Finite Group Databasebeta

← #32980394

Subgroups — #32980394

Order 18688.

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

normal subgroups only

This group has 336 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#22619037C211yesmaximal, characteristic
2#22610913C211yesmaximal, characteristic
3#22608698C212yesmaximal
4#22607825C212yesmaximal
5C73×#1526C211yesmaximal, characteristic
6#17988718—22no—
7#17988718—22no—
8#17988648—22no—
9#17988644C2212yes—
10#17988611C2211yescharacteristic
11#17988433C2211yescharacteristic
12#17988346—22no—
13#17988340—22no—
14#17988338—22no—
15#533×C73C2211yescharacteristic
16C4×M16×C73C2212yes—
17#330×C73C411yescharacteristic, complemented
18#330×C73C412yescomplemented
19#330×C73C411yescharacteristic, complemented
20C2×#11924012—21no—
21C2×C4×Dic73—21no—
22C4⋊#10596831—22no—
23C4⋊#10596831—22no—
24C4⋊#10596831—22no—
25C4⋊#10596831D412yes—
26C4⋊#10596831D412yes—
27C22⋊#10596831—22no—
28#14556502—22no—
29#14556502—22no—
30C4×#10596831—22no—
31#14556498—22no—
32C2×M16×C73—22no—
33C4⋊Q8×C73Q812yes—
34#125×C73C2×C411yescharacteristic
35#125×C73C2×C411yescharacteristic
36C22⋊Q8×C73—22no—
37C4×Q8×C73—22no—
38#117×C73D412yes—
39C2×C42×C73C2311yescharacteristic
40#97×C73C2×C412yes—
41#97×C73C2×C412yes—
42C4×C8×C73—22no—
43C22×Dic73—41no—
44#11924012—42no—
45C2×#10596831—22no—
46C2×#10596831—22no—
47C2×#10596831—41no—
48C4×Dic73—22no—
49C4×Dic73—41no—
50C2×Q8×C73—22no—
51C22×C4×C73C2×D411yescharacteristic
52C22×C4×C73#4112yes—
53M16×C73—44no—
54C2×C8×C73—22no—
55C2×C8×C73—22no—
56C4⋊C4×C73—22no—
57C4⋊C4×C73—22no—
58C4⋊C4×C73—22no—
59C4⋊C4×C73C4⋊C412yes—
60C4⋊C4×C73C4⋊C412yes—
61C22⋊C4×C73—22no—
62C42×C73—22no—
63C42×C73C22×C411yescharacteristic
64C42×C73C2×D412yes—
65C42×C73C2×Q811yescharacteristic
66C23×C73#12311yescharacteristic
67Q8×C73—82no—
68C2×C4×C73—21no—
69C2×C4×C73—21no—
70C2×C4×C73—22no—
71C2×C4×C73—22no—
72C2×C4×C73—42no—
73C2×C4×C73C2×C4⋊C411yescharacteristic
74C2×C4×C73C4×D412yes—
75C2×C4×C73#12011yescharacteristic
76C2×C4×C73C22⋊Q812yes—
77C2×Dic73—42no—
78C2×Dic73—82no—
79C8×C73—24no—
80C8×C73—42no—
81#10596831—42no—
82#10596831—42no—
83#10596831—81no—
84C22×C73—21no—
85C22×C73—22no—
86C22×C73#38511yescharacteristic
87C4×C73—21no—
88C4×C73—22no—
89C4×C73—22no—
90C4×C73—42no—
91C4×C73—42no—
92C4×C73#45411yescharacteristic
93C4×C73#45611yescharacteristic
94Dic73—82no—
95#8919—731nomaximal, Sylow
96C2×C73—41no—
97C2×C73#185711yescharacteristic
98C2×C73#186112yes—
99#2987—731no—
100#1995—731no—
101#1569—732no—
102#1526D7311yescharacteristic
103#1300—732no—
104C73#891911yescharacteristic, complemented, Sylow
105#533C2×D7311yescharacteristic
106#446—1462no—
107#446—1462no—
108#433—1462no—
109#432—732no—
110#423—731no—
111C4×M16C2×D7312yes—
112#376—731no—
113#339—1462no—
114#335—1462no—
115#334—1462no—
116#330Dic7311yescharacteristic, complemented
117#330Dic7312yescomplemented
118#330Dic7311yescharacteristic, complemented
119C2×M16—22no—
120C2×M16—1461no—
121C4⋊Q8Dic14612yes—
122#125C2×Dic7311yescharacteristic
123#125C2×Dic7311yescharacteristic
124C22⋊Q8—22no—
125C4×Q8—22no—
126#117D29212yes—
127C2×C42—1461no—
128C2×C42C22×D7311yescharacteristic
129C4⋊C8—732no—
130C4⋊C8—732no—
131C4⋊C8—1462no—
132C4⋊C8—1462no—
133C4⋊C8—1462no—
134C22⋊C8—1462no—
135#97—1462no—
136#97—1462no—
137#97C2×Dic7312yes—
138#97C2×Dic7312yes—
139C4×C8—22no—
140C4×C8—1462no—
141#95—1462no—
142C2×Q8—22no—
143C22×C4—2921no—
144C22×C4C2×D29211yescharacteristic
145C22×C4#1192402712yes—
146M16—44no—
147M16—2922no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—2921no—
151C2×C8—1462no—
152C2×C8—1462no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4C4⋊Dic7312yes—
157C4⋊C4C4⋊Dic7312yes—
158C22⋊C4—22no—
159C42—22no—
160C42—2921no—
161C42—1462no—
162C42C22×Dic7311yescharacteristic
163C42D4×D7312yes—
164C42C2×Dic14611yescharacteristic
165C23#1455657511yescharacteristic
166Q8—82no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—2922no—
173C2×C4—5842no—
174C2×C4C2×C4⋊Dic7311yescharacteristic
175C2×C4D4×Dic7312yes—
176C2×C4#1455656111yescharacteristic
177C2×C4#1455656812yes—
178C8—24no—
179C8—42no—
180C8—2922no—
181C8—2922no—
182C8—5841no—
183C22—21no—
184C22—22no—
185C22#1798846211yescharacteristic
186C4—21no—
187C4—22no—
188C4—22no—
189C4—42no—
190C4—42no—
191C4—5842no—
192C4#1798876211yescharacteristic
193C4#1798877211yescharacteristic
194C2—41no—
195C2#2261014811yescharacteristic
196C2#2261017312yes—
subgroupG/H
centerC2×C4#14556561
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1526C2
socleC22×C73#385
prime radicalC23×C73#123
cosocleC2×C42×C73C23
hypercenter#1526D73
hypocenterC73#8919
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
17. #330×C73C4semidirect
18. #330×C73C4semidirect
19. #330×C73C4semidirect
104. C73#8919semidirect
116. #330Dic73semidirect
117. #330Dic73semidirect
118. #330Dic73semidirect

Maximal subgroups

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

subgroupindex
#226190372
#226109132
#226086982
#226078252
C73×#15262
#891973