Finite Group Databasebeta

← #34010207

Subgroups — #34010207

Order 18688.

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

normal subgroups only

This group has 380 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#22624876C212yesmaximal
4#22624857C212yesmaximal
5#22614218C211yesmaximal, characteristic
6#17989607—21no—
7#17989603—21no—
8#17989432—21no—
9#17989432—21no—
10#17989430C2212yes—
11#17989430C2212yes—
12C2×#14556550C2211yescharacteristic
13#17988526—21no—
14#17988517—21no—
15#17988468—21no—
16#17988468—21no—
17#17988460C411yescharacteristic
18#17988460C2211yescharacteristic
19#17988450C411yescharacteristic
20#17988450C411yescharacteristic
21#17988450C2211yescharacteristic
22C22×C4⋊C4×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⋊Dic73D411yescharacteristic
32C2×C4⋊Dic73D411yescharacteristic
33C2×#11924007—21no—
34C2×#11924007—21no—
35C2×C22⋊Dic73C2×C411yescharacteristic
36C2×C22⋊Dic73C2×C411yescharacteristic
37C2×C22⋊Dic73C2×C411yescharacteristic
38C2×C22⋊Dic73C2311yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73C2×C411yescharacteristic
50C23×C4×C73C2×C411yescharacteristic
51C23×C4×C73C2×C411yescharacteristic
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—41no—
55C2×C4⋊C4×C73—41no—
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—22no—
69C22⋊Dic73—41no—
70C22⋊Dic73—41no—
71C22⋊Dic73—41no—
72C4×Dic73—21no—
73C4×Dic73—21no—
74C4×Dic73—21no—
75C4×Dic73—21no—
76C24×C73C22×C411yescharacteristic
77C22×C4×C73—21no—
78C22×C4×C73—22no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—22no—
83C22×C4×C73—41no—
84C22×C4×C73—42no—
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73—41no—
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C4⋊C4×C73—42no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73C2×C22⋊C411yescharacteristic
98C23×C73#13112yes—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—22no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—82no—
115C2×C4×C73#9911yescharacteristic
116C2×C4×C73#9911yescharacteristic
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—41no—
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—
131#10596833—82no—
132#10596833—82no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—22no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73#40811yescharacteristic
141C22×C73C2×#9911yescharacteristic
142C22×C73#41311yescharacteristic
143C4×C73—21no—
144C4×C73—21no—
145C4×C73—42no—
146C4×C73—41no—
147C4×C73—81no—
148Dic73—41no—
149Dic73—41no—
150Dic73—41no—
151Dic73—41no—
152Dic73—81no—
153Dic73—81no—
154#11544—731nomaximal, Sylow
155C2×C73—21no—
156C2×C73—21no—
157C2×C73—42no—
158C2×C73#146811yescharacteristic
159C2×C73#212011yescharacteristic
160C2×C73#212311yescharacteristic
161#2484—731no—
162#1464—731no—
163#1458—731no—
164#1326—732no—
165#1317—732no—
166C73#1154411yescharacteristic, complemented, Sylow
167C2×#117—731no—
168C22×C4⋊C4#6440811yescharacteristic
169#432—1461no—
170#431—1461no—
171#400—1461no—
172#398—1461no—
173#387—1461no—
174#387—1461no—
175#385—731no—
176#385—731no—
177#381—731no—
178#381—731no—
179#381—731no—
180#324—1461no—
181#324—1461no—
182#323—732no—
183#323—732no—
184C23×C4C2×#6440811yescharacteristic
185C23×C4C2×#6440811yescharacteristic
186C23×C4C2×#6440811yescharacteristic
187#117—2921no—
188#117—2921no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—731no—
194C2×C4⋊C4—731no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4—1461no—
197C2×C22⋊C4—731no—
198C2×C22⋊C4—731no—
199C2×C22⋊C4—731no—
200C2×C22⋊C4—731no—
201C2×C42—731no—
202C2×C42—731no—
203C4⋊C8—2921no—
204C4⋊C8—2921no—
205C22⋊C8—1462no—
206C22⋊C8—1462no—
207#97—2921no—
208C4×C8—2921no—
209#95—1461no—
210#95—1461no—
211#95—1461no—
212#95—1461no—
213#95—1461no—
214#95—1461no—
215#95—1461no—
216#95—1461no—
217C24C22×#6440811yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—731no—
230C22×C4—731no—
231C22×C4—731no—
232C22×C4—731no—
233C22×C4—1461no—
234C22×C4—1461no—
235C22×C4C22⋊#6440811yescharacteristic
236C22×C4C22⋊#6440811yescharacteristic
237C2×C8—2922no—
238C2×C8—2922no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2922no—
243C4⋊C4—2922no—
244C4⋊C4—2921no—
245C4⋊C4—2921no—
246C22⋊C4—2921no—
247C22⋊C4—2921no—
248C22⋊C4—2921no—
249C22⋊C4—1462no—
250C42—1461no—
251C42—1461no—
252C42—1461no—
253C42—1461no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊#6440811yescharacteristic
259C23#1455667812yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—2921no—
281C2×C4—2921no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—1461no—
287C2×C4—1461no—
288C2×C4—1461no—
289C2×C4—1461no—
290C2×C4#1455664911yescharacteristic
291C2×C4#1455664911yescharacteristic
292C8—5842no—
293C8—5842no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1798956011yescharacteristic
302C22C2×#1455664911yescharacteristic
303C22#1798957411yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—41no—
308C4—81no—
309C4—2921no—
310C4—2921no—
311C4—2921no—
312C4—2921no—
313C4—5841no—
314C4—5841no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2262519611yescharacteristic
319C2#2262618211yescharacteristic
320C2#2262619311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
cosocleC2×C22⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11544
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
166. C73#11544semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226251622
#226248762
#226248572
#226142182
#1154473