Finite Group Databasebeta

← #34058835

Subgroups — #34058835

Order 19072.

930 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C149⋊#564C211yesmaximal, characteristic, complemented
2#23750378C214yesmaximal, complemented
3#23750376C214yesmaximal, complemented
4#23750375C212yesmaximal, complemented
5#23750370C214yesmaximal, complemented
6#23750181C211yesmaximal, characteristic, complemented
7#23750177C214yesmaximal, complemented
8#23750156C214yesmaximal, complemented
9#23750154C212yesmaximal, complemented
10#23750152C214yesmaximal, complemented
11#562×C149C211yesmaximal, characteristic, complemented
12#18011910C2214yes—
13#18011910C2212yes—
14#18011909C2214yes—
15#18011906C2214yes—
16#18011904C2214yes—
17#18011904C2214yes—
18#18011904C2214yes—
19#18011904C2214yes—
20#18011903C2212yes—
21#18011903C2211yescharacteristic
22#18011902C2214yes—
23#18011901C2214yes—
24#18011901C2214yes—
25#18011901C2214yes—
26#18011898C2214yes—
27#18011897C2214yes—
28#18011894C2214yes—
29#18011893C2214yes—
30#18011892C2214yes—
31#18011892C2214yes—
32#18011890C2214yes—
33#18011890C2214yes—
34#18011889C2214yes—
35#18011888C2214yes—
36#18011888C2214yes—
37#18011888C2214yes—
38#18011887C2212yes—
39#18011883C2214yes—
40#18011882C2212yes—
41Q8×Dic149C2212yes—
42#18011876C2214yes—
43#18011876C2214yes—
44D4×Dic149C2214yes—
45D4×Dic149C2212yes—
46#18011871C2214yes—
47#18011869C2212yes—
48#18011867C2214yes—
49#18011867C2214yes—
50C2×C4×Dic149C2214yes—
51C42×D149C2211yescharacteristic
52#125×C149C2211yescharacteristic
53#124×C149C2212yes—
54#123×C149C2214yes—
55C22⋊Q8×C149C2214yes—
56#121×C149C2214yes—
57C2×Dic298C2314yes—
58C2×Dic298C2314yes—
59C2×C22⋊D149C2314yes—
60C22×Dic149C2314yes—
61C22×Dic149C2314yes—
62C2×C4×D149C2311yescharacteristic
63C2×C4×D149C2312yes—
64C4⋊Dic149C2312yes—
65C4⋊Dic149C2312yes—
66C4⋊Dic149C2314yes—
67C4⋊Dic149C2314yes—
68C4⋊Dic149C2312yes—
69#14558715C2314yes—
70#14558715C2314yes—
71#14558715C2314yes—
72#14558715C2314yes—
73#14558715C2314yes—
74#14558715C2314yes—
75#14558715C2314yes—
76#14558715C2314yes—
77#14558715C2314yes—
78C22⋊Dic149C2314yes—
79C22⋊Dic149C2314yes—
80C22⋊Dic149C2314yes—
81C22⋊Dic149C2314yes—
82C22⋊Dic149C2314yes—
83#14558713C2314yes—
84#14558713C2314yes—
85C4×Dic149C2311yescharacteristic
86C4×Dic149C2314yes—
87C4×Dic149C2312yes—
88C4×Dic149C2314yes—
89C4×Dic149C2314yes—
90C4×Dic149C2314yes—
91C2×Q8×C149C2312yes—
92C2×D4×C149C2314yes—
93C2×D4×C149C2312yes—
94C22×C4×C149C2314yes—
95C4⋊C4×C149C2314yes—
96C4⋊C4×C149C2312yes—
97C4⋊C4×C149C2314yes—
98C22⋊C4×C149C2314yes—
99C22⋊C4×C149C2314yes—
100C22⋊C4×C149C2314yes—
101C42×C149C2311yescharacteristic
102C22×D149C2411yescharacteristic
103Dic298—44no—
104Dic298—44no—
105C22⋊D149—44no—
106C2×Dic149—28no—
107C2×Dic149—44no—
108C2×Dic149C2412yes—
109C2×Dic149C2414yes—
110C2×Dic149C2414yes—
111C2×Dic149C2414yes—
112C2×Dic149C2411yescharacteristic
113C4×D149—22no—
114C4×D149—42no—
115C23×C149C2414yes—
116Q8×C149—42no—
117D4×C149—44no—
118D4×C149—42no—
119C2×C4×C149—44no—
120C2×C4×C149C2411yescharacteristic
121C2×C4×C149C2412yes—
122C2×C4×C149C2414yes—
123C2×C4×C149C2414yes—
124C22×C149—24no—
125C22×C149—24no—
126C22×C149—24no—
127C22×C149C2511yescharacteristic
128C2×D149—21no—
129C2×D149—22no—
130C4×C149—21no—
131C4×C149—24no—
132C4×C149—24no—
133C4×C149—22no—
134Dic149—24no—
135Dic149—22no—
136Dic149—24no—
137Dic149—24no—
138Dic1492−1+412yes—
139C2×C149—44no—
