Finite Group Databasebeta

← #32995257

Subgroups — #32995257

Order 18688.

460 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 89 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#22612470C211yesmaximal, characteristic
2#22608253C212yesmaximal, complemented
3#22607875C211yesmaximal, characteristic, complemented
4#22607862C212yesmaximal
5C73×#1324C211yesmaximal, characteristic, complemented
6C2×#14556579C411yescharacteristic
7C2×#14556556C2211yescharacteristic
8C2×#14556553C411yescharacteristic
9#17988637—22no—
10#17988492C411yescharacteristic
11#17988488C2211yescharacteristic
12#17988476—22no—
13#17988434—22no—
14C42×Dic73C411yescharacteristic
15#17988313C2212yes—
16#17988312—22no—
17#382×C73C2211yescharacteristic
18#324×C73C2212yes—
19C22×Dic146C2×C411yescharacteristic
20C22×D292C2×C411yescharacteristic
21C22×C4×D73—21no—
22#14556579—22no—
23#14556579C814yescomplemented
24#14556556—41no—
25#14556556—41no—
26#14556553—41no—
27#14556553—41no—
28C2×#11924007D412yes—
29C2×#11924005—22no—
30C2×#11924005—21no—
31C2×#11924005—22no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73D412yes—
37#14556520—42no—
38C22⋊#10596831—22no—
39C8×Dic73—42no—
40#14556498—22no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2311yescharacteristic
43C2×C42×C73C2×C411yescharacteristic
44C22⋊C8×C73—22no—
45#95×C73—22no—
46C23×D73—21no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146C2×C812yes—
50C2×Dic146M1612yes—
51C2×D292—21no—
52C2×D292—41no—
53C2×D292—41no—
54C2×D292C2×C812yes—
55C22×Dic73—21no—
56C22×Dic73C22⋊C412yes—
57C22×Dic73C22⋊C411yescharacteristic
58C22×Dic73C22⋊C411yescharacteristic
59C2×C4×D73—44no—
60C2×C4×D73—41no—
61C2×C4×D73—41no—
62C2×#10596831—42no—
63#11924007—42no—
64#11924005—28no—
65#11924005—44no—
66#11924005—42no—
67C4×Dic73—22no—
68C4×Dic73—22no—
69C4×Dic73—21no—
70C4×Dic73—21no—
71C4×Dic73—42no—
72C4×Dic73—42no—
73C4×Dic73—41no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C2×D412yes—
78C2×C8×C73—42no—
79C4⋊C4×C73—41no—
80C4⋊C4×C73—41no—
81C42×C73—21no—
82C42×C73C2×C812yes—
83C23×C73C2×C22⋊C411yescharacteristic
84C22×D73—22no—
85C22×D73—42no—
86C22×D73—42no—
87C22×D73—41no—
88C22×D73—41no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73—42no—
96C2×C4×C73—42no—
97C2×C4×C73—41no—
98C2×C4×C73C22×C811yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—22no—
103C2×Dic73—22no—
104C2×Dic73—21no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73—41no—
108C2×Dic73C22⋊C814yes—
109C2×Dic73#9911yescharacteristic
110C2×Dic73#9911yescharacteristic
111D292—42no—
112D292—42no—
113C4×D73—82no—
114C4×D73—82no—
115Dic146—42no—
116Dic146—42no—
117C8×C73—82no—
118#10596831—82no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73C2×C22⋊C811yescharacteristic
122C22×C73C2×#9911yescharacteristic
123C22×C73#41311yescharacteristic
124C2×D73—44no—
125C2×D73—42no—
126C2×D73—82no—
127C2×D73—81no—
128C2×D73—81no—
129C4×C73—21no—
130C4×C73—21no—
131C4×C73—42no—
132C4×C73—42no—
133C4×C73—81no—
134Dic73—21no—
135Dic73—24no—
136Dic73—42no—
137Dic73—41no—
138Dic73#35512yes—
139#11476—731nomaximal, Sylow
140C2×C73—21no—
141C2×C73—21no—
142C2×C73#145911yescharacteristic
