Finite Group Databasebeta

← #34078517

Subgroups — #34078517

Order 19328.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#23753048C211yesmaximal, characteristic
2#23752788C212yesmaximal, complemented
3#23752783C211yesmaximal, characteristic, complemented
4#23752403C218yesmaximal, complemented
5#23752185C212yesmaximal, complemented
6#410×C151C211yesmaximal, characteristic, complemented
7#18012787—21no—
8#18012786—22no—
9#18012785—22no—
10#18012784—21no—
11#18012784C2214yes—
12C2×#14559004—22no—
13C2×#14559004C2212yes—
14#41×D151—24no—
15C22×Dic302C2211yescharacteristic, complemented
16#18012739—22no—
17#18012737—21no—
18#18012727—22no—
19#18012726—21no—
20#18012723C2218yescomplemented
21#18012720C2214yescomplemented
22#18012719—24no—
23#18012718—22no—
24#18012698C2211yescharacteristic, complemented
25#18012653C2218yescomplemented
26C2×#41×C151C2211yescharacteristic
27#117×C151C2212yescomplemented
28#99×C151C2214yescomplemented
29#14559005—24no—
30#14559004—24no—
31#14559004—24no—
32#14559004—22no—
33#14559004—22no—
34#14559004—24no—
35#14559004—22no—
36#14559004—22no—
37#14559003—24no—
38#14559003—28no—
39C2×Dic302—22no—
40C2×Dic302—21no—
41C2×Dic302—22no—
42C2×Dic302—21no—
43C2×Dic302—22no—
44C2×Dic302D412yes—
45Q8×D151—24no—
46C2×D604—22no—
47C2×D604D412yes—
48C2×C22⋊D151—24no—
49C2×C22⋊D151C2314yes—
50D4×D151—24no—
51D4×D151—28no—
52C22×Dic151C2312yes—
53C2×C4×D151—24no—
54C2×C4×D151D414yes—
55C4⋊Dic151—28no—
56C4⋊Dic151—21no—
57Dic151⋊C4—24no—
58C22⋊Dic151—24no—
59C22⋊Dic151C2312yes—
60(C2×D151)⋊C4—28no—
61C4×Dic151—21no—
62#41×C151—22no—
63#41×C151—22no—
64C2×Q8×C151D411yescharacteristic
65C2×D4×C151D411yescharacteristic
66C2×D4×C151C2312yes—
67C22×C4×C151D412yes—
68C22×C4×C151C2311yescharacteristic
69C4⋊C4×C151—22no—
70C22⋊C4×C151C2314yes—
71C42×C151—22no—
72C22×D151—24no—
73C22×D151C2×D414yes—
74Dic302—24no—
75Dic302—22no—
76Dic302—22no—
77Dic302—41no—
78Dic302—41no—
79Dic302—42no—
80D604—24no—
81D604—22no—
82D604—22no—
83C22⋊D151—28no—
84C22⋊D151—24no—
85C22⋊D151—24no—
86C2×Dic151—24no—
87C2×Dic151—22no—
88C2×Dic151—24no—
89C2×Dic151—22no—
90C2×Dic151C2×D414yes—
91C4×D151—28no—
92C4×D151—24no—
93C4×D151—24no—
94C23×C151C2×D412yes—
95C23×C151C2411yescharacteristic
96Q8×C151—22no—
97D4×C151—24no—
98D4×C151—22no—
99C2×C4×C151—22no—
100C2×C4×C151—22no—
101C2×C4×C151—24no—
102C2×C4×C151C2×D412yes—
103C2×C4×C151C2×D412yes—
104C2×C4×C151C2×D412yes—
105C2×C4×C151C2×D411yescharacteristic
106C2×C4×C151C2×D411yescharacteristic
107C22×C151—22no—
108C22×C151—41no—
109C22×C151—42no—
110C22×C151C22×D412yes—
111C22×C151C22×D411yescharacteristic
112C2×D151—24no—
113C2×D151—44no—
114C2×D151—84no—
115C4×C151—22no—
116C4×C151—44no—
117C4×C151#12011yescharacteristic
118C4×C151#12011yescharacteristic
119C4×C151#12012yes—
120Dic151—24no—
121Dic151—42no—
122Dic151—42no—
123C2×C151—21no—
124C2×C151—22no—
125C2×C151—42no—
126C2×C151C2×#12011yescharacteristic
127D151—44no—
128C151#302811yescharacteristic, complemented, Sylow
129#3028—1511nomaximal, Sylow
130#585—1511no—
131#536—1511no—
132#536—1512no—
