Finite Group Databasebeta

← #32977718

Subgroups — #32977718

Order 18688.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17989307C211yesmaximal, characteristic
2#22608604C212yesmaximal, complemented
3#22608359C212yesmaximal, complemented
4#22608252C211yesmaximal, characteristic
5C73×#1800C211yesmaximal, characteristic
6#17989307—24no—
7#17989307—24no—
8C2×#14556591C2211yescharacteristic
9C2×#14556584—22no—
10C2×#14556578—21no—
11C2×#14556570C412yes—
12C2×C4×Dic146C2211yescharacteristic
13C2×#14556556—21no—
14C2×#14556550C412yes—
15#17988640C2212yes—
16#17988334—24no—
17#17988317—24no—
18#17988311—22no—
19C2×C4×D4×C73C2211yescharacteristic
20#423×C73C2212yes—
21C2×#100×C73—21no—
22#343×C73—21no—
23C22×Dic146—21no—
24C22×Dic146C2×C411yescharacteristic
25#14556591—22no—
26#14556591D414yes—
27#14556584—216no—
28#14556578—28no—
29#14556570—24no—
30#14556570—24no—
31C4×Dic146—22no—
32C4×Dic146D414yes—
33#14556556—28no—
34#14556550—24no—
35#14556550—24no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73C2×C412yes—
38C2×C4⋊Dic73C2×C411yescharacteristic
39C2×#11924007—22no—
40C2×#11924007—21no—
41C2×C22⋊Dic73—22no—
42C2×C22⋊Dic73—22no—
43C2×C4×Dic73—22no—
44C2×C4×Dic73—21no—
45C4⋊#10596831—22no—
46C22⋊#10596831—22no—
47C4×#10596831—22no—
48C22×D4×C73D411yescharacteristic
49C23×C4×C73C2×C412yes—
50C2×M16×C73—22no—
51C4×D4×C73—22no—
52C4×D4×C73—22no—
53C2×C4⋊C4×C73D411yescharacteristic
54C2×C22⋊C4×C73D412yes—
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—22no—
57C4⋊C8×C73—22no—
58#100×C73—42no—
59C2×Dic146—21no—
60C2×Dic146—24no—
61C2×Dic146—42no—
62C2×Dic146—42no—
63C2×Dic146—41no—
64C2×Dic146—41no—
65C2×Dic146C22⋊C412yes—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73—22no—
69C2×#10596831—42no—
70C4⋊Dic73—21no—
71C4⋊Dic73—22no—
72C4⋊Dic73—24no—
73C4⋊Dic73C22⋊C412yes—
74C4⋊Dic73C22⋊C414yes—
75#11924007—28no—
76#11924007—24no—
77C22⋊Dic73—28no—
78C22⋊Dic73—28no—
79C4×Dic73—28no—
80C4×Dic73—24no—
81C24×C73C22⋊C412yes—
82C2×D4×C73—22no—
83C2×D4×C73—22no—
84C2×D4×C73—21no—
85C2×D4×C73—21no—
86C22×C4×C73—22no—
87C22×C4×C73—22no—
88C22×C4×C73—44no—
89C22×C4×C73C22⋊C412yes—
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93M16×C73—42no—
94M16×C73—42no—
95C2×C8×C73—42no—
96C4⋊C4×C73—22no—
97C22⋊C4×C73—24no—
98C42×C73C2×D412yes—
99C42×C73C2×D412yes—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73—42no—
104C23×C73—42no—
105C23×C73C2×C22⋊C411yescharacteristic
106D4×C73—42no—
107D4×C73—42no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—41no—
111C2×C4×C73—44no—
112C2×C4×C73—44no—
113C2×C4×C73—42no—
114C2×C4×C73C2×C22⋊C411yescharacteristic
115C2×C4×C73C2×C22⋊C411yescharacteristic
116C2×C4×C73#12012yes—
117C2×C4×C73#12012yes—
118C2×C4×C73#13512yes—
119C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C73#13711yescharacteristic
121C2×Dic73—22no—
122C2×Dic73—22no—
123C2×Dic73—24no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128Dic146—24no—
129Dic146—42no—
130Dic146—44no—
131Dic146—44no—
132C8×C73—82no—
133#10596831—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—82no—
140C22×C73—82no—
141C22×C73#37911yescharacteristic
142C22×C73#41611yescharacteristic
143C22×C73#41711yescharacteristic
144C4×C73—22no—
145C4×C73—21no—
146C4×C73—42no—
147C4×C73#41312yes—
148C4×C73#44912yes—
149C4×C73#45212yes—
150Dic73—42no—
151Dic73—42no—
152Dic73—44no—
