Finite Group Databasebeta

← #34004140

Subgroups — #34004140

Order 18688.

836 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 146 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#22625786C211yesmaximal, characteristic
2C4×#14556648C211yesmaximal, characteristic
3C2×Q8⋊#10596833C214yesmaximal, complemented
4C2×#17989142C211yesmaximal, characteristic
5C2×C4⋊#10596833C2212yes—
6C2×#14556648—21no—
7C2×#14556648—21no—
8C2×C4×#10596833C2212yes—
9#17989453—21no—
10#17989453—21no—
11Q8⋊#10596833—28no—
12#17989142—28no—
13C2×#14556591C411yescharacteristic
14C2×#14556582—21no—
15C2×#14556571—21no—
16C2×#14556567C412yes—
17C2×Q8×Dic73C2212yes—
18C2×#14556556—21no—
19C2×#14556549—21no—
20C2×C42×D73C2211yescharacteristic
21C2×C4⋊Q8×C73C411yescharacteristic
22C22×#10596833—22no—
23C22×#10596833—22no—
24C4⋊#10596833—24no—
25#14556648—24no—
26#14556648—42no—
27C4×#10596833—24no—
28C22×Dic146—21no—
29C22×C4×D73C2×C411yescharacteristic
30C22×C4×D73D411yescharacteristic
31C22×C4×D73D411yescharacteristic
32#14556591—24no—
33#14556582—28no—
34#14556571—28no—
35#14556567—24no—
36#14556567—24no—
37Q8×Dic73—24no—
38Q8×Dic73D418yes—
39#14556556—28no—
40#14556549—28no—
41C2×C4⋊Dic73—22no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73C2×C412yes—
44C2×#11924007—22no—
45C2×#11924005—22no—
46C2×C4×Dic73—22no—
47C2×C4×Dic73D411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C2×C4×Dic73C2311yescharacteristic
50C42×D73—24no—
51C22×Q8×C73C2×C412yes—
52C4⋊Q8×C73—24no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C42×C73C2×C411yescharacteristic
56C2×#10596833—24no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C23×D73C22⋊C411yescharacteristic
63C2×Dic146—24no—
64C2×Dic146—42no—
65C2×Dic146—42no—
66C22×Dic73—22no—
67C22×Dic73C22⋊C411yescharacteristic
68C22×Dic73C2×D411yescharacteristic
69C22×Dic73C2×D411yescharacteristic
70C2×C4×D73—22no—
71C2×C4×D73—22no—
72C2×C4×D73—22no—
73C2×C4×D73—22no—
74C2×C4×D73—22no—
75C2×C4×D73—22no—
76C2×C4×D73—22no—
77C2×C4×D73—41no—
78C2×C4×D73SD1612yes—
79C2×C4×D73Q1612yes—
80C4⋊Dic73—24no—
81C4⋊Dic73—22no—
82C4⋊Dic73—28no—
83C4⋊Dic73C22⋊C414yes—
84#11924007—28no—
85#11924005—28no—
86C4×Dic73—22no—
87C4×Dic73—22no—
88C4×Dic73—28no—
89C4×Dic73C2×D414yes—
90C2×Q8×C73—22no—
91C2×Q8×C73—24no—
92C2×Q8×C73—24no—
93C2×Q8×C73C22⋊C414yes—
94C22×C4×C73—22no—
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C22×C411yescharacteristic
98C4⋊C4×C73—24no—
99C4⋊C4×C73—24no—
100C42×C73—22no—
101C23×C73C2×C22⋊C411yescharacteristic
102C22×D73—21no—
103C22×D73—21no—
104C22×D73—21no—
105C22×D73—21no—
106C22×D73—21no—
107C22×D73—21no—
108C22×D73#10312yes—
109Q8×C73—28no—
110Q8×C73—44no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—24no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73#10312yes—
119C2×C4×C73#10412yes—
120C2×C4×C73C2×C22⋊C412yes—
121C2×C4×C73#13112yes—
122C2×C4×C73#13112yes—
123C2×Dic73—21no—
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—24no—
127C2×Dic73—21no—
128C2×Dic73—21no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73#12012yes—
132C2×Dic73#12012yes—
133C2×Dic73C2×SD1611yescharacteristic
134C2×Dic73C2×Q1611yescharacteristic
135C2×Dic73#13712yes—
136C4×D73—24no—
137C4×D73—24no—
138C4×D73—44no—
139C4×D73—44no—
140Dic146—48no—
141#10596833—44no—
142#10596833—82no—
143C22×C73#37911yescharacteristic
144C22×C73#40811yescharacteristic
145C22×C73#40811yescharacteristic
146C22×C73C2×#10311yescharacteristic
147C22×C73#41911yescharacteristic
148C22×C73C2×#10411yescharacteristic
149C22×C73#42111yescharacteristic
150C2×D73—22no—
151C2×D73—22no—
152C2×D73—22no—
153C2×D73—41no—
154C2×D73—41no—
155C2×D73—41no—
156C2×D73—41no—
157C4×C73—22no—
158C4×C73—22no—
159C4×C73—44no—
160C4×C73#40814yes—
161Dic73—22no—
162Dic73—41no—
163Dic73—44no—
164Dic73#44812yes—
165Dic73#45112yes—
166#8314—731nomaximal, Sylow
167C2×C73#148111yescharacteristic
