Finite Group Databasebeta

← #32999006

Subgroups — #32999006

Order 18688.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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#22611281C211yesmaximal, characteristic
2#22608938C212yesmaximal
3#22607937C212yesmaximal
4#22607875C211yesmaximal, characteristic
5C73×#2822C211yesmaximal, characteristic
6#17989218C2211yescharacteristic
7#17988917—22no—
8#17988627—21no—
9#17988356C2211yescharacteristic
10#17988347—21no—
11#17988323—22no—
12#17988313C2212yes—
13C2×#125×C73—21no—
14#400×C73—22no—
15#398×C73—21no—
16#398×C73—22no—
17#398×C73C2212yes—
18#382×C73C2211yescharacteristic
19C2×#11924012—21no—
20#14556590—41no—
21#14556582—41no—
22#14556570—22no—
23#14556550—21no—
24#14556524—41no—
25#14556522—41no—
26#14556522—42no—
27#14556514—42no—
28C22⋊#10596831—22no—
29#125×C73—42no—
30#125×C73—42no—
31C2×C4⋊C4×C73—21no—
32C2×C4⋊C4×C73—22no—
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73—21no—
36C2×C4⋊C4×C73D411yescharacteristic
37C2×C4⋊C4×C73D412yes—
38C2×C4⋊C4×C73C2311yescharacteristic
39C2×C42×C73D411yescharacteristic
40#95×C73—22no—
41#95×C73—21no—
42#95×C73—22no—
43#95×C73—22no—
44#95×C73D412yes—
45C2×Dic146—22no—
46#11924012—22no—
47#11924012—41no—
48C2×#10596831—41no—
49C2×#10596831—42no—
50C4⋊Dic73—21no—
51C4⋊Dic73—22no—
52C4⋊Dic73—21no—
53#11924007—42no—
54C22⋊Dic73—41no—
55C4×Dic73—41no—
56C22×C4×C73—21no—
57C22×C4×C73—21no—
58C22×C4×C73—22no—
59C22×C4×C73—22no—
60C22×C4×C73C2×D411yescharacteristic
61C22×C4×C73C2×D412yes—
62C4⋊C4×C73—41no—
63C4⋊C4×C73—42no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—42no—
66C4⋊C4×C73—44no—
67C4⋊C4×C73—42no—
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C4⋊C4×C73—81no—
71C42×C73—21no—
72C42×C73SD1612yes—
73C23×C73#12011yescharacteristic
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—41no—
79C2×C4×C73—41no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73—42no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—41no—
89C2×C4×C73C2×SD1611yescharacteristic
90C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
91C2×Dic73—41no—
92C2×Dic73—41no—
93Dic146—82no—
94#10596831—42no—
95#10596831—82no—
96C22×C73—21no—
97C22×C73—21no—
98C22×C73#45011yescharacteristic
99C22×C73#45611yescharacteristic
100C22×C73B(4,𝔽2)11yescharacteristic
101C4×C73—21no—
102C4×C73—21no—
103C4×C73—42no—
104C4×C73—42no—
105C4×C73—82no—
106C4×C73—82no—
107C4×C73—81no—
108Dic73—81no—
109Dic73—81no—
110#12251—731nomaximal, Sylow
111C2×C73—21no—
112C2×C73—21no—
113C2×C73#160411yescharacteristic
114C2×C73#220111yescharacteristic
115C2×C73#220311yescharacteristic
116#2822D7311yescharacteristic
117#2061—731no—
118#1611—732no—
119#1346—732no—
120#1324—731no—
121C73#1225111yescharacteristic, complemented, Sylow
122#544—731no—
123C2×#125—21no—
124#483—1462no—
125#428—1461no—
126#400—22no—
127#398—21no—
128#398—22no—
129#398C2×D7312yes—
130#382C2×D7311yescharacteristic
131#343—731no—
132#339—1461no—
133#328—1462no—
134#324—732no—
135C2×M16—1461no—
136C4⋊Q8—2921no—
137#125—42no—
138#125—42no—
139#125—2921no—
140C22⋊Q8—1462no—
141#117—1461no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D7311yescharacteristic
148C2×C4⋊C4C22⋊D7312yes—
149C2×C4⋊C4C22×D7311yescharacteristic
150C2×C42C22⋊D7311yescharacteristic
151#107—2921no—
152#106—2921no—
153#106—2922no—
154#103—2922no—
155C22⋊C8—1462no—
156#95—22no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95C22⋊D7312yes—
161C2×Q8—1462no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D7311yescharacteristic
167C22×C4C2×C22⋊D7312yes—
168M16—2921no—
169M16—1462no—
170C2×C8—2921no—
171C2×C8—2922no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—2922no—
182C4⋊C4—1461no—
183C4⋊C4—1462no—
184C4⋊C4—1461no—
185C22⋊C4—2921no—
186C42—21no—
187C42—2921no—
188C42#1192401612yes—
189C23#1455656011yescharacteristic
190Q8—5842no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—2921no—
207C2×C4—2921no—
208C2×C4C2×#1192401611yescharacteristic
209C2×C4#1455661711yescharacteristic
210C8—2922no—
211C8—5842no—
212C22—21no—
213C22—21no—
214C22#1798873811yescharacteristic
215C22#1798877011yescharacteristic
216C22#1798877411yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—5841no—
225C4—5841no—
226C2—21no—
227C2—21no—
228C2#2260890011yescharacteristic
229C2#2261194411yescharacteristic
230C2#2261195211yescharacteristic
subgroupG/H
centerC22#17988774
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2822C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2822D73
hypocenterC73#12251
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
121. C73#12251semidirect

Maximal subgroups

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

subgroupindex
#226112812
#226089382
#226079372
#226078752
C73×#28222
#1225173