Finite Group Databasebeta

← #32995166

Subgroups — #32995166

Order 18688.

434 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 115 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#22612551C211yesmaximal, characteristic
2#22608245C211yesmaximal, characteristic
3#22607880C214yesmaximal
4C73×#1458C211yesmaximal, characteristic
5C2×#14556550C411yescharacteristic
6C22×C4⋊Dic73C411yescharacteristic
7#17988640—21no—
8#17988485—21no—
9#17988485—21no—
10#17988459C412yes—
11#17988459C2212yes—
12#17988456—21no—
13#17988452—21no—
14#17988313—21no—
15#17988311C2212yes—
16C2×#117×C73C2211yescharacteristic
17#431×C73—21no—
18#324×C73—21no—
19#323×C73C2212yes—
20C23×Dic73C2×C412yes—
21#14556550—22no—
22#14556550C814yescomplemented
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—41no—
27C2×C4⋊Dic73—41no—
28C2×#11924007—22no—
29C2×#11924007—22no—
30C2×C22⋊Dic73C2×C412yes—
31C2×C4×Dic73—21no—
32C4⋊#10596831—41no—
33C22⋊#10596831—22no—
34C4×#10596831—41no—
35#14556498—22no—
36#14556498—22no—
37C23×C4×C73C2×C411yescharacteristic
38#117×C73—41no—
39#117×C73—41no—
40C2×C4⋊C4×C73D411yescharacteristic
41C2×C4⋊C4×C73D411yescharacteristic
42C2×C22⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73C2311yescharacteristic
44C2×C42×C73D411yescharacteristic
45C2×C42×C73D411yescharacteristic
46C4⋊C8×C73—41no—
47C22⋊C8×C73—22no—
48C4×C8×C73—41no—
49C22×Dic73—22no—
50C22×Dic73—22no—
51C22×Dic73—42no—
52C22×Dic73—82no—
53C2×#10596831—42no—
54C4⋊Dic73—24no—
55C4⋊Dic73—42no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924007—82no—
59#11924007—82no—
60C22⋊Dic73—22no—
61C22⋊Dic73C2×C814yes—
62C4×Dic73—24no—
63C24×C73C22×C411yescharacteristic
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—22no—
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C2×C812yes—
72C22×C4×C73M1612yes—
73C22×C4×C73C2×D411yescharacteristic
74C22×C4×C73C2×D411yescharacteristic
75C2×C8×C73—42no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C22⋊C4×C73—41no—
79C22⋊C4×C73—41no—
80C42×C73—21no—
81C42×C73—21no—
82C42×C73—21no—
83C42×C73—21no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73C2×C22⋊C411yescharacteristic
89C23×C73C22×C811yescharacteristic
90C23×C73C2×M1611yescharacteristic
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73C22⋊C812yes—
106C2×C4×C73C22⋊C812yes—
107C2×C4×C73#9911yescharacteristic
108C2×C4×C73#9911yescharacteristic
109C2×C4×C73#10112yes—
110C2×Dic73—24no—
111C2×Dic73—42no—
112C2×Dic73—42no—
113C2×Dic73—42no—
114C2×Dic73—42no—
115C2×Dic73—42no—
116C2×Dic73—84no—
117C8×C73—82no—
118#10596831—82no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—41no—
124C22×C73—41no—
125C22×C73—42no—
126C22×C73—41no—
127C22×C73—41no—
128C22×C73C2×C22⋊C811yescharacteristic
129C22×C73C2×#9911yescharacteristic
130C22×C73C2×#10111yescharacteristic
131C4×C73—22no—
132C4×C73—41no—
133C4×C73—41no—
134C4×C73—41no—
135C4×C73—41no—
136C4×C73#32412yes—
137C4×C73#32412yes—
138Dic73—44no—
139Dic73—82no—
140#11454—731nomaximal, Sylow
141C2×C73—21no—
142C2×C73—21no—
143C2×C73—41no—
144C2×C73—41no—
145C2×C73C2×#32411yescharacteristic
146C2×C73#212211yescharacteristic
147C2×C73#212311yescharacteristic
