Finite Group Databasebeta

← #32974244

Subgroups — #32974244

Order 18688.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#22609783C212yesmaximal, complemented
3#22607757C211yesmaximal, characteristic, complemented
4#22607751C212yesmaximal
5C73×#1272C211yesmaximal, characteristic, complemented
6C2×#14556580C411yescharacteristic
7C2×Q8×Dic73C411yescharacteristic
8C2×#14556553C411yescharacteristic
9C2×#14556520—22no—
10C2×#14556501—22no—
11#17988489C2211yescharacteristic
12#17988479—22no—
13#17988452C411yescharacteristic
14#17988434—22no—
15C42×Dic73C2211yescharacteristic
16#17988342C2212yes—
17#391×C73C2211yescharacteristic
18#336×C73C2212yes—
19C22×D292C2×C411yescharacteristic
20C22×C4×D73—21no—
21C22×#10596831—22no—
22#14556580—22no—
23#14556580—22no—
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—28no—
37#14556501—44no—
38#14556498—22no—
39C22×Q8×C73C2×C411yescharacteristic
40C22×C8×C73—22no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C42×C73C2311yescharacteristic
43#95×C73—22no—
44C23×D73—21no—
45C2×D292—21no—
46C2×D292—41no—
47C2×D292—41no—
48C2×D292M1612yes—
49C22×Dic73—21no—
50C22×Dic73C22⋊C412yes—
51C22×Dic73C22⋊C411yescharacteristic
52C22×Dic73C22⋊C411yescharacteristic
53C2×C4×D73—42no—
54C2×C4×D73—44no—
55C2×#10596831—24no—
56C2×#10596831—42no—
57C2×#10596831—42no—
58C4⋊Dic73—42no—
59C4⋊Dic73—41no—
60#11924005—28no—
61#11924005—42no—
62#11924005—42no—
63#11924005—42no—
64C4×Dic73—21no—
65C4×Dic73—21no—
66C4×Dic73—22no—
67C4×Dic73—22no—
68C4×Dic73—22no—
69C4×Dic73—24no—
70C4×Dic73—42no—
71C2×Q8×C73—21no—
72C2×Q8×C73—41no—
73C2×Q8×C73—41no—
74C2×Q8×C73M1612yes—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73C22×C411yescharacteristic
78C22×C4×C73C2×D412yes—
79C2×C8×C73—24no—
80C2×C8×C73—42no—
81C2×C8×C73—42no—
82C4⋊C4×C73—41no—
83C42×C73—41no—
84C23×C73C2×C22⋊C411yescharacteristic
85C22×D73—22no—
86C22×D73—41no—
87C22×D73—41no—
88C22×D73—42no—
89C22×D73—42no—
90Q8×C73—42no—
91Q8×C73—42no—
92C2×C4×C73—22no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—41no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73C2×M1611yescharacteristic
102C2×C4×C73C2×M1611yescharacteristic
103C2×C4×C73AGL(1,ℤ/8)12yes—
104C2×C4×C73#13712yes—
105C2×Dic73—22no—
106C2×Dic73—21no—
107C2×Dic73—22no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—41no—
111C2×Dic73—42no—
112C2×Dic73#10012yes—
113C2×Dic73#10414yes—
114D292—42no—
115D292—42no—
116C4×D73—84no—
117C8×C73—44no—
118#10596831—44no—
119C22×C73#40711yescharacteristic
120C22×C73#41011yescharacteristic
121C22×C73C2×#10011yescharacteristic
122C22×C73#41712yes—
123C22×C73C2×#10412yes—
124C2×D73—42no—
125C2×D73—44no—
126C2×D73—81no—
127C2×D73—82no—
128C2×D73—81no—
129C4×C73—21no—
130C4×C73—21no—
131C4×C73—22no—
132C4×C73—22no—
133C4×C73—42no—
134C4×C73—81no—
135Dic73—41no—
136Dic73—42no—
137Dic73—81no—
138Dic73#32914yes—
139#7635—731nomaximal, Sylow
140C2×C73#146611yescharacteristic
141C2×C73#148912yes—
142C2×C73#150912yes—
143C2×C73C2×#32911yescharacteristic
144C2×C73#156011yescharacteristic
145D73—82no—
146C4×#124—731no—
