Finite Group Databasebeta

← #32983376

Subgroups — #32983376

Order 18688.

514 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 109 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#22618971C211yesmaximal, characteristic
2#22615105C211yesmaximal, characteristic
3#22609920C211yesmaximal, characteristic
4#22608612C211yesmaximal, characteristic, complemented
5#22607833C211yesmaximal, characteristic, complemented
6#22607832C211yesmaximal, characteristic, complemented
7C73×#1548C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9#17988666—22no—
10#17988664—22no—
11#17988640C2211yescharacteristic
12#17988611C2211yescharacteristic
13C2×#14556507—21no—
14#17988549—22no—
15#17988498C414yes—
16#17988488—21no—
17#17988452—21no—
18#17988444—22no—
19C4×C4⋊Dic73C2211yescharacteristic
20#17988355—21no—
21#17988336—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23#431×C73C2211yescharacteristic
24#423×C73C2211yescharacteristic
25#333×C73—22no—
26C22×D292D411yescharacteristic
27C22×C4×D73D412yes—
28C2×#11924012—21no—
29C2×#11924011—24no—
30C4×D292—21no—
31C4×D292—21no—
32C4×D292—21no—
33C4×D292—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73D411yescharacteristic
36C2×#11924005—24no—
37C2×#11924005C2×C412yes—
38C2×C4×Dic73—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831—21no—
41C4⋊#10596831D412yes—
42#14556507—42no—
43C22⋊#10596831—21no—
44#14556503—24no—
45C4×#10596831—21no—
46#14556498—24no—
47#14556498C2×C412yes—
48C22×Q8×C73C2×C412yes—
49C2×M16×C73—21no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C42×C73C2311yescharacteristic
57C4⋊C8×C73—21no—
58C4⋊C8×C73—21no—
59C4⋊C8×C73D412yes—
60C22⋊C8×C73—21no—
61C4×C8×C73—21no—
62C23×D73C22⋊C412yes—
63C2×D292—21no—
64C2×D292—21no—
65C2×D292—21no—
66C2×D292—21no—
67C2×D292—21no—
68C2×D292—21no—
69C22×Dic73—22no—
70C22×Dic73C22⋊C412yes—
71C2×C4×D73—22no—
72C2×C4×D73—22no—
73C2×C4×D73—44no—
74#11924012—41no—
75#11924012—41no—
76#11924011—84no—
77C2×#10596831—22no—
78C2×#10596831—41no—
79C4⋊Dic73—21no—
80C4⋊Dic73—21no—
81C4⋊Dic73—42no—
82#11924005—22no—
83#11924005—22no—
84#11924005—84no—
85C4×Dic73—41no—
86C4×Dic73—41no—
87C2×Q8×C73—22no—
88C2×Q8×C73—22no—
89C2×Q8×C73—82no—
90C22×C4×C73—22no—
91C22×C4×C73C22×C411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94M16×C73—41no—
95M16×C73—41no—
96C2×C8×C73—22no—
97C2×C8×C73—41no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C42×C73C2×D411yescharacteristic
103C42×C73C2×D411yescharacteristic
104C42×C73#4111yescharacteristic
105C42×C73#4111yescharacteristic
106C23×C73C2×C22⋊C411yescharacteristic
107C22×D73—22no—
108C22×D73—22no—
109C22×D73—22no—
110C22×D73—42no—
111C22×D73—42no—
112Q8×C73—82no—
113Q8×C73—82no—
114C2×C4×C73—22no—
115C2×C4×C73—41no—
116C2×C4×C73—44no—
117C2×C4×C73—42no—
118C2×C4×C73C4×D411yescharacteristic
119C2×C4×C73C4×D411yescharacteristic
120C2×C4×C73#12011yescharacteristic
121C2×C4×C73#12111yescharacteristic
122C2×C4×C73#12111yescharacteristic
123C2×C4×C73#12311yescharacteristic
124C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C73#13711yescharacteristic
126C2×C4×C73#13711yescharacteristic
127C2×C4×C73#13711yescharacteristic
128C2×Dic73—22no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133D292—41no—
134D292—41no—
135D292—41no—
136D292—41no—
137C4×D73—44no—
138C4×D73—82no—
139C8×C73—42no—
140C8×C73—81no—
141#10596831—42no—
142#10596831—81no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73#38611yescharacteristic
146C22×C73#41711yescharacteristic
147C22×C73#41911yescharacteristic
148C2×D73—44no—
149C2×D73—42no—
150C2×D73—42no—
151C2×D73—42no—
152C2×D73—82no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—21no—
156C4×C73—82no—
157C4×C73#45512yes—
158C4×C73#46811yescharacteristic
159C4×C73#46811yescharacteristic
160C4×C73#47011yescharacteristic
161C4×C73#47011yescharacteristic
162Dic73—42no—
163Dic73—82no—
164#9399—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73#188511yescharacteristic
