Finite Group Databasebeta

← #32981727

Subgroups — #32981727

Order 18688.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#22618262C211yesmaximal, characteristic
2C2×#17988433C211yesmaximal, characteristic, complemented
3C2×#17988322C212yesmaximal
4#22607811C212yesmaximal, complemented
5C73×#1817C211yesmaximal, characteristic, complemented
6C2×#14556574—21no—
7C2×#14556565C2211yescharacteristic
8C22×C4×Dic73—21no—
9#17988489C412yescomplemented
10#17988485C412yescomplemented
11#17988478—22no—
12#17988466—21no—
13#17988447C2211yescharacteristic
14#17988433—24no—
15C2×#14556498—21no—
16C2×#14556498C2212yes—
17#17988322—24no—
18C22×C42×C73C2211yescharacteristic
19C2×C22⋊C8×C73C2212yes—
20#336×C73—22no—
21C22×D292C2×C411yescharacteristic
22C22×C22⋊D73—21no—
23C23×Dic73—22no—
24#14556574—44no—
25#14556565—22no—
26#14556565—22no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924007—22no—
30C2×C22⋊Dic73—21no—
31C2×#11924005—22no—
32C2×#11924005—22no—
33C2×#11924005—21no—
34C2×C4×Dic73—24no—
35C2×C4×Dic73—28no—
36#14556498—24no—
37#14556498—22no—
38#14556498—22no—
39#14556498—21no—
40#14556498D414yes—
41C23×C4×C73D411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73C2311yescharacteristic
44C22×C8×C73—22no—
45C2×C42×C73—22no—
46C2×C42×C73—22no—
47C2×C42×C73C2×C412yes—
48C2×C42×C73D412yes—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C23×D73—21no—
52C2×D292—22no—
53C2×D292—42no—
54C2×C22⋊D73—41no—
55C2×C22⋊D73—41no—
56C2×C22⋊D73—42no—
57C2×C22⋊D73—42no—
58C22×Dic73—28no—
59C22×Dic73—24no—
60C22×Dic73—22no—
61C22×Dic73—21no—
62C22×Dic73—44no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C4⋊Dic73—22no—
66C4⋊Dic73—42no—
67#11924007—44no—
68C22⋊Dic73—42no—
69#11924005—44no—
70#11924005—44no—
71#11924005—42no—
72C4×Dic73—216no—
73C24×C73C2×D411yescharacteristic
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—24no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—22no—
80C22×C4×C73—22no—
81C22×C4×C73—42no—
82C22×C4×C73C22⋊C412yes—
83C22×C4×C73C22⋊C412yes—
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C2×D412yes—
86C22×C4×C73C2×D411yescharacteristic
87C22×C4×C73#4112yes—
88C2×C8×C73—44no—
89C2×C8×C73—42no—
90C42×C73—24no—
91C42×C73—24no—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73C2×C22⋊C411yescharacteristic
95C23×C73#12011yescharacteristic
96C23×C73#12311yescharacteristic
97C23×C73#13512yes—
98C23×C73AGL(1,ℤ/8)12yes—
99C22×D73—41no—
100C22×D73—42no—
101C22×D73—41no—
102C22×D73—41no—
103C22×D73—42no—
104C2×C4×C73—24no—
105C2×C4×C73—24no—
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73#10414yes—
118C2×C4×C73C4×D412yes—
119C2×C4×C73#12112yes—
120C22⋊D73—82no—
121C22⋊D73—82no—
122C2×Dic73—28no—
123C2×Dic73—44no—
124C2×Dic73—48no—
125C2×Dic73—48no—
126C2×Dic73—44no—
127C2×Dic73—41no—
128C2×Dic73—42no—
129C2×Dic73—41no—
130C2×Dic73—42no—
131C2×Dic73—41no—
132C2×Dic73—42no—
133D292—82no—
134C8×C73—82no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—24no—
140C22×C73—24no—
141C22×C73#38611yescharacteristic
142C22×C73#41012yes—
143C22×C73#41611yescharacteristic
144C22×C73#41811yescharacteristic
145C22×C73C2×#10412yes—
146C22×C73#44912yes—
147C22×C73#48212yes—
148C2×D73—81no—
149C2×D73—81no—
150C2×D73—81no—
151C2×D73—82no—
152C2×D73—82no—
153C4×C73—24no—
154C4×C73—22no—
155C4×C73—22no—
156C4×C73—42no—
157Dic73—48no—
158Dic73—81no—
159Dic73—82no—
160#9149—731nomaximal, Sylow
161C2×C73—22no—
162C2×C73—22no—
163C2×C73#150012yes—
164C2×C73#150512yes—
165C2×C73#187511yescharacteristic
