Finite Group Databasebeta

← #33001200

Subgroups — #33001200

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic73C211yesmaximal, characteristic
2C2×#17988388C211yesmaximal, characteristic
3C2×#17988362C211yesmaximal, characteristic, complemented
4#107×Dic73C211yesmaximal, characteristic, complemented
5#22608023C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#357×C73C211yesmaximal, characteristic
8C2×D4×Dic73—21no—
9C22×#11924003C2211yescharacteristic
10D8×Dic73—22no—
11D8×Dic73C414yescomplemented
12C2×#14556525C2211yescharacteristic
13C2×#14556525C2211yescharacteristic
14C2×#14556512—21no—
15C2×C8×Dic73C2211yescharacteristic
16C4⋊C4×Dic73—21no—
17#17988388C414yescomplemented
18#17988362—22no—
19#17988349—21no—
20C22×D8×C73C2211yescharacteristic
21C22×C16×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23#357×C73C414yescomplemented
24C23×Dic73—41no—
25C22×#10596831D411yescharacteristic
26D4×Dic73—24no—
27D4×Dic73—42no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—21no—
30C2×C22⋊Dic73—41no—
31C2×C4×Dic73—41no—
32C2×C4×Dic73D411yescharacteristic
33C2×#11924003—21no—
34C2×#11924003—21no—
35C2×#11924003C2×C412yescomplemented
36#14556525—21no—
37#14556525—21no—
38#14556525C2×C412yes—
39#14556525C2×C412yescomplemented
40#14556512—24no—
41C8×Dic73C2×C412yes—
42C8×Dic73D412yes—
43#14556498—41no—
44C22×D4×C73—21no—
45C2×D8×C73—21no—
46C2×D8×C73—21no—
47C2×D8×C73—21no—
48C2×D8×C73C2×C412yes—
49C2×D8×C73C2×C412yes—
50C2×D8×C73C2×C412yes—
51C22×C8×C73C2311yescharacteristic
52C2×C4⋊C4×C73—21no—
53C2×C16×C73—21no—
54C2×C16×C73—21no—
55C2×C16×C73C2×C412yescomplemented
56#107×C73—21no—
57#107×C73C2×C412yescomplemented
58C22×Dic73—41no—
59C22×Dic73—41no—
60C22×Dic73—44no—
61C22×Dic73—82no—
62C22×Dic73—82no—
63C22×Dic73D811yescharacteristic
64C22×Dic73D811yescharacteristic
65C2×#10596831—21no—
66C2×#10596831—21no—
67C2×#10596831C22⋊C412yes—
68C4⋊Dic73—22no—
69C4⋊Dic73—22no—
70C4⋊Dic73—41no—
71C4⋊Dic73—41no—
72C22⋊Dic73—44no—
73C4×Dic73—82no—
74C4×Dic73C22⋊C412yes—
75C4×Dic73SD1612yes—
76#11924003—22no—
77C24×C73—41no—
78C2×D4×C73—22no—
79C2×D4×C73—22no—
80C2×D4×C73—22no—
81C2×D4×C73—41no—
82C2×D4×C73—41no—
83C2×D4×C73—41no—
84C22×C4×C73—41no—
85C22×C4×C73C2×D411yescharacteristic
86D8×C73—22no—
87D8×C73—22no—
88D8×C73—22no—
89D8×C73C4214yescomplemented
90C2×C8×C73C22×C411yescharacteristic
91C2×C8×C73C22×C411yescharacteristic
92C2×C8×C73C22×C411yescharacteristic
93C2×C8×C73C2×D411yescharacteristic
94C2×C8×C73#4111yescharacteristic
95C2×C8×C73#4111yescharacteristic
96C4⋊C4×C73—22no—
97C4⋊C4×C73—41no—
98C16×C73—22no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—42no—
102C23×C73—81no—
103C23×C73—81no—
104C23×C73—81no—
105C23×C73—81no—
106C23×C73C2×D811yescharacteristic
107D4×C73—24no—
108D4×C73—42no—
109D4×C73—42no—
110D4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—81no—
113C2×C4×C73—81no—
114C2×C4×C73C2×C22⋊C411yescharacteristic
115C2×C4×C73C4×D411yescharacteristic
116C2×C4×C73C4×D411yescharacteristic
117C2×C4×C73C2×SD1611yescharacteristic
118C2×C4×C73#13511yescharacteristic
119C2×C4×C73#13511yescharacteristic
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—88no—
127C2×Dic73—81no—
128C2×Dic73—81no—
129C2×Dic73—81no—
130C2×Dic73—81no—
131C2×Dic73#10212yes—
132C2×Dic73#10212yes—
133C8×C73C2×C4211yescharacteristic
134C8×C73#11711yescharacteristic
135C8×C73C4×D411yescharacteristic
136C8×C73C4×D411yescharacteristic
137#10596831—22no—
138C22×C73—44no—
139C22×C73—82no—
140C22×C73—82no—
141C22×C73—82no—
142C22×C73—82no—
143C22×C73—82no—
144C22×C73—82no—
145C22×C73C2×#10211yescharacteristic
146C22×C73C4×D811yescharacteristic
147C22×C73C4×D811yescharacteristic
148C22×C73#50911yescharacteristic
149C22×C73#50911yescharacteristic
150C22×C73#50911yescharacteristic
151C22×C73#51011yescharacteristic
152C4×C73—82no—
153C4×C73C4×C22⋊C411yescharacteristic
154C4×C73#41611yescharacteristic
155C4×C73C4×SD1611yescharacteristic
156C4×C73C4×SD1611yescharacteristic
157Dic73—22no—
158Dic73—22no—
159Dic73—82no—
160Dic73—82no—
161#12637—731nomaximal, Sylow
162C2×C73—84no—
163C2×C73C4×#10211yescharacteristic
164C2×C73#213911yescharacteristic
165C2×C73#214011yescharacteristic
166C2×C73#217511yescharacteristic
167C2×C73#217511yescharacteristic
168C2×C73#217711yescharacteristic
169C2×C73#217711yescharacteristic
