Finite Group Databasebeta

← #34013233

Subgroups — #34013233

Order 18688.

566 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 80 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#22625765C212yesmaximal
2#22625029C214yesmaximal, complemented
3#22612515C211yesmaximal, characteristic
4C2×#14556652—22no—
5C2×#14556652C2214yes—
6#17989557—22no—
7C23×Dic146C2211yescharacteristic
8C2×#14556550C411yescharacteristic
9#17988493—22no—
10#17988492C412yes—
11#17988492C2212yes—
12#17988448—22no—
13C2×#117×C73C411yescharacteristic
14C2×#11924037—24no—
15#14556652—28no—
16#14556652—44no—
17C22⋊#10596833—44no—
18C22×Dic146—24no—
19C22×Dic146—42no—
20C22×Dic146—42no—
21C22×Dic146C2×C411yescharacteristic
22C22×Dic146D412yes—
23C22×Dic146C2311yescharacteristic
24C23×Dic73D412yes—
25#14556550—42no—
26C2×C4⋊Dic73—21no—
27C2×C22⋊Dic73—21no—
28C2×C4×Dic73C2×C412yes—
29#14556498—24no—
30#14556498—24no—
31C23×C4×C73C2×C411yescharacteristic
32#117×C73—42no—
33C2×C4⋊C4×C73C2×C412yes—
34C2×C22⋊C4×C73—21no—
35C2×C42×C73—21no—
36#11924037—44no—
37#11924037—44no—
38C2×#10596833—44no—
39C2×Dic146—24no—
40C2×Dic146—22no—
41C2×Dic146—24no—
42C2×Dic146—41no—
43C2×Dic146—41no—
44C2×Dic146—42no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146—416no—
48C2×Dic146#4114yes—
49C22×Dic73—24no—
50C22×Dic73—24no—
51C22×Dic73—22no—
52C22×Dic73—42no—
53C22×Dic73C22⋊C412yes—
54C22×Dic73C2×D412yes—
55C4⋊Dic73—42no—
56C22⋊Dic73—42no—
57C4×Dic73—42no—
58C24×C73C22⋊C411yescharacteristic
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73C22⋊C411yescharacteristic
64C22×C4×C73C22×C411yescharacteristic
65C4⋊C4×C73—42no—
66C22⋊C4×C73—42no—
67C42×C73—42no—
68C23×C73—21no—
69C23×C73—21no—
70C23×C73—42no—
71C23×C73#9912yes—
72C23×C73C2×C22⋊C411yescharacteristic
73C2×C4×C73—28no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—44no—
79C2×C4×C73—42no—
80C2×C4×C73#9912yes—
81C2×C4×C73#11712yes—
82C2×Dic73—22no—
83C2×Dic73—22no—
84C2×Dic73—22no—
85C2×Dic73—22no—
86C2×Dic73—22no—
87C2×Dic73—48no—
88C2×Dic73—44no—
89C2×Dic73—42no—
90C2×Dic73#12314yes—
91Dic146—44no—
92Dic146—48no—
93Dic146—44no—
94#10596833—84no—
95C22×C73—22no—
96C22×C73—22no—
97C22×C73—24no—
98C22×C73—42no—
99C22×C73—42no—
100C22×C73#38111yescharacteristic
101C22×C73C2×#9911yescharacteristic
102C22×C73C2×#9911yescharacteristic
103C4×C73—22no—
104C4×C73—22no—
105C4×C73—82no—
106Dic73—24no—
107Dic73—42no—
108Dic73—82no—
109#12862—731nomaximal, Sylow
110C2×C73—21no—
111C2×C73—21no—
112C2×C73—42no—
113C2×C73#181911yescharacteristic
114C2×C73#213312yes—
115#2310—731no—
116#1784—732no—
117#1394—734no—
118C73#1286211yescharacteristic, complemented, Sylow
119C23×Q8—731no—
120C2×#117—731no—
121C2×#117#6440811yescharacteristic
122C2×#101—734no—
123C2×#101—1462no—
124#407—1462no—
125#391—732no—
126#391—732no—
127#391—1462no—
128#381—1462no—
129C22×Q8—2922no—
130C22×Q8—2922no—
131C22×Q8—732no—
132C22×Q8—731no—
133C22×Q8—731no—
134C22×Q8—1464no—
135C23×C4—732no—
136C23×C4C2×#6440811yescharacteristic
137C2×M16—1464no—
138#117—42no—
139#117—2922no—
140C2×C4⋊C4—1461no—
141C2×C4⋊C4C2×#6440812yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—1461no—
144C2×C42—21no—
145C2×C42—732no—
146#101—2924no—
147#101—1468no—
148C22⋊C8—2924no—
149#95—1464no—
150#95—1464no—
151C24C22⋊#6440811yescharacteristic
152C2×Q8—29216no—
153C2×Q8—2922no—
154C2×Q8—2921no—
155C2×Q8—2921no—
156C2×Q8—2921no—
157C2×Q8—2921no—
158C2×Q8—734no—
159C2×Q8—1464no—
160C2×Q8—1464no—
161C2×Q8—1462no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—2922no—
167C22×C4—732no—
168C22×C4—732no—
169C22×C4—1462no—
170C22×C4—1464no—
171C22×C4—1464no—
172C22×C4C22⋊#6440811yescharacteristic
173C22×C4C22×#6440811yescharacteristic
174M16—2924no—
175M16—2924no—
176C2×C8—2924no—
177C4⋊C4—42no—
178C4⋊C4—2922no—
179C22⋊C4—42no—
180C22⋊C4—2922no—
181C42—42no—
182C42—2922no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1455664912yes—
187C23C2×C22⋊#6440811yescharacteristic
188Q8—2924no—
189Q8—2928no—
190Q8—2924no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—2928no—
199C2×C4—2924no—
200C2×C4—2922no—
201C2×C4—734no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1462no—
205C2×C4—1462no—
206C2×C4—1462no—
207C2×C4#1455665012yes—
208C2×C4#1455667012yes—
209C8—5844no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1798953011yescharacteristic
216C22C2×#1455664911yescharacteristic
217C22C2×#1455665011yescharacteristic
218C4—22no—
219C4—22no—
220C4—82no—
221C4—2922no—
222C4—5842no—
223C4—1464no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#2262582611yescharacteristic
228C2#2262622612yes—
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC22×Dic146C23
hypercenterC2×#117#64408
hypocenterC73#12862
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22625029C2semidirect
118. C73#12862semidirect

Maximal subgroups

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

subgroupindex
#226257652
#226250292
#226125152
#1286273