Finite Group Databasebeta

← #32995473

Subgroups — #32995473

Order 18688.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612645C211yesmaximal, characteristic
2#22608268C212yesmaximal
3#22607876C212yesmaximal, complemented
4#22607865C211yesmaximal, characteristic, complemented
5C73×#1319C211yesmaximal, characteristic, complemented
6C2×#14556580C411yescharacteristic
7C2×Q8×Dic73C411yescharacteristic
8C2×#14556553C2211yescharacteristic
9#17988549—22no—
10#17988489C411yescharacteristic
11#17988476—22no—
12#17988461—22no—
13#17988456C2211yescharacteristic
14#17988433C411yescharacteristic
15#17988313C2212yes—
16#17988310—22no—
17#391×C73C2211yescharacteristic
18#324×C73C2212yes—
19C22×D292C2×C411yescharacteristic
20C22×C4×D73D412yes—
21C2×#11924011—42no—
22#14556580—22no—
23#14556580—22no—
24Q8×Dic73—42no—
25#14556553—42no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—22no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C2×#11924005D412yes—
32C2×C4×Dic73—21no—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35C22⋊#10596831—22no—
36#14556503—42no—
37#14556498—21no—
38#14556498—21no—
39C22×Q8×C73C2×C411yescharacteristic
40C2×C4⋊C4×C73C2311yescharacteristic
41C2×C42×C73C2×C411yescharacteristic
42C22⋊C8×C73—22no—
43#95×C73—22no—
44C23×D73C22⋊C412yes—
45C2×D292—21no—
46C2×D292—21no—
47C2×D292—41no—
48C2×D292—41no—
49C22×Dic73—21no—
50C22×Dic73—21no—
51C22×Dic73C22⋊C411yescharacteristic
52C22×Dic73C22⋊C411yescharacteristic
53C2×C4×D73—22no—
54C2×C4×D73—22no—
55C2×C4×D73—82no—
56#11924011—84no—
57C2×#10596831—42no—
58C4⋊Dic73—42no—
59C4⋊Dic73—41no—
60#11924007—82no—
61#11924005—28no—
62#11924005—42no—
63#11924005—82no—
64C4×Dic73—21no—
65C4×Dic73—22no—
66C4×Dic73—42no—
67C4×Dic73M1612yescomplemented
68C2×Q8×C73—21no—
69C2×Q8×C73—41no—
70C2×Q8×C73—41no—
71C2×Q8×C73M1612yescomplemented
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73C22×C411yescharacteristic
75C22×C4×C73C2×D412yes—
76C2×C8×C73—42no—
77C4⋊C4×C73—41no—
78C42×C73—41no—
79C23×C73C2×C22⋊C411yescharacteristic
80C22×D73—22no—
81C22×D73—22no—
82C22×D73—22no—
83C22×D73—42no—
84C22×D73—42no—
85Q8×C73—42no—
86Q8×C73—42no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
90C2×C4×C73—22no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73—42no—
94C2×C4×C73—41no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73C2×M1611yescharacteristic
98C2×C4×C73C2×M1611yescharacteristic
99C2×Dic73—22no—
100C2×Dic73—21no—
101C2×Dic73—41no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73—42no—
106C2×Dic73—41no—
107C2×Dic73#9912yes—
108D292—42no—
109D292—42no—
110C4×D73—44no—
111C4×D73—82no—
112C8×C73—82no—
113#10596831—82no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73#40711yescharacteristic
117C22×C73C2×#9911yescharacteristic
118C22×C73#41311yescharacteristic
119C2×D73—44no—
120C2×D73—42no—
121C2×D73—42no—
122C2×D73—42no—
123C2×D73—82no—
124C4×C73—21no—
125C4×C73—21no—
126C4×C73—42no—
127C4×C73—42no—
128C4×C73—81no—
129Dic73—22no—
130Dic73—42no—
131Dic73—41no—
132Dic73—81no—
133#11535—731nomaximal, Sylow
134C2×C73—21no—
135C2×C73—21no—
136C2×C73#146511yescharacteristic