168C2×C73#189411yescharacteristic
169C2×C73#191411yescharacteristic
170D73—82no—
171#2975—731no—
172#2563—731no—
173#1800—731no—
174#1548—731no—
175#1548D7311yescharacteristic, complemented
176#1303—731no—
177#1303—731no—
178C73#939911yescharacteristic, complemented, Sylow
179C2×C4⋊Q8C2×D7311yescharacteristic
180C2×C4×D4—731no—
181#436—1462no—
182#436—1462no—
183#431—731no—
184#431C2×D7311yescharacteristic
185#423—731no—
186#423C2×D7311yescharacteristic
187C2×#100—1461no—
188#407—1462no—
189#393—734no—
190#390—1461no—
191#382—1461no—
192#380—1462no—
193C4×C4⋊C4—731no—
194#343—1461no—
195#333—22no—
196#333—1462no—
197C22×Q8C4×D7312yes—
198C22×D4—731no—
199C23×C4—732no—
200C2×M16—21no—
201C2×M16—1464no—
202C2×M16—1461no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4×D4—1461no—
208C4×D4—1461no—
209C4×D4—1461no—
210C4×D4—1461no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—731no—
214C2×C4⋊C4—1461no—
215C2×C22⋊C4—732no—
216C2×C22⋊C4—1464no—
217C2×C42—1461no—
218C2×C42C22×D7311yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—732no—
222C4⋊C8—1461no—
223C4⋊C8—1461no—
224C4⋊C8C22⋊D7312yes—
225#100—2922no—
226C22⋊C8—21no—
227C22⋊C8—1464no—
228C22⋊C8—1461no—
229C4×C8—21no—
230C4×C8—1461no—
231#95—732no—
232#95—1464no—
233C24—732no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—82no—
237C2×D4—1461no—
238C2×D4—1461no—
239C2×D4—1461no—
240C2×D4—1461no—
241C2×D4—1461no—
242C2×D4—1461no—
243C22×C4—22no—
244C22×C4—2924no—
245C22×C4—732no—
246C22×C4—1462no—
247C22×C4—1462no—
248C22×C4—1462no—
249C22×C4C2×C4×D7311yescharacteristic
250C22×C4D4×D7311yescharacteristic
251C22×C4D4×D7311yescharacteristic
252M16—41no—
253M16—41no—
254M16—2921no—
255M16—2921no—
256M16—5844no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—2921no—
260C2×C8—1462no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2922no—
266C4⋊C4—1461no—
267C4⋊C4—1461no—
268C22⋊C4—5844no—
269C22⋊C4—1462no—
270C22⋊C4—1462no—
271C42—2921no—
272C42—2921no—
273C42D4×D7311yescharacteristic
274C42C2×C22⋊D7311yescharacteristic
275C42#1192402711yescharacteristic
276C42#1192402811yescharacteristic
277C23—2922no—
278C23—2922no—
279C23—1462no—
280C23—1462no—
281C23—1462no—
282C23C22⋊C4×D7311yescharacteristic
283Q8—82no—
284Q8—82no—
285D4—2921no—
286D4—2921no—
287D4—2921no—
288D4—2921no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—2924no—
294C2×C4—2922no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—5842no—
299C2×C4—1462no—
300C2×C4#1455655111yescharacteristic
301C2×C4C4×C22⋊D7311yescharacteristic
302C2×C4#1455655911yescharacteristic
303C2×C4#1455656211yescharacteristic
304C2×C4#1455656411yescharacteristic
305C2×C4#1455657311yescharacteristic
306C2×C4#1455661811yescharacteristic
307C2×C4#1455662011yescharacteristic
308C2×C4#1455662311yescharacteristic
309C2×C4#1455662311yescharacteristic
310C8—42no—
311C8—81no—
312C8—2922no—
313C8—5841no—
314C22—21no—
315C22—21no—
316C22—2924no—
317C22—2922no—
318C22—2922no—
319C22—2922no—
320C22—5842no—
321C22#1798846311yescharacteristic
322C22#1798858711yescharacteristic
323C22#1798859411yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—2922no—
329C4—5842no—
330C4#1798876312yes—
331C4#1798883011yescharacteristic
332C4#1798883611yescharacteristic
333C4#1798884211yescharacteristic
334C4#1798884411yescharacteristic
335C2—21no—
336C2—21no—
337C2—5842no—
338C2#2261031611yescharacteristic
339C2#2261036111yescharacteristic
340C2#2261047511yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1548C2
socleC22×C73#386
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1548D73
hypocenterC73#9399
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
4. #22608612C2semidirect
5. #22607833C2semidirect
6. #22607832C2semidirect
7. C73×#1548C2semidirect
175. #1548D73semidirect
178. C73#9399semidirect

Maximal subgroups

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

subgroupindex
#226189712
#226151052
#226099202
#226086122
#226078332
#226078322
C73×#15482
#939973