Finite Group Databasebeta

← #34001710

Subgroups — #34001710

Order 18688.

488 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 94 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#22625657C211yesmaximal, characteristic
2C4×C22⋊#10596833C211yesmaximal, characteristic
3#22624751C212yesmaximal, complemented
4#22624750C212yesmaximal, complemented
5#22618265C211yesmaximal, characteristic
6C2×C22⋊#10596833—21no—
7C2×C22⋊#10596833—21no—
8C2×#14556646—21no—
9C2×C4×#10596833—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C22×C4×Dic73C2211yescharacteristic
13#17988494C411yescharacteristic
14#17988494C2211yescharacteristic
15#17988483C411yescharacteristic
16#17988483C2211yescharacteristic
17#17988468—21no—
18#17988468—21no—
19#17988459—21no—
20#17988448—21no—
21#17988447C411yescharacteristic
22C2×C22⋊Q8×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25C22⋊#10596833—24no—
26C22⋊#10596833—24no—
27#14556646—42no—
28C4×#10596833—42no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×C4⋊Dic73C2×C411yescharacteristic
33C2×#11924007—21no—
34C2×#11924007—21no—
35C2×C22⋊Dic73—21no—
36C2×C4×Dic73—42no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73C2311yescharacteristic
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48C22×Q8×C73C2×C411yescharacteristic
49C23×C4×C73C2×C411yescharacteristic
50C22⋊Q8×C73—42no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73—21no—
54C2×#10596833—24no—
55C2×#10596833—24no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C22×Dic73—22no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C22×Dic73C2×D411yescharacteristic
72C4⋊Dic73—22no—
73C4⋊Dic73—41no—
74#11924007—42no—
75#11924007—42no—
76C22⋊Dic73—42no—
77C4×Dic73—22no—
78C4×Dic73—22no—
79C4×Dic73—41no—
80C4×Dic73—41no—
81C24×C73C22⋊C411yescharacteristic
82C2×Q8×C73—22no—
83C2×Q8×C73—41no—
84C2×Q8×C73—41no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—41no—
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22×C411yescharacteristic
92C4⋊C4×C73—41no—
93C4⋊C4×C73—42no—
94C22⋊C4×C73—42no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73#10412yes—
102C23×C73C2×C22⋊C411yescharacteristic
103Q8×C73—82no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—44no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#9912yes—
116C2×C4×C73#13112yes—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—44no—
127C2×Dic73—44no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73#10412yes—
135C2×Dic73#13512yes—
136C2×Dic73#13712yes—
137#10596833—44no—
138#10596833—44no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—22no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73#40811yescharacteristic
147C22×C73C2×#9911yescharacteristic
148C22×C73#41311yescharacteristic
149C22×C73#41611yescharacteristic
150C22×C73#41911yescharacteristic
151C22×C73C2×#10411yescharacteristic
152C22×C73C2×#10411yescharacteristic
153C4×C73—22no—
154C4×C73—22no—
155C4×C73—81no—
156C4×C73—81no—
157Dic73—22no—
158Dic73—22no—
159Dic73—22no—
160Dic73—41no—
161Dic73—81no—
162Dic73—81no—
163#7675—731nomaximal, Sylow
164C2×C73—42no—
165C2×C73#146811yescharacteristic
166C2×C73#148511yescharacteristic
167C2×C73#149411yescharacteristic
168C2×C73#150111yescharacteristic
169C2×C73#150811yescharacteristic
170C2×C73#152411yescharacteristic
171C2×C73#155411yescharacteristic
172#2855—731no—
173#1749—731no—
174C4×C22⋊C8—731no—
175#1272—732no—
176#1271—732no—
177C73#767511yescharacteristic, complemented, Sylow
178C2×C22⋊Q8#6440811yescharacteristic
179C22×C42—731no—
180C2×C22⋊C8—1461no—
181C2×C22⋊C8—1461no—
182C2×#97—1461no—
183C2×C4×C8—1461no—
184#391—731no—
185#391—731no—
186#389—731no—
187#389—731no—
188#387—1461no—
189#387—1461no—
190#385—1461no—
191#381—1461no—
192#380—731no—
193#336—732no—
194#336—732no—
195C22×Q8C2×#6440811yescharacteristic
196C23×C4—731no—
197C23×C4—731no—
198C23×C4C2×#6440811yescharacteristic
199C22×C8—1462no—
200C22×C8—1462no—
201C22⋊Q8—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—731no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4C2×#6440811yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—1461no—
210C2×C42—2922no—
211C2×C42—731no—
212C2×C42—731no—
213C2×C42—731no—
214C2×C42—731no—
215C22⋊C8—1464no—
216C22⋊C8—1464no—
217#97—2922no—
218C4×C8—2922no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222#95—1461no—
223#95—1461no—
224#95—1461no—
225#95—1461no—
226C24C22⋊#6440811yescharacteristic
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—41no—
235C22×C4—2921no—
236C22×C4—2921no—
237C22×C4—731no—
238C22×C4—731no—
239C22×C4—731no—
240C22×C4—731no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4—1462no—
244C22×C4—1462no—
245C22×C4—1462no—
246C22×C4—1462no—
247C22×C4C22⋊#6440811yescharacteristic
248C22×C4C22×#6440811yescharacteristic
249C2×C8—2922no—
250C2×C8—2922no—
251C2×C8—2922no—
252C2×C8—2922no—
253C2×C8—1464no—
254C2×C8—1464no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—2921no—
258C4⋊C4—2922no—
259C4⋊C4—2922no—
260C4⋊C4—1462no—
261C22⋊C4—42no—
262C22⋊C4—2922no—
263C42—2921no—
264C42—2921no—
265C42—1462no—
266C42—1462no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#1455665612yes—
274C23C2×C22⋊#6440811yescharacteristic
275Q8—82no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—2924no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2924no—
295C2×C4—732no—
296C2×C4—732no—
297C2×C4—732no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1461no—
303C2×C4—1461no—
304C2×C4—1461no—
305C2×C4—1461no—
306C2×C4—1461no—
307C2×C4#1455664912yes—
308C2×C4#1455667812yes—
309C8—2924no—
310C8—2924no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#1798956011yescharacteristic
319C22C2×#1455664911yescharacteristic
320C22#1798957411yescharacteristic
321C22#1798957711yescharacteristic
322C22#1798958011yescharacteristic
323C22D73⋊#10411yescharacteristic
324C22C2×#1455665611yescharacteristic
325C4—22no—
326C4—22no—
327C4—81no—
328C4—81no—
329C4—2921no—
330C4—5841no—
331C4—5841no—
332C4—1462no—
333C4—1462no—
334C4—1462no—
335C2—42no—
336C2#2262519611yescharacteristic
337C2#2262525311yescharacteristic
338C2#2262528411yescharacteristic
339C2#2262530611yescharacteristic
340C2#2262533111yescharacteristic
341C2#2262538411yescharacteristic
342C2#2262545711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C22⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C22⋊Q8#64408
hypocenterC73#7675
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22624751C2semidirect
4. #22624750C2semidirect
177. C73#7675semidirect

Maximal subgroups

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

subgroupindex
#226256572
C4×C22⋊#105968332
#226247512
#226247502
#226182652
#767573