Finite Group Databasebeta

← #34010156

Subgroups — #34010156

Order 18688.

460 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 76 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#22625187C212yesmaximal
2#22624871C212yesmaximal, complemented
3#22624861C212yesmaximal, complemented
4#124×Dic73C211yesmaximal, characteristic
5#17989557—22no—
6#17989432C2214yes—
7#17989430—22no—
8C2×Q8×Dic73C411yescharacteristic
9C2×D4×Dic73C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988490C411yescharacteristic
12#17988480—22no—
13#17988453C2211yescharacteristic
14C22⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic
16C2×#124×C73C411yescharacteristic
17C2×#11924037—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C23×Dic73D412yes—
21Q8×Dic73—41no—
22Q8×Dic73—41no—
23D4×Dic73—41no—
24D4×Dic73—41no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2311yescharacteristic
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73D412yes—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35#14556498—22no—
36#14556498—22no—
37C22×Q8×C73C2×C411yescharacteristic
38C22×D4×C73C2×C411yescharacteristic
39#124×C73—22no—
40#124×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C42×C73C2×C411yescharacteristic
43#11924037—84no—
44C2×#10596833—44no—
45C22×Dic73—21no—
46C22×Dic73—22no—
47C22×Dic73—22no—
48C22×Dic73—21no—
49C22×Dic73—44no—
50C22×Dic73C22⋊C411yescharacteristic
51C22×Dic73C22⋊C411yescharacteristic
52C22×Dic73C2×D412yes—
53C4⋊Dic73—41no—
54C4⋊Dic73—42no—
55C4⋊Dic73—41no—
56C22⋊Dic73—42no—
57C22⋊Dic73—44no—
58C4×Dic73—21no—
59C4×Dic73—21no—
60C4×Dic73—42no—
61C4×Dic73—42no—
62C4×Dic73—41no—
63C4×Dic73—41no—
64C4×Dic73—41no—
65C24×C73C22⋊C412yes—
66C2×Q8×C73—21no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×Q8×C73M1612yescomplemented
70C2×D4×C73—21no—
71C2×D4×C73—21no—
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73C22×C411yescharacteristic
77C22⋊C4×C73—28no—
78C42×C73—21no—
79C42×C73M1612yescomplemented
80C23×C73—22no—
81C23×C73—22no—
82C23×C73—22no—
83C23×C73—42no—
84C23×C73—42no—
85C23×C73C2×C22⋊C411yescharacteristic
86Q8×C73—42no—
87Q8×C73—42no—
88D4×C73—42no—
89D4×C73—42no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—41no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73C2×M1611yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—21no—
103C2×Dic73—22no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—44no—
107C2×Dic73—41no—
108C2×Dic73—42no—
109C2×Dic73—42no—
110C2×Dic73—41no—
111C2×Dic73—82no—
112C2×Dic73#9911yescharacteristic
113C2×Dic73#9911yescharacteristic
114#10596833—84no—
115C22×C73—21no—
116C22×C73—21no—
117C22×C73—44no—
118C22×C73—42no—
119C22×C73—42no—
120C22×C73—42no—
121C22×C73—82no—
122C22×C73#40711yescharacteristic
123C22×C73C2×#9911yescharacteristic
124C22×C73#41311yescharacteristic
125C4×C73—21no—
126C4×C73—21no—
127C4×C73—42no—
128C4×C73—42no—
129Dic73—21no—
130Dic73—42no—
131Dic73—42no—
132Dic73—41no—
133Dic73—81no—
134Dic73#35412yes—
135#11531—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—21no—
138C2×C73—82no—
139C2×C73#146511yescharacteristic
140C2×C73C2×#35411yescharacteristic
141C2×C73#213211yescharacteristic
142C4×#124—731no—
143#1465—732no—
144#1324—732no—
145#1319—732no—
146C73#1153111yescharacteristic, complemented, Sylow
147C2×#124#6440811yescharacteristic
148C2×C4×Q8—731no—
149C2×C4×D4—731no—
150#407—1462no—
151#391—731no—
152#390—731no—
153#388—1462no—
154#382—731no—
155C4×C22⋊C4—1462no—
156C43—731no—
157#324—734no—
158#323—1462no—
159C22×Q8C2×#6440811yescharacteristic
160C22×D4C2×#6440811yescharacteristic
161C23×C4—732no—
162C2×M16—2922no—
163#124—22no—
164#124—22no—
165C4×Q8—2921no—
166C4×Q8—2921no—
167C4×D4—2921no—
168C4×D4—2921no—
169C2×C4⋊C4—731no—
170C2×C4⋊C4—731no—
171C2×C4⋊C4—1461no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—732no—
174C2×C22⋊C4—1462no—
175C2×C42—731no—
176C2×C42—731no—
177C2×C42—1461no—
178C2×C42—1461no—
179C2×C42—1461no—
180C2×C42C2×#6440811yescharacteristic
181C22⋊C8—2922no—
182C22⋊C8—1464no—
183#95—1462no—
184#95—1462no—
185C24C22⋊#6440812yes—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1192403712yescomplemented
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—2924no—
197C22×C4—731no—
198C22×C4—732no—
199C22×C4—731no—
200C22×C4—1461no—
201C22×C4—1462no—
202C22×C4—1461no—
203C22×C4—1462no—
204C22×C4C22×#6440811yescharacteristic
205M16—5844no—
206C2×C8—2924no—
207C4⋊C4—2921no—
208C4⋊C4—2921no—
209C4⋊C4—2922no—
210C22⋊C4—28no—
211C22⋊C4—2924no—
212C22⋊C4—2922no—
213C42—21no—
214C42—2921no—
215C42—2921no—
216C42—2921no—
217C42—2922no—
218C42—2922no—
219C42—1461no—
220C42—1461no—
221C42#1192403712yescomplemented
222C23—22no—
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊#6440811yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—2921no—
241C2×C4—2921no—
242C2×C4—2924no—
243C2×C4—2921no—
244C2×C4—2922no—
245C2×C4—2922no—
246C2×C4—2921no—
247C2×C4—5842no—
248C2×C4—731no—
249C2×C4—731no—
250C2×C4—1462no—
251C2×C4—1461no—
252C2×C4—1462no—
253C2×C4—1462no—
254C2×C4C2×#1192403711yescharacteristic
255C2×C4C2×#1192403711yescharacteristic
256C8—5844no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22#1798955711yescharacteristic
265C22D73⋊#9911yescharacteristic
266C22#1798957411yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—2922no—
272C4—2922no—
273C4—2921no—
274C4—5841no—
275C4—732no—
276C4—1461no—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#2262518711yescharacteristic
281C2D73⋊#35411yescharacteristic
282C2#2262622511yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#124×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#124#64408
hypocenterC73#11531
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #22624871C2semidirect
3. #22624861C2semidirect
69. C2×Q8×C73M16semidirect
79. C42×C73M16semidirect
146. C73#11531semidirect
189. C2×Q8#11924037semidirect
221. C42#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251872
#226248712
#226248612
#124×Dic732
#1153173