Finite Group Databasebeta

← #32995487

Subgroups — #32995487

Order 18688.

334 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 71 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#22618637C211yesmaximal, characteristic
2#22611744C212yesmaximal, complemented
3#22607869C211yesmaximal, characteristic, complemented
4#22607866C212yesmaximal, complemented
5C73×#1321C211yesmaximal, characteristic, complemented
6C2×#14556598C411yescharacteristic
7C22×#11924011C411yescharacteristic
8#17989094C411yescharacteristic
9#17988662—22no—
10#17988655—22no—
11#17988613C411yescharacteristic
12#17988550C2211yescharacteristic
13#17988545C2211yescharacteristic
14#17988407—22no—
15#17988369C2212yes—
16#17988364—22no—
17#408×C73C2211yescharacteristic
18#349×C73C2212yes—
19C22×C4×D73C2×C411yescharacteristic
20#14556598—42no—
21C2×#11924012—21no—
22C2×#11924012C2×C411yescharacteristic
23C2×#11924011—21no—
24C2×#11924011—21no—
25C2×#11924011—42no—
26C22×#10596831—21no—
27C22×#10596831C2×C411yescharacteristic
28#14556572—21no—
29#14556567—21no—
30#14556562—21no—
31#14556559—21no—
32C4×C22⋊D73—22no—
33#14556550C2×C411yescharacteristic
34#14556531—22no—
35#14556520—22no—
36#14556503—22no—
37#14556501D412yes—
38C8×Dic73D412yes—
39C2×#41×C73C2×C411yescharacteristic
40C2×M16×C73C2311yescharacteristic
41C22×C8×C73C2×C411yescharacteristic
42M32×C73—22no—
43C22⋊C8×C73—22no—
44C23×D73M1611yescharacteristic
45C2×C22⋊D73—22no—
46C22×Dic73M1611yescharacteristic
47C2×C4×D73—21no—
48C2×C4×D73—21no—
49C2×C4×D73—41no—
50C2×C4×D73—41no—
51#11924012—42no—
52#11924012—41no—
53#11924011—42no—
54#11924011—42no—
55C2×#10596831—22no—
56C2×#10596831—21no—
57C2×#10596831—42no—
58C2×#10596831C22⋊C411yescharacteristic
59C2×#10596831C22⋊C411yescharacteristic
60C4⋊Dic73—21no—
61#11924007—22no—
62C22⋊Dic73—22no—
63C22⋊Dic73—21no—
64#11924005—22no—
65C4×Dic73C22⋊C412yes—
66#11924003—42no—
67#41×C73—41no—
68#41×C73—41no—
69C2×Q8×C73M1611yescharacteristic
70C2×D4×C73—21no—
71C2×D4×C73M1611yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73C22×C411yescharacteristic
74M16×C73—41no—
75C2×C8×C73—21no—
76C2×C8×C73—41no—
77C2×C8×C73C2×D412yes—
78C16×C73—42no—
79C23×C73—21no—
80C23×C73C2×M1611yescharacteristic
81C22×D73—21no—
82C22×D73—41no—
83C22×D73—81no—
84Q8×C73—41no—
85D4×C73—41no—
86D4×C73—42no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73C2×C22⋊C411yescharacteristic
93C2×C4×C73C2×M1611yescharacteristic
94C22⋊D73—82no—
95C2×Dic73—22no—
96C2×Dic73—21no—
97C2×Dic73—41no—
98C4×D73—42no—
99C4×D73—41no—
100C4×D73—41no—
101C8×C73—22no—
102C8×C73—41no—
103#10596831—21no—
104#10596831—41no—
105#10596831#10012yes—
106C22×C73—21no—
107C22×C73—41no—
108C22×C73—42no—
109C22×C73—41no—
110C22×C73#40711yescharacteristic
111C2×D73—41no—
112C2×D73—41no—
113C2×D73—41no—
114C2×D73—82no—
115C4×C73—21no—
116C4×C73—41no—
117C4×C73#41011yescharacteristic
118C4×C73C2×#10011yescharacteristic
119Dic73—42no—
120Dic73—41no—
121#11538—731nomaximal, Sylow
