Finite Group Databasebeta

← #34014190

Subgroups — #34014190

Order 18688.

420 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 84 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#22625797C211yesmaximal, characteristic
2C4×#14556652C211yesmaximal, characteristic
3#22625029C214yesmaximal
4#22615140C211yesmaximal, characteristic
5#17989590—21no—
6C2×#14556652C2212yes—
7C2×#14556652C2212yes—
8C4×#11924037—21no—
9#17989467—21no—
10#17989467—21no—
11C2×C4×Dic146C2211yescharacteristic
12#17988506—21no—
13#17988506—21no—
14#17988492C2212yes—
15#17988483—21no—
16#17988483—21no—
17#17988456C412yes—
18C4×C4⋊Dic73C411yescharacteristic
19C2×C4⋊Q8×C73C411yescharacteristic
20C2×#11924037—22no—
21C2×#11924037—22no—
22#14556658—41no—
23C4⋊#10596833—41no—
24#14556652—22no—
25#14556652—24no—
26#14556652Q814yescomplemented
27#14556646—41no—
28C4×#10596833—41no—
29C22×Dic146C2311yescharacteristic
30C4×Dic146—41no—
31C4×Dic146—41no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×#11924007—22no—
35C2×#11924007—22no—
36C2×#11924007D411yescharacteristic
37C2×#11924007D411yescharacteristic
38C2×C4×Dic73C2×C412yes—
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41#14556498—22no—
42#14556498—22no—
43#14556498—21no—
44C22×Q8×C73—21no—
45C4⋊Q8×C73—41no—
46C4⋊Q8×C73—41no—
47C2×C4⋊C4×C73—21no—
48C2×C4⋊C4×C73C2×C412yes—
49C2×C42×C73C2×C411yescharacteristic
50#11924037—24no—
51#11924037—42no—
52#11924037—42no—
53#11924037—42no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C2×Dic146—41no—
57C2×Dic146—41no—
58C2×Dic146C2×Q812yes—
59C2×Dic146#4112yes—
60C22×Dic73—22no—
61C22×Dic73C22⋊C411yescharacteristic
62C22×Dic73C22⋊C411yescharacteristic
63C22×Dic73C2×D411yescharacteristic
64C22×Dic73C2×D411yescharacteristic
65C4⋊Dic73—24no—
66C4⋊Dic73—41no—
67#11924007—41no—
68#11924007—41no—
69#11924007—82no—
70#11924007—82no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C4×Dic73—21no—
74C4×Dic73—21no—
75C4×Dic73—42no—
76C2×Q8×C73—42no—
77C2×Q8×C73—82no—
78C22×C4×C73—22no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82C4⋊C4×C73—42no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C42×C73—21no—
86C42×C73—21no—
87C23×C73C2×C22⋊C411yescharacteristic
88Q8×C73—84no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—22no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73—42no—
95C2×C4×C73—42no—
96C2×C4×C73—42no—
97C2×C4×C73#9911yescharacteristic
98C2×C4×C73#9911yescharacteristic
99C2×C4×C73C4×Q812yes—
100C2×Dic73—21no—
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—21no—
104C2×Dic73—24no—
105C2×Dic73—21no—
106C2×Dic73—21no—
107C2×Dic73—42no—
108C2×Dic73—42no—
109C2×Dic73—41no—
110C2×Dic73—41no—
111C2×Dic73C22⋊Q812yes—
112C2×Dic73#12411yescharacteristic
113C2×Dic73C4⋊Q811yescharacteristic
114Dic146—42no—
115Dic146—42no—
116#10596833—44no—
117#10596833—82no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73#39111yescharacteristic
121C22×C73C2×#9911yescharacteristic
122C22×C73#41011yescharacteristic
123C4×C73—21no—
124C4×C73—22no—
125C4×C73—41no—
126C4×C73—82no—
127C4×C73#35612yes—
128Dic73—22no—
129Dic73—21no—
130Dic73—21no—
131Dic73—44no—
