Finite Group Databasebeta

← #34004228

Subgroups — #34004228

Order 18688.

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

normal subgroups only

This group has 802 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#22625787C212yesmaximal
2C2×#17989446C214yesmaximal, complemented
3C2×#17989140C211yesmaximal, characteristic
4C2×#14556658C2214yes—
5C2×C22⋊#10596833—22no—
6#17989453—22no—
7#17989446—28no—
8#17989140—28no—
9C2×#14556590C2212yes—
10C2×#14556582C412yes—
11C2×#14556568—22no—
12C2×C4×Dic146—22no—
13C2×#14556550C411yescharacteristic
14C22×C4×Dic73C2211yescharacteristic
15C2×#117×C73C411yescharacteristic
16C22×#10596833—24no—
17#14556658—28no—
18C22⋊#10596833—44no—
19C22×Dic146—22no—
20C23×Dic73D412yes—
21#14556590—24no—
22#14556590D418yes—
23#14556582—24no—
24#14556582—24no—
25#14556568—216no—
26C4×Dic146—216no—
27#14556550—24no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73C2×C412yes—
30C2×#11924007—22no—
31C2×#11924007—22no—
32C2×C22⋊Dic73—21no—
33C2×C4×Dic73—24no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73D412yes—
37C2×C4×Dic73C2311yescharacteristic
38C23×C4×C73C2×C411yescharacteristic
39#117×C73—24no—
40C2×C4⋊C4×C73C2×C412yes—
41C2×C22⋊C4×C73—21no—
42C2×C42×C73—21no—
43C2×#10596833—44no—
44C2×#10596833—44no—
45C2×#10596833—44no—
46C2×Dic146—28no—
47C2×Dic146—44no—
48C2×Dic146—42no—
49C2×Dic146—42no—
50C22×Dic73—22no—
51C22×Dic73—24no—
52C22×Dic73—24no—
53C22×Dic73—22no—
54C22×Dic73—22no—
55C22×Dic73C22⋊C412yes—
56C22×Dic73C2×D412yes—
57C4⋊Dic73—28no—
58C4⋊Dic73—22no—
59C4⋊Dic73C22⋊C414yes—
60#11924007—28no—
61#11924007—28no—
62C22⋊Dic73—24no—
63C4×Dic73—24no—
64C4×Dic73—22no—
65C4×Dic73—24no—
66C4×Dic73C2×D414yes—
67C24×C73C22⋊C411yescharacteristic
68C22×C4×C73—22no—
69C22×C4×C73—22no—
70C22×C4×C73—22no—
71C22×C4×C73—22no—
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22×C411yescharacteristic
74C4⋊C4×C73—22no—
75C4⋊C4×C73C22⋊C414yes—
76C22⋊C4×C73—24no—
77C42×C73—24no—
78C23×C73—21no—
79C23×C73—21no—
80C23×C73—22no—
81C23×C73—22no—
82C23×C73#10112yes—
83C23×C73C2×C22⋊C411yescharacteristic
84C2×C4×C73—28no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—24no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73#10112yes—
93C2×C4×C73C2×C22⋊C412yes—
94C2×Dic73—22no—
95C2×Dic73—24no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—44no—
101C2×Dic73—44no—
102C2×Dic73#12014yes—
103C2×Dic73#13514yes—
104C2×Dic73AGL(1,ℤ/8)12yes—
105C2×Dic73#13712yes—
106Dic146—48no—
107Dic146—48no—
108#10596833—84no—
109C22×C73—24no—
110C22×C73—22no—
111C22×C73—41no—
112C22×C73—41no—
113C22×C73—42no—
114C22×C73#37911yescharacteristic
115C22×C73C2×#10111yescharacteristic
116C22×C73C2×#10111yescharacteristic
117C22×C73#41612yes—
118C22×C73#41712yes—
119C4×C73—22no—
120C4×C73—22no—
121C4×C73—44no—
122Dic73—44no—
123Dic73—42no—
124Dic73#44914yes—
125Dic73#45214yes—