137C2×C73#212711yescharacteristic
138C2×C73#213511yescharacteristic
139D73—82no—
140#2326—731no—
141#1465—732no—
142#1325—732no—
143#1319—731no—
144#1319D7311yescharacteristic, complemented
145C73#1153511yescharacteristic, complemented, Sylow
146C2×#124—731no—
147C2×C4×Q8—731no—
148C2×C4×D4—731no—
149#407—1462no—
150#391C2×D7311yescharacteristic
151#390—731no—
152#388—1462no—
153#385—1462no—
154#384—731no—
155#376—731no—
156#324—732no—
157#324C2×D7312yes—
158#323—1462no—
159C22×Q8C4×D7311yescharacteristic
160C22×D4—731no—
161C23×C4—732no—
162C2×M16—2922no—
163#124—1462no—
164#124—1462no—
165C4×Q8—2922no—
166C4×D4—2922no—
167C2×C4⋊C4—731no—
168C2×C4⋊C4—1461no—
169C2×C4⋊C4—1462no—
170C2×C4⋊C4C22×D7311yescharacteristic
171C2×C22⋊C4—732no—
172C2×C22⋊C4—1462no—
173C2×C22⋊C4—1462no—
174C2×C42—731no—
175C2×C42—731no—
176C2×C42—1461no—
177C2×C42C4×D7311yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—2922no—
180C22⋊C8—1462no—
181#95—22no—
182#95—1461no—
183#95—1461no—
184C24—732no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#1192401112yescomplemented
189C2×D4—2921no—
190C2×D4—2921no—
191C2×D4—1461no—
192C2×D4—1461no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—5842no—
196C22×C4—731no—
197C22×C4—731no—
198C22×C4—1462no—
199C22×C4—1462no—
200C22×C4—1461no—
201C22×C4—1461no—
202C22×C4C2×C4×D7311yescharacteristic
203C22×C4D4×D7312yes—
204M16—5844no—
205C2×C8—42no—
206C2×C8—2922no—
207C4⋊C4—41no—
208C4⋊C4—2921no—
209C4⋊C4—2922no—
210C4⋊C4—5842no—
211C22⋊C4—2922no—
212C22⋊C4—5842no—
213C22⋊C4—1468no—
214C42—41no—
215C42—2922no—
216C42—732no—
217C42—1462no—
218C42—1461no—
219C23—2922no—
220C23—2922no—
221C23—1462no—
222C23—1462no—
223C23—1462no—
224C23C22⋊C4×D7311yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—2922no—
228D4—2922no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—2924no—
240C2×C4—2921no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—2922no—
244C2×C4—2921no—
245C2×C4—2921no—
246C2×C4—5842no—
247C2×C4—732no—
248C2×C4—1462no—
249C2×C4—1461no—
250C2×C4M16×D7311yescharacteristic
251C2×C4C2×#1192401111yescharacteristic
252C8—82no—
253C8—5842no—
254C22—21no—
255C22—21no—
256C22—2924no—
257C22—2922no—
258C22—2922no—
259C22—2922no—
260C22—5842no—
261C22#1798854911yescharacteristic
262C22#99×D7311yescharacteristic
263C22#1798856811yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—2922no—
270C4—2921no—
271C4—5841no—
272C4—1462no—
273C2—21no—
274C2—21no—
275C2—5842no—
276C2#2260826611yescharacteristic
277C2#2261159911yescharacteristic
278C2#2261162411yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1319C2
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1319D73
hypocenterC73#11535
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #22607876C2semidirect
4. #22607865C2semidirect
5. C73×#1319C2semidirect
67. C4×Dic73M16semidirect
71. C2×Q8×C73M16semidirect
144. #1319D73semidirect
145. C73#11535semidirect
188. C2×Q8#11924011semidirect

Maximal subgroups

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

subgroupindex
#226126452
#226082682
#226078762
#226078652
C73×#13192
#1153573