Finite Group Databasebeta

← #34056493

Subgroups — #34056493

Order 19072.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750440C212yesmaximal, complemented
2#23750439C211yesmaximal, characteristic, complemented
3#23750003C211yesmaximal, characteristic, complemented
4#23749900C211yesmaximal, characteristic, complemented
5#23749898C212yesmaximal, complemented
6#23749897C211yesmaximal, characteristic, complemented
7#23749808C212yesmaximal, complemented
8#23749806C212yesmaximal, complemented
9#23749742C211yesmaximal, characteristic, complemented
10#23749740C211yesmaximal, characteristic, complemented
11#578×C149C211yesmaximal, characteristic, complemented
12#18011958C2212yes—
13#18011942—22no—
14#18011942C2212yes—
15#18011941—22no—
16#18011941C2212yes—
17#18011941C2212yes—
18#18011940—22no—
19#18011931—22no—
20#18011931C2211yescharacteristic
21#18011930—21no—
22C2×Dic596C2211yescharacteristic
23C2×#14558726—22no—
24C2×#14558726C2212yes—
25Q16×D149—21no—
26#18011918C2212yes—
27#18011917C2211yescharacteristic
28#18011909C2212yes—
29#18011847C2211yescharacteristic
30#18011846C2211yescharacteristic
31#18011838C2211yescharacteristic
32#18011838C2212yes—
33#18011837C2211yescharacteristic
34#18011830C2212yes—
35#18011829C2212yes—
36C8×Dic149C2211yescharacteristic
372−1+4×C149C2212yes—
38#137×C149—22no—
39#137×C149C2212yes—
40#135×C149—21no—
41#135×C149C2211yescharacteristic
42C2×Q16×C149—21no—
43C2×Q16×C149C2211yescharacteristic
44#131×C149C2211yescharacteristic
45#14558736—22no—
46#14558736C2311yescharacteristic
47#14558735—22no—
48#14558735—42no—
49C2×Dic298—22no—
50C2×Dic298C2312yes—
51Q8×D149—22no—
52Dic596—21no—
53Dic596—21no—
54#14558726—22no—
55#14558726—22no—
56#14558726—22no—
57#14558726—22no—
58#14558726—22no—
59#14558726D411yescharacteristic
60#14558724—22no—
61#14558724—22no—
62#14558724D411yescharacteristic
63#14558723—22no—
64#14558723D411yescharacteristic
65#14558721D411yescharacteristic
66#14558720—22no—
67#14558720C2311yescharacteristic
68#14558720C2312yes—
69#14558719—21no—
70C2×#11924746—22no—
71C2×#11924746C2311yescharacteristic
72C8×D149—21no—
73#14558715—42no—
74C4×Dic149C2311yescharacteristic
75#41×C149—22no—
76#41×C149—22no—
77#41×C149—42no—
78#41×C149C2312yes—
79#41×C149C2311yescharacteristic
80C2×Q8×C149—22no—
81C2×Q8×C149—22no—
82C2×Q8×C149C2312yes—
83Q16×C149—22no—
84Q16×C149—21no—
85Q16×C149—21no—
86Q16×C149D411yescharacteristic
87SD16×C149—22no—
88SD16×C149D412yes—
89D8×C149D411yescharacteristic
90M16×C149—21no—
91M16×C149C2311yescharacteristic
92C2×C8×C149—21no—
93C2×C8×C149C2311yescharacteristic
94Dic298—22no—
95Dic298—22no—
96Dic298—42no—
97Dic298C2×D411yescharacteristic
98D596C2×D411yescharacteristic
99C22⋊D149—21no—
100C22⋊D149—42no—
101C2×Dic149—21no—
102C2×Dic149—22no—
103C2×Dic149—42no—
104C4×D149—21no—
105C4×D149—42no—
106#11924746—22no—
107#11924746—21no—
108#11924746C2×D411yescharacteristic
109#11924746C2×D411yescharacteristic
110Q8×C149—22no—
111Q8×C149—22no—
112Q8×C149—42no—
113Q8×C149C2×D411yescharacteristic
114Q8×C149C2×D412yes—
115D4×C149—22no—
116D4×C149—21no—
117D4×C149—42no—
118D4×C149C2×D411yescharacteristic
119D4×C149C2×D412yes—
120C2×C4×C149—22no—
121C2×C4×C149—21no—
122C2×C4×C149—22no—
123C2×C4×C149—42no—
124C2×C4×C149—42no—
125C2×C4×C149C2411yescharacteristic
126C8×C149—21no—
127C8×C149C2×D411yescharacteristic
128C8×C149C2×D411yescharacteristic
129C22×C149—21no—
130C22×C149—22no—
131C22×C149H(5,𝔽2)11yescharacteristic
132C2×D149—21no—
133C4×C149—21no—
134C4×C149—22no—
135C4×C149—42no—
136C4×C149C22×D411yescharacteristic
137C4×C149C22×D411yescharacteristic
