Finite Group Databasebeta

← #34010082

Subgroups — #34010082

Order 18688.

510 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 76 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#22625182C212yesmaximal
2#22624862C214yesmaximal, complemented
3#22612515C211yesmaximal, characteristic
4#17989607—22no—
5#17989432C2214yes—
6#17989430—22no—
7C23×Dic146C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988493C412yes—
10#17988493C2212yes—
11#17988492—22no—
12#17988448—22no—
13C2×#117×C73C411yescharacteristic
14#14556658—42no—
15C22⋊#10596833—24no—
16#14556646—42no—
17C22×Dic146—24no—
18C22×Dic146—21no—
19C22×Dic146—42no—
20C22×Dic146—42no—
21C22×Dic146C2×C412yes—
22C23×Dic73—21no—
23#14556550—42no—
24C2×C4⋊Dic73C2×C411yescharacteristic
25C2×C4⋊Dic73C2311yescharacteristic
26C2×C22⋊Dic73D412yes—
27C2×C4×Dic73D412yes—
28#14556498—24no—
29#14556498—24no—
30C23×C4×C73C2×C411yescharacteristic
31#117×C73—42no—
32C2×C4⋊C4×C73—21no—
33C2×C22⋊C4×C73—21no—
34C2×C42×C73C2×C412yes—
35C2×#10596833—44no—
36C2×Dic146—24no—
37C2×Dic146—22no—
38C2×Dic146—42no—
39C2×Dic146—416no—
40C2×Dic146—42no—
41C2×Dic146—42no—
42C2×Dic146—42no—
43C2×Dic146M1614yes—
44C22×Dic73—24no—
45C22×Dic73—22no—
46C22×Dic73—42no—
47C22×Dic73—42no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—41no—
51C4⋊Dic73—41no—
52C22⋊Dic73—42no—
53C4×Dic73—22no—
54C4×Dic73—22no—
55C24×C73C22⋊C411yescharacteristic
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—22no—
59C22×C4×C73—22no—
60C22×C4×C73C22⋊C411yescharacteristic
61C22×C4×C73C22×C411yescharacteristic
62C4⋊C4×C73—42no—
63C22⋊C4×C73—42no—
64C42×C73—42no—
65C23×C73—21no—
66C23×C73—21no—
67C23×C73—41no—
68C23×C73—41no—
69C23×C73#10012yes—
70C23×C73C2×C22⋊C411yescharacteristic
71C2×C4×C73—28no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—42no—
77C2×C4×C73—42no—
78C2×C4×C73—42no—
79C2×C4×C73#9912yes—
80C2×C4×C73C2×M1612yes—
81C2×Dic73—24no—
82C2×Dic73—22no—
83C2×Dic73—22no—
84C2×Dic73—48no—
85C2×Dic73—42no—
86C2×Dic73—42no—
87C2×Dic73—42no—
88C2×Dic73—42no—
89Dic146—48no—
90Dic146—44no—
91Dic146—44no—
92#10596833—84no—
93C22×C73—21no—
94C22×C73—21no—
95C22×C73—22no—
96C22×C73—42no—
97C22×C73—42no—
98C22×C73—41no—
99C22×C73—41no—
100C22×C73#40711yescharacteristic
101C22×C73C2×#9911yescharacteristic
102C22×C73C2×#10011yescharacteristic
103C4×C73—22no—
104C4×C73—22no—
105C4×C73—82no—
106Dic73—42no—
107Dic73—42no—
108Dic73—44no—
109#11510—731nomaximal, Sylow
110C2×C73—21no—
111C2×C73—21no—
112C2×C73—42no—
113C2×C73#146211yescharacteristic
114C2×C73#212911yescharacteristic
115C2×C73#213511yescharacteristic
116#2310—731no—
117#1464—732no—
118#1319—734no—
119C73#1151011yescharacteristic, complemented, Sylow
120C23×Q8—731no—
121C2×#117—731no—
122C2×#117#6440811yescharacteristic
123#432—1462no—
124#391—732no—
125#391—732no—
126#391—1462no—
127#381—1462no—
128#324—734no—
129#323—1462no—
130C22×Q8—2922no—
131C22×Q8—2922no—
132C22×Q8—732no—
133C22×Q8—1464no—
134C22×Q8—1461no—
135C23×C4—1461no—
136C23×C4C2×#6440811yescharacteristic
137#117—42no—
138#117—2922no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—731no—
141C2×C4⋊C4—731no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—732no—
144C2×C42—732no—
145C2×C42C2×#6440812yes—
146C4⋊C8—2922no—
147C22⋊C8—1464no—
148#97—2922no—
149#95—1464no—
150#95—1464no—
151C24C22⋊#6440811yescharacteristic
152C2×Q8—29216no—
153C2×Q8—2922no—
154C2×Q8—2922no—
155C2×Q8—2922no—
156C2×Q8—2922no—
157C2×Q8—734no—
158C2×Q8—1464no—
159C2×Q8—1462no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—2922no—
165C22×C4—2922no—
166C22×C4—732no—
167C22×C4—732no—
168C22×C4—1464no—
169C22×C4—1462no—
170C22×C4C22⋊#6440811yescharacteristic
171C22×C4C22×#6440811yescharacteristic
172C2×C8—2924no—
173C4⋊C4—42no—
174C4⋊C4—2921no—
175C4⋊C4—2921no—
176C22⋊C4—42no—
177C22⋊C4—2922no—
178C42—42no—
179C42—1462no—
180C42—1462no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#1455665112yes—
186C23C2×C22⋊#6440811yescharacteristic
187Q8—2924no—
188Q8—2924no—
189Q8—2928no—
190C2×C4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—2928no—
199C2×C4—2922no—
200C2×C4—2922no—
201C2×C4—2922no—
202C2×C4—2922no—
203C2×C4—1464no—
204C2×C4—1462no—
205C2×C4—1462no—
206C2×C4#1455665012yes—
207C2×C4D73⋊M1612yes—
208C8—5844no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22#1798955911yescharacteristic
217C22C2×#1455665011yescharacteristic
218C22C2×#1455665111yescharacteristic
219C4—22no—
220C4—22no—
221C4—82no—
222C4—2924no—
223C4—2922no—
224C4—2922no—
225C2—21no—
226C2—21no—
227C2—42no—
228C2#2262517811yescharacteristic
229C2#2262621511yescharacteristic
230C2#2262623511yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#117#64408
hypocenterC73#11510
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624862C2semidirect
119. C73#11510semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226248622
#226125152
#1151073