Finite Group Databasebeta

← #34057615

Subgroups — #34057615

Order 19072.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750351C211yesmaximal, characteristic, complemented
2#23750290C211yesmaximal, characteristic
3#23750033C211yesmaximal, characteristic, complemented
4#23750031C211yesmaximal, characteristic, complemented
5#23750029C211yesmaximal, characteristic, complemented
6#23749928C211yesmaximal, characteristic
7#23749926C211yesmaximal, characteristic
8#23749918C211yesmaximal, characteristic, complemented
9#23749908C211yesmaximal, characteristic, complemented
10#23749904C211yesmaximal, characteristic
11#23749754C211yesmaximal, characteristic, complemented
12#23749728C211yesmaximal, characteristic
13#23749723C211yesmaximal, characteristic
14#23749710C211yesmaximal, characteristic
15#477×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149—21no—
17C2×#14558723C2211yescharacteristic
18#18011910—21no—
19#18011910C2211yescharacteristic
20#18011910C2211yescharacteristic
21#18011910C2211yescharacteristic
22#18011903—21no—
23#18011903C2211yescharacteristic
24#18011902—21no—
25#18011902C2211yescharacteristic
26#18011889—21no—
27#18011889—21no—
28#18011887—21no—
29Q8×Dic149C2211yescharacteristic
30Q8×Dic149C2211yescharacteristic
31C4×Dic298—21no—
32C4×Dic298C2211yescharacteristic
33C4×D596C2211yescharacteristic
34#18011873C2211yescharacteristic
35#18011873C2211yescharacteristic
36#18011845C2211yescharacteristic
37#18011844C2211yescharacteristic
38#18011844C2211yescharacteristic
39#18011844C2211yescharacteristic
40#18011843C2211yescharacteristic
41#18011842C2211yescharacteristic
42#18011842C2211yescharacteristic
43#18011840C2211yescharacteristic
44#18011840C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011833C2211yescharacteristic
47#18011833C2211yescharacteristic
48#18011831C2211yescharacteristic
49#18011822C2211yescharacteristic
50#18011821C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C4⋊Q8×C149C2211yescharacteristic
53C4×Q8×C149C2211yescharacteristic
54#107×C149C2211yescharacteristic
55#107×C149C2211yescharacteristic
56C4⋊C8×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58#103×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×Dic298—21no—
61C2×Dic298—21no—
62C2×Dic298D411yescharacteristic
63Q8×D149—44no—
64C2×D596C2311yescharacteristic
65C2×C4×D149—21no—
66C2×C4×D149—21no—
67C2×C4×D149—21no—
68#14558723Q814yescomplemented
69C2×#11924746C2311yescharacteristic
70C2×#11924746C2311yescharacteristic
71C4⋊Dic149—21no—
72C4⋊Dic149—21no—
73C4⋊Dic149—21no—
74C4⋊Dic149D411yescharacteristic
75C4⋊Dic149C2311yescharacteristic
76C4⋊Dic149C2311yescharacteristic
77C4⋊Dic149C2311yescharacteristic
78#14558715—21no—
79#14558715—21no—
80#14558715—21no—
81#14558715—21no—
82#14558715—21no—
83#14558715—21no—
84#14558715D411yescharacteristic
85#14558715D411yescharacteristic
86#14558713—21no—
87#14558713—21no—
88#14558713—21no—
89C4×Dic149—21no—
90C4×Dic149—21no—
91C4×Dic149C2311yescharacteristic
92C4×Dic149C2311yescharacteristic
93C2×Q8×C149—21no—
94C2×Q8×C149C2311yescharacteristic
95C2×C8×C149C2311yescharacteristic
96C2×C8×C149C2311yescharacteristic
97C4⋊C4×C149—21no—
98C4⋊C4×C149—21no—
99C4⋊C4×C149C2311yescharacteristic
100C4⋊C4×C149C2311yescharacteristic
101C4⋊C4×C149C2311yescharacteristic
102C42×C149—21no—
103C42×C149C2311yescharacteristic
104C22×D149—21no—
105Dic298—22no—
106Dic298—42no—
107Dic298—42no—
108Dic298—42no—
109D596—21no—
110D596C2×Q812yes—
111C2×Dic149—21no—
112C2×Dic149—21no—
113C2×Dic149—21no—
114C2×Dic149C2×D411yescharacteristic
115C2×Dic149C2×D411yescharacteristic
116C2×Dic149C2×D411yescharacteristic
117C2×Dic149C2×D411yescharacteristic
118C4×D149—42no—
119C4×D149—42no—
120C4×D149—42no—
121#11924746—21no—
122#11924746C2×Q812yescomplemented
123Q8×C149—21no—
124Q8×C149—42no—
125Q8×C149C2×Q812yescomplemented
126C2×C4×C149—21no—
127C2×C4×C149—21no—
128C2×C4×C149—21no—
129C2×C4×C149—21no—
130C2×C4×C149C2×D411yescharacteristic
131C2×C4×C149C2×D411yescharacteristic
132C2×C4×C149C2411yescharacteristic
133C8×C149—21no—
134C8×C149—21no—
135C22×C149C22×D411yescharacteristic
136C2×D149—22no—
137C2×D149—41no—
138C2×D149—41no—
139C4×C149—22no—
