Finite Group Databasebeta

← #33004187

Subgroups — #33004187

Order 18688.

438 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 111 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#22618627C211yesmaximal, characteristic
2#22611560C212yesmaximal, complemented
3#22611541C212yesmaximal, complemented
4#22611536C211yesmaximal, characteristic, complemented
5C73×#2153C211yesmaximal, characteristic, complemented
6C2×#14556597C2211yescharacteristic
7C22×C8×D73C2211yescharacteristic
8#17989099C411yescharacteristic
9C2×#14556531C2212yes—
10#17988660—22no—
11C8×D292—22no—
12#17988614C411yescharacteristic
13#17988555C411yescharacteristic
14#17988550C411yescharacteristic
15#17988372—24no—
16#17988366—24no—
17#17988365—22no—
18C2×M32×C73C2212yes—
19#405×C73C2211yescharacteristic
20#364×C73—22no—
21C2×#11924028C2×C411yescharacteristic
22C22×C4×D73C2×C411yescharacteristic
23#14556597—22no—
24#14556597—22no—
25C2×#11924012C2×C411yescharacteristic
26C2×#11924011—21no—
27C22×#10596831C2×C411yescharacteristic
28C2×C8×D73—22no—
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73—22no—
33#14556575—21no—
34#14556571—21no—
35#14556566—21no—
36#14556561—21no—
37C4×D292—22no—
38#14556531—22no—
39#14556531—22no—
40C2×#11924003D414yescomplemented
41C4⋊#10596831—22no—
42#14556503—22no—
43#14556503—22no—
44C2×M16×C73C2×C411yescharacteristic
45C22×C8×C73C2311yescharacteristic
46#117×C73C2×C411yescharacteristic
47M32×C73—22no—
48M32×C73—22no—
49C2×C16×C73—22no—
50#97×C73D412yes—
51C4×C8×C73D412yes—
52C23×D73M1611yescharacteristic
53#11924028—22no—
54#11924028—22no—
55C2×Dic146M1611yescharacteristic
56C2×D292—22no—
57C2×D292M1611yescharacteristic
58C2×C22⋊D73—21no—
59C22×Dic73M1611yescharacteristic
60C2×C4×D73—21no—
61C2×C4×D73—21no—
62C2×C4×D73—22no—
63C2×C4×D73—21no—
64C2×C4×D73—22no—
65#11924012—22no—
66#11924011—42no—
67C2×#10596831—21no—
68C2×#10596831—21no—
69C2×#10596831—22no—
70C8×D73—42no—
71C8×D73—41no—
72C8×D73—41no—
73C8×D73—41no—
74C8×D73—41no—
75C4⋊Dic73—22no—
76#11924005—22no—
77#11924005—22no—
78#11924003—24no—
79C22×C4×C73C22×C411yescharacteristic
80M16×C73—22no—
81C2×C8×C73C22⋊C412yes—
82C2×C8×C73C2×D412yes—
83C2×C8×C73C2×D411yescharacteristic
84C2×C8×C73C2×D411yescharacteristic
85C2×C8×C73#4112yes—
86C4⋊C4×C73—21no—
87C22⋊C4×C73—21no—
88C42×C73C22⋊C412yes—
89C16×C73—42no—
90C23×C73C2×M1611yescharacteristic
91C22×D73—21no—
92C22×D73—22no—
93C22×D73—21no—
94C22×D73—42no—
95C2×C4×C73—22no—
96C2×C4×C73C2×C22⋊C411yescharacteristic
97C2×C4×C73C4×D412yes—
98C2×C4×C73C2×M1611yescharacteristic
99C2×C4×C73#13112yes—
100C22⋊D73—42no—
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—22no—
104D292—22no—
105D292—44no—
106C4×D73—41no—
107C4×D73—41no—
108C4×D73—41no—
109C4×D73—41no—
110C4×D73—42no—
111Dic146—22no—
112C8×C73—22no—
113C8×C73#12112yes—
114C8×C73#12411yescharacteristic
115C8×C73#12711yescharacteristic
116#10596831—41no—
117#10596831—41no—
118C22×C73—41no—
119C22×C73#40711yescharacteristic
120C22×C73#43512yes—
121C2×D73—22no—
122C2×D73—24no—
123C2×D73—41no—
124C2×D73—42no—
125C2×D73—41no—
126C2×D73—42no—
127C2×D73—41no—
128C4×C73—42no—
129C4×C73#39011yescharacteristic
130C4×C73#40811yescharacteristic
131C4×C73#43512yes—
132Dic73—41no—
133Dic73—41no—
134#13200—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—22no—
