Finite Group Databasebeta

← #32977717

Subgroups — #32977717

Order 18688.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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#22609923C211yesmaximal, characteristic
2#22608603C212yesmaximal, complemented
3#22608360C212yesmaximal, complemented
4#22608252C211yesmaximal, characteristic
5C2×#556×C73C211yesmaximal, characteristic
6#17988640C2212yes—
7#17988611C2212yes—
8C2×#14556508—21no—
9#17988356—21no—
10#17988333—24no—
11Q8⋊#10596831—24no—
12#17988311—22no—
13#556×C73—24no—
14#556×C73—24no—
15C2×C4⋊Q8×C73C2211yescharacteristic
16C2×#126×C73—22no—
17C2×#124×C73—21no—
18C2×C22⋊Q8×C73C412yes—
19C2×C4×Q8×C73—21no—
20C2×C4×Q8×C73C2211yescharacteristic
21C2×C4×D4×C73C2211yescharacteristic
22C2×#117×C73C412yes—
23C2×#11924012—22no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26C4⋊#10596831—22no—
27#14556508—42no—
28C22⋊#10596831—22no—
29C4×#10596831—22no—
30C22×Q8×C73—21no—
31C22×Q8×C73C2×C411yescharacteristic
32C22×D4×C73D411yescharacteristic
33C23×C4×C73C2×C412yes—
34C4⋊Q8×C73—22no—
35C4⋊Q8×C73D414yes—
36#126×C73—216no—
37#124×C73—28no—
38C22⋊Q8×C73—24no—
39C22⋊Q8×C73—24no—
40C4×Q8×C73—28no—
41C4×Q8×C73—22no—
42C4×Q8×C73D414yes—
43C4×D4×C73—22no—
44C4×D4×C73—22no—
45#117×C73—24no—
46#117×C73—24no—
47C2×C4⋊C4×C73—22no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73—21no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C412yes—
52C2×C4⋊C4×C73D411yescharacteristic
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73D412yes—
56C2×C42×C73—22no—
57C2×C42×C73—21no—
58C2×C42×C73C2311yescharacteristic
59#11924012—42no—
60#11924012—42no—
61C2×#10596831—42no—
62C2×#10596831—42no—
63C24×C73C22⋊C412yes—
64C2×Q8×C73—21no—
65C2×Q8×C73—24no—
66C2×Q8×C73—42no—
67C2×Q8×C73—42no—
68C2×Q8×C73—41no—
69C2×Q8×C73—41no—
70C2×Q8×C73C22⋊C412yes—
71C2×D4×C73—22no—
72C2×D4×C73—22no—
73C2×D4×C73—21no—
74C2×D4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—22no—
78C22×C4×C73—22no—
79C22×C4×C73—22no—
80C22×C4×C73—44no—
81C22×C4×C73C22⋊C412yes—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C4⋊C4×C73—28no—
86C4⋊C4×C73—21no—
87C4⋊C4×C73—22no—
88C4⋊C4×C73—24no—
89C4⋊C4×C73—24no—
90C4⋊C4×C73—22no—
91C4⋊C4×C73C22⋊C412yes—
92C4⋊C4×C73C22⋊C414yes—
93C22⋊C4×C73—28no—
94C22⋊C4×C73—28no—
95C22⋊C4×C73—24no—
96C42×C73—28no—
97C42×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
106Q8×C73—24no—
107Q8×C73—42no—
108Q8×C73—44no—
109Q8×C73—44no—
110D4×C73—42no—
111D4×C73—42no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—24no—
117C2×C4×C73—41no—
118C2×C4×C73—44no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—44no—
124C2×C4×C73—42no—
125C2×C4×C73C2×C22⋊C411yescharacteristic
126C2×C4×C73C2×C22⋊C411yescharacteristic
127C2×C4×C73#12012yes—
128C2×C4×C73#12012yes—
129C2×C4×C73#13512yes—
130C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C73#13711yescharacteristic
132#10596831—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—42no—
148C4×C73—42no—
149C4×C73—44no—
150C4×C73#41312yes—
151C4×C73#44912yes—
152C4×C73#45212yes—
153#8389—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73—82no—
157C2×C73#178511yescharacteristic
