Finite Group Databasebeta

← #34056559

Subgroups — #34056559

Order 19072.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#23750290C211yesmaximal, characteristic, complemented
2#23750280C211yesmaximal, characteristic
3#23749975C211yesmaximal, characteristic, complemented
4#23749954C212yesmaximal, complemented
5#23749911C211yesmaximal, characteristic, complemented
6#23749851C212yesmaximal, complemented
7#23749762C212yesmaximal, complemented
8C4×#14558724C211yesmaximal, characteristic, complemented
9#23749690C211yesmaximal, characteristic, complemented
10#103×D149C212yesmaximal, complemented
11#461×C149C211yesmaximal, characteristic, complemented
12C2×#14558736—21no—
13C2×Q8×D149C2212yes—
14C2×#14558724C2211yescharacteristic
15C2×C8×D149C2212yes—
16#18011911C2211yescharacteristic
17#18011910—21no—
18#18011893—22no—
19#18011890—21no—
20#18011889—22no—
21#18011887C2212yes—
22#18011882C2212yes—
23C4×Dic298C2211yescharacteristic
24C4×D596C2211yescharacteristic
25#18011873—21no—
26C4⋊C4×D149C2212yes—
27#18011843C2212yes—
28#18011842C2211yescharacteristic
29#18011836C2212yes—
30#18011835C2212yes—
31#18011835C2212yes—
32#18011834C2211yescharacteristic
33#18011832C2211yescharacteristic
34#18011823C2212yes—
35C4×#11924746C2211yescharacteristic
36C2×SD16×C149C2212yes—
37C4⋊Q8×C149C2211yescharacteristic
38C4×D4×C149C2211yescharacteristic
39C4⋊C8×C149C2211yescharacteristic
40#103×C149C2212yes—
41#14558736—44no—
42C2×Dic298—22no—
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45Q8×D149—24no—
46Q8×D149—42no—
47C2×D596D411yescharacteristic
48C2×C22⋊D149—22no—
49C2×C4×D149—22no—
50C2×C4×D149—22no—
51C2×C4×D149C2312yes—
52#14558724—22no—
53C2×#11924746C2311yescharacteristic
54C2×#11924746C2311yescharacteristic
55C8×D149—42no—
56C4⋊Dic149—21no—
57C4⋊Dic149D411yescharacteristic
58C4⋊Dic149C2312yes—
59C4⋊Dic149C2311yescharacteristic
60#14558715—22no—
61#14558715—22no—
62#14558715—21no—
63C22⋊Dic149—22no—
64#14558713—22no—
65#14558713D412yes—
66C4×Dic149—21no—
67C2×Q8×C149C2312yes—
68C2×D4×C149C2311yescharacteristic
69C22×C4×C149—21no—
70SD16×C149—42no—
71C2×C8×C149C2312yes—
72C4⋊C4×C149—21no—
73C4⋊C4×C149—21no—
74C4⋊C4×C149C2311yescharacteristic
75C22⋊C4×C149—21no—
76C42×C149C2311yescharacteristic
77C22×D149C2×D412yes—
78Dic298—21no—
79Dic298—44no—
80Dic298—42no—
81Dic298#4112yescomplemented
82D596SD1614yescomplemented
83C22⋊D149—44no—
84C2×Dic149—21no—
85C2×Dic149—22no—
86C2×Dic149—21no—
87C2×Dic149C2×D412yes—
88C4×D149—22no—
89C4×D149—22no—
90C4×D149—44no—
91C4×D149—44no—
92#11924746—21no—
93#11924746—21no—
94C23×C149—21no—
95Q8×C149—22no—
96Q8×C149—22no—
97D4×C149—21no—
98D4×C149—21no—
99C2×C4×C149—21no—
100C2×C4×C149—22no—
101C2×C4×C149—42no—
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2411yescharacteristic
105C8×C149—22no—
106C22×C149—41no—
107C22×C149—41no—
108C22×C149—41no—
109C22×C149C22×D411yescharacteristic
110C2×D149—22no—
111C2×D149—22no—
112C2×D149C2×SD1614yes—
113C4×C149—21no—
114C4×C149—42no—
115C4×C149—41no—
116C4×C149C2×SD1612yes—
117C4×C149C2×#4111yescharacteristic
118C4×C149H(5,𝔽2)11yescharacteristic
119Dic149—22no—
120Dic149—22no—
121Dic149—42no—
122Dic149—41no—
123C2×C149—81no—
124C2×C149#54611yescharacteristic
125C2×C149C22×SD1611yescharacteristic
