Finite Group Databasebeta

← #1615×C233 #44447647

Subgroups — #1615×C233 #44447647

Order 29824.

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

normal subgroups only

This group has 208 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#539×C233C211yesmaximal, characteristic
2#484×C233C211yesmaximal, characteristic, complemented
3#481×C233C211yesmaximal, characteristic
4#400×C233C211yesmaximal, characteristic, complemented
5#328×C233C211yesmaximal, characteristic, complemented
6#327×C233C211yesmaximal, characteristic
7#324×C233C211yesmaximal, characteristic, complemented
8#126×C233—21no—
9#125×C233—21no—
10#123×C233—21no—
11#123×C233—21no—
12C22⋊Q8×C233—21no—
13C22⋊Q8×C233C2211yescharacteristic
14#121×C233C2211yescharacteristic
15C4×D4×C233—21no—
16C2×C4⋊C4×C233C2211yescharacteristic
17C2×C4⋊C4×C233C2211yescharacteristic
18#107×C233—21no—
19#106×C233—21no—
20#103×C233—21no—
21#102×C233—21no—
22C22⋊C8×C233C2211yescharacteristic
23C22⋊C8×C233C2211yescharacteristic
24#95×C233—21no—
25#95×C233—21no—
26#95×C233C2211yescharacteristic
27C2×Q8×C233D411yescharacteristic
28C2×D4×C233—21no—
29C2×D4×C233D411yescharacteristic
30C22×C4×C233—21no—
31C22×C4×C233—21no—
32C22×C4×C233—21no—
33C22×C4×C233D411yescharacteristic
34C22×C4×C233D411yescharacteristic
35C22×C4×C233C2311yescharacteristic
36C2×C8×C233—21no—
37C2×C8×C233—21no—
38C4⋊C4×C233—21no—
39C4⋊C4×C233—21no—
40C4⋊C4×C233—21no—
41C4⋊C4×C233—21no—
42C4⋊C4×C233—21no—
43C4⋊C4×C233—21no—
44C4⋊C4×C233—21no—
45C4⋊C4×C233—21no—
46C4⋊C4×C233D411yescharacteristic
47C4⋊C4×C233D411yescharacteristic
48C22⋊C4×C233—21no—
49C22⋊C4×C233—21no—
50C22⋊C4×C233—21no—
51C22⋊C4×C233—21no—
52C22⋊C4×C233—21no—
53C42×C233—21no—
54C23×C233—21no—
55C23×C233C2×D411yescharacteristic
56Q8×C233—41no—
57D4×C233—41no—
58D4×C233—41no—
59D4×C233—41no—
60C2×C4×C233—21no—
61C2×C4×C233—21no—
62C2×C4×C233—21no—
63C2×C4×C233—21no—
64C2×C4×C233—21no—
65C2×C4×C233—21no—
66C2×C4×C233—21no—
67C2×C4×C233—41no—
68C2×C4×C233—41no—
69C2×C4×C233—41no—
70C2×C4×C233—41no—
71C2×C4×C233—42no—
72C2×C4×C233—41no—
73C2×C4×C233—41no—
74C2×C4×C233—81no—
75C2×C4×C233C2×D411yescharacteristic
76C2×C4×C233C2×D411yescharacteristic
77C8×C233—41no—
78C8×C233—41no—
79C22×C233—21no—
80C22×C233—21no—
81C22×C233—41no—
82C22×C233—41no—
83C22×C233—41no—
84C22×C233#12011yescharacteristic
85C22×C233#13511yescharacteristic
86C22×C233#13711yescharacteristic
87C4×C233—21no—
88C4×C233—21no—
89C4×C233—41no—
90C4×C233—41no—
91C4×C233—41no—
92C4×C233—41no—
93C4×C233—41no—
94C4×C233—81no—
95C4×C233—81no—
96C2×C233—21no—
97C2×C233—21no—
98C2×C233—81no—
99C2×C233#45211yescharacteristic
100C2×C233#45411yescharacteristic
101C2×C233#45811yescharacteristic
102C233#161511yescharacteristic, direct product factor, Sylow
103#1615C23311yesmaximal, characteristic, direct product factor, Sylow
104#539C2×C23311yescharacteristic
105#484C2×C23311yescharacteristic, complemented
106#481C2×C23311yescharacteristic
107#400C2×C23311yescharacteristic, complemented
108#328C2×C23311yescharacteristic, complemented
109#327C2×C23311yescharacteristic
110#324C2×C23311yescharacteristic, complemented
111#126—21no—
112#125—21no—
113#123—21no—
114#123—21no—
115C22⋊Q8—21no—
116C22⋊Q8C22×C23311yescharacteristic
117#121C22×C23311yescharacteristic
118C4×D4—21no—
119C2×C4⋊C4C22×C23311yescharacteristic
120C2×C4⋊C4C22×C23311yescharacteristic
121#107—21no—
122#106—21no—
123#103—21no—
124#102—21no—
125C22⋊C8C22×C23311yescharacteristic
126C22⋊C8C22×C23311yescharacteristic
127#95—21no—
128#95—21no—
129#95C22×C23311yescharacteristic
130C2×Q8D4×C23311yescharacteristic
131C2×D4—21no—
132C2×D4D4×C23311yescharacteristic
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4D4×C23311yescharacteristic
137C22×C4D4×C23311yescharacteristic
138C22×C4C23×C23311yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4D4×C23311yescharacteristic
150C4⋊C4D4×C23311yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C42—21no—
157C23—21no—
158C23C2×D4×C23311yescharacteristic
159Q8—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4C2×D4×C23311yescharacteristic
179C2×C4C2×D4×C23311yescharacteristic
180C8—41no—
181C8—41no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#120×C23311yescharacteristic
188C22#135×C23311yescharacteristic
189C22#137×C23311yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—41no—
197C4—81no—
198C4—81no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#452×C23311yescharacteristic
203C2#454×C23311yescharacteristic
204C2#458×C23311yescharacteristic
subgroupG/H
center, socleC22×C233#120
commutator, Frattini, cosocleC22×C4C23×C233
prime radical#121×C233C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C233 ◁ C22×C4×C233 ◁ G

Complements

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

NH ≅ G/Ntype
2. #484×C233C2semidirect
4. #400×C233C2semidirect
5. #328×C233C2semidirect
7. #324×C233C2semidirect
102. C233#1615direct (N × H)
103. #1615C233direct (N × H)
105. #484C2×C233semidirect
107. #400C2×C233semidirect
108. #328C2×C233semidirect
110. #324C2×C233semidirect

Maximal subgroups

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

subgroupindex
#539×C2332
#484×C2332
#481×C2332
#400×C2332
#328×C2332
#327×C2332
#324×C2332
#1615233