Finite Group Databasebeta

← #32981308

Subgroups — #32981308

Order 18688.

426 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 103 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#22620256C211yesmaximal, characteristic
2#22619033C211yesmaximal, characteristic, complemented
3#22608349C212yesmaximal
4#22607769C212yesmaximal, complemented
5C73×#1526C211yesmaximal, characteristic, complemented
6#17989137C2211yescharacteristic
7#17988944—22no—
8#17988939C412yes—
9#17988935C412yes—
10#17988718—22no—
11Q8×#10596831—22no—
12#17988648—22no—
13#17988640C2212yes—
14#17988581—22no—
15C2×C4×#10596831C2211yescharacteristic
16#17988316—22no—
17#17988316—22no—
18#533×C73C2211yescharacteristic
19C4×M16×C73C2212yes—
20#330×C73—22no—
21C2×#11924028—21no—
22C22×#10596831—21no—
23#14556591C2×C411yescharacteristic
24#14556588C2×C411yescharacteristic
25#14556579—21no—
26#14556565—22no—
27C4×D292—22no—
28C4⋊#10596831—24no—
29C4⋊#10596831D412yes—
30C4⋊#10596831D412yes—
31#14556514—24no—
32#14556511—24no—
33C22⋊#10596831—22no—
34#14556502—22no—
35C4×#10596831—22no—
36C4×#10596831—22no—
37C4×#10596831—21no—
38C4×#10596831C2×C412yes—
39C2×M16×C73—22no—
40C4⋊Q8×C73C2×C412yes—
41#125×C73D412yes—
42C22⋊Q8×C73—22no—
43C4×Q8×C73—22no—
44#117×C73D412yes—
45C2×C42×C73C2311yescharacteristic
46#97×C73—22no—
47C4×C8×C73—22no—
48#11924028—44no—
49C2×Dic146—21no—
50C2×D292—21no—
51C2×D292—22no—
52C2×C22⋊D73—41no—
53C2×C4×D73—41no—
54C2×#10596831—22no—
55C2×#10596831—22no—
56C2×#10596831—22no—
57C2×#10596831—42no—
58C4⋊Dic73—22no—
59#11924005—42no—
60C2×Q8×C73—22no—
61C22×C4×C73C2×D411yescharacteristic
62C22×C4×C73#4112yes—
63M16×C73—24no—
64M16×C73—42no—
65C2×C8×C73—42no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—24no—
68C4⋊C4×C73C22⋊C414yes—
69C22⋊C4×C73—22no—
70C42×C73—22no—
71C42×C73C22×C411yescharacteristic
72C42×C73C2×D412yes—
73C42×C73C2×D411yescharacteristic
74C23×C73#12311yescharacteristic
75C22×D73—41no—
76Q8×C73—42no—
77Q8×C73—42no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—22no—
81C2×C4×C73—24no—
82C2×C4×C73C2×C22⋊C411yescharacteristic
83C2×C4×C73C4×D412yes—
84C2×C4×C73#12011yescharacteristic
85C2×C4×C73#12112yes—
86C22⋊D73—82no—
87C2×Dic73—41no—
88D292—42no—
89D292—42no—
90D292—42no—
91C4×D73—82no—
92Dic146—42no—
93C8×C73—44no—
94#10596831—42no—
95#10596831—42no—
96#10596831—42no—
97C22×C73—22no—
98C22×C73—21no—
99C22×C73#38611yescharacteristic
100C2×D73—82no—
101C2×D73—81no—
102C4×C73—22no—
103C4×C73—21no—
104C4×C73—44no—
105C4×C73#44312yes—
106C4×C73#44412yes—
107C4×C73#45412yes—
108Dic73—81no—
109#9070—731nomaximal, Sylow
110C2×C73—41no—
111C2×C73#187612yes—
112C2×C73#188811yescharacteristic
113D73—161no—
114#3131—731no—
115#2986—731no—
116#1526D7311yescharacteristic, complemented
117#1484—732no—
118#1278—732no—
119C73#907011yescharacteristic, complemented, Sylow
120#533C2×D7311yescharacteristic
121#532—731no—
122#488—1462no—
123#487—732no—
124#486—732no—
125#446—1462no—
126C8×Q8—1462no—
127#433—1462no—
128#431—732no—
129#416—1462no—
130C4×M16C2×D7312yes—
131C2×C4×C8—731no—
132#330—22no—
133D4⋊C8—1462no—
134D4⋊C8—1462no—
135C2×#41—1461no—
136C2×M16—22no—
137C22×C8—1461no—
138C4⋊Q8—731no—
139C4⋊Q8C4×D7312yes—
140#127—731no—
141#125C22⋊D7312yes—
142#124—1461no—
143C22⋊Q8—22no—
144#121—1462no—
145C4×Q8—22no—
146C4×D4—1462no—
147#117D29212yes—
148C2×C42C22×D7311yescharacteristic
149C4⋊C8—732no—
150C4⋊C8—732no—
151C4⋊C8—1464no—
152#103—1464no—
153#102—1464no—
154C22⋊C8—1462no—
155#97—22no—
156#97—1462no—
157C4×C8—22no—
158C4×C8—732no—
159C4×C8—1462no—
160C4×C8—1462no—
161C4×C8—1461no—
162#41—2924no—
163C2×Q8—22no—
164C2×Q8—1461no—
165C2×D4—2921no—
166C2×D4—1461no—
167C2×D4—1462no—
168C22×C4—2921no—
169C22×C4C2×D29211yescharacteristic
170C22×C4#1192402712yes—
171M16—24no—
172M16—42no—
173C2×C8—42no—
174C2×C8—2922no—
175C2×C8—1462no—
176C2×C8—1462no—
177C2×C8—1462no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—1462no—
181C4⋊C4#1192400514yes—
182C22⋊C4—22no—
183C22⋊C4—2922no—
184C42—22no—
185C42C2×C4×D7311yescharacteristic
186C42D4×D7312yes—
187C42C2×C22⋊D7311yescharacteristic
188C23—2921no—
189C23#1455657511yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—2922no—
193D4—2922no—
194D4—2922no—
195D4—2922no—
196D4—5842no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—2921no—
202C2×C4—5842no—
203C2×C4C2×#1192400511yescharacteristic
204C2×C4#1455655112yes—
205C2×C4#1455656111yescharacteristic
206C2×C4#1455656312yes—
207C8—44no—
208C8—2922no—
209C8—2922no—
210C8—2922no—
211C22—22no—
212C22—21no—
213C22—5842no—
214C22—5841no—
215C22#1798846411yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—5841no—
220C4#1798870112yes—
221C4#1798870612yes—
222C4#1798876212yes—
223C2—41no—
224C2—11681no—
225C2#2261025612yes—
226C2#2261033211yescharacteristic
subgroupG/H
centerC2×C4#14556561
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1526C2
socleC22×C73#386
prime radical#17989137C22
cosocleC2×C42×C73C23
hypercenter#1526D73
hypocenterC73#9070
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #22619033C2semidirect
4. #22607769C2semidirect
5. C73×#1526C2semidirect
116. #1526D73semidirect
119. C73#9070semidirect

Maximal subgroups

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

subgroupindex
#226202562
#226190332
#226083492
#226077692
C73×#15262
#907073