Finite Group Databasebeta

← #33001969

Subgroups — #33001969

Order 18688.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#124×Dic73C211yesmaximal, characteristic
2C2×#17988382C211yesmaximal, characteristic
3#22609917C212yesmaximal
4#22608055C212yesmaximal
5C2×#354×C73C211yesmaximal, characteristic
6C2×Q8×Dic73C411yescharacteristic, complemented
7C2×D4×Dic73C2211yescharacteristic
8C2×#14556506C2212yes—
9#17988494C411yescharacteristic, complemented
10#17988490C2211yescharacteristic
11#17988480—22no—
12#17988453C411yescharacteristic, complemented
13#17988445—22no—
14C22⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic, complemented
16#17988382—22no—
17#17988353—22no—
18C2×#124×C73C2211yescharacteristic
19C2×#99×C73C2212yes—
20#354×C73—22no—
21C23×Dic73D412yes—
22Q8×Dic73—41no—
23Q8×Dic73—41no—
24D4×Dic73—41no—
25D4×Dic73—41no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—42no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73D412yes—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73C2×C411yescharacteristic
37C2×C4×Dic73C2×C411yescharacteristic
38#14556506—22no—
39#14556506—22no—
40#14556498—22no—
41#14556498—22no—
42#14556498—42no—
43C22×Q8×C73C2×C411yescharacteristic
44C22×D4×C73C2311yescharacteristic
45#124×C73—22no—
46#124×C73D412yes—
47#124×C73Q812yes—
48C2×C22⋊C4×C73—22no—
49C2×C22⋊C4×C73—22no—
50C2×C42×C73C2×C411yescharacteristic
51#99×C73—22no—
52#99×C73—22no—
53C22×Dic73—21no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—21no—
57C22×Dic73—44no—
58C22×Dic73—42no—
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C22⋊C412yes—
61C22×Dic73C22⋊C411yescharacteristic
62C4⋊Dic73—41no—
63C4⋊Dic73—42no—
64C4⋊Dic73—41no—
65#11924007—84no—
66C22⋊Dic73—42no—
67C22⋊Dic73—44no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C4×Dic73—21no—
71C4×Dic73—21no—
72C4×Dic73—42no—
73C4×Dic73—42no—
74C4×Dic73—41no—
75C4×Dic73—41no—
76C4×Dic73—41no—
77C24×C73C2×D412yes—
78C2×Q8×C73—21no—
79C2×Q8×C73—41no—
80C2×Q8×C73—41no—
81C2×Q8×C73C4⋊C412yescomplemented
82C2×D4×C73—41no—
83C2×D4×C73—41no—
84C2×D4×C73C2×D411yescharacteristic
85C2×D4×C73C2×Q811yescharacteristic
86C2×D4×C73#4112yes—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—42no—
90C22×C4×C73C22×C411yescharacteristic
91C22⋊C4×C73—24no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—42no—
95C42×C73—21no—
96C42×C73C4⋊C412yescomplemented
97C23×C73—22no—
98C23×C73—22no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73C2×C22⋊C411yescharacteristic
102C23×C73#12112yes—
103C23×C73C22⋊Q812yes—
104Q8×C73—42no—
105Q8×C73—42no—
106D4×C73—41no—
107D4×C73—41no—
108D4×C73—42no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—82no—
118C2×C4×C73—82no—
119C2×C4×C73—82no—
120C2×C4×C73C2×C4⋊C411yescharacteristic
121C2×C4×C73#11711yescharacteristic
122C2×Dic73—22no—
123C2×Dic73—22no—
124C2×Dic73—21no—
125C2×Dic73—22no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—44no—
129C2×Dic73—41no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—41no—
133C2×Dic73—82no—
134C2×Dic73—82no—
135C2×Dic73—82no—
136C2×Dic73—82no—
137C2×Dic73#9911yescharacteristic
138C2×Dic73#9911yescharacteristic
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—22no—
142C22×C73—22no—
143C22×C73—42no—
144C22×C73—42no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—82no—
148C22×C73#38011yescharacteristic
149C22×C73C2×#9911yescharacteristic
150C22×C73#41011yescharacteristic
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—42no—
154C4×C73—42no—
155C4×C73—162no—
156Dic73—21no—
157Dic73—42no—
158Dic73—42no—
159Dic73—41no—
160Dic73—81no—
161Dic73—162no—
162Dic73#35412yescomplemented
