Finite Group Databasebeta

← #1618×C239 #44480388

Subgroups — #1618×C239 #44480388

Order 30592.

240 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 64 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#538×C239C211yesmaximal, characteristic
2#484×C239C211yesmaximal, characteristic, complemented
3#481×C239C211yesmaximal, characteristic, complemented
4#399×C239C211yesmaximal, characteristic, complemented
5#328×C239C211yesmaximal, characteristic, complemented
6#327×C239C211yesmaximal, characteristic, complemented
7#323×C239C211yesmaximal, characteristic, complemented
8#126×C239—21no—
9#124×C239—21no—
10#123×C239—21no—
11C22⋊Q8×C239C2211yescharacteristic
12#121×C239—21no—
13#121×C239C2211yescharacteristic
14#120×C239—21no—
15C4×D4×C239—21no—
16C2×C4⋊C4×C239—21no—
17C2×C4⋊C4×C239C2211yescharacteristic
18C2×C22⋊C4×C239—21no—
19C2×C22⋊C4×C239C2211yescharacteristic
20#107×C239—21no—
21#106×C239—21no—
22#103×C239—21no—
23#102×C239—21no—
24C22⋊C8×C239C2211yescharacteristic
25C22⋊C8×C239C2211yescharacteristic
26#95×C239C2211yescharacteristic
27C24×C239D411yescharacteristic
28C2×Q8×C239D411yescharacteristic
29C2×D4×C239—21no—
30C2×D4×C239—21no—
31C2×D4×C239—21no—
32C2×D4×C239D411yescharacteristic
33C22×C4×C239—21no—
34C22×C4×C239—21no—
35C22×C4×C239—21no—
36C22×C4×C239D411yescharacteristic
37C22×C4×C239C2311yescharacteristic
38C2×C8×C239—21no—
39C2×C8×C239—21no—
40C4⋊C4×C239—21no—
41C4⋊C4×C239—21no—
42C4⋊C4×C239—21no—
43C4⋊C4×C239—21no—
44C4⋊C4×C239—42no—
45C4⋊C4×C239D411yescharacteristic
46C4⋊C4×C239D411yescharacteristic
47C22⋊C4×C239—21no—
48C22⋊C4×C239—21no—
49C22⋊C4×C239—21no—
50C22⋊C4×C239—21no—
51C22⋊C4×C239—21no—
52C22⋊C4×C239—21no—
53C22⋊C4×C239—21no—
54C22⋊C4×C239—42no—
55C42×C239—21no—
56C23×C239—21no—
57C23×C239—21no—
58C23×C239—21no—
59C23×C239—21no—
60C23×C239—41no—
61C23×C239—41no—
62C23×C239C2×D411yescharacteristic
63Q8×C239—41no—
64D4×C239—41no—
65D4×C239—41no—
66D4×C239—41no—
67D4×C239—81no—
68D4×C239—81no—
69C2×C4×C239—22no—
70C2×C4×C239—21no—
71C2×C4×C239—21no—
72C2×C4×C239—21no—
73C2×C4×C239—21no—
74C2×C4×C239—41no—
75C2×C4×C239—41no—
76C2×C4×C239—41no—
77C2×C4×C239—41no—
78C2×C4×C239—41no—
79C2×C4×C239—41no—
80C2×C4×C239—41no—
81C2×C4×C239—81no—
82C2×C4×C239C2×D411yescharacteristic
83C2×C4×C239C2×D411yescharacteristic
84C8×C239—41no—
85C8×C239—41no—
86C22×C239—21no—
87C22×C239—21no—
88C22×C239—42no—
89C22×C239—41no—
90C22×C239—41no—
91C22×C239—41no—
92C22×C239—41no—
93C22×C239—41no—
94C22×C239—41no—
95C22×C239—81no—
96C22×C239#12011yescharacteristic
97C22×C239#13511yescharacteristic
98C22×C239#13711yescharacteristic
99C4×C239—21no—
100C4×C239—21no—
101C4×C239—42no—
102C4×C239—41no—
103C4×C239—41no—
104C4×C239—41no—
105C4×C239—41no—
106C4×C239—81no—
107C2×C239—21no—
108C2×C239—21no—
109C2×C239—81no—
110C2×C239—81no—
111C2×C239#45211yescharacteristic
112C2×C239#45311yescharacteristic
113C2×C239#45811yescharacteristic
114C239#161811yescharacteristic, direct product factor, Sylow
115#1618C23911yesmaximal, characteristic, direct product factor, Sylow
116#538C2×C23911yescharacteristic
117#484C2×C23911yescharacteristic, complemented
118#481C2×C23911yescharacteristic, complemented
119#399C2×C23911yescharacteristic, complemented
120#328C2×C23911yescharacteristic, complemented
121#327C2×C23911yescharacteristic, complemented
122#323C2×C23911yescharacteristic, complemented
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C22×C23911yescharacteristic
127#121—21no—
128#121C22×C23911yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C22×C23911yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4C22×C23911yescharacteristic
135#107—21no—
136#106—21no—
137#103—21no—
138#102—21no—
139C22⋊C8C22×C23911yescharacteristic
140C22⋊C8C22×C23911yescharacteristic
141#95C22×C23911yescharacteristic
142C24D4×C23911yescharacteristic
143C2×Q8D4×C23911yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4D4×C23911yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4D4×C23911yescharacteristic
152C22×C4C23×C23911yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—42no—
160C4⋊C4D4×C23911yescharacteristic
161C4⋊C4D4×C23911yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—42no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×D4×C23911yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4C2×D4×C23911yescharacteristic
198C2×C4C2×D4×C23911yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#120×C23911yescharacteristic
212C22#135×C23911yescharacteristic
213C22#137×C23911yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—81no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#452×C23911yescharacteristic
227C2#453×C23911yescharacteristic
228C2#458×C23911yescharacteristic
subgroupG/H
center, socleC22×C239#120
commutator, Frattini, cosocleC22×C4C23×C239
prime radical#538×C239C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C239 ◁ C22×C4×C239 ◁ G

Complements

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

NH ≅ G/Ntype
2. #484×C239C2semidirect
3. #481×C239C2semidirect
4. #399×C239C2semidirect
5. #328×C239C2semidirect
6. #327×C239C2semidirect
7. #323×C239C2semidirect
114. C239#1618direct (N × H)
115. #1618C239direct (N × H)
117. #484C2×C239semidirect
118. #481C2×C239semidirect
119. #399C2×C239semidirect
120. #328C2×C239semidirect
121. #327C2×C239semidirect
122. #323C2×C239semidirect

Maximal subgroups

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

subgroupindex
#538×C2392
#484×C2392
#481×C2392
#399×C2392
#328×C2392
#327×C2392
#323×C2392
#1618239