133#458—1518no—
134#410—1512no—
135#410D15111yescharacteristic, complemented
1362−1+4—3022no—
1372−1+4—3021no—
138H(5,𝔽2)—3022no—
139H(5,𝔽2)—3021no—
140H(5,𝔽2)—1514no—
141C2×#41—3024no—
142C2×#41—3022no—
143C2×#41—1512no—
144C2×#41C2×D15111yescharacteristic
145C22×Q8—1511no—
146C4⋊Q8—3022no—
147C4⋊Q8—3021no—
148#124—3022no—
149#124—3021no—
150#123—1514no—
151#123—1518no—
152C22⋊Q8—3024no—
153C22⋊Q8—3022no—
154#117—1511no—
155#117C2×D15112yescomplemented
156#99—1518no—
157#99C2×D15114yescomplemented
158#41—22no—
159#41—22no—
160#41—3024no—
161#41—3024no—
162#41—3024no—
163#41—3024no—
164#41—3028no—
165#41—3022no—
166#41—3022no—
167#41—3022no—
168#41—3024no—
169#41—3022no—
170C2×Q8—3022no—
171C2×Q8—3024no—
172C2×Q8—3022no—
173C2×Q8—3022no—
174C2×Q8—3021no—
175C2×Q8—3021no—
176C2×Q8—1512no—
177C2×Q8D60411yescharacteristic
178C2×D4—3022no—
179C2×D4—3024no—
180C2×D4—3028no—
181C2×D4—3024no—
182C2×D4—1514no—
183C2×D4—1512no—
184C2×D4D60411yescharacteristic
185C2×D4C22×D15112yes—
186C22×C4—3024no—
187C22×C4—1512no—
188C22×C4—1514no—
189C22×C4D60412yes—
190C22×C4C22×D15111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3028no—
193C4⋊C4—3024no—
194C4⋊C4—3021no—
195C22⋊C4—3028no—
196C22⋊C4—3024no—
197C22⋊C4—1512no—
198C22⋊C4C22×D15114yes—
199C42—22no—
200C42—3021no—
201C23—3024no—
202C23—1514no—
203C23C2×D60412yes—
204C23C23×D15111yescharacteristic
205Q8—22no—
206Q8—3024no—
207Q8—3022no—
208Q8—3022no—
209Q8—6041no—
210Q8—6041no—
211Q8—6042no—
212D4—24no—
213D4—22no—
214D4—3024no—
215D4—3028no—
216D4—3024no—
217D4—3024no—
218D4—3022no—
219D4—3022no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—3028no—
224C2×C4—3024no—
225C2×C4—3024no—
226C2×C4—3024no—
227C2×C4—3024no—
228C2×C4—3022no—
229C2×C4—3022no—
230C2×C4—1514no—
231C2×C4D4×D15112yes—
232C2×C4D4×D15112yes—
233C2×C4C2×D60412yes—
234C2×C4C2×D60411yescharacteristic
235C2×C4C2×D60411yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—3024no—
240C22—6044no—
241C22—12084no—
242C22C2×D4×D15112yes—
243C22C22×D60411yescharacteristic
244C4—22no—
245C4—44no—
246C4—3024no—
247C4—6042no—
248C4—6042no—
249C4#1801270911yescharacteristic
250C4#1801270911yescharacteristic
251C4#1801270912yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—6044no—
256C2C2×#1801270911yescharacteristic
subgroupG/H
centerC2C2×#18012709
commutator, cosocleC23×C151C24
FrattiniC23C23×D151
Fitting#410×C151C2
socleC2×C151C2×#120
prime radical#23753048C2
hypercenter#410D151
hypocenterC151#3028
LCSG ▷ C23×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23752788C2semidirect
3. #23752783C2semidirect
4. #23752403C2semidirect
5. #23752185C2semidirect
6. #410×C151C2semidirect
15. C22×Dic302C22semidirect
20. #18012723C22semidirect
21. #18012720C22semidirect
24. #18012698C22semidirect
25. #18012653C22semidirect
27. #117×C151C22semidirect
28. #99×C151C22semidirect
128. C151#3028semidirect
135. #410D151semidirect
155. #117C2×D151semidirect
157. #99C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530482
#237527882
#237527832
#237524032
#237521852
#410×C1512
#3028151