126#8345—731nomaximal, Sylow
127C2×C73—42no—
128C2×C73#154411yescharacteristic
129C2×C73#155212yes—
130C2×C73#156111yescharacteristic
131C2×C73#178411yescharacteristic
132C2×C73#178812yes—
133C2×#533—731no—
134#1797—732no—
135C2×#332—734no—
136C73#834511yescharacteristic, complemented, Sylow
137#533—1468no—
138C2×C4⋊Q8—732no—
139C2×#125—732no—
140C2×C22⋊Q8—1462no—
141C2×C4×Q8—1462no—
142C2×#117—731no—
143C2×#117#6440811yescharacteristic
144C22×C42—731no—
145C2×C4⋊C8—734no—
146C2×C22⋊C8—1462no—
147#336—1462no—
148#332—1468no—
149C22×Q8—1462no—
150C23×C4—732no—
151C23×C4C2×#6440811yescharacteristic
152C22×C8—1464no—
153C4⋊Q8—738no—
154C4⋊Q8—1464no—
155#125—1464no—
156#125—1464no—
157C22⋊Q8—14616no—
158C4×Q8—14616no—
159#117—24no—
160#117—1464no—
161C2×C4⋊C4—732no—
162C2×C4⋊C4—1462no—
163C2×C4⋊C4—1462no—
164C2×C4⋊C4—1462no—
165C2×C4⋊C4C2×#6440812yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—1461no—
168C2×C42—21no—
169C2×C42—731no—
170C2×C42—732no—
171C2×C42—731no—
172C2×C42—1464no—
173C2×C42—1461no—
174C4⋊C8—1468no—
175C22⋊C8—2924no—
176C24C22⋊#6440811yescharacteristic
177C2×Q8—2924no—
178C2×Q8—2922no—
179C2×Q8—2922no—
180C2×Q8—1468no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—732no—
186C22×C4—732no—
187C22×C4—1462no—
188C22×C4—1464no—
189C22×C4—1464no—
190C22×C4—1462no—
191C22×C4—1462no—
192C22×C4C22⋊#6440811yescharacteristic
193C22×C4C22×#6440811yescharacteristic
194C2×C8—2924no—
195C2×C8—2924no—
196C2×C8—2924no—
197C4⋊C4—22no—
198C4⋊C4—734no—
199C4⋊C4—1468no—
200C4⋊C4—1468no—
201C4⋊C4—1468no—
202C4⋊C4—1462no—
203C4⋊C4C22⋊#6440814yes—
204C22⋊C4—24no—
205C22⋊C4—1464no—
206C42—24no—
207C42—734no—
208C42—1464no—
209C42—1464no—
210C42—1462no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23#1455665212yes—
216C23C2×C22⋊#6440811yescharacteristic
217Q8—2928no—
218Q8—2928no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—2924no—
230C2×C4—2924no—
231C2×C4—734no—
232C2×C4—734no—
233C2×C4—732no—
234C2×C4—732no—
235C2×C4—1462no—
236C2×C4—1464no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4#1455665212yes—
240C2×C4C2×C22⋊#6440812yes—
241C8—5844no—
242C22—24no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22#1798952811yescharacteristic
248C22C2×#1455665211yescharacteristic
249C22C2×#1455665211yescharacteristic
250C22#1798957712yes—
251C22#1798957812yes—
252C4—22no—
253C4—22no—
254C4—44no—
255C4—2922no—
256C4—2924no—
257C4—734no—
258C4—734no—
259C2—42no—
260C2#2262543611yescharacteristic
261C2#2262545212yes—
262C2#2262547011yescharacteristic
263C2#2262576511yescharacteristic
264C2#2262577112yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#8345
LCSG ▷ C22×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. C2×#17989446C2semidirect
136. C73#8345semidirect

Maximal subgroups

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

subgroupindex
#226257872
C2×#179894462
C2×#179891402
#834573