Finite Group Databasebeta

← #34010096

Subgroups — #34010096

Order 18688.

434 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 74 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#22625180C212yesmaximal
2#22624873C212yesmaximal, complemented
3#22624872C212yesmaximal, complemented
4#22612556C211yesmaximal, characteristic
5#17989558—22no—
6#17989433—22no—
7#17989432C2214yes—
8C2×#14556549C411yescharacteristic
9C2×#14556548C411yescharacteristic
10C2×C4⋊C4×D73C2211yescharacteristic
11#17988484—22no—
12#17988461—22no—
13#17988457C2211yescharacteristic
14#17988456C411yescharacteristic
15#17988452C2211yescharacteristic
16#382×C73C411yescharacteristic
17C2×#11924038—42no—
18#14556648—42no—
19C22⋊#10596833—24no—
20C22×C4×D73C2×C411yescharacteristic
21C22×C4×D73D412yes—
22#14556549—42no—
23#14556548—22no—
24#14556548—22no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2311yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—22no—
30C2×#11924007D412yes—
31C4⋊C4×D73—42no—
32C2×#11924005—21no—
33C2×#11924005—21no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36#14556498—22no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C42×C73C2×C411yescharacteristic
40#95×C73—22no—
41#11924038—84no—
42C2×#10596833—44no—
43C23×D73C22⋊C411yescharacteristic
44C22×Dic73—21no—
45C22×Dic73—21no—
46C22×Dic73C22⋊C411yescharacteristic
47C22×Dic73C2×D412yes—
48C2×C4×D73—22no—
49C2×C4×D73—21no—
50C2×C4×D73—21no—
51C2×C4×D73—22no—
52C2×C4×D73—22no—
53C2×C4×D73—22no—
54C2×C4×D73—82no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—82no—
58#11924007—24no—
59#11924007—42no—
60#11924007—82no—
61#11924005—24no—
62#11924005—42no—
63C4×Dic73—21no—
64C4×Dic73—42no—
65C4×Dic73M1612yescomplemented
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73C22⋊C412yes—
69C22×C4×C73C22×C411yescharacteristic
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C42×C73—21no—
73C42×C73M1612yescomplemented
74C23×C73C2×C22⋊C411yescharacteristic
75C22×D73—21no—
76C22×D73—21no—
77C22×D73—41no—
78C22×D73—41no—
79C22×D73#9912yes—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—22no—
83C2×C4×C73—22no—
84C2×C4×C73—41no—
85C2×C4×C73—41no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—41no—
89C2×C4×C73C2×M1611yescharacteristic
90C2×C4×C73C2×M1611yescharacteristic
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—22no—
94C2×Dic73—22no—
95C2×Dic73—41no—
96C2×Dic73—41no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—41no—
100C2×Dic73#10112yes—
101C4×D73—24no—
102C4×D73—24no—
103C4×D73—44no—
104C4×D73—82no—
105#10596833—84no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73#40711yescharacteristic
109C22×C73C2×#9911yescharacteristic
110C22×C73C2×#10111yescharacteristic
111C2×D73—22no—
112C2×D73—22no—
113C2×D73—22no—
114C2×D73—42no—
115C2×D73—42no—
116C4×C73—21no—
117C4×C73—21no—
118C4×C73—42no—
119C4×C73—42no—
120C4×C73—81no—
121Dic73—22no—
122Dic73—42no—
123Dic73—42no—
124Dic73—81no—
125#11514—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—21no—
128C2×C73#146311yescharacteristic
129C2×C73#212711yescharacteristic
130C2×C73#213311yescharacteristic
131D73—42no—
132#2317—731no—
133#1463—732no—
134#1325—732no—
135#1324—732no—
136C73#1151411yescharacteristic, complemented, Sylow
137C2×#117—731no—
138C2×#117—731no—
139C22×C4⋊C4—731no—
140#407—1462no—
141#389—1462no—
142#385—1462no—
143#384—731no—
144#384—731no—
145#382—731no—
146#382#6440811yescharacteristic
147#324—734no—
148#324—1462no—
149C23×C4—732no—
150C23×C4—731no—
151C2×M16—2922no—
152#117—2922no—
153#117—1462no—
154#117—1462no—
155C2×C4⋊C4—2922no—
156C2×C4⋊C4—732no—
157C2×C4⋊C4—731no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—1462no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4—1461no—
162C2×C4⋊C4C2×#6440811yescharacteristic
163C2×C4⋊C4C2×#6440811yescharacteristic
164C2×C22⋊C4—1461no—
165C2×C22⋊C4—1461no—
166C2×C42—731no—
167C2×C42—1461no—
168C2×C42C2×#6440811yescharacteristic
169C22⋊C8—2922no—
170C22⋊C8—1464no—
171#95—22no—
172#95—1462no—
173C24—731no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—5842no—
177C22×C4—732no—
178C22×C4—731no—
179C22×C4—1462no—
180C22×C4—1462no—
181C22×C4—1462no—
182C22×C4—1462no—
183C22×C4—1461no—
184C22×C4—1461no—
185C22×C4—1461no—
186C22×C4—1461no—
187C22×C4C22⋊#6440812yes—
188C22×C4C22×#6440811yescharacteristic
189M16—5844no—
190C2×C8—2924no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—2921no—
194C4⋊C4—2922no—
195C4⋊C4—2921no—
196C4⋊C4—5842no—
197C4⋊C4—5842no—
198C4⋊C4—1464no—
199C22⋊C4—2922no—
200C22⋊C4—1464no—
201C42—21no—
202C42—2922no—
203C42—732no—
204C42—1461no—
205C42#1192403812yescomplemented
206C23—2921no—
207C23—2921no—
208C23—732no—
209C23—1461no—
210C23—1461no—
211C23C2×C22⋊#6440811yescharacteristic
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—2924no—
222C2×C4—2921no—
223C2×C4—2921no—
224C2×C4—2922no—
225C2×C4—2922no—
226C2×C4—2921no—
227C2×C4—5842no—
228C2×C4—732no—
229C2×C4—1464no—
230C2×C4—1464no—
231C2×C4—1462no—
232C2×C4—1462no—
233C2×C4—1461no—
234C2×C4—1461no—
235C2×C4D73⋊M1611yescharacteristic
236C2×C4C2×#1192403811yescharacteristic
237C8—5844no—
238C22—21no—
239C22—21no—
240C22—2922no—
241C22—2922no—
242C22—1462no—
243C22—1462no—
244C22—1462no—
245C22#1798955811yescharacteristic
246C22D73⋊#9911yescharacteristic
247C22D73⋊#10111yescharacteristic
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—81no—
253C4—2922no—
254C4—2922no—
255C4—5841no—
256C4—1462no—
257C2—21no—
258C2—21no—
259C2—2922no—
260C2#2262518011yescharacteristic
261C2#2262620911yescharacteristic
262C2#2262622711yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#382×C73C4
socleC22×C73C2×#99
prime radicalC23×D73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenter#382#64408
hypocenterC73#11514
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #22624873C2semidirect
3. #22624872C2semidirect
65. C4×Dic73M16semidirect
73. C42×C73M16semidirect
136. C73#11514semidirect
205. C42#11924038semidirect

Maximal subgroups

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

subgroupindex
#226251802
#226248732
#226248722
#226125562
#1151473