Finite Group Databasebeta

← #33002234

Subgroups — #33002234

Order 18688.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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#22609859C211yesmaximal, characteristic
2#22609849C211yesmaximal, characteristic
3#22608049C214yesmaximal, complemented
4C73×#2311C211yesmaximal, characteristic
5C2×#14556510—21no—
6C2×#14556508C2212yes—
7C2×#14556506—21no—
8C2×#14556506C2212yes—
9#17988551—21no—
10#17988450—21no—
11C22×#41×C73C2211yescharacteristic
12C2×#117×C73C411yescharacteristic, complemented
13C22×C4⋊C4×C73C411yescharacteristic, complemented
14#391×C73—21no—
15#390×C73—21no—
16#386×C73C412yescomplemented
17#386×C73C2212yes—
18#380×C73—21no—
19#380×C73—21no—
20C2×#11924012—22no—
21C2×C22⋊Dic73—22no—
22#14556510—42no—
23#14556508—24no—
24#14556506—24no—
25#14556506—42no—
26C22⋊#10596831—42no—
27#14556498—42no—
28C2×#41×C73—24no—
29C2×#41×C73—42no—
30C2×#41×C73—42no—
31C22×Q8×C73D411yescharacteristic
32C22×D4×C73C2×C411yescharacteristic
33C22×D4×C73D411yescharacteristic
34C22×D4×C73C2311yescharacteristic
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73C2×C412yes—
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39#117×C73—42no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—42no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73C2×C412yes—
47C2×C42×C73—21no—
48#95×C73—22no—
49#95×C73—22no—
50C22×Dic73—42no—
51#11924012—42no—
52#11924012—42no—
53C2×#10596831—42no—
54C22⋊Dic73—42no—
55C22⋊Dic73—42no—
56C24×C73C22⋊C411yescharacteristic
57C24×C73C22×C411yescharacteristic
58C24×C73C2×D411yescharacteristic
59#41×C73—416no—
60C2×Q8×C73—22no—
61C2×Q8×C73—22no—
62C2×Q8×C73—41no—
63C2×D4×C73—22no—
64C2×D4×C73—22no—
65C2×D4×C73—22no—
66C2×D4×C73—41no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73#4114yes—
72C22×C4×C73—22no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73—22no—
78C22×C4×C73—22no—
79C22×C4×C73—22no—
80C22×C4×C73—22no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73—42no—
84C22×C4×C73—82no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C4⋊C4×C73—42no—
90C4⋊C4×C73—42no—
91C4⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—82no—
94C22⋊C4×C73—82no—
95C42×C73—42no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73C2×C22⋊C411yescharacteristic
110C23×C73#11712yes—
111C23×C73#12312yes—
112Q8×C73—44no—
113D4×C73—44no—
114D4×C73—44no—
115D4×C73—44no—
116C2×C4×C73—28no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—44no—
127C2×C4×C73—44no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—84no—
135C2×C4×C73#9912yes—
136C2×C4×C73#9912yes—
137C2×C4×C73#12312yes—
138C2×Dic73—82no—
139C2×Dic73—82no—
140C2×Dic73—82no—
141#10596831—82no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—41no—
154C22×C73—41no—
155C22×C73—41no—
156C22×C73—41no—
157C22×C73—81no—
158C22×C73—81no—
159C22×C73#38111yescharacteristic
160C22×C73C2×#9911yescharacteristic
161C22×C73C2×#9911yescharacteristic
162C4×C73—22no—
163C4×C73—22no—
164C4×C73—22no—
165C4×C73—41no—
166C4×C73—82no—
167C4×C73—82no—
168Dic73—162no—
169#12855—731nomaximal, Sylow
170C2×C73—21no—
171C2×C73—21no—
172C2×C73—42no—
173C2×C73—42no—
174C2×C73—81no—
175C2×C73#181911yescharacteristic
176C2×C73#212211yescharacteristic
177C2×C73#212311yescharacteristic
178#2311D7311yescharacteristic
179#1785—731no—
180#1782—731no—
181#1390—734no—
182C73#1285511yescharacteristic, complemented, Sylow
183C22×#41C2×D7311yescharacteristic
184C2×#117Dic7311yescharacteristic, complemented
185C22×C4⋊C4Dic7311yescharacteristic, complemented
186C2×#101—1461no—
187C2×#100—732no—
188C2×#99—732no—
189C2×#99—1461no—
190#407—1461no—
191#391—21no—
192#390—21no—
193#386Dic7312yescomplemented
194#386C2×D7312yes—
195#381—1461no—
196#380—21no—
197#380—21no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8C22⋊D7311yescharacteristic
202C22×D4C2×Dic7311yescharacteristic
203C22×D4C22⋊D7311yescharacteristic
204C22×D4C22×D7311yescharacteristic
205C23×C4C2×Dic7311yescharacteristic
206C23×C4C2×Dic7312yes—
207C23×C4C22⋊D7311yescharacteristic
208C23×C4C22⋊D7311yescharacteristic
209C2×M16—1462no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4C2×Dic7312yes—
219C2×C42—21no—
220#101—2922no—
221#100—1464no—
222#99—2922no—
223#99—1464no—
224C22⋊C8—2922no—
225#95—22no—
226#95—22no—
227#95—2922no—
228C24C22⋊Dic7311yescharacteristic
229C24C22×Dic7311yescharacteristic
230C24C2×C22⋊D7311yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1192402814yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—2922no—
258C22×C4C22⋊Dic7311yescharacteristic
259C22×C4C22⋊Dic7311yescharacteristic
260C22×C4C22⋊Dic7311yescharacteristic
261C22×C4C2×C22⋊D7311yescharacteristic
262M16—2922no—
263M16—2922no—
264C2×C8—2922no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—2922no—
272C22⋊C4—2922no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊Dic7311yescharacteristic
288C23#1455655012yes—
289C23#1455657412yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—5842no—
314C2×C4—5842no—
315C2×C4—5842no—
316C2×C4#1455650612yes—
317C2×C4#1455650612yes—
318C2×C4#1455657412yes—
319C8—5842no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1798845011yescharacteristic
338C22C2×#1455650611yescharacteristic
339C22C2×#1455650611yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—11682no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#2261000511yescharacteristic
353C2#2261158111yescharacteristic
354C2#2261158511yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC24×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#2311C2
socleC22×C73C2×#99
prime radicalC22×#41×C73C22
cosocleC22×D4×C73C23
hypercenter#2311D73
hypocenterC73#12855
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #22608049C2semidirect
12. C2×#117×C73C4semidirect
13. C22×C4⋊C4×C73C4semidirect
16. #386×C73C4semidirect
182. C73#12855semidirect
184. C2×#117Dic73semidirect
185. C22×C4⋊C4Dic73semidirect
193. #386Dic73semidirect

Maximal subgroups

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

subgroupindex
#226098592
#226098492
#226080492
C73×#23112
#1285573