140C2×C149C2×2−1+411yescharacteristic
141C2×C149#58512yes—
142D149—41no—
143C149#353111yescharacteristic, complemented, Sylow
144#3531—1491nomaximal, Sylow
145#564—1491no—
146#563—1492no—
147#563—1494no—
148#563—1494no—
149#562—1494no—
150#562D14911yescharacteristic, complemented
151#533—1491no—
152#533—1494no—
153#529—1492no—
154#529—1494no—
155#529—1494no—
156C4⋊Q8—1494no—
157C4⋊Q8—1494no—
158C4⋊Q8—1492no—
159#126—1494no—
160#126—1494no—
161#126—1494no—
162#126—1494no—
163#126—1494no—
164#125—1494no—
165#125—1494no—
166#125—1494no—
167#125—1494no—
168#125—1492no—
169#125—1491no—
170#125C2×D14911yescharacteristic
171#124—1494no—
172#124—1494no—
173#124C2×D14912yes—
174#123—1494no—
175#123—1494no—
176#123—1494no—
177#123—1494no—
178#123C2×D14914yes—
179C22⋊Q8—1492no—
180C22⋊Q8—1494no—
181C22⋊Q8—1494no—
182C22⋊Q8—1494no—
183C22⋊Q8—1494no—
184C22⋊Q8—1494no—
185C22⋊Q8—1494no—
186C22⋊Q8C2×D14914yes—
187#121—1492no—
188#121—1494no—
189#121C2×D14914yes—
190C4×Q8—1492no—
191C4×Q8—1494no—
192C4×Q8—1494no—
193C4×D4—1492no—
194C4×D4—1494no—
195C4×D4—1494no—
196#117—1492no—
197#117—1494no—
198#117—1494no—
199C2×C42—1494no—
200C2×C42—1491no—
201C2×Q8—1494no—
202C2×Q8—1494no—
203C2×Q8C22×D14912yes—
204C2×D4—1494no—
205C2×D4C22×D14912yes—
206C2×D4C22×D14914yes—
207C22×C4—1494no—
208C22×C4—1491no—
209C22×C4—1494no—
210C22×C4—1492no—
211C22×C4C22×D14914yes—
212C4⋊C4—1494no—
213C4⋊C4—1494no—
214C4⋊C4—1492no—
215C4⋊C4—1494no—
216C4⋊C4—1494no—
217C4⋊C4—1494no—
218C4⋊C4—1494no—
219C4⋊C4—1492no—
220C4⋊C4—1494no—
221C4⋊C4—1492no—
222C4⋊C4—1494no—
223C4⋊C4—1494no—
224C4⋊C4—1494no—
225C4⋊C4—1494no—
226C4⋊C4C22×D14912yes—
227C4⋊C4C22×D14914yes—
228C4⋊C4C22×D14914yes—
229C22⋊C4—1494no—
230C22⋊C4—1494no—
231C22⋊C4—1494no—
232C22⋊C4—1494no—
233C22⋊C4—1494no—
234C22⋊C4—1494no—
235C22⋊C4—1494no—
236C22⋊C4C22×D14914yes—
237C22⋊C4C22×D14914yes—
238C22⋊C4C22×D14914yes—
239C42—1494no—
240C42—1492no—
241C42—1494no—
242C42—1494no—
243C42—1494no—
244C42—1491no—
245C42C22×D14911yescharacteristic
246C23—1491no—
247C23C23×D14914yes—
248Q8—42no—
249Q8—5964no—
250Q8—5964no—
251D4—44no—
252D4—42no—
253D4—5964no—
254C2×C4—44no—
255C2×C4—2982no—
256C2×C4—2988no—
257C2×C4—5962no—
258C2×C4—5964no—
259C2×C4—1494no—
260C2×C4—1492no—
261C2×C4—1494no—
262C2×C4—1494no—
263C2×C4—1491no—
264C2×C4C23×D14914yes—
265C2×C4C23×D14914yes—
266C2×C4C23×D14911yescharacteristic
267C2×C4C23×D14912yes—
268C22—24no—
269C22—24no—
270C22—24no—
271C22—2981no—
272C22—2982no—
273C22C24×D14911yescharacteristic
274C4—21no—
275C4—24no—
276C4—24no—
277C4—22no—
278C4—2984no—
279C4—2984no—
280C4—2982no—
281C4—2984no—
282C4—1492no—
283C2—44no—
284C2—5961no—
285C22−1+4×D14911yescharacteristic
286C2#2375046012yes—
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#562×C149C2
hypercenter#562D149
hypocenterC149#3531
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. C149⋊#564C2semidirect
2. #23750378C2semidirect
3. #23750376C2semidirect
4. #23750375C2semidirect
5. #23750370C2semidirect
6. #23750181C2semidirect
7. #23750177C2semidirect
8. #23750156C2semidirect
9. #23750154C2semidirect
10. #23750152C2semidirect
11. #562×C149C2semidirect
143. C149#3531semidirect
150. #562D149semidirect

Maximal subgroups

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

subgroupindex
C149⋊#5642
#237503782
#237503762
#237503752
#237503702
#237501812
#237501772
#237501562
#237501542
#237501522
#562×C1492
#3531149