Finite Group Databasebeta

← #34012523

Subgroups — #34012523

Order 18688.

532 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#17989495C212yesmaximal
2#22625684C212yesmaximal, complemented
3#22625028C212yesmaximal, complemented
4#22612478C211yesmaximal, characteristic
5C2×#14556652C2214yes—
6C4×#11924037—22no—
7#17989495C418yescomplemented
8#17989467—22no—
9C2×#14556578C2211yescharacteristic
10C2×#14556556C2211yescharacteristic
11C2×D4×Dic73C411yescharacteristic
12#17988493C2211yescharacteristic
13#17988475—22no—
14#17988452C411yescharacteristic
15C22⋊C4×Dic73—22no—
16C42×Dic73C411yescharacteristic
17#390×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556652—24no—
20#14556652C2×C418yescomplemented
21#14556646—42no—
22C4×#10596833—42no—
23C22×Dic146C2311yescharacteristic
24C23×Dic73—21no—
25#14556578—22no—
26#14556578C2×C414yes—
27#14556556—41no—
28#14556556—41no—
29D4×Dic73—41no—
30D4×Dic73—41no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×#11924007D412yes—
33C2×C22⋊Dic73—22no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73D412yes—
40#14556498—22no—
41#14556498—22no—
42C22×D4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C22⋊C4×C73—22no—
45C2×C42×C73C2×C411yescharacteristic
46#11924037—28no—
47#11924037—44no—
48C2×#10596833—44no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146C22×C412yes—
52C2×Dic146#4112yes—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73—44no—
56C22×Dic73—41no—
57C22×Dic73—41no—
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C2×D412yes—
61C4⋊Dic73—41no—
62#11924007—42no—
63C22⋊Dic73—28no—
64C22⋊Dic73—42no—
65C4×Dic73—21no—
66C4×Dic73—21no—
67C4×Dic73—21no—
68C4×Dic73—22no—
69C4×Dic73—22no—
70C4×Dic73—42no—
71C4×Dic73—42no—
72C4×Dic73C4212yes—
73C24×C73—21no—
74C2×D4×C73—21no—
75C2×D4×C73—41no—
76C2×D4×C73—41no—
77C2×D4×C73C4212yes—
78C22×C4×C73—21no—
79C22×C4×C73C22⋊C412yes—
80C22×C4×C73C22×C411yescharacteristic
81C4⋊C4×C73—41no—
82C22⋊C4×C73—44no—
83C42×C73—41no—
84C23×C73—22no—
85C23×C73—42no—
86C23×C73—42no—
87C23×C73—41no—
88C23×C73—41no—
89C23×C73C2×C22⋊C411yescharacteristic
90D4×C73—42no—
91D4×C73—42no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73—42no—
99C2×C4×C73C2×C4211yescharacteristic
100C2×C4×C73#11711yescharacteristic
101C2×Dic73—22no—
102C2×Dic73—22no—
103C2×Dic73—22no—
104C2×Dic73—22no—
105C2×Dic73—21no—
106C2×Dic73—41no—
107C2×Dic73—41no—
108C2×Dic73—41no—
109C2×Dic73—42no—
110C2×Dic73—41no—
111C2×Dic73—82no—
112C2×Dic73—82no—
113C2×Dic73#9911yescharacteristic
114C2×Dic73#9911yescharacteristic
115C2×Dic73C4×D414yes—
116Dic146—42no—
117Dic146—42no—
118#10596833—48no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—44no—
122C22×C73—42no—
123C22×C73—82no—
124C22×C73—81no—
125C22×C73—81no—
126C22×C73C4×C22⋊C411yescharacteristic
127C22×C73C2×#9911yescharacteristic
128C22×C73#41011yescharacteristic
129C4×C73—21no—
130C4×C73—21no—
131C4×C73—42no—
132C4×C73—42no—
133Dic73—21no—
134Dic73—24no—
135Dic73—42no—
136Dic73—41no—
137Dic73—81no—
138Dic73#35512yescomplemented
139C4×#355—731nomaximal, Sylow
140C2×C73—21no—
141C2×C73—21no—
