Finite Group Databasebeta

← #32989640

Subgroups — #32989640

Order 18688.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#17988874C211yesmaximal, characteristic
2#22609920C211yesmaximal, characteristic
3#22608706C211yesmaximal, characteristic, complemented
4#22608616C211yesmaximal, characteristic, complemented
5#22608321C211yesmaximal, characteristic
6#22607826C211yesmaximal, characteristic, complemented
7C73×#1548C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9#17988874—22no—
10#17988874—22no—
11#17988640C2211yescharacteristic
12C2×#14556522—21no—
13#17988611C2211yescharacteristic
14C2×C4⋊#10596831C2211yescharacteristic
15C2×#14556511—21no—
16C2×#14556507—21no—
17#17988355—21no—
18#17988340—22no—
19#17988333—22no—
20#17988314—22no—
21C2×C4⋊Q8×C73C2211yescharacteristic
22#431×C73C2211yescharacteristic
23#423×C73C2211yescharacteristic
24#333×C73—22no—
25C22×D292D411yescharacteristic
26C22×C4×D73—21no—
27C2×#11924012—21no—
28C22×#10596831—21no—
29C4×D292—21no—
30C4×D292—21no—
31C4×D292—21no—
32C4×D292—21no—
33C2×C4⋊Dic73D411yescharacteristic
34C2×#11924005—21no—
35#14556522—24no—
36C4⋊#10596831—21no—
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831D412yes—
41#14556511—24no—
42#14556507—42no—
43C22⋊#10596831—21no—
44C4×#10596831—21no—
45C22×Q8×C73—21no—
46C2×M16×C73—21no—
47C4⋊Q8×C73—21no—
48C4⋊Q8×C73—21no—
49C4⋊Q8×C73—21no—
50C4⋊Q8×C73D412yes—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73—21no—
53C2×C42×C73C2311yescharacteristic
54C4⋊C8×C73—21no—
55C4⋊C8×C73—21no—
56C4⋊C8×C73D412yes—
57C22⋊C8×C73—21no—
58C4×C8×C73—21no—
59C23×D73—21no—
60C2×D292—21no—
61C2×D292—21no—
62C2×D292—21no—
63C2×D292—21no—
64C2×D292—21no—
65C2×D292Q1612yes—
66C22×Dic73—21no—
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69C2×C4×D73—82no—
70#11924012—41no—
71#11924012—41no—
72C2×#10596831—22no—
73C2×#10596831—41no—
74C2×#10596831—41no—
75C2×#10596831—41no—
76C2×#10596831—41no—
77C4⋊Dic73—21no—
78C4⋊Dic73—21no—
79#11924005—41no—
80#11924005—41no—
81C2×Q8×C73—22no—
82C2×Q8×C73—41no—
83C2×Q8×C73—82no—
84C22×C4×C73—41no—
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73#4111yescharacteristic
87C22×C4×C73#4111yescharacteristic
88M16×C73—41no—
89M16×C73—41no—
90C2×C8×C73—22no—
91C2×C8×C73—41no—
92C4⋊C4×C73—22no—
93C4⋊C4×C73—22no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—41no—
96C42×C73C2×D411yescharacteristic
97C42×C73C2×D411yescharacteristic
98C42×C73C2×D411yescharacteristic
99C42×C73#4111yescharacteristic
100C23×C73#12311yescharacteristic
101C22×D73—22no—
102C22×D73—41no—
103C22×D73—41no—
104C22×D73—81no—
105C22×D73—81no—
106Q8×C73—82no—
107Q8×C73—82no—
108C2×C4×C73—22no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—82no—
112C2×C4×C73#12011yescharacteristic
113C2×C4×C73#12111yescharacteristic
114C2×C4×C73#12111yescharacteristic
115C2×C4×C73#12111yescharacteristic
116C2×C4×C73#12311yescharacteristic
117C2×C4×C73#12411yescharacteristic
118C2×C4×C73C2×Q1611yescharacteristic
119C2×C4×C73#13511yescharacteristic
120C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C73#13711yescharacteristic
122C2×Dic73—41no—
123C2×Dic73—81no—
124D292—22no—
125D292—22no—
126D292—41no—
127D292—41no—
128C4×D73—82no—
129C4×D73—82no—
130C8×C73—42no—
131C8×C73—81no—
132#10596831—42no—
133#10596831—81no—
134#10596831—81no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73#39411yescharacteristic
138C22×C73#48411yescharacteristic
139C22×C73#48511yescharacteristic
140C2×D73—42no—
141C2×D73—82no—
142C2×D73—81no—
143C2×D73—81no—
144C2×D73—82no—
145C4×C73—21no—
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—82no—
149C4×C73#44911yescharacteristic
150C4×C73#45111yescharacteristic
151C4×C73#46711yescharacteristic
152C4×C73#46811yescharacteristic
153C4×C73#47112yes—
154Dic73—81no—
155#10453—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73#203611yescharacteristic
