Finite Group Databasebeta

← #32974098

Subgroups — #32974098

Order 18688.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#22612474C211yesmaximal, characteristic
2#22610046C212yesmaximal, complemented
3#22607757C211yesmaximal, characteristic, complemented
4#22607751C212yesmaximal
5C73×#1272C211yesmaximal, characteristic, complemented
6C2×#14556580C411yescharacteristic
7C2×Q8×Dic73C411yescharacteristic
8C2×#14556553C411yescharacteristic
9C2×#14556520—22no—
10#17988489C2211yescharacteristic
11#17988479—22no—
12#17988452C411yescharacteristic
13#17988434—22no—
14C42×Dic73C2211yescharacteristic
15#17988342C2212yes—
16#17988341—22no—
17#391×C73C2211yescharacteristic
18#336×C73C2212yes—
19C22×D292C2×C411yescharacteristic
20C22×C4×D73—21no—
21C22×#10596831—22no—
22#14556580—22no—
23#14556580C814yescomplemented
24Q8×Dic73—42no—
25#14556553—42no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924005—22no—
29C2×#11924005—21no—
30C2×#11924005—22no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
35C2×C4×Dic73D412yes—
36#14556520—44no—
37#14556498—22no—
38C22×Q8×C73C2×C411yescharacteristic
39C22×C8×C73—22no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C42×C73C2311yescharacteristic
42#95×C73—22no—
43C23×D73—21no—
44C2×D292—21no—
45C2×D292—41no—
46C2×D292—41no—
47C2×D292C2×C812yes—
48C22×Dic73—21no—
49C22×Dic73C22⋊C411yescharacteristic
50C22×Dic73C22⋊C412yes—
51C22×Dic73C22⋊C411yescharacteristic
52C2×C4×D73—44no—
53C2×C4×D73—42no—
54C2×#10596831—42no—
55C2×#10596831—42no—
56C2×#10596831—42no—
57C4⋊Dic73—42no—
58C4⋊Dic73—41no—
59#11924005—28no—
60#11924005—42no—
61#11924005—42no—
62#11924005—42no—
63C4×Dic73—22no—
64C4×Dic73—22no—
65C4×Dic73—22no—
66C4×Dic73—42no—
67C4×Dic73—42no—
68C4×Dic73C2×C812yes—
69C4×Dic73M1612yes—
70C2×Q8×C73—21no—
71C2×Q8×C73—41no—
72C2×Q8×C73—41no—
73C2×Q8×C73C2×C812yes—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C2×D412yes—
78C2×C8×C73—42no—
79C2×C8×C73—42no—
80C2×C8×C73—42no—
81C4⋊C4×C73—41no—
82C42×C73—41no—
83C23×C73C2×C22⋊C411yescharacteristic
84C22×D73—22no—
85C22×D73—41no—
86C22×D73—41no—
87C22×D73—42no—
88C22×D73—42no—
89Q8×C73—42no—
90Q8×C73—42no—
91C2×C4×C73—22no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73—41no—
99C2×C4×C73—42no—
100C2×C4×C73C22×C811yescharacteristic
101C2×C4×C73C2×M1611yescharacteristic
102C2×C4×C73AGL(1,ℤ/8)12yes—
103C2×C4×C73#13712yes—
104C2×Dic73—22no—
105C2×Dic73—22no—
106C2×Dic73—22no—
107C2×Dic73—21no—
108C2×Dic73—22no—
109C2×Dic73—41no—
110C2×Dic73—42no—
111C2×Dic73C22⋊C814yes—
112C2×Dic73#10012yes—
113D292—42no—
114D292—42no—
115C4×D73—84no—
116C8×C73—82no—
117#10596831—82no—
118C22×C73C2×C22⋊C811yescharacteristic
119C22×C73#41011yescharacteristic
120C22×C73C2×#10011yescharacteristic
121C22×C73#41712yes—
122C22×C73#42112yes—
123C2×D73—42no—
124C2×D73—44no—
125C2×D73—81no—
126C2×D73—82no—
127C2×D73—81no—
128C4×C73—21no—
129C4×C73—21no—
130C4×C73—22no—
131C4×C73—22no—
132C4×C73—42no—
133C4×C73—81no—
134Dic73—24no—
135Dic73—22no—
136Dic73—41no—
137Dic73—81no—
138#7601—731nomaximal, Sylow
139C2×C73#146011yescharacteristic
140C2×C73#147712yes—
141C2×C73#151212yes—
142C2×C73#153311yescharacteristic