168C2×C73#148511yescharacteristic
169C2×C73#149311yescharacteristic
170C2×C73#149711yescharacteristic
171C2×C73#178011yescharacteristic
172C2×C73#178711yescharacteristic
173C2×C73#178911yescharacteristic
174D73—42no—
175C2×#533—731no—
176#1797—731no—
177C4×C22⋊C8—731no—
178C2×Q8⋊C8—734no—
179C73#831411yescharacteristic, complemented, Sylow
180#533—1468no—
181C2×C4⋊Q8—731no—
182C2×C4⋊Q8#6440811yescharacteristic
183C2×#125—1461no—
184C2×C22⋊Q8—732no—
185C2×C22⋊Q8—1461no—
186C2×C4×Q8—732no—
187C2×C4×Q8—1461no—
188C2×#117—1461no—
189C22×C42—731no—
190C2×C4⋊C8—732no—
191C2×C22⋊C8—1461no—
192C2×C22⋊C8—1461no—
193C2×C4×C8—732no—
194#336—1461no—
195#336—1461no—
196Q8⋊C8—1468no—
197C22×Q8—1461no—
198C22×Q8C2×#6440812yes—
199C23×C4—731no—
200C23×C4—731no—
201C23×C4—731no—
202C22×C8—1462no—
203C22×C8—1462no—
204C4⋊Q8—24no—
205C4⋊Q8—1464no—
206#125—1468no—
207C22⋊Q8—1468no—
208C22⋊Q8—1464no—
209C22⋊Q8—1464no—
210C4×Q8—738no—
211C4×Q8—1464no—
212C4×Q8—1468no—
213#117—1468no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—732no—
217C2×C4⋊C4—1462no—
218C2×C4⋊C4—1462no—
219C2×C4⋊C4—1461no—
220C2×C22⋊C4—1462no—
221C2×C42—731no—
222C2×C42—731no—
223C2×C42—731no—
224C2×C42—1464no—
225C2×C42—1462no—
226C2×C42C2×#6440811yescharacteristic
227C4⋊C8—1464no—
228C22⋊C8—2922no—
229C22⋊C8—1464no—
230C4×C8—1464no—
231C24—731no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—2922no—
236C2×Q8—2922no—
237C2×Q8—1464no—
238C2×Q8C22⋊#6440814yes—
239C22×C4—22no—
240C22×C4—2921no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—731no—
246C22×C4—1462no—
247C22×C4—1462no—
248C22×C4—1462no—
249C22×C4—1462no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4—1462no—
253C22×C4—1462no—
254C22×C4C22⋊#6440811yescharacteristic
255C22×C4C22⋊#6440811yescharacteristic
256C22×C4C22×#6440811yescharacteristic
257C2×C8—2922no—
258C2×C8—2922no—
259C2×C8—2922no—
260C2×C8—2922no—
261C2×C8—2922no—
262C2×C8—1464no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—734no—
266C4⋊C4—1468no—
267C4⋊C4—1468no—
268C4⋊C4—1464no—
269C4⋊C4—1462no—
270C22⋊C4—1468no—
271C42—22no—
272C42—734no—
273C42—1462no—
274C42—1462no—
275C42—1468no—
276C23—732no—
277C23—1461no—
278C23—1461no—
279C23—1461no—
280C23—1461no—
281C23—1461no—
282C23—1461no—
283C23C2×C22⋊#6440811yescharacteristic
284Q8—28no—
285Q8—44no—
286Q8—2928no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—2924no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2924no—
298C2×C4—732no—
299C2×C4—731no—
300C2×C4—731no—
301C2×C4—732no—
302C2×C4—732no—
303C2×C4—1464no—
304C2×C4—1464no—
305C2×C4—1461no—
306C2×C4—1464no—
307C2×C4—1461no—
308C2×C4—1461no—
309C2×C4—1461no—
310C2×C4—1461no—
311C2×C4#1455665412yes—
312C2×C4#1455665612yes—
313C2×C4C2×C22⋊#6440812yes—
314C2×C4#1455667812yes—
315C2×C4#1455667812yes—
316C8—2924no—
317C8—5842no—
318C22—2921no—
319C22—2921no—
320C22—2921no—
321C22—2921no—
322C22—1462no—
323C22—1462no—
324C22—1462no—
325C22#1798952811yescharacteristic
326C22#1798956111yescharacteristic
327C22#1798956111yescharacteristic
328C22C2×#1455665411yescharacteristic
329C22#1798958011yescharacteristic
330C22C2×#1455665611yescharacteristic
331C22#1798958611yescharacteristic
332C4—22no—
333C4—22no—
334C4—44no—
335C4—2921no—
336C4—2924no—
337C4—732no—
338C4—732no—
339C4—1462no—
340C4#1798956014yes—
341C2—2922no—
342C2#2262524011yescharacteristic
343C2#2262525411yescharacteristic
344C2#2262528011yescharacteristic
345C2#2262529411yescharacteristic
346C2#2262575511yescharacteristic
347C2#2262577011yescharacteristic
348C2#2262577411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#8314
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#10596833C2semidirect
179. C73#8314semidirect

Maximal subgroups

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

subgroupindex
#226257862
C4×#145566482
C2×Q8⋊#105968332
C2×#179891422
#831473