132Dic73—41no—
133Dic73—41no—
134#13302—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—21no—
137C2×C73#199611yescharacteristic
138C2×C73C2×#35611yescharacteristic
139C2×C73#213411yescharacteristic
140#2567—731no—
141#1801—731no—
142C4×#101—731no—
143#1394—734no—
144C73#1330211yescharacteristic, complemented, Sylow
145C2×C4⋊Q8#6440811yescharacteristic
146C2×C4×Q8—731no—
147#423—1461no—
148C2×#101—732no—
149C2×#101—732no—
150C4×M16—1461no—
151#395—1461no—
152#395—1461no—
153#391—732no—
154#389—1461no—
155#389—1461no—
156#384—732no—
157C4×C4⋊C4—731no—
158#343—1461no—
159#343—1461no—
160C22×Q8—21no—
161C22×Q8—731no—
162C2×M16—1462no—
163C2×M16—1462no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4×Q8—2921no—
167C4×Q8—2921no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—731no—
170C2×C4⋊C4—731no—
171C2×C4⋊C4—731no—
172C2×C4⋊C4—1462no—
173C2×C4⋊C4—1462no—
174C2×C4⋊C4—1461no—
175C2×C4⋊C4C2×#6440812yes—
176C2×C42—732no—
177C2×C42—731no—
178C2×C42—731no—
179C2×C42C2×#6440811yescharacteristic
180C4⋊C8—2921no—
181C4⋊C8—2921no—
182#101—734no—
183#101—1464no—
184#101—1462no—
185#97—2921no—
186C4×C8—2921no—
187#95—1462no—
188#95—1462no—
189#95—1461no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—2921no—
193C2×Q8—2921no—
194C2×Q8—732no—
195C2×Q8—732no—
196C22×C4—22no—
197C22×C4—731no—
198C22×C4—731no—
199C22×C4—731no—
200C22×C4—731no—
201C22×C4—1462no—
202C22×C4C22⋊#6440811yescharacteristic
203C22×C4C22⋊#6440811yescharacteristic
204C22×C4C22×#6440811yescharacteristic
205M16—2922no—
206M16—2922no—
207M16—2922no—
208M16—1464no—
209C2×C8—2922no—
210C2×C8—2922no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—2921no—
215C4⋊C4—2921no—
216C4⋊C4—2921no—
217C4⋊C4—5842no—
218C4⋊C4—5842no—
219C4⋊C4—1464no—
220C42—21no—
221C42—21no—
222C42—2922no—
223C42—1461no—
224C42—1461no—
225C42—1461no—
226C42—1461no—
227C23C2×C22⋊#6440811yescharacteristic
228Q8—84no—
229Q8—2922no—
230Q8—2922no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—2922no—
240C2×C4—2922no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—731no—
244C2×C4—732no—
245C2×C4—731no—
246C2×C4—1464no—
247C2×C4—1461no—
248C2×C4—1461no—
249C2×C4—1461no—
250C2×C4—1461no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4#1455664911yescharacteristic
254C2×C4#1455664911yescharacteristic
255C2×C4Q8×#6440812yes—
256C8—2924no—
257C8—5842no—
258C22—21no—
259C22—21no—
260C22(C2×#64408)⋊Q811yescharacteristic
261C22C2×#1455664911yescharacteristic
262C22#1798956811yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—82no—
267C4—2924no—
268C4—2921no—
269C4—2921no—
270C4—1462no—
271C4—1461no—
272C4—1461no—
273C4#1798949712yes—
274C2—21no—
275C2—21no—
276C2#2262612011yescharacteristic
277C2C2×#1798949711yescharacteristic
278C2#2262623011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Dic146C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#13302
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
26. #14556652Q8semidirect
144. C73#13302semidirect

Maximal subgroups

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

subgroupindex
#226257972
C4×#145566522
#226250292
#226151402
#1330273