163#12798—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—42no—
167C2×C73—42no—
168C2×C73#179911yescharacteristic
169C2×C73C2×#35411yescharacteristic
170C2×C73#212611yescharacteristic
171C4×#124—731no—
172C2×#354—731no—
173C2×#354D7311yescharacteristic
174#1799—732no—
175#1393—732no—
176C73#1279811yescharacteristic, complemented, Sylow
177C2×#124C2×D7311yescharacteristic
178C2×C4×Q8—731no—
179C2×C4×D4—731no—
180C2×#99—732no—
181C2×#99C2×D7312yes—
182#391—731no—
183#390—731no—
184#388—1462no—
185#382—731no—
186#380—1462no—
187C4×C22⋊C4—1462no—
188C43—731no—
189#354—22no—
190#354—1462no—
191#342—1462no—
192C22×Q8C4×D7311yescharacteristic
193C22×D4C22×D7311yescharacteristic
194C23×C4—732no—
195#124—22no—
196#124Dic14612yes—
197#124C22⋊D7312yes—
198C4×Q8—2921no—
199C4×Q8—2921no—
200C4×D4—2921no—
201C4×D4—2921no—
202C2×C4⋊C4—2922no—
203C2×C4⋊C4—731no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—1461no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—732no—
209C2×C22⋊C4—1462no—
210C2×C22⋊C4—1462no—
211C2×C42—731no—
212C2×C42—731no—
213C2×C42—1461no—
214C2×C42—1461no—
215C2×C42—1461no—
216C2×C42C4×D7311yescharacteristic
217#99—22no—
218#99—22no—
219#99—1462no—
220#99—1462no—
221#95—2922no—
222#95—1462no—
223#95—1462no—
224C24D4×D7312yes—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8#1192400712yescomplemented
229C2×D4—41no—
230C2×D4—41no—
231C2×D4C2×C22⋊D7311yescharacteristic
232C2×D4C2×Dic14611yescharacteristic
233C2×D4#1192402712yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—2924no—
238C22×C4—2922no—
239C22×C4—732no—
240C22×C4—731no—
241C22×C4—731no—
242C22×C4—1461no—
243C22×C4—1462no—
244C22×C4—1461no—
245C22×C4—1462no—
246C22×C4C2×C4×D7311yescharacteristic
247C4⋊C4—2921no—
248C4⋊C4—2921no—
249C4⋊C4—2922no—
250C4⋊C4—5844no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—2924no—
256C22⋊C4—2922no—
257C22⋊C4—2922no—
258C22⋊C4—2922no—
259C42—21no—
260C42—2921no—
261C42—2921no—
262C42—2921no—
263C42—2922no—
264C42—2922no—
265C42—1461no—
266C42—1461no—
267C42#1192400712yescomplemented
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23C22⋊C4×D7311yescharacteristic
273C23#1455656312yes—
274C23#1455656812yes—
275Q8—42no—
276Q8—42no—
277D4—41no—
278D4—41no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2924no—
294C2×C4—2921no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2921no—
298C2×C4—5842no—
299C2×C4—5842no—
300C2×C4—5842no—
301C2×C4—5842no—
302C2×C4—731no—
303C2×C4—731no—
304C2×C4—1462no—
305C2×C4—1461no—
306C2×C4—1462no—
307C2×C4—1462no—
308C2×C4C2×#1192400711yescharacteristic
309C2×C4#1455654711yescharacteristic
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22#1798844511yescharacteristic
320C22#99×D7311yescharacteristic
321C22#1798855911yescharacteristic
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—162no—
327C4—2922no—
328C4—2922no—
329C4—2921no—
330C4—5841no—
331C4—732no—
332C4—11682no—
333C4—1461no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2#2260991711yescharacteristic
339C2#354×D7311yescharacteristic
340C2#2261159511yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC2×#354×C73C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenterC2×#354D73
hypocenterC73#12798
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic73C4semidirect
9. #17988494C4semidirect
12. #17988453C4semidirect
15. C42×Dic73C4semidirect
81. C2×Q8×C73C4⋊C4semidirect
96. C42×C73C4⋊C4semidirect
162. Dic73#354semidirect
176. C73#12798semidirect
228. C2×Q8#11924007semidirect
267. C42#11924007semidirect

Maximal subgroups

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

subgroupindex
#124×Dic732
C2×#179883822
#226099172
#226080552
C2×#354×C732
#1279873