147#1767—732no—
148#1272—731no—
149#1272D7311yescharacteristic, complemented
150#1270—732no—
151C73#763511yescharacteristic, complemented, Sylow
152C2×#124—731no—
153C2×C4×Q8—731no—
154C2×C4×D4—731no—
155C2×C4⋊C8—1462no—
156C2×#97—1462no—
157#391C2×D7311yescharacteristic
158#390—731no—
159#388—1462no—
160#382—731no—
161C4×C22⋊C4—1462no—
162C43—731no—
163#336—732no—
164#336C2×D7312yes—
165C22×Q8C4×D7311yescharacteristic
166C22×D4—731no—
167C23×C4—1461no—
168C22×C8—22no—
169C22×C8—1462no—
170#124—1462no—
171#124—1462no—
172C4×Q8—2922no—
173C4×D4—2922no—
174C2×C4⋊C4—731no—
175C2×C4⋊C4—1461no—
176C2×C4⋊C4C4×D7311yescharacteristic
177C2×C22⋊C4—1461no—
178C2×C22⋊C4—1462no—
179C2×C22⋊C4—1462no—
180C2×C42—731no—
181C2×C42—731no—
182C2×C42—732no—
183C2×C42—732no—
184C2×C42—1461no—
185C2×C42C22×D7311yescharacteristic
186C4⋊C8—1468no—
187#97—2924no—
188#95—22no—
189#95—1462no—
190C24—1461no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8#1192401112yes—
195C2×D4—2921no—
196C2×D4—2921no—
197C2×D4—732no—
198C2×D4—1461no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—2922no—
202C22×C4—2924no—
203C22×C4—731no—
204C22×C4—732no—
205C22×C4—731no—
206C22×C4—1461no—
207C22×C4C2×C4×D7311yescharacteristic
208C22×C4D4×D7312yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—2922no—
213C2×C8—2922no—
214C2×C8—1464no—
215C4⋊C4—41no—
216C4⋊C4—2921no—
217C4⋊C4—2922no—
218C22⋊C4—2922no—
219C22⋊C4—2922no—
220C22⋊C4—2922no—
221C22⋊C4—1468no—
222C42—41no—
223C42—2922no—
224C42—1464no—
225C42—1462no—
226C42—1461no—
227C42—1461no—
228C42—1462no—
229C42—1462no—
230C23—2921no—
231C23—2921no—
232C23—2922no—
233C23—2922no—
234C23—1462no—
235C23C22⋊C4×D7311yescharacteristic
236Q8—42no—
237Q8—42no—
238D4—2922no—
239D4—2922no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—2921no—
250C2×C4—2922no—
251C2×C4—5844no—
252C2×C4—732no—
253C2×C4—734no—
254C2×C4—1462no—
255C2×C4—1461no—
256C2×C4—1462no—
257C2×C4—1462no—
258C2×C4—1462no—
259C2×C4M16×D7311yescharacteristic
260C2×C4C2×#1192401111yescharacteristic
261C2×C4#1455661812yes—
262C2×C4#1455662312yes—
263C8—44no—
264C8—2924no—
265C22—2922no—
266C22—2924no—
267C22—5841no—
268C22—5842no—
269C22—5841no—
270C22#1798854911yescharacteristic
271C22#1798855911yescharacteristic
272C22#100×D7311yescharacteristic
273C22#1798858712yes—
274C22#104×D7312yes—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—42no—
280C4—81no—
281C4—2921no—
282C4—2922no—
283C4—5841no—
284C4—734no—
285C2—5842no—
286C2#2260827011yescharacteristic
287C2#2260836912yes—
288C2#2260845512yes—
289C2#329×D7311yescharacteristic
290C2#2260866511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1272C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1272D73
hypocenterC73#7635
LCSG ▷ C22×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. #22609783C2semidirect
3. #22607757C2semidirect
5. C73×#1272C2semidirect
149. #1272D73semidirect
151. C73#7635semidirect

Maximal subgroups

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

subgroupindex
#226124742
#226097832
#226077572
#226077512
C73×#12722
#763573