153#8389—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73—82no—
157C2×C73#178511yescharacteristic
158C2×C73#178811yescharacteristic
159C2×C73#179611yescharacteristic
160C2×#556—731no—
161#1800D7311yescharacteristic
162#1546—732no—
163#1487—732no—
164#1460—731no—
165C73#838911yescharacteristic, complemented, Sylow
166#556—1464no—
167#556—1464no—
168C2×C4⋊Q8—731no—
169C2×#126—1462no—
170C2×#124—1461no—
171C2×C22⋊Q8—732no—
172C2×C4×Q8—731no—
173C2×C4×Q8—1461no—
174C2×C4×D4C2×D7311yescharacteristic
175C2×#117—732no—
176#431—732no—
177#423C2×D7312yes—
178C2×#100—21no—
179#343—21no—
180#332—1464no—
181Q8⋊C8—1464no—
182#323—1462no—
183C22×Q8—731no—
184C22×Q8—1461no—
185C22×D4C22⋊D7311yescharacteristic
186C23×C4C4×D7312yes—
187C2×M16—22no—
188C4⋊Q8—734no—
189C4⋊Q8—1462no—
190#126—14616no—
191#124—1468no—
192C22⋊Q8—1464no—
193C22⋊Q8—1464no—
194C4×Q8—734no—
195C4×Q8—1468no—
196C4×Q8—1462no—
197C4×D4—22no—
198C4×D4—22no—
199#117—1464no—
200#117—1464no—
201C2×C4⋊C4—731no—
202C2×C4⋊C4—732no—
203C2×C4⋊C4—1462no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4C22⋊D7311yescharacteristic
207C2×C22⋊C4—1462no—
208C2×C22⋊C4—1462no—
209C2×C22⋊C4D29212yes—
210C2×C42—1462no—
211C2×C42—1461no—
212C2×C42C22×D7311yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—1462no—
216#100—42no—
217C22⋊C8—1462no—
218C4×C8—1462no—
219C24#1192400512yes—
220C2×Q8—2922no—
221C2×Q8—2922no—
222C2×Q8—2921no—
223C2×Q8—2921no—
224C2×Q8—732no—
225C2×Q8—1461no—
226C2×Q8—1464no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—1461no—
235C22×C4—1462no—
236C22×C4—1461no—
237C22×C4#1192400512yes—
238C22×C4C2×C4×D7311yescharacteristic
239C22×C4C2×C22⋊D7311yescharacteristic
240C22×C4C2×D29211yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—2922no—
245C4⋊C4—22no—
246C4⋊C4—732no—
247C4⋊C4—734no—
248C4⋊C4—1468no—
249C4⋊C4—1461no—
250C4⋊C4—1462no—
251C4⋊C4—1464no—
252C4⋊C4—1464no—
253C22⋊C4—24no—
254C22⋊C4—1468no—
255C22⋊C4—1468no—
256C42—1468no—
257C42—1464no—
258C42D4×D7312yes—
259C42D4×D7312yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×#1192400511yescharacteristic
266Q8—2922no—
267Q8—2924no—
268Q8—2924no—
269Q8—1464no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—2922no—
279C2×C4—2922no—
280C2×C4—2922no—
281C2×C4—2922no—
282C2×C4—1462no—
283C2×C4—1464no—
284C2×C4—1462no—
285C2×C4C22⋊C4×D7311yescharacteristic
286C2×C4C22⋊C4×D7311yescharacteristic
287C2×C4#1455655912yes—
288C2×C4#1455656112yes—
289C2×C4#1455661112yes—
290C2×C4#1455661711yescharacteristic
291C2×C4#1455662211yescharacteristic
292C8—82no—
293C8—5842no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1798844211yescharacteristic
302C22#1798858111yescharacteristic
303C22#1798858611yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—2922no—
308C4—2924no—
309C4—2922no—
310C4#1798856812yes—
311C4#1798873012yes—
312C4#1798874712yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2260985611yescharacteristic
317C2#2260986911yescharacteristic
318C2#2260990711yescharacteristic
subgroupG/H
centerC22#17988442
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1800C2
socleC22×C73#379
prime radicalC22×D4×C73D4
cosocleC2×C42×C73C23
hypercenter#1800D73
hypocenterC73#8389
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #22608604C2semidirect
3. #22608359C2semidirect
165. C73#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#179893072
#226086042
#226083592
#226082522
C73×#18002
#838973