Finite Group Databasebeta

← #34010152

Subgroups — #34010152

Order 18688.

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

normal subgroups only

This group has 444 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#22625194C212yesmaximal, complemented
2#22624871C212yesmaximal, complemented
3#22624859C212yesmaximal
4#124×Dic73C211yesmaximal, characteristic
5#17989607—22no—
6#17989432—22no—
7#17989432C2214yes—
8C2×Q8×Dic73C2211yescharacteristic
9C2×D4×Dic73C411yescharacteristic
10#17988494C411yescharacteristic
11#17988490C2211yescharacteristic
12#17988480—22no—
13#17988453C2211yescharacteristic
14C22⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic
16C2×#124×C73C411yescharacteristic
17#14556658—42no—
18C22⋊#10596833—24no—
19#14556646—42no—
20C23×Dic73—21no—
21Q8×Dic73—41no—
22Q8×Dic73—41no—
23D4×Dic73—41no—
24D4×Dic73—41no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73C2311yescharacteristic
28C2×C22⋊Dic73—21no—
29C2×C22⋊Dic73—22no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
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
43C2×#10596833—44no—
44C22×Dic73—21no—
45C22×Dic73—21no—
46C22×Dic73—44no—
47C22×Dic73—41no—
48C22×Dic73—41no—
49C22×Dic73C22⋊C411yescharacteristic
50C22×Dic73C22⋊C411yescharacteristic
51C22×Dic73C2×D412yes—
52C4⋊Dic73—42no—
53C4⋊Dic73—41no—
54C4⋊Dic73—41no—
55C22⋊Dic73—44no—
56C22⋊Dic73—42no—
57C4×Dic73—21no—
58C4×Dic73—21no—
59C4×Dic73—22no—
60C4×Dic73—22no—
61C4×Dic73—42no—
62C4×Dic73—42no—
63C4×Dic73—41no—
64C24×C73—21no—
65C2×Q8×C73—21no—
66C2×Q8×C73—21no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×D4×C73—21no—
70C2×D4×C73—41no—
71C2×D4×C73—41no—
72C2×D4×C73M1612yes—
73C22×C4×C73—21no—
74C22×C4×C73C22⋊C412yes—
75C22×C4×C73C22×C411yescharacteristic
76C22⋊C4×C73—28no—
77C42×C73—21no—
78C42×C73M1612yes—
79C23×C73—22no—
80C23×C73—42no—
81C23×C73—42no—
82C23×C73—41no—
83C23×C73—41no—
84C23×C73C2×C22⋊C411yescharacteristic
85Q8×C73—42no—
86Q8×C73—42no—
87D4×C73—42no—
88D4×C73—42no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73C2×M1611yescharacteristic
98C2×C4×C73C2×M1611yescharacteristic
99C2×Dic73—22no—
100C2×Dic73—22no—
101C2×Dic73—21no—
102C2×Dic73—22no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—42no—
107C2×Dic73—42no—
108C2×Dic73—41no—
109C2×Dic73—82no—
110C2×Dic73—82no—
111C2×Dic73#9911yescharacteristic
112C2×Dic73#9911yescharacteristic
113#10596833—84no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73—44no—
117C22×C73—42no—
118C22×C73—82no—
119C22×C73—81no—
120C22×C73—81no—
121C22×C73#40711yescharacteristic
122C22×C73C2×#9911yescharacteristic
123C22×C73#41311yescharacteristic
124C4×C73—21no—
125C4×C73—21no—
126C4×C73—42no—
127C4×C73—42no—
128Dic73—21no—
129Dic73—42no—
130Dic73—42no—
131Dic73—41no—
132Dic73—81no—
133Dic73#35512yes—
134#11530—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—21no—
137C2×C73—82no—
138C2×C73#146511yescharacteristic
139C2×C73#212611yescharacteristic
140C2×C73C2×#35511yescharacteristic
141C4×#124—731no—
142#1467—732no—
143#1324—732no—
144#1318—732no—
145C73#1153011yescharacteristic, complemented, Sylow
146C2×#124#6440811yescharacteristic
147C2×C4×Q8—731no—
148C2×C4×D4—731no—
149#432—1462no—
150#391—731no—
151#390—731no—
152#388—1462no—
153#382—731no—
154C4×C22⋊C4—1462no—
155C43—731no—
156#324—734no—
157#324—1462no—
158C22×Q8C2×#6440811yescharacteristic
159C22×D4C2×#6440811yescharacteristic
160C23×C4—1461no—
161#124—22no—
162#124—22no—
163C4×Q8—2921no—
164C4×Q8—2921no—
165C4×D4—2921no—
166C4×D4—2921no—
167C2×C4⋊C4—732no—
168C2×C4⋊C4—731no—
169C2×C4⋊C4—731no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—1461no—
172C2×C22⋊C4—1462no—
173C2×C42—731no—
174C2×C42—732no—
175C2×C42—731no—
176C2×C42—1461no—
177C2×C42—1461no—
178C2×C42C2×#6440811yescharacteristic
179C4⋊C8—2922no—
180C22⋊C8—1464no—
181#97—2922no—
182#95—1462no—
183#95—1462no—
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4#1192403712yes—
193C22×C4—21no—
194C22×C4—2921no—
195C22×C4—2921no—
196C22×C4—2924no—
197C22×C4—731no—
198C22×C4—732no—
199C22×C4—731no—
200C22×C4—1461no—
201C22×C4—1461no—
202C22×C4C22⋊#6440812yes—
203C22×C4C22×#6440811yescharacteristic
204C2×C8—2924no—
205C4⋊C4—2921no—
206C4⋊C4—2921no—
207C4⋊C4—2922no—
208C22⋊C4—28no—
209C22⋊C4—2924no—
210C22⋊C4—2922no—
211C42—21no—
212C42—2921no—
213C42—2922no—
214C42—2922no—
215C42—1462no—
216C42—1462no—
217C42—1461no—
218C42—1461no—
219C42#1192403712yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊#6440811yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2921no—
239C2×C4—2921no—
240C2×C4—2921no—
241C2×C4—2922no—
242C2×C4—2922no—
243C2×C4—2921no—
244C2×C4—5842no—
245C2×C4—5842no—
246C2×C4—731no—
247C2×C4—731no—
248C2×C4—1462no—
249C2×C4—1461no—
250C2×C4—1462no—
251C2×C4—1462no—
252C2×C4C2×#1192403711yescharacteristic
253C2×C4C2×#1192403711yescharacteristic
254C8—5844no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22#1798955711yescharacteristic
263C22D73⋊#9911yescharacteristic
264C22#1798957411yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—42no—
269C4—2922no—
270C4—2922no—
271C4—2921no—
272C4—5841no—
273C4—732no—
274C4—1461no—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#2262518711yescharacteristic
279C2#2262620311yescharacteristic
280C2D73⋊#35511yescharacteristic
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#11530
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22625194C2semidirect
2. #22624871C2semidirect
145. C73#11530semidirect

Maximal subgroups

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

subgroupindex
#226251942
#226248712
#226248592
#124×Dic732
#1153073