Finite Group Databasebeta

← #32024865

Subgroups — #32024865

Order 18176.

1402 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 275 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#22590920C213yesmaximal
2#22589503C218yesmaximal, complemented
3#22588073C213yesmaximal, complemented
4C71×#3458C211yesmaximal, characteristic, complemented
5C23×D284C2211yescharacteristic
6C2×#14556087C2213yes—
7#17985901—26no—
8#17985900—26no—
9#17985854C22112yes—
10#17985766C22112yes—
11#544×C71—26no—
12#543×C71C2214yes—
13#533×C71—26no—
14C2×#117×C71C2213yes—
15C24×D71C2312yes—
16C22×D284—23no—
17C22×D284—23no—
18C22×D284—26no—
19C22×D284Q814yescomplemented
20#14556087—212no—
21C2×C4⋊Dic71—23no—
22C2×C22⋊Dic71C2316yes—
23C2×(C2×D71)⋊C4—224no—
24C2×C4×Dic71—23no—
25C23×C4×C71C2311yescharacteristic
26C4⋊Q8×C71—212no—
27#125×C71—212no—
28C22⋊Q8×C71—224no—
29C4×Q8×C71—28no—
30#117×C71—26no—
31#117×C71—26no—
32C2×C4⋊C4×C71D416yes—
33C2×C22⋊C4×C71C2316yes—
34C2×C42×C71D416yes—
35C23×D71—26no—
36C23×D71—26no—
37C23×D71—84no—
38C23×D71C2×Q814yes—
39C2×D284—212no—
40C2×D284—212no—
41C2×D284—41no—
42C2×D284—43no—
43C2×D284—46no—
44C2×D284—46no—
45C22×Dic71—26no—
46C4⋊Dic71—46no—
47C22⋊Dic71—212no—
48(C2×D71)⋊C4—824no—
49C4×Dic71—46no—
50C24×C71C2411yescharacteristic
51C2×Q8×C71—28no—
52C22×C4×C71C2×D4112yes—
53C22×C4×C71C2×D413yes—
54C22×C4×C71C2×D413yes—
55C22×C4×C71C2×Q812yes—
56C22×C4×C71#4116yes—
57C4⋊C4×C71—224no—
58C4⋊C4×C71—212no—
59C22⋊C4×C71—212no—
60C42×C71—26no—
61C42×C71—26no—
62C23×C71—42no—
63C23×C71C22×D413yes—
64C23×C71C22×Q811yescharacteristic
65C23×C71C2×#4113yes—
66C22×D71—212no—
67C22×D71—424no—
68C22×D71—42no—
69C22×D71—46no—
70C22×D71—84no—
71C22×D71—812no—
72Q8×C71—88no—
73C2×C4×C71—212no—
74C2×C4×C71—412no—
75C2×C4×C71#12011yescharacteristic
76C2×C4×C71#12013yes—
77C2×C4×C71C22⋊Q816yes—
78C2×C4×C71C22⋊Q816yes—
79C2×C4×C71#12416yes—
80C2×C4×C71#12416yes—
81C2×Dic71—412no—
82C2×Dic71—46no—
83D284—44no—
84D284—412no—
85C22×C71—26no—
86C22×C71—41no—
87C22×C71—43no—
88C22×C71C2×#12011yescharacteristic
89C22×C71C2×C22⋊Q813yes—
90C22×C71C2×#12413yes—
91C2×D71—424no—
92C2×D71—44no—
93C2×D71—812no—
94C2×D71—84no—
95C4×C71—412no—
96C4×C71#39312yes—
97C4×C71#39316yes—
98Dic71—86no—
99#23469—711nomaximal, Sylow
100C2×C71—23no—
101C2×C71—23no—
102C2×C71C2×#39311yescharacteristic
103C2×C71#302712yes—
104D71—84no—
105#3458D7111yescharacteristic, complemented
106#2309—713no—
107#2029—718no—
108#1740—713no—
109C71#2346911yescharacteristic, complemented, Sylow
110C23×D4—711no—
111#544—26no—
112#543C2×D7114yes—
113#533—26no—
114C2×#117—713no—
115C2×#117C2×D7113yes—
116#390—1426no—
117#390—1426no—
118#379—7112no—
119#342—7112no—
120C25—712no—
121C22×D4—714no—
122C22×D4—1423no—
123C22×D4—1423no—
124C22×D4—1426no—
125C23×C4C22×D7111yescharacteristic
126C4⋊Q8—212no—
127#125—212no—
128C22⋊Q8—224no—
129C4×Q8—28no—
130#117—26no—
131#117—26no—
132#117—14212no—
133C2×C4⋊C4—1423no—
134C2×C4⋊C4C22⋊D7116yes—
135C2×C22⋊C4—716no—
136C2×C22⋊C4—14224no—
137C2×C22⋊C4C22×D7116yes—
138C2×C42—1423no—
139C2×C42C22⋊D7116yes—
140C24—5684no—
141C24—714no—
142C24—1426no—
143C24—1426no—
144C24C23×D7111yescharacteristic
145C2×Q8—28no—
146C2×D4—2841no—
147C2×D4—2843no—
148C2×D4—2846no—
149C2×D4—2846no—
150C2×D4—14212no—
151C2×D4—14212no—
152C22×C4—1426no—
153C22×C4C2×C22⋊D71112yes—
154C22×C4C2×C22⋊D7113yes—
155C22×C4C2×C22⋊D7113yes—
156C22×C4Q8×D7112yes—
157C22×C4#1176586616yes—
158C4⋊C4—224no—
159C4⋊C4—212no—
160C4⋊C4—2846no—
161C22⋊C4—212no—
162C22⋊C4—56824no—
163C22⋊C4—14212no—
164C42—26no—
165C42—26no—
166C42—2846no—
167C23—42no—
168C23—28424no—
169C23—2842no—
170C23—2846no—
171C23—5684no—
172C23—56812no—
173C23—14212no—
174C23C22×C22⋊D7113yes—
175C23C2×Q8×D7111yescharacteristic
176C23C2×#1176586613yes—
177Q8—88no—
178D4—2844no—
179D4—28412no—
180C2×C4—212no—
181C2×C4—412no—
182C2×C4—28412no—
183C2×C4—2846no—
184C2×C4#1455609711yescharacteristic
185C2×C4#1455609713yes—
186C2×C4#1455610416yes—
187C2×C4#1455610416yes—
188C2×C4#1455611716yes—
189C2×C4#1455611716yes—
190C22—26no—
191C22—41no—
192C22—43no—
193C22—28424no—
194C22—2844no—
195C22—56812no—
196C22—5684no—
197C22C2×#1455609711yescharacteristic
198C22C2×#1455610413yes—
199C22C2×#1455611713yes—
200C4—412no—
201C4—5686no—
202C4#1798591012yes—
203C4#1798591016yes—
204C2—23no—
205C2—23no—
206C2—5684no—
207C2C2×#1798591011yescharacteristic
208C2#2259774212yes—
subgroupG/H
centerC22C2×#14556097
commutator, cosocleC24×C71C24
FrattiniC24C23×D71
FittingC71×#3458C2
socleC22×C71C2×#120
prime radicalC23×D284C22
hypercenter#3458D71
hypocenterC71#23469
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #22589503C2semidirect
3. #22588073C2semidirect
4. C71×#3458C2semidirect
19. C22×D284Q8semidirect
105. #3458D71semidirect
109. C71#23469semidirect

Maximal subgroups

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

subgroupindex
#225909202
#225895032
#225880732
C71×#34582
#2346971