Finite Group Databasebeta

← #34012526

Subgroups — #34012526

Order 18688.

500 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#17989498C212yesmaximal
2#22625684C212yesmaximal
3#22625029C212yesmaximal
4#22612498C211yesmaximal, characteristic
5C2×#14556652C2214yes—
6C4×#11924037—22no—
7#17989498C418yescomplemented
8#17989467—22no—
9C2×#14556591C2211yescharacteristic
10C2×#14556556C411yescharacteristic
11C2×#14556556C2211yescharacteristic
12#17988492C411yescharacteristic
13#17988492C2211yescharacteristic
14#17988483—22no—
15C4⋊C4×Dic73—22no—
16C42×Dic73C411yescharacteristic
17#384×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556652—24no—
20#14556652C2×C418yes—
21#14556646—42no—
22C4×#10596833—42no—
23C22×Dic146C2×C411yescharacteristic
24C22×Dic146C2311yescharacteristic
25#14556591—22no—
26#14556591C2×C414yes—
27#14556556—41no—
28#14556556—41no—
29#14556556—41no—
30#14556556—41no—
31C2×C4⋊Dic73—22no—
32C2×#11924007—21no—
33C2×#11924007D412yes—
34C2×C4×Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73D412yes—
40#14556498—22no—
41#14556498—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C42×C73C2×C411yescharacteristic
46#11924037—28no—
47#11924037—44no—
48C2×#10596833—44no—
49C2×Dic146—21no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C2×Dic146—41no—
53C2×Dic146—41no—
54C2×Dic146C4212yes—
55C2×Dic146C22×C412yes—
56C2×Dic146#4112yes—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C2×D412yes—
62C4⋊Dic73—28no—
63#11924007—42no—
64#11924007—42no—
65C4×Dic73—24no—
66C4×Dic73—21no—
67C4×Dic73—21no—
68C4×Dic73—22no—
69C4×Dic73—22no—
70C4×Dic73—42no—
71C4×Dic73—41no—
72C4×Dic73—41no—
73C4×Dic73—41no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73C22⋊C412yes—
77C22×C4×C73C22×C411yescharacteristic
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C4⋊C4×C73—44no—
81C42×C73—21no—
82C42×C73C4212yes—
83C23×C73C2×C22⋊C411yescharacteristic
84C2×C4×C73—22no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73—41no—
93C2×C4×C73C2×C4211yescharacteristic
94C2×C4×C73#11711yescharacteristic
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—21no—
101C2×Dic73—41no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73#9911yescharacteristic
107C2×Dic73#9911yescharacteristic
108C2×Dic73C4×D414yes—
109Dic146—42no—
110Dic146—42no—
111Dic146—42no—
112Dic146—42no—
113#10596833—48no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73C4×C22⋊C411yescharacteristic
117C22×C73C2×#9911yescharacteristic
118C22×C73#41011yescharacteristic
119C4×C73—24no—
120C4×C73—21no—
121C4×C73—21no—
122C4×C73—42no—
123C4×C73—81no—
124Dic73—21no—
125Dic73—24no—
126Dic73—42no—
127Dic73—41no—
128Dic73—42no—
129Dic73#35612yescomplemented
130C4×#356—731nomaximal, Sylow
131C2×C73—21no—
132C2×C73—21no—
133C2×C73C4×#9911yescharacteristic
134C2×C73C2×#35611yescharacteristic
135C2×C73#213211yescharacteristic
136C4×C4⋊Q8—731no—
137C2×#356—732no—
138C4×#101—732no—
139#1394—732no—
140C73C4×#35611yescharacteristic, complemented, Sylow
141C2×C4⋊Q8—731no—
142C2×C4×Q8—731no—
143C2×C4×Q8—731no—
144C2×#101—734no—
145C4×M16—1462no—
146#391—731no—
147#391—731no—
148#389—1462no—
149#384#6440811yescharacteristic
150C4×C4⋊C4—1462no—
151C43—731no—
152#356—738no—
153#343—1462no—
154C22×Q8—731no—
155C22×Q8—731no—
156C2×M16—1464no—
157C4⋊Q8—734no—
158C4⋊Q8—1462no—
159C4×Q8—2921no—
160C4×Q8—2921no—
161C4×Q8—2921no—
162C4×Q8—2921no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—732no—
165C2×C4⋊C4—1461no—
166C2×C4⋊C4—1462no—
167C2×C4⋊C4C2×#6440811yescharacteristic
168C2×C4⋊C4C2×#6440811yescharacteristic
169C2×C42—731no—
170C2×C42—732no—
171C2×C42—731no—
172C2×C42—1461no—
173C2×C42—1461no—
174C2×C42—1461no—
175C2×C42C2×#6440811yescharacteristic
176#101—738no—
177#101—1464no—
178#97—2922no—
179C4×C8—2922no—
180#95—1462no—
181#95—1462no—
182C2×Q8—2921no—
183C2×Q8—2921no—
184C2×Q8—2921no—
185C2×Q8—2921no—
186C2×Q8—732no—
187C2×Q8—732no—
188C2×Q8—732no—
189C2×Q8—1461no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—731no—
193C22×C4—731no—
194C22×C4—732no—
195C22×C4—1461no—
196C22×C4—1461no—
197C22×C4C22⋊#6440812yes—
198C22×C4C22×#6440811yescharacteristic
199M16—2924no—
200M16—1468no—
201C2×C8—2924no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—2922no—
206C4⋊C4—2922no—
207C4⋊C4—1468no—
208C42—21no—
209C42—2921no—
210C42—2922no—
211C42—2921no—
212C42—2921no—
213C42—1462no—
214C42—1462no—
215C42—1461no—
216C42—1461no—
217C42—1464no—
218C42C4×#6440812yes—
219C23C2×C22⋊#6440811yescharacteristic
220Q8—2922no—
221Q8—2922no—
222Q8—2922no—
223Q8—2922no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—2921no—
234C2×C4—2921no—
235C2×C4—2921no—
236C2×C4—2921no—
237C2×C4—2921no—
238C2×C4—731no—
239C2×C4—734no—
240C2×C4—731no—
241C2×C4—1462no—
242C2×C4—1462no—
243C2×C4—1461no—
244C2×C4—1462no—
245C2×C4—1462no—
246C2×C4—1462no—
247C2×C4C2×C4×#6440811yescharacteristic
248C2×C4#1455667011yescharacteristic
249C8—2928no—
250C22—21no—
251C22—21no—
252C22C4×C22⋊#6440811yescharacteristic
253C22D73⋊#9911yescharacteristic
254C22#1798956811yescharacteristic
255C4—24no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—81no—
260C4—2922no—
261C4—2922no—
262C4—2921no—
263C4—732no—
264C4—1464no—
265C4—1461no—
266C2—21no—
267C2—21no—
268C2#2262567611yescharacteristic
269C2D73⋊#35611yescharacteristic
270C2#2262622311yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Dic146C23
hypercenter#384#64408
hypocenterC73C4×#356
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
7. #17989498C4semidirect
129. Dic73#356semidirect
140. C73C4×#356semidirect

Maximal subgroups

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

subgroupindex
C2×#179894982
#226256842
#226250292
#226124982
C4×#35673