138Dic149—22no—
139Dic149—21no—
140Dic149—42no—
141C2×C149—21no—
142C2×C149—41no—
143C2×C149—42no—
144C2×C149D4211yescharacteristic
145D149—41no—
146C149#329311yescharacteristic, complemented, Sylow
147#3293—1491nomaximal, Sylow
148#578—1492no—
149#578—1491no—
150#578D14911yescharacteristic, complemented
151#494—1491no—
152#473—1492no—
153#473—1491no—
154#473—1491no—
155#456—1492no—
156#456—1492no—
157#443—1491no—
158#443—1491no—
1592−1+4—1492no—
1602−1+4C2×D14912yes—
161#137—22no—
162#137—2982no—
163#137—2982no—
164#137—2982no—
165#137—1492no—
166#137—1492no—
167#137—1492no—
168#137C2×D14912yes—
169#135—21no—
170#135—2982no—
171#135—2981no—
172#135—1491no—
173#135C2×D14911yescharacteristic
174C2×Q16—21no—
175C2×Q16—2982no—
176C2×Q16—2981no—
177C2×Q16—1492no—
178C2×Q16—1491no—
179C2×Q16C2×D14911yescharacteristic
180#131—1492no—
181#131—1491no—
182#131C2×D14911yescharacteristic
183C4⋊Q8—1492no—
184#108—1491no—
185#108—1491no—
186#104—1491no—
187#104—1492no—
188#104—1491no—
189#101—1492no—
190#101—1492no—
191C4×C8—1491no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—2982no—
196#41—2982no—
197#41—5962no—
198#41—1491no—
199#41C22×D14911yescharacteristic
200#41C22×D14912yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—2982no—
204C2×Q8—2982no—
205C2×Q8—1492no—
206C2×Q8C22×D14912yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—2982no—
211Q16—2982no—
212Q16—2982no—
213Q16—2982no—
214Q16—2982no—
215Q16—2981no—
216Q16—2981no—
217Q16—1491no—
218Q16C22⋊D14911yescharacteristic
219SD16—22no—
220SD16—2982no—
221SD16—2982no—
222SD16—2982no—
223SD16—1491no—
224SD16—1491no—
225SD16C22⋊D14912yes—
226D8—1491no—
227D8C22⋊D14911yescharacteristic
228M16—21no—
229M16—2982no—
230M16—2981no—
231M16—1492no—
232M16—1491no—
233M16C22×D14911yescharacteristic
234C2×C8—21no—
235C2×C8—2982no—
236C2×C8—2981no—
237C2×C8—1491no—
238C2×C8C22×D14911yescharacteristic
239C4⋊C4—5962no—
240C42—1491no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—2982no—
245Q8—2982no—
246Q8—5962no—
247Q8—1491no—
248Q8D4×D14911yescharacteristic
249Q8C2×C22⋊D14912yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—2981no—
254D4—5962no—
255D4—1491no—
256D4D4×D14911yescharacteristic
257D4C2×C22⋊D14912yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—2981no—
264C2×C4—2981no—
265C2×C4—2982no—
266C2×C4—5962no—
267C2×C4—5962no—
268C2×C4C23×D14911yescharacteristic
269C8—21no—
270C8—2982no—
271C8—2981no—
272C8—1491no—
273C8—1491no—
274C8C2×C22⋊D14911yescharacteristic
275C8C2×C22⋊D14911yescharacteristic
276C22—21no—
277C22—22no—
278C22—2981no—
279C22#1801195611yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—2982no—
284C4—2981no—
285C4—5962no—
286C4C2×D4×D14911yescharacteristic
287C4C22×C22⋊D14911yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—5961no—
292C2D4×C22⋊D14911yescharacteristic
subgroupG/H
centerC2D4×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#578×C149C2
socleC2×C149D42
hypercenter#578D149
hypocenterC149#3293
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #23750440C2semidirect
2. #23750439C2semidirect
3. #23750003C2semidirect
4. #23749900C2semidirect
5. #23749898C2semidirect
6. #23749897C2semidirect
7. #23749808C2semidirect
8. #23749806C2semidirect
9. #23749742C2semidirect
10. #23749740C2semidirect
11. #578×C149C2semidirect
146. C149#3293semidirect
150. #578D149semidirect

Maximal subgroups

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

subgroupindex
#237504402
#237504392
#237500032
#237499002
#237498982
#237498972
#237498082
#237498062
#237497422
#237497402
#578×C1492
#3293149