Finite Group Databasebeta

← #34010060

Subgroups — #34010060

Order 18688.

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

normal subgroups only

This group has 510 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#22625182C212yesmaximal
2#22624857C214yesmaximal
3#22612517C211yesmaximal, characteristic
4#17989607—22no—
5#17989432—22no—
6#17989430C2214yes—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988494—22no—
10#17988494—22no—
11#17988450C412yes—
12#17988450C2212yes—
13C23×Q8×C73C411yescharacteristic
14#14556658—42no—
15C22⋊#10596833—24no—
16#14556646—42no—
17#14556550—41no—
18#14556550—41no—
19#14556550—42no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73D412yes—
22C2×C22⋊Dic73C2×C412yes—
23C2×C22⋊Dic73C2×C411yescharacteristic
24C2×C22⋊Dic73C2311yescharacteristic
25C2×C4×Dic73—21no—
26C2×C4×Dic73D412yes—
27#14556498—24no—
28#14556498—24no—
29C22×Q8×C73—21no—
30C22×Q8×C73—21no—
31C22×Q8×C73—24no—
32C22×Q8×C73—41no—
33C22×Q8×C73—41no—
34C22×Q8×C73—42no—
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73C2×C412yes—
37C2×#10596833—44no—
38C22×Dic73—22no—
39C22×Dic73C22⋊C412yes—
40C22×Dic73C2×D412yes—
41C4⋊Dic73—42no—
42C4⋊Dic73—42no—
43C22⋊Dic73—41no—
44C22⋊Dic73—41no—
45C22⋊Dic73—42no—
46C4×Dic73—22no—
47C4×Dic73—22no—
48C4×Dic73—42no—
49C24×C73C22×C411yescharacteristic
50C2×Q8×C73—24no—
51C2×Q8×C73—24no—
52C2×Q8×C73—48no—
53C2×Q8×C73—48no—
54C2×Q8×C73—42no—
55C2×Q8×C73—42no—
56C2×Q8×C73—41no—
57C2×Q8×C73—41no—
58C2×Q8×C73—42no—
59C22×C4×C73—22no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—22no—
65C22×C4×C73—42no—
66C22×C4×C73—42no—
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73M1614yes—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73—42no—
73C23×C73C2×C22⋊C411yescharacteristic
74C23×C73C2×M1612yes—
75Q8×C73—44no—
76Q8×C73—44no—
77Q8×C73—48no—
78C2×C4×C73—24no—
79C2×C4×C73—24no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—24no—
85C2×C4×C73—44no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—44no—
89C2×C4×C73#9911yescharacteristic
90C2×C4×C73#9911yescharacteristic
91C2×C4×C73#10012yes—
92C2×Dic73—22no—
93C2×Dic73—22no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—44no—
97C2×Dic73—42no—
98#10596833—84no—
99C22×C73—22no—
100C22×C73—22no—
101C22×C73—42no—
102C22×C73—42no—
103C22×C73—42no—
104C22×C73#40711yescharacteristic
105C22×C73C2×#9911yescharacteristic
106C22×C73C2×#10011yescharacteristic
107C4×C73—21no—
108C4×C73—21no—
109C4×C73—22no—
110C4×C73—44no—
111Dic73—42no—
112Dic73—42no—
113Dic73—82no—
114#11504—731nomaximal, Sylow
115C2×C73—21no—
116C2×C73—21no—
117C2×C73—42no—
118C2×C73#146211yescharacteristic
119C2×C73#212312yes—
120#2310—731no—
121#1464—732no—
122#1317—734no—
123C73#1150411yescharacteristic, complemented, Sylow
124C23×Q8#6440811yescharacteristic
125C2×#117—731no—
126C2×#117—731no—
127#432—1462no—
128#391—1462no—
129#391—1462no—
130#381—732no—
131#381—732no—
132#324—1462no—
133#323—734no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—42no—
140C23×C4C2×#6440812yes—
141C23×C4C2×#6440811yescharacteristic
142#117—2922no—
143#117—2921no—
144#117—2921no—
145C2×C4⋊C4—732no—
146C2×C4⋊C4—1461no—
147C2×C22⋊C4—731no—
148C2×C22⋊C4—731no—
149C2×C22⋊C4—732no—
150C2×C42—732no—
151C2×C42—1461no—
152C4⋊C8—2922no—
153C22⋊C8—1464no—
154#97—2922no—
155#95—1464no—
156#95—1464no—
157C24C22×#6440811yescharacteristic
158C2×Q8—24no—
159C2×Q8—24no—
160C2×Q8—48no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—732no—
176C22×C4—732no—
177C22×C4—1462no—
178C22×C4C22⋊#6440811yescharacteristic
179C22×C4C22⋊#6440811yescharacteristic
180C22×C4#1192403714yes—
181C2×C8—2924no—
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C22⋊C4—2922no—
185C22⋊C4—2921no—
186C22⋊C4—2921no—
187C42—2922no—
188C42—1462no—
189C42—1462no—
190C23—21no—
191C23—21no—
192C23—42no—
193C23C2×C22⋊#6440811yescharacteristic
194C23C2×#1192403712yes—
195Q8—44no—
196Q8—44no—
197Q8—48no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—2924no—
212C2×C4—2922no—
213C2×C4—1462no—
214C2×C4—1462no—
215C2×C4#1455664911yescharacteristic
216C2×C4#1455664911yescharacteristic
217C2×C4#1455665112yes—
218C8—5844no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22#1798955711yescharacteristic
225C22C2×#1455664911yescharacteristic
226C22C2×#1455665111yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—44no—
231C4—2922no—
232C4—2922no—
233C4—5842no—
234C2—21no—
235C2—21no—
236C2—42no—
237C2#2262517611yescharacteristic
238C2#2262619312yes—
subgroupG/H
centerC22C2×#14556649
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC23×Q8×C73C4
socleC22×C73C2×#99
cosocleC2×C22⋊Dic73C23
hypercenterC23×Q8#64408
hypocenterC73#11504
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
123. C73#11504semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226248572
#226125172
#1150473