Finite Group Databasebeta

← #34056555

Subgroups — #34056555

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#23750278C211yesmaximal, characteristic
3#23749954C212yesmaximal, complemented
4#23749907C211yesmaximal, characteristic, complemented
5#23749857C212yesmaximal, complemented
6#23749827C211yesmaximal, characteristic, complemented
7#23749765C212yesmaximal, complemented
8C4×#14558725C211yesmaximal, characteristic, complemented
9C8×D596C211yesmaximal, characteristic, complemented
10#103×D149C212yesmaximal, complemented
11#487×C149C211yesmaximal, characteristic, complemented
12C2×#14558737—21no—
13C2×Q8×D149C2212yes—
14C2×#14558725C2211yescharacteristic
15C2×#14558723C2212yes—
16C2×C8×D149C2212yes—
17#18011910—21no—
18#18011896—22no—
19#18011889—22no—
20#18011887—21no—
21#18011887C2212yes—
22#18011886C2212yes—
23C4×Dic298C2211yescharacteristic
24C4×D596C2211yescharacteristic
25C4×D596C2211yescharacteristic
26#18011873—21no—
27C4⋊C4×D149C2212yes—
28#18011843C2211yescharacteristic
29#18011842C2212yes—
30C4⋊#11924746C2211yescharacteristic
31#18011836C2211yescharacteristic
32#18011835C2212yes—
33#18011834C2212yes—
34#18011833C2211yescharacteristic
35#18011823C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4⋊Q8×C149C2211yescharacteristic
38#103×C149C2212yes—
39#103×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41#14558737—44no—
42C2×Dic298—22no—
43C2×Dic298C2311yescharacteristic
44Q8×D149—24no—
45Q8×D149—42no—
46C2×D596—22no—
47C2×D596D411yescharacteristic
48C2×D596C2311yescharacteristic
49C2×C4×D149—22no—
50C2×C4×D149—22no—
51C2×C4×D149—21no—
52C2×C4×D149C2312yes—
53#14558725—22no—
54#14558723—42no—
55C2×#11924746C2312yes—
56C8×D149—42no—
57C4⋊Dic149D411yescharacteristic
58C4⋊Dic149C2311yescharacteristic
59C4⋊Dic149C2311yescharacteristic
60#14558715—22no—
61#14558715—22no—
62#14558715—21no—
63#14558713—22no—
64#14558713—22no—
65#14558713—21no—
66#14558713D412yes—
67C4×Dic149—21no—
68C2×Q8×C149—21no—
69C2×Q8×C149C2312yes—
70C2×C8×C149C2311yescharacteristic
71C2×C8×C149C2311yescharacteristic
72C4⋊C4×C149—21no—
73C4⋊C4×C149—21no—
74C4⋊C4×C149—21no—
75C4⋊C4×C149C2312yes—
76C42×C149C2311yescharacteristic
77C22×D149—21no—
78C22×D149C2×D412yes—
79Dic298—21no—
80Dic298—44no—
81Dic298#4112yescomplemented
82D596—21no—
83D596—21no—
84D596—44no—
85D596SD1614yescomplemented
86C2×Dic149—21no—
87C2×Dic149—21no—
88C2×Dic149—21no—
89C2×Dic149C2×D412yes—
90C4×D149—22no—
91C4×D149—22no—
92C4×D149—44no—
93C4×D149—44no—
94C4×D149—42no—
95#11924746—22no—
96Q8×C149—22no—
97Q8×C149—22no—
98Q8×C149—42no—
99C2×C4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149C2×D411yescharacteristic
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2411yescharacteristic
104C8×C149—21no—
105C8×C149—21no—
106C22×C149C22×D411yescharacteristic
107C2×D149—22no—
108C2×D149—22no—
109C2×D149—41no—
110C2×D149—41no—
111C2×D149—41no—
112C2×D149C2×SD1614yes—
113C4×C149—21no—
114C4×C149—42no—
115C4×C149—42no—
116C4×C149C2×SD1612yes—
117C4×C149C2×#4111yescharacteristic
118C4×C149H(5,𝔽2)11yescharacteristic
119Dic149—22no—
120Dic149—22no—
121Dic149—41no—
122Dic149—41no—
123C2×C149#54611yescharacteristic