170C2×C4×D8—731no—
171C2×#357—731no—
172C2×#357D7311yescharacteristic
173C2×#346—731no—
174C4×#107—731no—
175#1380—731no—
176#1380—731no—
177C73#1263711yescharacteristic, complemented, Sylow
178C22×D8C2×D7311yescharacteristic
179C2×C4×D4—1461no—
180C22×C16—731no—
181C22×C16C2×D7311yescharacteristic
182C4×D8—734no—
183C4×D8—1462no—
184C2×#107—731no—
185C2×#107—731no—
186C2×#107C2×D7311yescharacteristic
187C2×#102—1461no—
188C2×C4×C8—731no—
189C4×C4⋊C4—1461no—
190#357—734no—
191#357Dic7314yescomplemented
192#346—1462no—
193#340—1461no—
194C22×D4—21no—
195C23×C4—2921no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8C4×D7312yes—
200C2×D8C4×D7312yes—
201C2×D8C2×Dic7312yes—
202C22×C8—731no—
203C22×C8C22×D7311yescharacteristic
204C4×D4—2922no—
205C4×D4—1464no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4—1461no—
209C2×C22⋊C4—2921no—
210C2×C42—2921no—
211C2×C42—731no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—732no—
215C2×C16—1461no—
216C2×C16—1461no—
217C2×C16C2×Dic7312yescomplemented
218#107—21no—
219#107—732no—
220#107—732no—
221#107—1461no—
222#107—1461no—
223#107C2×Dic7312yescomplemented
224#102—1464no—
225C4×C8—732no—
226C4×C8—732no—
227#95—2921no—
228C24—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—41no—
236C22×C4—2921no—
237C22×C4—2921no—
238C22×C4—2924no—
239C22×C4—5842no—
240C22×C4—5842no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4D4×D7311yescharacteristic
244D8—22no—
245D8—22no—
246D8—22no—
247D8C4×Dic7314yescomplemented
248C2×C8—732no—
249C2×C8—1461no—
250C2×C8—1461no—
251C2×C8C2×C4×D7311yescharacteristic
252C2×C8C2×C4×D7311yescharacteristic
253C2×C8C22×Dic7311yescharacteristic
254C2×C8D4×D7311yescharacteristic
255C2×C8#1192402711yescharacteristic
256C2×C8#1192402711yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—2921no—
260C4⋊C4—2921no—
261C4⋊C4—1462no—
262C4⋊C4—1462no—
263C22⋊C4—2924no—
264C42—5842no—
265C42—732no—
266C42—732no—
267C16—22no—
268C16—1462no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23D8×D7311yescharacteristic
277D4—24no—
278D4—42no—
279D4—42no—
280D4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—5841no—
287C2×C4—5841no—
288C2×C4—5841no—
289C2×C4—5841no—
290C2×C4—5848no—
291C2×C4—732no—
292C2×C4—732no—
293C2×C4—1461no—
294C2×C4—1461no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4C22⋊C4×D7311yescharacteristic
298C2×C4#1455655111yescharacteristic
299C2×C4D4×Dic7311yescharacteristic
300C2×C4SD16×D7311yescharacteristic
301C2×C4#1455661011yescharacteristic
302C2×C4#1455661211yescharacteristic
303C8—1462no—
304C8C2×C4×Dic7311yescharacteristic
305C8#1455654711yescharacteristic
306C8#1455655111yescharacteristic
307C8D4×Dic7311yescharacteristic
308C22—44no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22#102×D7311yescharacteristic
316C22#1798866711yescharacteristic
317C22D8×Dic7311yescharacteristic
318C22#1798902111yescharacteristic
319C22#1798902111yescharacteristic
320C22#1798902411yescharacteristic
321C22#1798902611yescharacteristic
322C4—82no—
323C4—5842no—
324C4—5842no—
325C4—1462no—
326C4—1462no—
327C4C22⋊C4×Dic7311yescharacteristic
328C4#1798858011yescharacteristic
329C4SD16×Dic7311yescharacteristic
330C4#1798867811yescharacteristic
331C2—84no—
332C2#102×Dic7311yescharacteristic
333C2#2261164011yescharacteristic
334C2#2261164511yescharacteristic
335C2#2261179211yescharacteristic
336C2#2261179611yescharacteristic
337C2#2261180411yescharacteristic
338C2#2261180711yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC8×C73C2×C42
FrattiniC22×C8C22×D73
FittingC2×#357×C73C2
socleC23×C73C2×D8
prime radicalC22×D8×C73C22
cosocleC22×C8×C73C23
hypercenterC2×#357D73
hypocenterC73#12637
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
3. C2×#17988362C2semidirect
4. #107×Dic73C2semidirect
5. #22608023C2semidirect
6. #22608022C2semidirect
11. D8×Dic73C4semidirect
17. #17988388C4semidirect
23. #357×C73C4semidirect
35. C2×#11924003C2×C4semidirect
39. #14556525C2×C4semidirect
55. C2×C16×C73C2×C4semidirect
57. #107×C73C2×C4semidirect
89. D8×C73C42semidirect
177. C73#12637semidirect
191. #357Dic73semidirect
217. C2×C16C2×Dic73semidirect
223. #107C2×Dic73semidirect
247. D8C4×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic732
C2×#179883882
C2×#179883622
#107×Dic732
#226080232
#226080222
C2×#357×C732
#1263773