148#2317—731no—
149#1458—731no—
150#1458D7311yescharacteristic
151#1326—734no—
152C73#1145411yescharacteristic, complemented, Sylow
153C2×#117—731no—
154C2×#117C2×D7311yescharacteristic
155C22×C4⋊C4—731no—
156#431—21no—
157#431—1461no—
158#389—1461no—
159#389—1461no—
160#385—732no—
161#385—732no—
162#384—1461no—
163#382—1461no—
164#324—21no—
165#324—1461no—
166#323—732no—
167#323C2×D7312yes—
168C23×C4—732no—
169C23×C4C4×D7311yescharacteristic
170#117—41no—
171#117—41no—
172#117—734no—
173#117—1462no—
174C2×C4⋊C4—2921no—
175C2×C4⋊C4—2921no—
176C2×C4⋊C4—1461no—
177C2×C4⋊C4—1461no—
178C2×C4⋊C4—1462no—
179C2×C4⋊C4—1462no—
180C2×C4⋊C4—1461no—
181C2×C4⋊C4D29211yescharacteristic
182C2×C4⋊C4C22⋊D7311yescharacteristic
183C2×C22⋊C4—732no—
184C2×C22⋊C4C4×D7311yescharacteristic
185C2×C22⋊C4C22×D7311yescharacteristic
186C2×C42—1461no—
187C2×C42D29211yescharacteristic
188C2×C42C22⋊D7311yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—2921no—
191C22⋊C8—22no—
192C22⋊C8—1462no—
193C4×C8—41no—
194C4×C8—2921no—
195#95—1462no—
196#95—1462no—
197C24C2×C4×D7311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—2922no—
202C22×C4—5842no—
203C22×C4—1462no—
204C22×C4—1462no—
205C22×C4#1192400511yescharacteristic
206C22×C4#1192400511yescharacteristic
207C22×C4#1192400511yescharacteristic
208C22×C4#1192400511yescharacteristic
209C22×C4C8×D7312yes—
210C22×C4#1192401112yes—
211C22×C4C2×C22⋊D7311yescharacteristic
212C22×C4C2×D29211yescharacteristic
213C2×C8—42no—
214C2×C8—2922no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—2922no—
218C4⋊C4—2921no—
219C4⋊C4—2921no—
220C4⋊C4—5842no—
221C4⋊C4—5842no—
222C4⋊C4—1464no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—734no—
226C22⋊C4—1462no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—1464no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×#1192400511yescharacteristic
237C23C2×C8×D7311yescharacteristic
238C23C2×#1192401111yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—2922no—
254C2×C4—2922no—
255C2×C4—2922no—
256C2×C4—2922no—
257C2×C4—2922no—
258C2×C4—5844no—
259C2×C4—1464no—
260C2×C4#1455650312yes—
261C2×C4#1455650312yes—
262C2×C4#1455650511yescharacteristic
263C2×C4#1455650511yescharacteristic
264C2×C4#1455650912yes—
265C8—82no—
266C8—5842no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×#1455650311yescharacteristic
277C22C2×#1455650511yescharacteristic
278C22C2×#1455650911yescharacteristic
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—2924no—
285C4—5842no—
286C4#1798831212yes—
287C4#1798831212yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1798831211yescharacteristic
293C2#2261158211yescharacteristic
294C2#2261158411yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC23×C73C22×C8
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1458C2
socleC22×C73C2×#99
prime radicalC24×C73C22×C4
cosocleC2×C22⋊C4×C73C23
hypercenter#1458D73
hypocenterC73#11454
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
22. #14556550C8semidirect
152. C73#11454semidirect

Maximal subgroups

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

subgroupindex
#226125512
#226082452
#226078802
C73×#14582
#1145473