122C2×C73—21no—
123C2×C73—41no—
124C2×C73—81no—
125C2×C73#146611yescharacteristic
126D73—81no—
127#2930—731no—
128#2166—732no—
129#1321—731no—
130#1321D7311yescharacteristic, complemented
131#1320—732no—
132C73#1153811yescharacteristic, complemented, Sylow
133C2×#131—731no—
134C22×M16—731no—
135#525—731no—
136#436—1462no—
137#435—1462no—
138#424—731no—
139#408C2×D7311yescharacteristic
140#407—731no—
141#405—731no—
142#364—1462no—
143#349—732no—
144#349C2×D7312yes—
145#347—1462no—
146C2×#41C4×D7311yescharacteristic
147C23×C4—731no—
148#131—2922no—
149C2×M16—2922no—
150C2×M16—731no—
151C2×M16—1461no—
152C2×M16—1461no—
153C2×M16—1461no—
154C2×M16C22×D7311yescharacteristic
155C22×C8—731no—
156C22×C8—1461no—
157C22×C8C4×D7311yescharacteristic
158#123—1461no—
159C22⋊Q8—1461no—
160#121—1461no—
161#120—1461no—
162C4×D4—1462no—
163#117—731no—
164M32—22no—
165M32—1462no—
166C4⋊C8—1462no—
167C22⋊C8—22no—
168C22⋊C8—1462no—
169#97—732no—
170C4×C8—732no—
171C24—731no—
172#41—41no—
173#41—41no—
174C2×Q8#1192401111yescharacteristic
175C2×D4—21no—
176C2×D4—1462no—
177C2×D4#1192401111yescharacteristic
178C22×C4—21no—
179C22×C4—2921no—
180C22×C4—2921no—
181C22×C4—731no—
182C22×C4—1461no—
183C22×C4—1461no—
184C22×C4C2×C4×D7311yescharacteristic
185M16—41no—
186M16—2922no—
187M16—2922no—
188M16—2922no—
189M16—2921no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—2922no—
193C2×C8—731no—
194C2×C8—731no—
195C2×C8—1461no—
196C2×C8—1462no—
197C2×C8D4×D7312yes—
198C4⋊C4—1462no—
199C4⋊C4—1461no—
200C22⋊C4—1462no—
201C22⋊C4—1462no—
202C22⋊C4—1461no—
203C42—732no—
204C16—42no—
205C16—2922no—
206C23—21no—
207C23—2921no—
208C23—5841no—
209C23—1461no—
210C23M16×D7311yescharacteristic
211Q8—41no—
212D4—41no—
213D4—42no—
214D4—5842no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—2922no—
221C2×C4—2921no—
222C2×C4—2921no—
223C2×C4—2921no—
224C2×C4—1462no—
225C2×C4—1461no—
226C2×C4C22⋊C4×D7311yescharacteristic
227C2×C4C2×#1192401111yescharacteristic
228C8—22no—
229C8—41no—
230C8—2921no—
231C8—732no—
232C8—1461no—
233C22—21no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—2921no—
238C22—2921no—
239C22—2921no—
240C22—5842no—
241C22#1798854911yescharacteristic
242C4—21no—
243C4—41no—
244C4—2922no—
245C4—2921no—
246C4#1798855911yescharacteristic
247C4#100×D7311yescharacteristic
248C2—21no—
249C2—41no—
250C2—81no—
251C2—5841no—
252C2#2260827011yescharacteristic
subgroupG/H
centerC4#17988559
commutatorC22×C4×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#1321C2
socleC2×C73#1466
prime radical#17989094C4
cosocleC2×M16×C73C23
hypercenter#1321D73
hypocenterC73#11538
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #22611744C2semidirect
3. #22607869C2semidirect
4. #22607866C2semidirect
5. C73×#1321C2semidirect
130. #1321D73semidirect
132. C73#11538semidirect

Maximal subgroups

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

subgroupindex
#226186372
#226117442
#226078692
#226078662
C73×#13212
#1153873