Finite Group Databasebeta

← #34009986

Subgroups — #34009986

Order 18688.

460 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 90 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#22625167C212yesmaximal
2#22624874C212yesmaximal, complemented
3#22624862C212yesmaximal, complemented
4#22612641C211yesmaximal, characteristic
5C2×C22⋊#10596833—22no—
6#17989432C2214yes—
7#17989430—22no—
8C2×#14556578C411yescharacteristic
9C2×#14556556C411yescharacteristic
10C2×D4×Dic73C2211yescharacteristic
11#17988493C2211yescharacteristic
12#17988481—22no—
13#17988459—22no—
14#17988456C2211yescharacteristic
15#17988433C411yescharacteristic
16#390×C73C411yescharacteristic
17C22×#10596833—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C22×Dic146C2×C411yescharacteristic
21C23×Dic73D412yes—
22#14556578—22no—
23#14556578C814yescomplemented
24#14556556—42no—
25D4×Dic73—42no—
26C2×C4⋊Dic73C2311yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—22no—
29C2×C22⋊Dic73—22no—
30C2×C22⋊Dic73D412yes—
31C2×C4×Dic73—21no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34#14556498—21no—
35#14556498—21no—
36#14556498—22no—
37C22×D4×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C22⋊C4×C73—22no—
40C2×C42×C73C2×C411yescharacteristic
41C2×#10596833—44no—
42C2×#10596833—84no—
43C2×Dic146—21no—
44C2×Dic146—41no—
45C2×Dic146—41no—
46C2×Dic146C2×C812yescomplemented
47C22×Dic73—21no—
48C22×Dic73—21no—
49C22×Dic73—22no—
50C22×Dic73—22no—
51C22×Dic73—82no—
52C22×Dic73C22⋊C411yescharacteristic
53C22×Dic73C22⋊C411yescharacteristic
54C22×Dic73C2×D412yes—
55C4⋊Dic73—41no—
56#11924007—42no—
57#11924007—82no—
58C22⋊Dic73—28no—
59C22⋊Dic73—42no—
60C4×Dic73—21no—
61C4×Dic73—22no—
62C4×Dic73—42no—
63C4×Dic73C2×C812yescomplemented
64C24×C73C22⋊C412yes—
65C2×D4×C73—41no—
66C2×D4×C73—41no—
67C2×D4×C73C2×C812yes—
68C2×D4×C73M1612yes—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73C22×C411yescharacteristic
72C4⋊C4×C73—41no—
73C22⋊C4×C73—82no—
74C42×C73—41no—
75C23×C73—22no—
76C23×C73—22no—
77C23×C73—42no—
78C23×C73—42no—
79C23×C73C22⋊C814yes—
80C23×C73C2×C22⋊C411yescharacteristic
81D4×C73—42no—
82D4×C73—42no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—41no—
86C2×C4×C73—42no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73C22×C811yescharacteristic
91C2×C4×C73C2×M1611yescharacteristic
92C2×Dic73—22no—
93C2×Dic73—22no—
94C2×Dic73—21no—
95C2×Dic73—22no—
96C2×Dic73—41no—
97C2×Dic73—44no—
98C2×Dic73—41no—
99C2×Dic73—41no—
100C2×Dic73—41no—
101C2×Dic73—41no—
102C2×Dic73—42no—
103C2×Dic73—41no—
104C2×Dic73—82no—
105C2×Dic73#9912yes—
106Dic146—42no—
107Dic146—42no—
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73—24no—
112C22×C73—44no—
113C22×C73—42no—
114C22×C73—42no—
115C22×C73—82no—
116C22×C73C2×C22⋊C811yescharacteristic
117C22×C73C2×#9911yescharacteristic
118C22×C73#41311yescharacteristic
119C4×C73—21no—
120C4×C73—21no—
121C4×C73—81no—
122C4×C73—81no—
123Dic73—22no—
124Dic73—42no—
125Dic73—42no—
126Dic73—41no—
127Dic73—42no—
128#11481—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73—44no—
132C2×C73#145911yescharacteristic
133C2×C73#212711yescharacteristic
134C2×C73#213511yescharacteristic
135#2326—731no—
136#1459—732no—
137#1325—732no—
138#1319—732no—
139C73#1148111yescharacteristic, complemented, Sylow
140C2×#124—731no—
141C2×C4×Q8—731no—
142C2×C4×D4—731no—
143C2×C22⋊C8—1462no—
144#391—731no—
145#390#6440811yescharacteristic
146#388—1462no—
147#385—1462no—
148#384—731no—
149#376—731no—
150#324—734no—
151#323—1462no—
152C22×Q8—731no—
153C22×D4C2×#6440811yescharacteristic
154C23×C4—732no—
155C22×C8—2922no—
156#124—734no—
157#124—1462no—
158C4×Q8—2922no—
159C4×D4—2922no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4—1461no—
162C2×C4⋊C4—1462no—
163C2×C4⋊C4C2×#6440811yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—732no—
166C2×C22⋊C4—1462no—
167C2×C42—731no—
168C2×C42—731no—
169C2×C42—1461no—
170C2×C42C2×#6440811yescharacteristic
171C22⋊C8—2922no—
172C22⋊C8—1464no—
173#95—1461no—
174#95—1461no—
175#95—1462no—
176C24C22⋊#6440812yes—
177C2×Q8—2921no—
178C2×Q8—2921no—
179C2×Q8—732no—
180C2×Q8—1461no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#1192403612yes—
184C2×D4#1192403812yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—5842no—
188C22×C4—731no—
189C22×C4—731no—
190C22×C4—732no—
191C22×C4—1462no—
192C22×C4—1461no—
193C22×C4—1461no—
194C22×C4—1462no—
195C22×C4C22×#6440811yescharacteristic
196C2×C8—2924no—
197C2×C8—5844no—
198C4⋊C4—41no—
199C4⋊C4—2921no—
200C4⋊C4—2922no—
201C4⋊C4—5842no—
202C22⋊C4—82no—
203C22⋊C4—2922no—
204C22⋊C4—1468no—
205C42—41no—
206C42—2922no—
207C42—732no—
208C42—1462no—
209C42—1461no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23#1455664814yes—
215C23C2×C22⋊#6440811yescharacteristic
216Q8—2922no—
217Q8—2922no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—2921no—
228C2×C4—2921no—
229C2×C4—2921no—
230C2×C4—2921no—
231C2×C4—2924no—
232C2×C4—2922no—
233C2×C4—2921no—
234C2×C4—2921no—
235C2×C4—5842no—
236C2×C4—732no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4—1461no—
240C2×C4—1462no—
241C2×C4C2×#1192403611yescharacteristic
242C2×C4C2×#1192403811yescharacteristic
243C8—5844no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×#1455664811yescharacteristic
252C22D73⋊#9911yescharacteristic
253C22#1798957411yescharacteristic
254C4—21no—
255C4—21no—
256C4—81no—
257C4—81no—
258C4—2922no—
259C4—2922no—
260C4—2922no—
261C4—2921no—
262C4—1462no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2262516911yescharacteristic
267C2#2262620711yescharacteristic
268C2#2262623411yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC2×C4⋊Dic73C23
hypercenter#390#64408
hypocenterC73#11481
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #22624874C2semidirect
3. #22624862C2semidirect
23. #14556578C8semidirect
46. C2×Dic146C2×C8semidirect
63. C4×Dic73C2×C8semidirect
139. C73#11481semidirect

Maximal subgroups

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

subgroupindex
#226251672
#226248742
#226248622
#226126412
#1148173