Finite Group Databasebeta

← #34001437

Subgroups — #34001437

Order 18688.

514 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 132 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#22625785C212yesmaximal
2#22624749C214yesmaximal, complemented
3#22612415C211yesmaximal, characteristic
4C2×C4⋊#10596833—22no—
5C2×C22⋊#10596833—22no—
6#17989453C2214yes—
7C2×#14556550C411yescharacteristic
8C22×C4×Dic73C2211yescharacteristic
9#17988486C412yes—
10#17988486C2212yes—
11#17988462—22no—
12C4×C4⋊Dic73—22no—
13C2×#117×C73C411yescharacteristic
14C22×#10596833—24no—
15C4⋊#10596833—44no—
16C22⋊#10596833—44no—
17C23×Dic73D412yes—
18#14556550—42no—
19C2×C4⋊Dic73—22no—
20C2×C4⋊Dic73—22no—
21C2×C4⋊Dic73C2×C412yes—
22C2×C22⋊Dic73—21no—
23C2×C4×Dic73—21no—
24C2×C4×Dic73—42no—
25C2×C4×Dic73C2×C411yescharacteristic
26C2×C4×Dic73D412yes—
27C2×C4×Dic73C2311yescharacteristic
28#14556498—22no—
29#14556498—22no—
30C23×C4×C73C2×C411yescharacteristic
31#117×C73—22no—
32#117×C73C814yescomplemented
33C2×C4⋊C4×C73C2×C412yes—
34C2×C22⋊C4×C73—21no—
35C2×C42×C73—21no—
36C2×#10596833—44no—
37C2×#10596833—44no—
38C2×#10596833—44no—
39C22×Dic73—22no—
40C22×Dic73—22no—
41C22×Dic73—22no—
42C22×Dic73—42no—
43C22×Dic73—42no—
44C22×Dic73C22⋊C412yes—
45C22×Dic73C2×D412yes—
46C4⋊Dic73—44no—
47C4⋊Dic73—42no—
48C4⋊Dic73—42no—
49C4⋊Dic73—42no—
50C22⋊Dic73—42no—
51C4×Dic73—24no—
52C4×Dic73—41no—
53C4×Dic73—41no—
54C4×Dic73—42no—
55C24×C73C22⋊C411yescharacteristic
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—22no—
59C22×C4×C73C22⋊C411yescharacteristic
60C22×C4×C73C2×C812yes—
61C22×C4×C73M1612yes—
62C22×C4×C73C22×C411yescharacteristic
63C4⋊C4×C73—22no—
64C4⋊C4×C73C2×C814yes—
65C22⋊C4×C73—24no—
66C42×C73—24no—
67C23×C73—21no—
68C23×C73—21no—
69C23×C73—22no—
70C23×C73—22no—
71C23×C73C22⋊C812yes—
72C23×C73C2×C22⋊C411yescharacteristic
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—24no—
76C2×C4×C73—24no—
77C2×C4×C73—24no—
78C2×C4×C73—42no—
79C2×C4×C73—42no—
80C2×C4×C73C22⋊C812yes—
81C2×C4×C73#9912yes—
82C2×C4×C73#10112yes—
83C2×C4×C73C22×C811yescharacteristic
84C2×C4×C73C2×M1611yescharacteristic
85C2×Dic73—22no—
86C2×Dic73—22no—
87C2×Dic73—22no—
88C2×Dic73—22no—
89C2×Dic73—22no—
90C2×Dic73—44no—
91C2×Dic73—44no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73AGL(1,ℤ/8)12yes—
96C2×Dic73#13712yes—
97#10596833—84no—
98C22×C73—22no—
99C22×C73—24no—
100C22×C73—41no—
101C22×C73—41no—
102C22×C73—42no—
103C22×C73C2×C22⋊C811yescharacteristic
104C22×C73C2×#9911yescharacteristic
105C22×C73C2×#10111yescharacteristic
106C22×C73#41712yes—
107C22×C73#42112yes—
108C4×C73—21no—
109C4×C73—21no—
110C4×C73—44no—
111C4×C73#32414yes—
112Dic73—22no—
113Dic73—22no—
114Dic73—42no—
115Dic73—82no—
116#7598—731nomaximal, Sylow
117C2×C73—42no—