158C2×C73#178811yescharacteristic
159C2×C73#179611yescharacteristic
160C2×#556D7311yescharacteristic
161#1800—731no—
162#1546—732no—
163#1487—732no—
164#1460—731no—
165C73#838911yescharacteristic, complemented, Sylow
166#556—24no—
167#556—24no—
168C2×C4⋊Q8C2×D7311yescharacteristic
169C2×#126—22no—
170C2×#124—21no—
171C2×C22⋊Q8Dic7312yes—
172C2×C4×Q8—21no—
173C2×C4×Q8C2×D7311yescharacteristic
174C2×C4×D4C2×D7311yescharacteristic
175C2×#117Dic7312yes—
176#431—732no—
177#423—732no—
178C2×#100—1461no—
179#343—1461no—
180#332—1464no—
181Q8⋊C8—1464no—
182#323—1462no—
183C22×Q8—21no—
184C22×Q8C2×Dic7311yescharacteristic
185C22×D4C22⋊D7311yescharacteristic
186C23×C4C2×Dic7312yes—
187C2×M16—1462no—
188C4⋊Q8—22no—
189C4⋊Q8C22⋊D7314yes—
190#126—216no—
191#124—28no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—28no—
195C4×Q8—22no—
196C4×Q8C22⋊D7314yes—
197C4×D4—22no—
198C4×D4—22no—
199#117—24no—
200#117—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic7311yescharacteristic
205C2×C4⋊C4C2×Dic7312yes—
206C2×C4⋊C4C22⋊D7311yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊D7312yes—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C22×D7311yescharacteristic
213C4⋊C8—1462no—
214C4⋊C8—1462no—
215C4⋊C8—1462no—
216#100—2922no—
217C22⋊C8—1462no—
218C4×C8—1462no—
219C24C22⋊Dic7312yes—
220C2×Q8—21no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8C22⋊Dic7312yes—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4C22⋊Dic7312yes—
238C22×C4C22×Dic7311yescharacteristic
239C22×C4C2×C22⋊D7311yescharacteristic
240C22×C4C2×C22⋊D7311yescharacteristic
241M16—2922no—
242M16—2922no—
243C2×C8—2922no—
244C2×C8—2922no—
245C4⋊C4—28no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4C22⋊Dic7312yes—
252C4⋊C4C22⋊Dic7314yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—24no—
258C42C2×C22⋊D7312yes—
259C42C2×C22⋊D7312yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×C22⋊Dic7311yescharacteristic
266Q8—24no—
267Q8—42no—
268Q8—44no—
269Q8—44no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4C2×C22⋊Dic7311yescharacteristic
286C2×C4C2×C22⋊Dic7311yescharacteristic
287C2×C4#1455656012yes—
288C2×C4#1455656012yes—
289C2×C4#1455661112yes—
290C2×C4#1455661611yescharacteristic
291C2×C4#1455662111yescharacteristic
292C8—5842no—
293C8—5842no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1798844311yescharacteristic
302C22#1798858311yescharacteristic
303C22#1798858811yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—44no—
309C4—42no—
310C4#1798857112yes—
311C4#1798873212yes—
312C4#1798874912yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2260985911yescharacteristic
317C2#2260987111yescharacteristic
318C2#2260990611yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#556×C73C2
socleC22×C73#379
prime radicalC22×D4×C73D4
cosocleC2×C42×C73C23
hypercenterC2×#556D73
hypocenterC73#8389
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. #22608603C2semidirect
3. #22608360C2semidirect
165. C73#8389semidirect

Maximal subgroups

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

subgroupindex
#226099232
#226086032
#226083602
#226082522
C2×#556×C732
#838973