143C2×C73#156011yescharacteristic
144D73—82no—
145C4×#124—731no—
146#1831—732no—
147#1272—731no—
148#1272D7311yescharacteristic, complemented
149#1270—732no—
150C73#760111yescharacteristic, complemented, Sylow
151C2×#124—731no—
152C2×C4×Q8—731no—
153C2×C4×D4—731no—
154C2×C4⋊C8—1462no—
155#391C2×D7311yescharacteristic
156#390—731no—
157#388—1462no—
158#382—731no—
159C4×C22⋊C4—1462no—
160C43—731no—
161#336—732no—
162#336—1462no—
163#336C2×D7312yes—
164C22×Q8C4×D7311yescharacteristic
165C22×D4—731no—
166C23×C4—1461no—
167C22×C8—22no—
168C22×C8—1462no—
169#124—734no—
170#124—1462no—
171C4×Q8—2922no—
172C4×D4—2922no—
173C2×C4⋊C4—731no—
174C2×C4⋊C4—1461no—
175C2×C4⋊C4C4×D7311yescharacteristic
176C2×C22⋊C4—1461no—
177C2×C22⋊C4—1462no—
178C2×C22⋊C4—1462no—
179C2×C42—732no—
180C2×C42—731no—
181C2×C42—731no—
182C2×C42—732no—
183C2×C42—1461no—
184C2×C42C22×D7311yescharacteristic
185C4⋊C8—2924no—
186#95—22no—
187#95—1462no—
188C24—1461no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D7312yes—
193C2×D4—2921no—
194C2×D4—2921no—
195C2×D4—732no—
196C2×D4—1461no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—2922no—
200C22×C4—2924no—
201C22×C4—732no—
202C22×C4—731no—
203C22×C4—731no—
204C22×C4—1461no—
205C22×C4C2×C4×D7311yescharacteristic
206C22×C4D4×D7312yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—2922no—
211C2×C8—2922no—
212C2×C8—2922no—
213C4⋊C4—41no—
214C4⋊C4—2921no—
215C4⋊C4—2922no—
216C22⋊C4—2922no—
217C22⋊C4—2922no—
218C22⋊C4—2922no—
219C22⋊C4—1468no—
220C42—41no—
221C42—2922no—
222C42—2922no—
223C42—732no—
224C42—732no—
225C42—1462no—
226C42—1462no—
227C42—1462no—
228C23—2921no—
229C23—2921no—
230C23—2922no—
231C23—2922no—
232C23—1462no—
233C23C22⋊C4×D7311yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—2922no—
237D4—2922no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—2921no—
248C2×C4—2922no—
249C2×C4—5844no—
250C2×C4—734no—
251C2×C4—732no—
252C2×C4—1461no—
253C2×C4—1462no—
254C2×C4—1462no—
255C2×C4—1462no—
256C2×C4—1462no—
257C2×C4C2×C8×D7311yescharacteristic
258C2×C4M16×D7311yescharacteristic
259C2×C4#1455661812yes—
260C2×C4#1455662312yes—
261C8—82no—
262C8—5842no—
263C22—2922no—
264C22—2924no—
265C22—5841no—
266C22—5842no—
267C22—5841no—
268C22C22⋊C8×D7311yescharacteristic
269C22#1798855911yescharacteristic
270C22#100×D7311yescharacteristic
271C22#1798858712yes—
272C22#1798860212yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—2921no—
280C4—5841no—
281C4—1462no—
282C4—1464no—
283C2—5842no—
284C2#2260825011yescharacteristic
285C2#2260831712yes—
286C2#2260846812yes—
287C2#2260855311yescharacteristic
288C2#2260866511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×C42C22×D73
FittingC73×#1272C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1272D73
hypocenterC73#7601
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #22610046C2semidirect
3. #22607757C2semidirect
5. C73×#1272C2semidirect
23. #14556580C8semidirect
148. #1272D73semidirect
150. C73#7601semidirect

Maximal subgroups

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

subgroupindex
#226124742
#226100462
#226077572
#226077512
C73×#12722
#760173