Finite Group Databasebeta

← #34013078

Subgroups — #34013078

Order 18688.

588 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 86 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×#17989491C212yesmaximal
2#22625792C212yesmaximal, complemented
3#22625026C212yesmaximal, complemented
4#22612653C211yesmaximal, characteristic
5C2×#14556650C2214yes—
6#17989529—22no—
7#17989491—24no—
8#17989465—22no—
9C2×#14556587C2211yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988490C411yescharacteristic, complemented
12#17988488C2211yescharacteristic
13#17988466—22no—
14#17988463—22no—
15#17988452C411yescharacteristic, complemented
16#17988433C411yescharacteristic, complemented
17C2×C4×D4×C73C411yescharacteristic, complemented
18C2×C4⋊#64408—42no—
19C2×C22⋊#64408—24no—
20#14556650—24no—
21#14556650—24no—
22(C2×#64408)⋊C4—42no—
23C22×D292C2311yescharacteristic
24C22×C22⋊D73—22no—
25C22×C4×D73D412yes—
26#14556587—22no—
27#14556587D412yes—
28#14556587Q812yes—
29C4×D292—42no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×C22⋊Dic73—22no—
32C2×#11924005—22no—
33C2×#11924005D412yes—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73C2×C411yescharacteristic
36#14556498—22no—
37#14556498—21no—
38#14556498—21no—
39C22×D4×C73C2×C411yescharacteristic
40C23×C4×C73—21no—
41C4×D4×C73—42no—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73—21no—
44C2×C42×C73C2×C411yescharacteristic
45C22×#64408—44no—
46C4⋊#64408—84no—
47C22⋊#64408—44no—
48C22⋊#64408—44no—
49C23×D73C2×D412yes—
50C2×D292—41no—
51C2×D292—41no—
52C2×D292C2×D411yescharacteristic
53C2×D292C2×Q811yescharacteristic
54C2×D292#4112yes—
55C2×C22⋊D73—24no—
56C2×C22⋊D73—24no—
57C2×C22⋊D73—82no—
58C2×C22⋊D73—82no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73C22⋊C412yes—
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—82no—
65C4⋊Dic73—41no—
66C22⋊Dic73—82no—
67#11924005—42no—
68#11924005—82no—
69C4×Dic73—21no—
70C4×Dic73—41no—
71C4×Dic73C4⋊C412yescomplemented
72C24×C73—21no—
73C2×D4×C73—21no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C2×D4×C73C4⋊C412yescomplemented
77C22×C4×C73—21no—
78C22×C4×C73—42no—
79C22×C4×C73—82no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C4⋊C4×C73—41no—
84C22⋊C4×C73—42no—
85C42×C73—22no—
86C23×C73—22no—
87C23×C73—42no—
88C23×C73—42no—
89C23×C73—41no—
90C23×C73—41no—
91C23×C73C2×C22⋊C411yescharacteristic
92C22×D73—22no—
93C22×D73—22no—
94C22×D73—42no—
95C22×D73—42no—
96C22×D73#12112yes—
97C22×D73C22⋊Q812yes—
98C2×#64408—84no—
99C2×#64408—84no—
100C2×#64408—84no—
101D4×C73—42no—
102D4×C73—42no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—84no—
111C2×C4×C73#9912yes—
112C2×C4×C73C2×C4⋊C411yescharacteristic
113C2×C4×C73#11711yescharacteristic
114C22⋊D73—84no—
115C22⋊D73—84no—
116C2×Dic73—22no—
117C2×Dic73—22no—
118C2×Dic73—41no—
119C2×Dic73—41no—
120C2×Dic73—42no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124D292—41no—
125D292—41no—
126D292—42no—
127C4×D73—44no—
128C4×D73—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—44no—
132C22×C73—42no—
133C22×C73—82no—
134C22×C73—81no—
135C22×C73—81no—
136C22×C73#38011yescharacteristic
137C22×C73C2×#9911yescharacteristic
138C22×C73#41011yescharacteristic
139C2×D73—22no—
140C2×D73—22no—
141C2×D73—42no—
142C2×D73—42no—
143C2×D73—42no—
144C2×D73—42no—
145C2×D73—82no—
146#64408—164no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—22no—
150C4×C73—41no—
151C4×C73—81no—
152Dic73—42no—
153Dic73—42no—
154Dic73—81no—
155#12799—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73—82no—
159C2×C73#179911yescharacteristic
160C2×C73#212711yescharacteristic
