Finite Group Databasebeta

← #34010235

Subgroups — #34010235

Order 18688.

398 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 72 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#22625179C211yesmaximal, characteristic
2C2×#17989432C211yesmaximal, characteristic
3#22624872C212yesmaximal, complemented
4#22624861C212yesmaximal, complemented
5#22614417C211yesmaximal, characteristic
6#17989557—21no—
7C2×C22⋊#10596833—21no—
8#17989432—21no—
9#17989432—21no—
10#17989432C2212yes—
11#17989432C2212yes—
12C22×C4⋊Dic73C2211yescharacteristic
13#17988526—21no—
14#17988526—21no—
15#17988494C411yescharacteristic
16#17988494C2211yescharacteristic
17#17988452C411yescharacteristic
18#17988452C2211yescharacteristic
19#17988448—21no—
20#17988448—21no—
21C4×C22⋊Dic73C411yescharacteristic
22C2×C22⋊Q8×C73C411yescharacteristic
23C2×#11924037—41no—
24C22×#10596833—41no—
25C22⋊#10596833—22no—
26C22⋊#10596833—22no—
27C22⋊#10596833—41no—
28C22⋊#10596833—41no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31C2×C4⋊Dic73—41no—
32C2×C4⋊Dic73—41no—
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×C4⋊Dic73D411yescharacteristic
35C2×C4⋊Dic73D411yescharacteristic
36C2×C4⋊Dic73C2311yescharacteristic
37C2×C22⋊Dic73—21no—
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73C2×C411yescharacteristic
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C22×Q8×C73C2×C411yescharacteristic
50C23×C4×C73C2×C411yescharacteristic
51C22⋊Q8×C73—41no—
52C22⋊Q8×C73—41no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73C2×C411yescharacteristic
55C2×C22⋊C4×C73—21no—
56#11924037—82no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—82no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—42no—
67C22×Dic73—42no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C22×Dic73C2×D411yescharacteristic
72C4⋊Dic73—41no—
73C4⋊Dic73—41no—
74C4⋊Dic73—41no—
75C4⋊Dic73—41no—
76C22⋊Dic73—42no—
77C4×Dic73—22no—
78C4×Dic73—41no—
79C24×C73C22⋊C411yescharacteristic
80C2×Q8×C73—22no—
81C2×Q8×C73—41no—
82C2×Q8×C73—41no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—22no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—41no—
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C4⋊C4×C73—41no—
92C4⋊C4×C73—42no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C23×C73—22no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—41no—
99C23×C73#9911yescharacteristic
100C23×C73#9911yescharacteristic
101C23×C73C2×C22⋊C411yescharacteristic
102Q8×C73—82no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#13112yes—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—81no—
133C2×Dic73—81no—
134#10596833—82no—
135#10596833—82no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—42no—
144C22×C73—42no—
145C22×C73#35412yes—
146C22×C73#40811yescharacteristic
147C22×C73C2×#9911yescharacteristic
148C22×C73#41311yescharacteristic
149C4×C73—22no—
150C4×C73—41no—
151C4×C73—81no—
152C4×C73—81no—
153Dic73—41no—
154Dic73—41no—
155Dic73—41no—
156Dic73—41no—
157Dic73—81no—
158Dic73—81no—
159#11551—731nomaximal, Sylow
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—22no—
163C2×C73—41no—
164C2×C73#146811yescharacteristic
165C2×C73C2×#35411yescharacteristic
166C2×C73#213311yescharacteristic
167#2503—731no—
168#1463—731no—
169C2×#324—731no—
170#1324—732no—
171#1319—732no—
172C73#1155111yescharacteristic, complemented, Sylow
173C2×C22⋊Q8#6440811yescharacteristic
174C22×C4⋊C4—731no—
175#407—1461no—
176C2×C22⋊C8—1461no—
177#400—1461no—
178#400—1461no—
179#391—731no—
180#391—731no—
181#382—731no—
182#382—731no—
183#381—1461no—
184#381—1461no—
185C4×C22⋊C4—731no—
186#324—732no—
187#324—732no—
188#324—1461no—
189#324—1461no—
190C22×Q8C2×#6440811yescharacteristic
191C23×C4—731no—
192C23×C4—731no—
193C23×C4C2×#6440811yescharacteristic
194C2×M16—2921no—
195C22×C8—2921no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—2921no—
200C2×C4⋊C4—2921no—
201C2×C4⋊C4—731no—
202C2×C4⋊C4—731no—
203C2×C4⋊C4—731no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4C2×#6440811yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1461no—
208C2×C42—731no—
209C2×C42—731no—
210C22⋊C8—2921no—
211C22⋊C8—2921no—
212C22⋊C8—1462no—
213C22⋊C8—1462no—
214#95—1461no—
215#95—1461no—
216#95—1461no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222#95—1461no—
223C24C22⋊#6440811yescharacteristic
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—2922no—
234C22×C4—2922no—
235C22×C4—731no—
236C22×C4—731no—
237C22×C4—731no—
238C22×C4—731no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4—1461no—
244C22×C4—1461no—
245C22×C4C22⋊#6440811yescharacteristic
246C22×C4C22×#6440811yescharacteristic
247M16—5842no—
248C2×C8—2922no—
249C2×C8—2922no—
250C2×C8—5842no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—2921no—
254C4⋊C4—2921no—
255C4⋊C4—2921no—
256C4⋊C4—2921no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—2922no—
260C42—2921no—
261C42—1462no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23#1455664911yescharacteristic
267C23#1455664911yescharacteristic
268C23C2×C22⋊#6440811yescharacteristic
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—2922no—
283C2×C4—2922no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—5841no—
295C2×C4—5841no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4#1455667812yes—
301C8—5842no—
302C8—5842no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1798949312yes—
313C22#1798956011yescharacteristic
314C22C2×#1455664911yescharacteristic
315C22#1798957411yescharacteristic
316C4—22no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—2921no—
321C4—2921no—
322C4—2921no—
323C4—2921no—
324C4—5841no—
325C4—5841no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#2262519611yescharacteristic
331C2C2×#1798949311yescharacteristic
332C2#2262622611yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×C22⋊Q8#64408
hypocenterC73#11551
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22624872C2semidirect
4. #22624861C2semidirect
172. C73#11551semidirect

Maximal subgroups

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

subgroupindex
#226251792
C2×#179894322
#226248722
#226248612
#226144172
#1155173