Finite Group Databasebeta

← #32987275

Subgroups — #32987275

Order 18688.

552 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#17988953C212yesmaximal
2#22610754C211yesmaximal, characteristic
3C2×#17988329C212yesmaximal, complemented
4#22607812C211yesmaximal, characteristic, complemented
5C73×#2822C211yesmaximal, characteristic, complemented
6C2×#14556588C2211yescharacteristic
7#17988953—24no—
8#17988953—24no—
9C2×#14556511—22no—
10C2×#14556511—22no—
11C2×#14556511—22no—
12C2×#14556502C2211yescharacteristic
13C2×#14556502C2212yes—
14#17988329—24no—
15C2×#125×C73C2212yes—
16#400×C73—22no—
17#398×C73—22no—
18#398×C73—22no—
19#382×C73C2211yescharacteristic
20C22×D292—22no—
21C22×D292—21no—
22C22×#10596831—22no—
23C22×#10596831—21no—
24#14556588—22no—
25#14556588—21no—
26#14556588D412yes—
27#14556511—28no—
28#14556511—28no—
29#14556511—44no—
30#14556502—22no—
31#14556502—22no—
32#14556502D414yes—
33#125×C73—22no—
34#125×C73—22no—
35#125×C73—22no—
36#125×C73—22no—
37C2×C4⋊C4×C73—22no—
38C2×C4⋊C4×C73—22no—
39C2×C4⋊C4×C73—22no—
40C2×C4⋊C4×C73D412yes—
41C2×C4⋊C4×C73D412yes—
42C2×C4⋊C4×C73D412yes—
43C2×C42×C73C2311yescharacteristic
44#95×C73—24no—
45#95×C73—22no—
46#95×C73—22no—
47C23×D73—41no—
48C2×D292—24no—
49C2×D292—22no—
50C2×D292—42no—
51C2×D292—48no—
52C2×D292—42no—
53C2×D292—42no—
54C2×D292—41no—
55C2×#10596831—22no—
56C2×#10596831—24no—
57C2×#10596831—44no—
58C2×#10596831—41no—
59C2×#10596831—41no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73C2×D412yes—
64C22×C4×C73C2×D411yescharacteristic
65C4⋊C4×C73—22no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—22no—
68C4⋊C4×C73—22no—
69C4⋊C4×C73—42no—
70C4⋊C4×C73—42no—
71C4⋊C4×C73—42no—
72C4⋊C4×C73—42no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—82no—
75C42×C73C2×D411yescharacteristic
76C42×C73C2×D412yes—
77C42×C73#4111yescharacteristic
78C23×C73#12011yescharacteristic
79C22×D73—42no—
80C22×D73—81no—
81C22×D73—84no—
82C22×D73—81no—
83C2×C4×C73—21no—
84C2×C4×C73—24no—
85C2×C4×C73—21no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73—44no—
91C2×C4×C73—44no—
92C2×C4×C73—42no—
93C2×C4×C73#12012yes—
94C2×C4×C73#12111yescharacteristic
95C2×C4×C73#12112yes—
96C2×C4×C73#12711yescharacteristic
97D292—48no—
98D292—44no—
99D292—84no—
100D292—82no—
101#10596831—44no—
102#10596831—42no—
103C22×C73#39211yescharacteristic
104C22×C73#45311yescharacteristic
105C22×C73#45414yes—
106C22×C73#45411yescharacteristic
107C2×D73—84no—
108C2×D73—82no—
109C2×D73—162no—
110C2×D73—162no—
111C4×C73—22no—
112C4×C73—22no—
113C4×C73—21no—
114C4×C73—21no—
115C4×C73—82no—
116C4×C73—82no—
117C4×C73—82no—
118#10080—731nomaximal, Sylow
119C2×C73#165712yes—
120C2×C73#165812yes—
121C2×C73#200312yes—
122C2×C73#200611yescharacteristic
123D73—162no—
124C2×#490—732no—
125#2822D7311yescharacteristic, complemented
126#1961—731no—
127C2×#330—732no—
128#1295—731no—
129C73#1008011yescharacteristic, complemented, Sylow
130C2×#127—731no—
131C2×#125C2×D7312yes—
132#490—1464no—
133#490—1464no—
134C2×#102—1462no—
135C2×#102—1462no—
136C2×#102—1462no—
137C2×#97—732no—
138C2×#97—731no—
139#400—22no—
140#398—22no—
141#398—22no—
142#382C2×D7311yescharacteristic
143#330—1464no—
144C22×D4—1462no—
145C22×D4—1461no—
146C22×C8—1462no—
147C22×C8—1461no—
148#127—732no—
149#127—1462no—
150#127—1461no—
151#125—22no—
152#125—22no—
153#125—22no—
154#125—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊D7312yes—
159C2×C4⋊C4C22⋊D7312yes—
160C2×C4⋊C4C22⋊D7312yes—
161C2×C42C22×D7311yescharacteristic
162#102—2924no—
163#102—1468no—
164#102—1468no—
165#97—734no—
166#97—1462no—
167#97—1462no—
168#95—24no—
169#95—22no—
170#95—22no—
171C24—2921no—
172C2×D4—2922no—
173C2×D4—2928no—
174C2×D4—2922no—
175C2×D4—2922no—
176C2×D4—2921no—
177C2×D4—1464no—
178C2×D4—1462no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×C22⋊D7312yes—
183C22×C4C2×C22⋊D7311yescharacteristic
184C2×C8—2924no—
185C2×C8—2921no—
186C2×C8—2921no—
187C2×C8—1462no—
188C2×C8—1464no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C42D4×D7311yescharacteristic
200C42D4×D7312yes—
201C42#1192402711yescharacteristic
202C23—2922no—
203C23—5841no—
204C23—5844no—
205C23—5841no—
206C23#1455656011yescharacteristic
207D4—2928no—
208D4—2924no—
209D4—5844no—
210D4—5842no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4#1455655912yes—
222C2×C4#1455656211yescharacteristic
223C2×C4#1455656312yes—
224C2×C4#1455658711yescharacteristic
225C8—2924no—
226C8—2922no—
227C22—5844no—
228C22—5842no—
229C22—11682no—
230C22—11682no—
231C22#1798849511yescharacteristic
232C22#1798875511yescharacteristic
233C22#1798876014yes—
234C22#1798876011yescharacteristic
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—82no—
240C4—82no—
241C4—82no—
242C2—11682no—
243C2#2260922912yes—
244C2#2260923312yes—
245C2#2261095112yes—
246C2#2261097311yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2822C2
socleC23×C73#120
prime radicalC2×#14556588C22
hypercenter#2822D73
hypocenterC73#10080
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
3. C2×#17988329C2semidirect
4. #22607812C2semidirect
5. C73×#2822C2semidirect
125. #2822D73semidirect
129. C73#10080semidirect

Maximal subgroups

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

subgroupindex
C2×#179889532
#226107542
C2×#179883292
#226078122
C73×#28222
#1008073