Finite Group Databasebeta

← #34010173

Subgroups — #34010173

Order 18688.

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

normal subgroups only

This group has 440 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#22625188C212yesmaximal
2#22624873C212yesmaximal, complemented
3#22624862C212yesmaximal, complemented
4#22612642C211yesmaximal, characteristic
5#17989558—22no—
6#17989432C2214yes—
7#17989431—22no—
8C2×#14556579C411yescharacteristic
9C2×C4×Dic146C411yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988493C2211yescharacteristic
12#17988489C411yescharacteristic
13#17988479—22no—
14#17988461—22no—
15#17988457C2211yescharacteristic
16#376×C73C411yescharacteristic
17C2×#11924038—42no—
18#14556648—42no—
19C22⋊#10596833—24no—
20C22×Dic146C2×C411yescharacteristic
21C22×D292C2×C411yescharacteristic
22C22×C4×D73D412yes—
23#14556579—22no—
24#14556579—22no—
25C4×Dic146—42no—
26C4×D292—42no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×C4⋊Dic73C2311yescharacteristic
30C2×#11924007—21no—
31C2×#11924005—22no—
32C2×#11924005—22no—
33C2×#11924005D412yes—
34C2×C4×Dic73—21no—
35#14556498—22no—
36C2×C42×C73C2×C411yescharacteristic
37C2×C42×C73C2×C411yescharacteristic
38C2×C42×C73C2×C411yescharacteristic
39#95×C73—21no—
40#95×C73—21no—
41#11924038—84no—
42C2×#10596833—44no—
43C23×D73C22⋊C412yes—
44C2×Dic146—21no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146M1612yescomplemented
48C2×D292—21no—
49C2×D292—21no—
50C2×D292—41no—
51C2×D292—41no—
52C22×Dic73—21no—
53C22×Dic73—21no—
54C22×Dic73C2×D412yes—
55C2×C4×D73—22no—
56C2×C4×D73—22no—
57C2×C4×D73—82no—
58C4⋊Dic73—41no—
59C4⋊Dic73—41no—
60C4⋊Dic73—82no—
61#11924007—42no—
62#11924005—28no—
63#11924005—42no—
64#11924005—82no—
65C4×Dic73—42no—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22×C411yescharacteristic
71C42×C73—22no—
72C42×C73—21no—
73C42×C73—41no—
74C42×C73M1612yescomplemented
75C23×C73C2×C22⋊C411yescharacteristic
76C22×D73—22no—
77C22×D73—22no—
78C22×D73—22no—
79C22×D73—42no—
80C22×D73—42no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—22no—
84C2×C4×C73—21no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73#9912yes—
93C2×C4×C73C2×M1611yescharacteristic
94C2×C4×C73C2×M1611yescharacteristic
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—41no—
98C2×Dic73—41no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—41no—
102D292—42no—
103D292—42no—
104C4×D73—44no—
105C4×D73—82no—
106Dic146—42no—
107Dic146—42no—
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73#40711yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73#41311yescharacteristic
114C2×D73—44no—
115C2×D73—42no—
116C2×D73—42no—
117C2×D73—42no—
118C2×D73—82no—
119C4×C73—22no—
120C4×C73—21no—
121C4×C73—21no—
122C4×C73—42no—
123C4×C73—41no—
124C4×C73—81no—
125Dic73—42no—
126Dic73—42no—
127Dic73—81no—
128#11535—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73#146511yescharacteristic
132C2×C73#212711yescharacteristic
133C2×C73#213511yescharacteristic
134D73—82no—
135#2326—731no—
136#1465—732no—
137#1325—732no—
138#1319—732no—
139C73#1153511yescharacteristic, complemented, Sylow
140C2×#124—731no—
141C2×C4×Q8—731no—
142C2×C4×D4—731no—
143#407—1462no—
144#391—731no—
145#390—731no—
146#388—1462no—
147#385—1462no—
148#384—731no—
149#376#6440811yescharacteristic
150#324—734no—
151#323—1462no—
152C22×Q8—731no—
153C22×D4—731no—
154C23×C4—732no—
155C2×M16—2922no—
156#124—1462no—
157#124—1462no—
158C4×Q8—2922no—
159C4×D4—2922no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4—1462no—
164C2×C22⋊C4—732no—
165C2×C22⋊C4—1462no—
166C2×C22⋊C4—1462no—
167C2×C42—1461no—
168C2×C42C2×#6440811yescharacteristic
169C2×C42C2×#6440811yescharacteristic
170C2×C42C2×#6440811yescharacteristic
171C22⋊C8—2922no—
172C22⋊C8—1464no—
173#95—21no—
174#95—21no—
175#95—1462no—
176C24—732no—
177C2×Q8—2921no—
178C2×Q8—2921no—
179C2×Q8—732no—
180C2×Q8—1461no—
181C2×D4—2921no—
182C2×D4—2921no—
183C2×D4—1461no—
184C2×D4—1461no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—5842no—
188C22×C4—732no—
189C22×C4—1461no—
190C22×C4—1461no—
191C22×C4—1462no—
192C22×C4—1462no—
193C22×C4C22⋊#6440811yescharacteristic
194C22×C4C22⋊#6440811yescharacteristic
195C22×C4C22×#6440811yescharacteristic
196M16—5844no—
197C2×C8—2924no—
198C4⋊C4—2921no—
199C4⋊C4—2921no—
200C4⋊C4—2922no—
201C4⋊C4—5842no—
202C22⋊C4—2922no—
203C22⋊C4—5842no—
204C22⋊C4—1468no—
205C42—21no—
206C42—22no—
207C42—41no—
208C42—2922no—
209C42#1192403812yescomplemented
210C23—2922no—
211C23—2922no—
212C23—1462no—
213C23—1462no—
214C23—1462no—
215C23C2×C22⋊#6440811yescharacteristic
216Q8—2922no—
217Q8—2922no—
218D4—2922no—
219D4—2922no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—2924no—
232C2×C4—2921no—
233C2×C4—2921no—
234C2×C4—2922no—
235C2×C4—2922no—
236C2×C4—2921no—
237C2×C4—5842no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4#1455665012yes—
241C2×C4D73⋊M1611yescharacteristic
242C2×C4C2×#1192403811yescharacteristic
243C8—5844no—
244C22—21no—
245C22—21no—
246C22—2924no—
247C22—2922no—
248C22—2922no—
249C22—2922no—
250C22—5842no—
251C22#1798955811yescharacteristic
252C22C2×#1455665011yescharacteristic
253C22#1798957311yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—2922no—
261C4—2922no—
262C4—5841no—
263C2—21no—
264C2—21no—
265C2—5842no—
266C2#2262518811yescharacteristic
267C2#2262620811yescharacteristic
268C2#2262623511yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#376×C73C4
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊Dic73C23
hypercenter#376#64408
hypocenterC73#11535
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #22624873C2semidirect
3. #22624862C2semidirect
47. C2×Dic146M16semidirect
74. C42×C73M16semidirect
139. C73#11535semidirect
209. C42#11924038semidirect

Maximal subgroups

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

subgroupindex
#226251882
#226248732
#226248622
#226126422
#1153573