118C2×C73C2×#32411yescharacteristic
119C2×C73#147712yes—
120C2×C73#151312yes—
121C2×C73#153711yescharacteristic
122C2×C73#156111yescharacteristic
123#2296—731no—
124#1797—732no—
125#1271—734no—
126C73#759811yescharacteristic, complemented, Sylow
127C2×#117—731no—
128C2×#117#6440811yescharacteristic
129C22×C42—731no—
130C2×C4⋊C8—1462no—
131C2×C22⋊C8—1462no—
132#389—732no—
133#389—732no—
134#385—1462no—
135C4×C4⋊C4—1462no—
136#336—734no—
137C23×C4—732no—
138C23×C4C2×#6440811yescharacteristic
139C22×C8—1464no—
140#117—22no—
141#117—2922no—
142#117#1059683314yescomplemented
143C2×C4⋊C4—732no—
144C2×C4⋊C4—1462no—
145C2×C4⋊C4—1462no—
146C2×C4⋊C4C2×#6440812yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—1461no—
149C2×C42—21no—
150C2×C42—2922no—
151C2×C42—731no—
152C2×C42—731no—
153C2×C42—732no—
154C2×C42—1461no—
155C4⋊C8—2924no—
156C22⋊C8—2924no—
157#95—1462no—
158#95—1462no—
159C24C22⋊#6440811yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—2922no—
164C22×C4—2922no—
165C22×C4—732no—
166C22×C4—732no—
167C22×C4—1462no—
168C22×C4—1462no—
169C22×C4—1462no—
170C22×C4C22⋊#6440811yescharacteristic
171C22×C4C2×#1059683312yes—
172C22×C4#1192403712yes—
173C22×C4C22×#6440811yescharacteristic
174C2×C8—2924no—
175C2×C8—2924no—
176C2×C8—2924no—
177C4⋊C4—22no—
178C4⋊C4—2922no—
179C4⋊C4—2922no—
180C4⋊C4—2922no—
181C4⋊C4—2924no—
182C4⋊C4C2×#1059683314yes—
183C22⋊C4—24no—
184C22⋊C4—2922no—
185C42—24no—
186C42—2921no—
187C42—2921no—
188C42—2922no—
189C42—1464no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C22⋊#1059683312yes—
195C23C2×C22⋊#6440811yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—2924no—
204C2×C4—2924no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—732no—
209C2×C4—732no—
210C2×C4—1462no—
211C2×C4—1462no—
212C2×C4—1462no—
213C2×C4—1462no—
214C2×C4—1462no—
215C2×C4C22⋊#1059683312yes—
216C2×C4#1455665012yes—
217C2×C4#1455665212yes—
218C2×C4C22×#1059683311yescharacteristic
219C2×C4C2×#1192403711yescharacteristic
220C8—5844no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×C22⋊#1059683311yescharacteristic
227C22C2×#1455665011yescharacteristic
228C22C2×#1455665211yescharacteristic
229C22#1798957812yes—
230C22#1798958412yes—
231C4—21no—
232C4—21no—
233C4—44no—
234C4—2922no—
235C4—5842no—
236C4—1462no—
237C4—1462no—
238C4#1798943214yes—
239C2—42no—
240C2C2×#1798943211yescharacteristic
241C2#2262522612yes—
242C2#2262534812yes—
243C2#2262542211yescharacteristic
244C2#2262547011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#7598
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624749C2semidirect
32. #117×C73C8semidirect
126. C73#7598semidirect
142. #117#10596833semidirect

Maximal subgroups

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

subgroupindex
#226257852
#226247492
#226124152
#759873