161C2×C73#212811yescharacteristic
162D73—42no—
163D73—42no—
164#2328—731no—
165C2×#353—732no—
166#1799—732no—
167#1393—732no—
168C73#1279911yescharacteristic, complemented, Sylow
169C2×#127—731no—
170C2×C4×D4—731no—
171C2×C4×D4#6440811yescharacteristic, complemented
172C2×#99—734no—
173#390—731no—
174#390—731no—
175#386—1462no—
176#386—1462no—
177#382—731no—
178#380—1462no—
179#376—731no—
180#353—1464no—
181#342—1462no—
182C22×D4—731no—
183C22×D4—1462no—
184C22×D4C2×#6440811yescharacteristic
185C23×C4—21no—
186C23×C4—732no—
187#127—732no—
188#127—732no—
189#127—1462no—
190C4×D4—42no—
191C4×D4—2922no—
192C2×C4⋊C4—2922no—
193C2×C4⋊C4—731no—
194C2×C4⋊C4C2×#6440811yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—732no—
197C2×C22⋊C4—1464no—
198C2×C22⋊C4—1462no—
199C2×C22⋊C4—1462no—
200C2×C42—731no—
201C2×C42—731no—
202C2×C42C2×#6440811yescharacteristic
203#99—1464no—
204#99—1464no—
205#95—2922no—
206#95—1462no—
207#95—1461no—
208#95—1461no—
209C24—21no—
210C24—732no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—2921no—
215C2×D4—2921no—
216C2×D4—5842no—
217C2×D4—5842no—
218C2×D4—731no—
219C2×D4—731no—
220C2×D4—732no—
221C2×D4—1464no—
222C2×D4—1464no—
223C2×D4C4⋊#6440812yescomplemented
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4—2924no—
228C22×C4—5842no—
229C22×C4—732no—
230C22×C4—1461no—
231C22×C4—1461no—
232C22×C4—1462no—
233C22×C4—1462no—
234C22×C4C22⋊#6440811yescharacteristic
235C22×C4C22⋊#6440811yescharacteristic
236C22×C4C22×#6440811yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—2921no—
239C4⋊C4—5844no—
240C22⋊C4—42no—
241C22⋊C4—2922no—
242C22⋊C4—2924no—
243C22⋊C4—2924no—
244C22⋊C4—5842no—
245C22⋊C4—5842no—
246C42—22no—
247C42—2921no—
248C42—732no—
249C42—1461no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—2922no—
256C23—2922no—
257C23—732no—
258C23—732no—
259C23—1462no—
260C23—1462no—
261C23C2×C22⋊#6440811yescharacteristic
262D4—42no—
263D4—42no—
264D4—2921no—
265D4—2921no—
266D4—2922no—
267D4—5844no—
268D4—5844no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—84no—
277C2×C4—2924no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—2922no—
281C2×C4—2921no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—5842no—
285C2×C4—5844no—
286C2×C4—5844no—
287C2×C4—5844no—
288C2×C4—1462no—
289C2×C4—1462no—
290C2×C4#1455664912yes—
291C2×C4C2×C4⋊#6440811yescharacteristic
292C2×C4#1455667011yescharacteristic
293C22—21no—
294C22—21no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—81no—
299C22—81no—
300C22—2922no—
301C22—2922no—
302C22—2922no—
303C22—2922no—
304C22—5842no—
305C22—1462no—
306C22—1462no—
307C22#1798952911yescharacteristic
308C22C2×#1455664911yescharacteristic
309C22#1798956611yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—81no—
315C4—2922no—
316C4—2922no—
317C4—5841no—
318C4—11684no—
319C2—21no—
320C2—21no—
321C2—82no—
322C2—2922no—
323C2—2922no—
324C2#2262579111yescharacteristic
325C2#2262620611yescharacteristic
326C2#2262621011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4×D4×C73C4
socleC22×C73C2×#99
prime radicalC2×#14556587C22
cosocleC22×D292C23
hypercenterC2×C4×D4#64408
hypocenterC73#12799
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #22625792C2semidirect
3. #22625026C2semidirect
11. #17988490C4semidirect
15. #17988452C4semidirect
16. #17988433C4semidirect
17. C2×C4×D4×C73C4semidirect
71. C4×Dic73C4⋊C4semidirect
76. C2×D4×C73C4⋊C4semidirect
168. C73#12799semidirect
171. C2×C4×D4#64408semidirect
223. C2×D4C4⋊#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894912
#226257922
#226250262
#226126532
#1279973