159C2×C73#204611yescharacteristic
160C2×C73#204811yescharacteristic
161D73—82no—
162C2×#476—731no—
163#1800—731no—
164#1571—731no—
165#1549—731no—
166#1548D7311yescharacteristic, complemented
167#1478—731no—
168#1301—731no—
169C73#1045311yescharacteristic, complemented, Sylow
170C2×C4⋊Q8C2×D7311yescharacteristic
171C2×C4×D4—731no—
172#476—1462no—
173#476—1462no—
174#431—731no—
175#431C2×D7311yescharacteristic
176C2×#106—1461no—
177#423—731no—
178#423C2×D7311yescharacteristic
179C2×C4⋊C8—731no—
180C2×#102—1461no—
181C2×#100—1461no—
182#343—1461no—
183#335—1462no—
184#333—22no—
185#332—1462no—
186D4⋊C8—1462no—
187C22×Q8—21no—
188C22×D4—731no—
189C23×C4—1461no—
190C2×M16—21no—
191C2×M16—1461no—
192C22×C8—1461no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8D29212yes—
197C4×D4—1461no—
198C4×D4—1461no—
199C4×D4—1461no—
200C4×D4—1461no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—731no—
204C2×C22⋊C4—1461no—
205C2×C42C22×D7311yescharacteristic
206#106—1464no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—732no—
210C4⋊C8—1461no—
211C4⋊C8—1461no—
212C4⋊C8—1461no—
213C4⋊C8—1461no—
214C4⋊C8C22⋊D7312yes—
215#102—1464no—
216#100—2922no—
217C22⋊C8—21no—
218C22⋊C8—1461no—
219C4×C8—21no—
220C4×C8—1461no—
221C24—1461no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—732no—
226C2×D4—1461no—
227C2×D4—1461no—
228C2×D4—1461no—
229C2×D4—1461no—
230C2×D4—1461no—
231C22×C4—41no—
232C22×C4—2921no—
233C22×C4—2921no—
234C22×C4—5842no—
235C22×C4—1461no—
236C22×C4D4×D7311yescharacteristic
237C22×C4#1192402711yescharacteristic
238C22×C4#1192402811yescharacteristic
239M16—41no—
240M16—41no—
241M16—2921no—
242M16—2921no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—2921no—
246C2×C8—2921no—
247C2×C8—2921no—
248C2×C8—2921no—
249C2×C8—1462no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1461no—
255C4⋊C4—1461no—
256C22⋊C4—2921no—
257C22⋊C4—2921no—
258C42D4×D7311yescharacteristic
259C42C2×C22⋊D7311yescharacteristic
260C42C2×D29211yescharacteristic
261C42#1192402711yescharacteristic
262C23—2921no—
263C23—2921no—
264C23—5841no—
265C23—5841no—
266C23—1462no—
267C23#1455657311yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—2921no—
271D4—2921no—
272D4—1462no—
273D4—1462no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—2921no—
279C2×C4—5842no—
280C2×C4—5842no—
281C2×C4—5841no—
282C2×C4#1455656111yescharacteristic
283C2×C4#1455656211yescharacteristic
284C2×C4#1455656311yescharacteristic
285C2×C4#1455656511yescharacteristic
286C2×C4#1455657511yescharacteristic
287C2×C4#1455657711yescharacteristic
288C2×C4Q16×D7311yescharacteristic
289C2×C4#1455661411yescharacteristic
290C2×C4#1455661811yescharacteristic
291C2×C4#1455662011yescharacteristic
292C8—42no—
293C8—81no—
294C8—2922no—
295C8—5841no—
296C8—5841no—
297C22—21no—
298C22—21no—
299C22—2922no—
300C22—5842no—
301C22—5842no—
302C22—5841no—
303C22—5841no—
304C22#1798850311yescharacteristic
305C22#1798892111yescharacteristic
306C22#1798892711yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—5841no—
312C4#1798873511yescharacteristic
313C4#1798874411yescharacteristic
314C4#1798882511yescharacteristic
315C4#1798883011yescharacteristic
316C4#1798885212yes—
317C2—21no—
318C2—21no—
319C2—5842no—
320C2#2261113311yescharacteristic
321C2#2261119111yescharacteristic
322C2#2261120311yescharacteristic
subgroupG/H
centerC22#17988503
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1548C2
socleC22×C73#394
prime radicalC22×D292D4
hypercenter#1548D73
hypocenterC73#10453
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. #22608706C2semidirect
4. #22608616C2semidirect
6. #22607826C2semidirect
7. C73×#1548C2semidirect
166. #1548D73semidirect
169. C73#10453semidirect

Maximal subgroups

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

subgroupindex
C2×#179888742
#226099202
#226087062
#226086162
#226083212
#226078262
C73×#15482
#1045373