126C2×C149#57811yescharacteristic
127D149—24no—
128C149#329811yescharacteristic, complemented, Sylow
129#3298—1491nomaximal, Sylow
130D4×Q8—1491no—
131#547—1491no—
132#487—1491no—
133#483—1492no—
134#475—1491no—
135#465—1492no—
136#461D14911yescharacteristic, complemented
137#448—1492no—
138C4×SD16—1491no—
139C8×D4—1491no—
140C2×#103—1492no—
141C2×#41—2981no—
142C22×Q8—1492no—
143C2×SD16—1491no—
144C2×SD16C2×D14912yes—
145C22×C8—1492no—
146C4⋊Q8—2981no—
147C4⋊Q8—1491no—
148C4⋊Q8C2×D14911yescharacteristic
149#123—2982no—
150C22⋊Q8—2981no—
151C22⋊Q8—2982no—
152C22⋊Q8—1492no—
153#121—1492no—
154C4×Q8—1491no—
155C4×D4—2981no—
156C4×D4—1491no—
157C4×D4C2×D14911yescharacteristic
158C2×C4⋊C4—1492no—
159#106—1492no—
160#106—1491no—
161C4⋊C8C2×D14911yescharacteristic
162#103—1492no—
163#103—1492no—
164#103—1492no—
165#103—1491no—
166#103C2×D14912yes—
167#102—1491no—
168C22⋊C8—1492no—
169C4×C8—1491no—
170#41—5964no—
171C2×Q8—2982no—
172C2×Q8—2981no—
173C2×Q8—2984no—
174C2×Q8—5962no—
175C2×Q8—1491no—
176C2×Q8C22×D14912yes—
177C2×D4—2982no—
178C2×D4—1491no—
179C2×D4C22×D14911yescharacteristic
180C22×C4—21no—
181C22×C4—2982no—
182C22×C4—2982no—
183C22×C4—1492no—
184SD16—42no—
185SD16—2982no—
186C2×C8—5962no—
187C2×C8—1491no—
188C2×C8—1491no—
189C2×C8C22×D14912yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—2982no—
193C4⋊C4—2982no—
194C4⋊C4—2981no—
195C4⋊C4—2981no—
196C4⋊C4—1491no—
197C4⋊C4—1492no—
198C4⋊C4—1491no—
199C4⋊C4C22×D14911yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—2982no—
202C22⋊C4—2982no—
203C22⋊C4—1492no—
204C42—2981no—
205C42C22×D14911yescharacteristic
206C23—21no—
207C23—1492no—
208Q8—22no—
209Q8—22no—
210Q8—2981no—
211Q8—5964no—
212Q8—5962no—
213Q8—1492no—
214D4—21no—
215D4—21no—
216D4—5964no—
217D4—1494no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—2981no—
222C2×C4—2982no—
223C2×C4—2982no—
224C2×C4—2981no—
225C2×C4—2982no—
226C2×C4—5964no—
227C2×C4—5964no—
228C2×C4—1492no—
229C2×C4D4×D14911yescharacteristic
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—22no—
233C8—2981no—
234C8—2981no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—2982no—
239C22—2982no—
240C22—1494no—
241C22C2×D4×D14911yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—2982no—
246C4—2982no—
247C4—5961no—
248C4—5962no—
249C4SD16×D14912yes—
250C4#41×D14911yescharacteristic
251C4#1801195611yescharacteristic
252C2—81no—
253C2—2984no—
254C2#2375027211yescharacteristic
255C2C2×SD16×D14911yescharacteristic
256C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#461×C149C2
socleC22×C149C22×D4
prime radical#23750280C2
hypercenter#461D149
hypocenterC149#3298
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23750290C2semidirect
3. #23749975C2semidirect
4. #23749954C2semidirect
5. #23749911C2semidirect
6. #23749851C2semidirect
7. #23749762C2semidirect
8. C4×#14558724C2semidirect
9. #23749690C2semidirect
10. #103×D149C2semidirect
11. #461×C149C2semidirect
81. Dic298#41semidirect
82. D596SD16semidirect
128. C149#3298semidirect
136. #461D149semidirect

Maximal subgroups

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

subgroupindex
#237502902
#237502802
#237499752
#237499542
#237499112
#237498512
#237497622
C4×#145587242
#237496902
#103×D1492
#461×C1492
#3298149