. C149#3172semidirect
135. #491D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237501602
#237500352
#237500172
#237499852
#237498822
#237498742
#237497172
#237497102
#237495772
#491×C1492
#3172149