143C2×C73#212611yescharacteristic
144C2×C73C2×#35511yescharacteristic
145D73—82no—
146C4×#124—731no—
147#1461—732no—
148#1324—731no—
149#1324D7311yescharacteristic, complemented
150#1318—732no—
151C73#1147611yescharacteristic, complemented, Sylow
152C2×#124—731no—
153C2×C4×Q8—731no—
154C2×C4×D4—731no—
155#431—1462no—
156#391—731no—
157#390—731no—
158#388—1462no—
159#382C2×D7311yescharacteristic
160C4×C22⋊C4—1462no—
161C43—731no—
162#324—732no—
163#324—1462no—
164#324C2×D7312yes—
165C22×Q8—731no—
166C22×D4—731no—
167C23×C4—1461no—
168#124—734no—
169#124—1462no—
170C4×Q8—2921no—
171C4×Q8—2921no—
172C4×D4—2921no—
173C4×D4—2921no—
174C2×C4⋊C4—732no—
175C2×C4⋊C4C4×D7311yescharacteristic
176C2×C4⋊C4C22×D7311yescharacteristic
177C2×C22⋊C4—1461no—
178C2×C22⋊C4—1462no—
179C2×C22⋊C4—1462no—
180C2×C42—731no—
181C2×C42—731no—
182C2×C42—732no—
183C2×C42—1461no—
184C2×C42—1461no—
185C2×C42C4×D7311yescharacteristic
186C4⋊C8—2922no—
187C22⋊C8—22no—
188C22⋊C8—1462no—
189C4×C8—2922no—
190#95—22no—
191#95—1462no—
192C24—1461no—
193C2×Q8—2921no—
194C2×Q8—2921no—
195C2×Q8—732no—
196C2×Q8—732no—
197C2×D4—2921no—
198C2×D4—2921no—
199C2×D4—732no—
200C2×D4—1461no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—2921no—
204C22×C4—2921no—
205C22×C4—2924no—
206C22×C4—731no—
207C22×C4—732no—
208C22×C4—731no—
209C22×C4—1461no—
210C22×C4C2×C4×D7311yescharacteristic
211C22×C4D4×D7312yes—
212C2×C8—42no—
213C2×C8—2922no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—2922no—
217C22⋊C4—2924no—
218C22⋊C4—2922no—
219C22⋊C4—1468no—
220C42—21no—
221C42—2921no—
222C42—2922no—
223C42—2922no—
224C42—1462no—
225C42—1462no—
226C42—1461no—
227C42—1461no—
228C42C8×D7312yes—
229C23—2922no—
230C23—2922no—
231C23—2921no—
232C23—2921no—
233C23—1462no—
234C23C22⋊C4×D7311yescharacteristic
235Q8—2922no—
236Q8—2922no—
237D4—2922no—
238D4—2922no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—2921no—
249C2×C4—2921no—
250C2×C4—2921no—
251C2×C4—5842no—
252C2×C4—5842no—
253C2×C4—731no—
254C2×C4—734no—
255C2×C4—731no—
256C2×C4—1462no—
257C2×C4—1461no—
258C2×C4—1462no—
259C2×C4—1462no—
260C2×C4—1462no—
261C2×C4C2×C8×D7311yescharacteristic
262C2×C4M16×D7311yescharacteristic
263C8—82no—
264C8—5842no—
265C22—21no—
266C22—21no—
267C22—2924no—
268C22—2922no—
269C22—5842no—
270C22—5841no—
271C22—5841no—
272C22C22⋊C8×D7311yescharacteristic
273C22#99×D7311yescharacteristic
274C22#1798857011yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—2922no—
281C4—2921no—
282C4—732no—
283C4—1464no—
284C4—1461no—
285C2—21no—
286C2—21no—
287C2—5842no—
288C2#2260824811yescharacteristic
289C2#2261159511yescharacteristic
290C2#355×D7311yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1324C2
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1324D73
hypocenterC73#11476
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
2. #22608253C2semidirect
3. #22607875C2semidirect
5. C73×#1324C2semidirect
23. #14556579C8semidirect
149. #1324D73semidirect
151. C73#11476semidirect

Maximal subgroups

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

subgroupindex
#226124702
#226082532
#226078752
#226078622
C73×#13242
#1147673