166C2×C73#189611yescharacteristic
167C2×C73#189911yescharacteristic
168D73—161no—
169#2854—731no—
170C2×#376—731no—
171#1817D7311yescharacteristic, complemented
172C2×#327—732no—
173#1294—732no—
174C73#914911yescharacteristic, complemented, Sylow
175C2×#123—1461no—
176C2×#121—731no—
177C22×C42—1461no—
178C22×C42C2×D7311yescharacteristic
179C2×C22⋊C8C2×D7312yes—
180#390—732no—
181#389—732no—
182#388—1462no—
183#386—1461no—
184#380—731no—
185#376—1464no—
186C2×#95—732no—
187C2×#95—1461no—
188#336—22no—
189#327—1464no—
190C22×D4—731no—
191C22×D4—1461no—
192C23×C4—1462no—
193C23×C4D29211yescharacteristic
194C23×C4D29211yescharacteristic
195C23×C4C22×D7311yescharacteristic
196C22×C8—22no—
197#123—2924no—
198#121—1462no—
199#121—1462no—
200C2×C4⋊C4—732no—
201C2×C4⋊C4—731no—
202C2×C4⋊C4—1462no—
203C2×C22⋊C4—1461no—
204C2×C22⋊C4—1462no—
205C2×C22⋊C4—1462no—
206C2×C22⋊C4—1461no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—1464no—
210C2×C42—1468no—
211C2×C42C4×D7312yes—
212C2×C42C22⋊D7312yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—734no—
216#95—1464no—
217#95—1462no—
218#95—1462no—
219#95—1461no—
220C24—1461no—
221C24C2×D29211yescharacteristic
222C2×D4—2922no—
223C2×D4—2921no—
224C2×D4—2921no—
225C2×D4—2922no—
226C2×D4—2922no—
227C2×D4—1462no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—2924no—
237C22×C4—2922no—
238C22×C4—2922no—
239C22×C4—1462no—
240C22×C4—1461no—
241C22×C4—1468no—
242C22×C4—1464no—
243C22×C4#1192400512yes—
244C22×C4#1192400512yes—
245C22×C4C2×C4×D7311yescharacteristic
246C22×C4D4×D7312yes—
247C22×C4C2×C22⋊D7311yescharacteristic
248C22×C4#1192402712yes—
249C2×C8—44no—
250C2×C8—42no—
251C4⋊C4—2922no—
252C4⋊C4—2924no—
253C4⋊C4—1462no—
254C22⋊C4—2924no—
255C22⋊C4—2922no—
256C22⋊C4—2924no—
257C22⋊C4—2922no—
258C42—24no—
259C42—24no—
260C42—14616no—
261C23—22no—
262C23—22no—
263C23—2921no—
264C23—2922no—
265C23—2921no—
266C23—2921no—
267C23—2922no—
268C23C2×#1192400511yescharacteristic
269C23#1455656111yescharacteristic
270C23#1455657511yescharacteristic
271C23#1455661312yes—
272C23#1455661912yes—
273D4—5842no—
274D4—5842no—
275D4—5842no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—2928no—
290C2×C4—2928no—
291C2×C4—2924no—
292C2×C4—2924no—
293C2×C4—2921no—
294C2×C4—2922no—
295C2×C4—2921no—
296C2×C4—2922no—
297C2×C4—2921no—
298C2×C4—2922no—
299C2×C4—1468no—
300C2×C4#1455651714yes—
301C2×C4#1455655112yes—
302C2×C4#1455656312yes—
303C8—82no—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—5842no—
311C22—5842no—
312C22—5841no—
313C22—5841no—
314C22—5841no—
315C22#1798846411yescharacteristic
316C22#1798856012yes—
317C22#1798858411yescharacteristic
318C22#1798859311yescharacteristic
319C22C2×#1455651712yes—
320C22#1798873412yes—
321C22#1798891212yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—2928no—
327C4—5842no—
328C4—5841no—
329C2—22no—
330C2—22no—
331C2—11681no—
332C2#2260841812yes—
333C2#2260844012yes—
334C2#2261025111yescharacteristic
335C2#2261037211yescharacteristic
336C2#2261038711yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1817C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenter#1817D73
hypocenterC73#9149
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#17988433C2semidirect
4. #22607811C2semidirect
5. C73×#1817C2semidirect
9. #17988489C4semidirect
10. #17988485C4semidirect
171. #1817D73semidirect
174. C73#9149semidirect

Maximal subgroups

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

subgroupindex
#226182622
C2×#179884332
C2×#179883222
#226078112
C73×#18172
#914973