140C4×C149—21no—
141C4×C149—21no—
142C4×C149—41no—
143C4×C149—41no—
144C4×C149—41no—
145C4×C149C22×Q811yescharacteristic
146C4×C1492−1+411yescharacteristic
147Dic149—21no—
148Dic149—21no—
149Dic149—21no—
150Dic149—41no—
151Dic149—41no—
152Dic149—41no—
153Dic149AGL(1,ℤ/8)12yes—
154C2×C149D4×Q811yescharacteristic
155C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C149#57811yescharacteristic
157D149—42no—
158C149#338611yescharacteristic, complemented, Sylow
159#3386—1491nomaximal, Sylow
160#557—1491no—
161D4×Q8—1491no—
162#501—1491no—
163#501—1491no—
164#500—1491no—
165#478—1491no—
166#478—1491no—
167#477—1491no—
168#477D14911yescharacteristic, complemented
169#475—1491no—
170#474—1491no—
171#446—1491no—
172#440—1491no—
173#440—1491no—
174C4×SD16—1491no—
175C22×Q8—2981no—
176C2×SD16—1491no—
177C4⋊Q8—2981no—
178C4⋊Q8—1491no—
179C4⋊Q8—1491no—
180C4⋊Q8—1491no—
181C4⋊Q8C2×D14911yescharacteristic
182#125—2981no—
183#125—2981no—
184#125—1491no—
185#125—1491no—
186C22⋊Q8—2981no—
187C22⋊Q8—2981no—
188C22⋊Q8—2981no—
189C4×Q8—2981no—
190C4×Q8—1491no—
191C4×Q8—1491no—
192C4×Q8—1491no—
193C4×Q8C2×D14911yescharacteristic
194C4×D4—1491no—
195C4×D4—1491no—
196C4×D4—1491no—
197#107—1491no—
198#107—1491no—
199#107—1491no—
200#107—1491no—
201#107C2×D14911yescharacteristic
202#107C2×D14911yescharacteristic
203#106—1491no—
204#106—1491no—
205#106—1491no—
206C4⋊C8—1491no—
207C4⋊C8—1491no—
208C4⋊C8C2×D14911yescharacteristic
209#103—1491no—
210#103C2×D14911yescharacteristic
211#103C2×D14911yescharacteristic
212#102—1491no—
213#102—1491no—
214#102—1491no—
215#97—1491no—
216#97—1491no—
217C4×C8—1491no—
218C2×Q8—21no—
219C2×Q8—2981no—
220C2×Q8—2981no—
221C2×Q8—2981no—
222C2×Q8—5964no—
223C2×Q8—1491no—
224C2×Q8C22×D14911yescharacteristic
225C2×D4—1491no—
226C22×C4—2981no—
227C22×C4—2981no—
228C22×C4—2981no—
229SD16—1494no—
230C2×C8—1491no—
231C2×C8—1491no—
232C2×C8C22×D14911yescharacteristic
233C2×C8C22×D14911yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—2981no—
237C4⋊C4—2981no—
238C4⋊C4—2981no—
239C4⋊C4—2981no—
240C4⋊C4—2981no—
241C4⋊C4—2981no—
242C4⋊C4—2981no—
243C4⋊C4—2981no—
244C4⋊C4—2981no—
245C4⋊C4—1491no—
246C4⋊C4—1491no—
247C4⋊C4—1491no—
248C4⋊C4—1491no—
249C4⋊C4—1491no—
250C4⋊C4—1491no—
251C4⋊C4C22×D14911yescharacteristic
252C4⋊C4C22×D14911yescharacteristic
253C4⋊C4C22×D14911yescharacteristic
254C22⋊C4—2981no—
255C22⋊C4—2981no—
256C22⋊C4—2981no—
257C42—21no—
258C42—2981no—
259C42—2981no—
260C42—1491no—
261C42—1491no—
262C42C22×D14911yescharacteristic
263C23—2981no—
264Q8—21no—
265Q8—42no—
266Q8—2982no—
267Q8—5962no—
268Q8—5962no—
269Q8—5962no—
270Q8Q8×D14912yescomplemented
271D4—2981no—
272D4—1492no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—2981no—
278C2×C4—2981no—
279C2×C4—2981no—
280C2×C4—5962no—
281C2×C4—5962no—
282C2×C4—5962no—
283C2×C4—1491no—
284C2×C4—1491no—
285C2×C4—1491no—
286C2×C4—1491no—
287C2×C4D4×D14911yescharacteristic
288C2×C4D4×D14911yescharacteristic
289C2×C4C23×D14911yescharacteristic
290C8—21no—
291C8—21no—
292C8—2981no—
293C8—1492no—
294C22—2982no—
295C22—5961no—
296C22—5961no—
297C22C2×D4×D14911yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—2981no—
305C4—2981no—
306C4—2981no—
307C4—5961no—
308C4—5961no—
309C4—5961no—
310C4—1492no—
311C4C2×Q8×D14911yescharacteristic
312C4#1801195811yescharacteristic
313C2—5962no—
314C2#2375028811yescharacteristic
315C2AGL(1,ℤ/8)×D14911yescharacteristic
316C2#2375043811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#477×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#477D149
hypocenterC149#3386
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
3. #23750033C2semidirect
4. #23750031C2semidirect
5. #23750029C2semidirect
8. #23749918C2semidirect
9. #23749908C2semidirect
11. #23749754C2semidirect
15. #477×C149C2semidirect
68. #14558723Q8semidirect
122. #11924746C2×Q8semidirect
125. Q8×C149C2×Q8semidirect
158. C149#3386semidirect
168. #477D149semidirect
270. Q8Q8×D149semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237502902
#237500332
#237500312
#237500292
#237499282
#237499262
#237499182
#237499082
#237499042
#237497542
#237497282
#237497232
#237497102
#477×C1492
#3386149