137C2×C73#195511yescharacteristic
138D73—42no—
139D73—81no—
140#2929—731no—
141#2153D7311yescharacteristic, complemented
142#2117—732no—
143#2112—732no—
144#2110—731no—
145C73#1320011yescharacteristic, complemented, Sylow
146C2×#131—731no—
147C23×C8—731no—
148#525—731no—
149C2×M32—732no—
150C2×M32C2×D7312yes—
151#435—1462no—
152C8×D4—1462no—
153#424—731no—
154#408—731no—
155#407—731no—
156#405C2×D7311yescharacteristic
157#364—22no—
158#350—1464no—
159C22⋊C16—1464no—
160#347—1462no—
161C2×#41—731no—
162C23×C4—731no—
163#131—1462no—
164#131—1462no—
165C2×M16—731no—
166C2×M16—1461no—
167C2×M16C4×D7311yescharacteristic
168C22×C8—731no—
169C22×C8—1462no—
170C22×C8—1461no—
171C22×C8—1461no—
172C22×C8—1461no—
173C22×C8—1462no—
174C22×C8C22×D7311yescharacteristic
175#123—1461no—
176C22⋊Q8—1461no—
177#121—1461no—
178#120—1461no—
179C4×D4—1462no—
180#117C4×D7311yescharacteristic
181M32—22no—
182M32—22no—
183M32—1462no—
184M32—1462no—
185C2×C16—22no—
186C2×C16—734no—
187C4⋊C8—1462no—
188C22⋊C8—1462no—
189C22⋊C8—1462no—
190#97D29212yes—
191C4×C8C22⋊D7312yes—
192C24—731no—
193#41—1462no—
194#41—1462no—
195C2×Q8—731no—
196C2×D4—731no—
197C2×D4—1462no—
198C2×D4—1461no—
199C22×C4—731no—
200C22×C4—1461no—
201C22×C4—1461no—
202C22×C4—1462no—
203C22×C4—1461no—
204C22×C4—1462no—
205C22×C4C2×C4×D7311yescharacteristic
206M16—22no—
207M16—2922no—
208M16—1462no—
209C2×C8—2922no—
210C2×C8—2921no—
211C2×C8—2921no—
212C2×C8—2921no—
213C2×C8—2921no—
214C2×C8—1461no—
215C2×C8—1461no—
216C2×C8—1462no—
217C2×C8#1192400512yes—
218C2×C8D4×D7312yes—
219C2×C8C2×C22⋊D7311yescharacteristic
220C2×C8C2×D29211yescharacteristic
221C2×C8#1192402912yes—
222C4⋊C4—21no—
223C4⋊C4—1462no—
224C22⋊C4—21no—
225C22⋊C4—1462no—
226C22⋊C4—1462no—
227C42#1192400512yes—
228C16—42no—
229C16—1464no—
230C23—2922no—
231C23—1461no—
232C23—1462no—
233C23—1461no—
234C23M16×D7311yescharacteristic
235Q8—1462no—
236D4—2924no—
237D4—2922no—
238D4—1462no—
239C2×C4—22no—
240C2×C4—2922no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—2921no—
245C2×C4—1462no—
246C2×C4—1461no—
247C2×C4—1461no—
248C2×C4C2×#1192400511yescharacteristic
249C2×C4#1455655312yes—
250C2×C4M16×D7311yescharacteristic
251C2×C4#1455659712yes—
252C8—22no—
253C8—2921no—
254C8—2921no—
255C8#1455656612yes—
256C8#1455658011yescharacteristic
257C8#1455658711yescharacteristic
258C22—41no—
259C22—2921no—
260C22—2922no—
261C22—2922no—
262C22—2921no—
263C22—2921no—
264C22—1462no—
265C22—1464no—
266C22#1798855011yescharacteristic
267C22#1798865712yes—
268C4—42no—
269C4—2921no—
270C4—2921no—
271C4#1798848811yescharacteristic
272C4#1798855311yescharacteristic
273C4#1798865712yes—
274C2—21no—
275C2—22no—
276C2—2922no—
277C2—5841no—
278C2#2261072811yescharacteristic
subgroupG/H
centerC8#14556587
commutatorC22×C4×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2153C2
socleC2×C73#1955
prime radical#17989099C4
cosocleC22×C8×C73C23
hypercenter#2153D73
hypocenterC73#13200
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #22611560C2semidirect
3. #22611541C2semidirect
4. #22611536C2semidirect
5. C73×#2153C2semidirect
40. C2×#11924003D4semidirect
141. #2153D73semidirect
145. C73#13200semidirect

Maximal subgroups

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

subgroupindex
#226186272
#226115602
#226115412
#226115362
C73×#21532
#1320073