Finite Group Databasebeta

← #34010239

Subgroups — #34010239

Order 18688.

366 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 68 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#22625182C211yesmaximal, characteristic
2#22625162C211yesmaximal, characteristic
3#22624874C212yesmaximal, complemented
4#22624861C212yesmaximal, complemented
5#22614385C211yesmaximal, characteristic
6#17989607—21no—
7#17989603—21no—
8#17989432C2212yes—
9#17989432C2212yes—
10#17989430—21no—
11#17989430—21no—
12C2×#14556550C2211yescharacteristic
13#17988515—21no—
14#17988515—21no—
15#17988494C411yescharacteristic
16#17988494C2211yescharacteristic
17#17988468—21no—
18#17988468—21no—
19#17988456C2211yescharacteristic
20#17988452C411yescharacteristic
21C4×C22⋊Dic73C411yescharacteristic
22C2×C22⋊Q8×C73C411yescharacteristic
23#14556658—41no—
24C4⋊#10596833—41no—
25C22⋊#10596833—22no—
26C22⋊#10596833—22no—
27#14556646—41no—
28C4×#10596833—41no—
29#14556550—41no—
30#14556550—41no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×C4⋊Dic73C2311yescharacteristic
33C2×#11924007—21no—
34C2×#11924007—21no—
35C2×C22⋊Dic73—21no—
36C2×C22⋊Dic73D411yescharacteristic
37C2×C22⋊Dic73D411yescharacteristic
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73C2×C411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
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—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C22⋊C411yescharacteristic
62C22×Dic73C2×D411yescharacteristic
63C22×Dic73C2×D411yescharacteristic
64C4⋊Dic73—41no—
65C4⋊Dic73—41no—
66#11924007—42no—
67#11924007—42no—
68C22⋊Dic73—41no—
69C22⋊Dic73—41no—
70C22⋊Dic73—42no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C4×Dic73—21no—
74C4×Dic73—21no—
75C4×Dic73—22no—
76C4×Dic73—41no—
77C24×C73C22⋊C411yescharacteristic
78C2×Q8×C73—22no—
79C2×Q8×C73—41no—
80C2×Q8×C73—41no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—22no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—41no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C4⋊C4×C73—41no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73C2×C22⋊C411yescharacteristic
99Q8×C73—82no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73#9911yescharacteristic
113C2×C4×C73#9911yescharacteristic
114C2×C4×C73#13112yes—
115C2×Dic73—22no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129#10596833—82no—
130#10596833—82no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—41no—
134C22×C73—41no—
135C22×C73—41no—
136C22×C73—81no—
137C22×C73—81no—
138C22×C73#40811yescharacteristic
139C22×C73C2×#9911yescharacteristic
140C22×C73#41311yescharacteristic
141C4×C73—21no—
142C4×C73—21no—
143C4×C73—22no—
144C4×C73—81no—
145C4×C73—81no—
146Dic73—42no—
147Dic73—41no—
148Dic73—41no—
149Dic73—41no—
150Dic73—41no—
151Dic73—81no—
152#11552—731nomaximal, Sylow
153C2×C73—21no—
154C2×C73—21no—
155C2×C73—81no—
156C2×C73#146811yescharacteristic
157C2×C73#212611yescharacteristic
158C2×C73#213511yescharacteristic
159#2499—731no—
160#1464—731no—
161#1458—731no—
162#1325—732no—
163#1319—732no—
164C73#1155211yescharacteristic, complemented, Sylow
165C2×C22⋊Q8#6440811yescharacteristic
166C2×#117—731no—
167#432—1461no—
168#431—1461no—
169#397—1461no—
170#397—1461no—
171#391—731no—
172#391—731no—
173#387—1461no—
174#387—1461no—
175#384—731no—
176#382—731no—
177C4×C22⋊C4—731no—
178#324—732no—
179#324—732no—
180#323—1461no—
181#323—1461no—
182C22×Q8C2×#6440811yescharacteristic
183C23×C4C2×#6440811yescharacteristic
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#117—2921no—
187#117—2921no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—731no—
190C2×C4⋊C4—731no—
191C2×C4⋊C4—1461no—
192C2×C4⋊C4—1461no—
193C2×C4⋊C4C2×#6440811yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—731no—
196C2×C22⋊C4—731no—
197C2×C22⋊C4—1461no—
198C2×C42—731no—
199C2×C42—731no—
200C2×C42—731no—
201C2×C42—731no—
202C4⋊C8—2921no—
203C4⋊C8—2921no—
204C22⋊C8—1462no—
205C22⋊C8—1462no—
206#97—2921no—
207C4×C8—2921no—
208#95—1461no—
209#95—1461no—
210#95—1461no—
211#95—1461no—
212#95—1461no—
213#95—1461no—
214#95—1461no—
215C24C22⋊#6440811yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—731no—
226C22×C4—731no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—1461no—
230C22×C4—1461no—
231C22×C4C22⋊#6440811yescharacteristic
232C22×C4C22×#6440811yescharacteristic
233C2×C8—2922no—
234C2×C8—2922no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—2921no—
239C4⋊C4—2921no—
240C4⋊C4—2922no—
241C4⋊C4—2922no—
242C22⋊C4—42no—
243C22⋊C4—2922no—
244C22⋊C4—2921no—
245C22⋊C4—2921no—
246C42—2921no—
247C42—1462no—
248C42—1461no—
249C42—1461no—
250C42—1461no—
251C42—1461no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊#6440811yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—2921no—
272C2×C4—2921no—
273C2×C4—2921no—
274C2×C4—2921no—
275C2×C4—2921no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—1462no—
281C2×C4—1461no—
282C2×C4—1461no—
283C2×C4—1461no—
284C2×C4—1461no—
285C2×C4#1455664911yescharacteristic
286C2×C4#1455664911yescharacteristic
287C2×C4#1455667812yes—
288C8—5842no—
289C8—5842no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1798956011yescharacteristic
298C22C2×#1455664911yescharacteristic
299C22#1798957411yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—81no—
304C4—81no—
305C4—2922no—
306C4—2921no—
307C4—2921no—
308C4—2921no—
309C4—2921no—
310C4—5841no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2262519611yescharacteristic
315C2#2262620211yescharacteristic
316C2#2262623311yescharacteristic
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#11552
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22624874C2semidirect
4. #22624861C2semidirect
164. C73#11552semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226251622
#226248742
#226248612
#226143852
#1155273