Finite Group Databasebeta

← #34001317

Subgroups — #34001317

Order 18688.

550 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×C22⋊#10596833C212yesmaximal
2#22624749C214yesmaximal, complemented
3#22612415C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5C2×C4×#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—
15C22⋊#10596833—28no—
16C4×#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—28no—
37C2×#10596833—44no—
38C2×#10596833—44no—
39C22×Dic73—24no—
40C22×Dic73—24no—
41C22×Dic73—22no—
42C22×Dic73—42no—
43C22×Dic73C22⋊C412yes—
44C22×Dic73C2×D412yes—
45C4⋊Dic73—44no—
46C4⋊Dic73—44no—
47C4⋊Dic73—42no—
48C22⋊Dic73—42no—
49C4×Dic73—24no—
50C4×Dic73—41no—
51C4×Dic73—41no—
52C4×Dic73—42no—
53C24×C73C22⋊C411yescharacteristic
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—22no—
57C22×C4×C73C22⋊C411yescharacteristic
58C22×C4×C73C2×C812yes—
59C22×C4×C73M1612yes—
60C22×C4×C73C22×C411yescharacteristic
61C4⋊C4×C73—22no—
62C4⋊C4×C73C2×C814yes—
63C22⋊C4×C73—24no—
64C42×C73—24no—
65C23×C73—21no—
66C23×C73—21no—
67C23×C73—22no—
68C23×C73—22no—
69C23×C73C22⋊C812yes—
70C23×C73C2×C22⋊C411yescharacteristic
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—24no—
74C2×C4×C73—24no—
75C2×C4×C73—24no—
76C2×C4×C73—42no—
77C2×C4×C73—42no—
78C2×C4×C73C22⋊C812yes—
79C2×C4×C73#9912yes—
80C2×C4×C73#10112yes—
81C2×C4×C73C22×C811yescharacteristic
82C2×C4×C73C2×M1611yescharacteristic
83C2×Dic73—22no—
84C2×Dic73—22no—
85C2×Dic73—22no—
86C2×Dic73—22no—
87C2×Dic73—48no—
88C2×Dic73—42no—
89C2×Dic73—42no—
90C2×Dic73—42no—
91C2×Dic73#10414yes—
92C2×Dic73#13514yes—
93#10596833—48no—
94C22×C73—22no—
95C22×C73—24no—
96C22×C73—41no—
97C22×C73—42no—
98C22×C73—41no—
99C22×C73C2×C22⋊C811yescharacteristic
100C22×C73C2×#9911yescharacteristic
101C22×C73C2×#10111yescharacteristic
102C22×C73#41612yes—
103C22×C73C2×#10412yes—
104C4×C73—21no—
105C4×C73—21no—
106C4×C73—44no—
107C4×C73#32414yes—
108Dic73—24no—
109Dic73—24no—
110Dic73—82no—
111#7564—731nomaximal, Sylow
112C2×C73—42no—
113C2×C73C2×#32411yescharacteristic
114C2×C73#147612yes—
115C2×C73#150112yes—
116C2×C73#152611yescharacteristic
117C2×C73#154411yescharacteristic
118#2296—731no—
119C4×C22⋊C8—732no—
120#1271—734no—
121C73#756411yescharacteristic, complemented, Sylow
122C2×#117—731no—
123C2×#117#6440811yescharacteristic
124C22×C42—731no—
125C2×C22⋊C8—1462no—
126C2×C4×C8—1462no—
127#389—732no—
128#389—732no—
129#385—1462no—
130C4×C4⋊C4—1462no—
131#336—734no—
132C23×C4—732no—
133C23×C4C2×#6440811yescharacteristic
134C22×C8—1464no—
135#117—22no—
136#117—2922no—
137#117#1059683314yescomplemented
138C2×C4⋊C4—732no—
139C2×C4⋊C4—1462no—
140C2×C4⋊C4—1462no—
141C2×C4⋊C4C2×#6440812yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—1461no—
144C2×C42—21no—
145C2×C42—2922no—
146C2×C42—731no—
147C2×C42—731no—
148C2×C42—732no—
149C2×C42—1461no—
150C22⋊C8—1468no—
151C4×C8—2924no—
152#95—1462no—
153#95—1462no—
154C24C22⋊#6440811yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—2922no—
159C22×C4—732no—
160C22×C4—732no—
161C22×C4—1462no—
162C22×C4—1464no—
163C22×C4—1464no—
164C22×C4C22⋊#6440811yescharacteristic
165C22×C4C2×#1059683312yes—
166C22×C4#1192403712yes—
167C22×C4C22×#6440811yescharacteristic
168C2×C8—2924no—
169C2×C8—2924no—
170C2×C8—1468no—
171C4⋊C4—22no—
172C4⋊C4—2922no—
173C4⋊C4—2924no—
174C4⋊C4—2924no—
175C4⋊C4C2×#1059683314yes—
176C22⋊C4—24no—
177C22⋊C4—2922no—
178C42—24no—
179C42—2921no—
180C42—2921no—
181C42—2922no—
182C42—1464no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23C22⋊#1059683312yes—
188C23C2×C22⋊#6440811yescharacteristic
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2922no—
197C2×C4—2922no—
198C2×C4—2922no—
199C2×C4—2928no—
200C2×C4—734no—
201C2×C4—734no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1462no—
205C2×C4—1462no—
206C2×C4C22⋊#1059683312yes—
207C2×C4#1455665012yes—
208C2×C4#1455665212yes—
209C2×C4C22×#1059683311yescharacteristic
210C2×C4C2×#1192403711yescharacteristic
211C8—2928no—
212C22—22no—
213C22—24no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22C2×C22⋊#1059683311yescharacteristic
218C22C2×#1455665011yescharacteristic
219C22C2×#1455665211yescharacteristic
220C22#1798957712yes—
221C22D73⋊#10412yes—
222C4—21no—
223C4—21no—
224C4—44no—
225C4—5842no—
226C4—1464no—
227C4—1464no—
228C4#1798943214yes—
229C2—42no—
230C2C2×#1798943211yescharacteristic
231C2#2262522212yes—
232C2#2262530812yes—
233C2#2262538811yescharacteristic
234C2#2262543611yescharacteristic
subgroupG/H
centerC22×C4C22⋊#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#7564
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624749C2semidirect
32. #117×C73C8semidirect
121. C73#7564semidirect
137. #117#10596833semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#105968332
#226247492
#226124152
#756473