142C2×C73—82no—
143C2×C73C4×#9911yescharacteristic
144C2×C73C2×#35511yescharacteristic
145C2×C73#213211yescharacteristic
146C4×#124—731no—
147C2×#355—732no—
148C4×#101—732no—
149#1394—732no—
150C73C4×#35511yescharacteristic, complemented, Sylow
151C2×#124—731no—
152C2×C4×Q8—731no—
153C2×C4×D4—731no—
154C2×#101—734no—
155C4×M16—1462no—
156#391—731no—
157#390#6440811yescharacteristic
158#388—1462no—
159#382—731no—
160C4×C22⋊C4—1462no—
161C43—731no—
162#355—738no—
163#343—1462no—
164C22×Q8—731no—
165C22×D4C2×#6440811yescharacteristic
166C23×C4—1461no—
167C2×M16—1464no—
168#124—734no—
169#124—1462no—
170C4×Q8—2921no—
171C4×Q8—2921no—
172C4×D4—2921no—
173C4×D4—2921no—
174C2×C4⋊C4—732no—
175C2×C4⋊C4—731no—
176C2×C4⋊C4C2×#6440811yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1461no—
179C2×C22⋊C4—1462no—
180C2×C42—731no—
181C2×C42—732no—
182C2×C42—731no—
183C2×C42—1461no—
184C2×C42—1461no—
185C2×C42C2×#6440811yescharacteristic
186#101—738no—
187#101—1464no—
188#97—2922no—
189C4×C8—2922no—
190#95—1462no—
191#95—1462no—
192C24—21no—
193C2×Q8—2921no—
194C2×Q8—2921no—
195C2×Q8—732no—
196C2×Q8—732no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4C4×#6440812yes—
201C22×C4—21no—
202C22×C4—2921no—
203C22×C4—2921no—
204C22×C4—2924no—
205C22×C4—731no—
206C22×C4—731no—
207C22×C4—732no—
208C22×C4—1461no—
209C22×C4—1461no—
210C22×C4C22⋊#6440812yes—
211C22×C4C22×#6440811yescharacteristic
212M16—2924no—
213M16—1468no—
214C2×C8—2924no—
215C4⋊C4—41no—
216C4⋊C4—2921no—
217C4⋊C4—2922no—
218C22⋊C4—44no—
219C22⋊C4—2922no—
220C22⋊C4—1468no—
221C42—41no—
222C42—2922no—
223C42—2922no—
224C42—732no—
225C42—1462no—
226C42—1462no—
227C42—1461no—
228C42—1461no—
229C42—1461no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊#6440811yescharacteristic
236Q8—2922no—
237Q8—2922no—
238D4—42no—
239D4—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—2921no—
248C2×C4—2921no—
249C2×C4—2921no—
250C2×C4—2922no—
251C2×C4—2921no—
252C2×C4—5842no—
253C2×C4—5842no—
254C2×C4—731no—
255C2×C4—734no—
256C2×C4—731no—
257C2×C4—1462no—
258C2×C4—1461no—
259C2×C4—1462no—
260C2×C4—1462no—
261C2×C4—1462no—
262C2×C4C2×C4×#6440811yescharacteristic
263C2×C4#1455667011yescharacteristic
264C8—2928no—
265C22—21no—
266C22—21no—
267C22—44no—
268C22—42no—
269C22—82no—
270C22—81no—
271C22—81no—
272C22C4×C22⋊#6440811yescharacteristic
273C22D73⋊#9911yescharacteristic
274C22#1798956711yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—2922no—
280C4—2921no—
281C4—5841no—
282C4—732no—
283C4—1464no—
284C4—1461no—
285C2—21no—
286C2—21no—
287C2—82no—
288C2#2262567811yescharacteristic
289C2D73⋊#35511yescharacteristic
290C2#2262622311yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic146C23
hypercenter#390#64408
hypocenterC73C4×#355
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #22625684C2semidirect
3. #22625028C2semidirect
7. #17989495C4semidirect
20. #14556652C2×C4semidirect
138. Dic73#355semidirect
150. C73C4×#355semidirect

Maximal subgroups

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

subgroupindex
C2×#179894952
#226256842
#226250282
#226124782
C4×#35573