124C2×C149C22×SD1611yescharacteristic
125C2×C149#57811yescharacteristic
126D149—24no—
127D149—81no—
128C149#329811yescharacteristic, complemented, Sylow
129#3298—1491nomaximal, Sylow
130D4×Q8—1491no—
131#547—1491no—
132#487D14911yescharacteristic, complemented
133#483—1492no—
134#475—1491no—
135#465—1492no—
136#461—1491no—
137#448—1492no—
138C4×SD16—1491no—
139C8×D4—1491no—
140C2×#103—1492no—
141C2×#41—2981no—
142C22×Q8—1492no—
143C2×SD16—1492no—
144C2×SD16—1491no—
145C22×C8—1492no—
146C4⋊Q8—2981no—
147C4⋊Q8C2×D14911yescharacteristic
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×D4—1491no—
158C2×C4⋊C4—1492no—
159#106—1491no—
160#106—1492no—
161C4⋊C8—1491no—
162#103—1492no—
163#103—1491no—
164#103—1492no—
165#103C2×D14912yes—
166#103C2×D14912yes—
167#102—1491no—
168C22⋊C8—1492no—
169C4×C8C2×D14911yescharacteristic
170#41—5964no—
171C2×Q8—21no—
172C2×Q8—2982no—
173C2×Q8—2984no—
174C2×Q8—5962no—
175C2×Q8—1491no—
176C2×Q8C22×D14912yes—
177C2×D4—2982no—
178C2×D4—1491no—
179C2×D4—1491no—
180C22×C4—2982no—
181C22×C4—2982no—
182C22×C4—2981no—
183C22×C4—1492no—
184SD16—2982no—
185SD16—5962no—
186C2×C8—5962no—
187C2×C8—1492no—
188C2×C8C22×D14911yescharacteristic
189C2×C8C22×D14911yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—2982no—
194C4⋊C4—2982no—
195C4⋊C4—2981no—
196C4⋊C4—1491no—
197C4⋊C4—1491no—
198C4⋊C4—1491no—
199C4⋊C4C22×D14912yes—
200C22⋊C4—2982no—
201C22⋊C4—2982no—
202C22⋊C4—2981no—
203C22⋊C4—1492no—
204C42—2981no—
205C42C22×D14911yescharacteristic
206C23—2981no—
207C23—1492no—
208Q8—22no—
209Q8—22no—
210Q8—42no—
211Q8—2981no—
212Q8—5964no—
213Q8—1492no—
214D4—2981no—
215D4—2981no—
216D4—5964no—
217D4—1494no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—2981no—
221C2×C4—2982no—
222C2×C4—2982no—
223C2×C4—2981no—
224C2×C4—2981no—
225C2×C4—5964no—
226C2×C4—5964no—
227C2×C4—5962no—
228C2×C4—1492no—
229C2×C4D4×D14911yescharacteristic
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—21no—
233C8—21no—
234C8—2982no—
235C22—2982no—
236C22—2982no—
237C22—5961no—
238C22—5961no—
239C22—5961no—
240C22—1494no—
241C22C2×D4×D14911yescharacteristic
242C4—21no—
243C4—42no—
244C4—42no—
245C4—2982no—
246C4—2982no—
247C4—5961no—
248C4—5961no—
249C4SD16×D14912yes—
250C4#41×D14911yescharacteristic
251C4#1801195711yescharacteristic
252C2—2984no—
253C2—11921no—
254C2#2375026811yescharacteristic
255C2C2×SD16×D14911yescharacteristic
256C2#2375043811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#487×C149C2
socleC22×C149C22×D4
prime radical#23750278C2
hypercenter#487D149
hypocenterC149#3298
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #23750290C2semidirect
3. #23749954C2semidirect
4. #23749907C2semidirect
5. #23749857C2semidirect
6. #23749827C2semidirect
7. #23749765C2semidirect
8. C4×#14558725C2semidirect
9. C8×D596C2semidirect
10. #103×D149C2semidirect
11. #487×C149C2semidirect
81. Dic298#41semidirect
85. D596SD16semidirect
128. C149#3298semidirect
132. #487D149semidirect

Maximal subgroups

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

subgroupindex
#237502902
#237502782
#237499542
#237499072
#237498572
#237498272
#237497652
C4×#145587252
C8×D5962
#103×D1492
#487×C1492
#3298149