Finite Group Databasebeta

← #34003828

Subgroups — #34003828

Order 18688.

436 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 104 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#22625434C211yesmaximal, characteristic
2#22625385C211yesmaximal, characteristic
3#22625140C212yesmaximal, complemented
4#22624815C212yesmaximal, complemented
5#22618639C211yesmaximal, characteristic
6#17989604C2212yes—
7C4×#11924037C2212yes—
8#17989447C412yes—
9#17989445C412yes—
10#17989443C412yes—
11#17989442C412yes—
12#17989429—22no—
13#17989428—22no—
14C2×#14556599C411yescharacteristic
15C2×#14556598C411yescharacteristic
16#17989133C2211yescharacteristic
17#17988662—22no—
18#17988650—22no—
19#17988610C2211yescharacteristic
20#17988553C411yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22#408×C73C411yescharacteristic
23C2×#11924037—22no—
24#14556658C2×C414yes—
25C22⋊#10596833—22no—
26#14556646C2×C414yes—
27C4×#10596833—22no—
28C4×#10596833—22no—
29C2×#11924028C2×C411yescharacteristic
30#14556599—42no—
31#14556598—42no—
32C2×#11924012—21no—
33C2×#11924012C2×C411yescharacteristic
34C2×#11924011—21no—
35C22×#10596831—21no—
36C22×#10596831C2×C411yescharacteristic
37C2×C8×D73—21no—
38#14556589C2×C411yescharacteristic
39#14556587C2×C411yescharacteristic
40#14556580C2×C411yescharacteristic
41#14556578C2×C411yescharacteristic
42#14556563—22no—
43C4×C22⋊D73—22no—
44C2×C4×Dic73C2311yescharacteristic
45#14556520—22no—
46#14556503—22no—
47C8×Dic73D412yes—
48C8×Dic73D412yes—
49C2×#41×C73C2×C411yescharacteristic
50C2×M16×C73C2×C411yescharacteristic
51C22×C8×C73C2×C411yescharacteristic
52C22⋊C8×C73—22no—
53#11924037—44no—
54C2×#10596833—24no—
55C2×#10596833—24no—
56#11924028—41no—
57#11924028—41no—
58C2×Dic146C4211yescharacteristic
59C2×D292C4211yescharacteristic
60C2×C22⋊D73—21no—
61C2×C22⋊D73—22no—
62C22×Dic73#4112yes—
63C2×C4×D73—21no—
64#11924012—42no—
65#11924012—41no—
66#11924011—42no—
67C2×#10596831—22no—
68C2×#10596831—21no—
69C2×#10596831—42no—
70C2×#10596831C22⋊C411yescharacteristic
71C2×#10596831C22⋊C411yescharacteristic
72C8×D73—42no—
73#11924007—22no—
74C22⋊Dic73—22no—
75#11924005—22no—
76C4×Dic73C22×C411yescharacteristic
77C4×Dic73C22×C411yescharacteristic
78C4×Dic73C2×D412yes—
79#41×C73—41no—
80#41×C73—41no—
81C2×Q8×C73C4211yescharacteristic
82C2×D4×C73—21no—
83C2×D4×C73C4211yescharacteristic
84C22×C4×C73—21no—
85C22×C4×C73C22×C411yescharacteristic
86M16×C73—41no—
87C2×C8×C73—21no—
88C2×C8×C73—41no—
89C2×C8×C73C22⋊C412yes—
90C23×C73—21no—
91C23×C73#11711yescharacteristic
92C22×D73—21no—
93Q8×C73—41no—
94D4×C73—41no—
95D4×C73—42no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73C2×C4211yescharacteristic
102C2×C4×C73C2×C22⋊C411yescharacteristic
103C22⋊D73—42no—
104C22⋊D73—44no—
105C2×Dic73—24no—
106C2×Dic73—21no—
107C2×Dic73C4×D412yes—
108C2×Dic73C4×D412yes—
109D292—41no—
110C4×D73—41no—
111C4×D73—41no—
112Dic146—41no—
113#10596833—44no—
114#10596833—44no—
115C8×C73—22no—
116C8×C73—41no—
117#10596831—21no—
118#10596831—41no—
119#10596831#10012yes—
120C22×C73—21no—
121C22×C73—22no—
122C22×C73—41no—
123C22×C73—42no—
124C22×C73C4×C22⋊C411yescharacteristic
125C2×D73—41no—
126C2×D73—42no—
127C4×C73—21no—
128C4×C73—41no—
129C4×C73C2×#10011yescharacteristic
130C4×C73#41311yescharacteristic
131Dic73—22no—
132Dic73—41no—
133Dic73#44414yes—
134#8243—731nomaximal, Sylow
135C2×C73—41no—
136C2×C73—81no—
137C2×C73C4×#10011yescharacteristic
138C2×C73#176512yes—
139D73—81no—
140#2931—731no—
141#1543—731no—
142#1525—731no—
143#1451—732no—
144#1299—732no—
145C73#824311yescharacteristic, complemented, Sylow
146C2×#131—731no—
147C2×#131—731no—
148#532—731no—
149#436—1462no—
150C8×D4—1462no—
151#431—732no—
152#423—731no—
153#408—731no—
154#408#6440811yescharacteristic
155C4×M16—732no—
156C2×C4×C8—731no—
157#333—732no—
158#331—732no—
159#329—732no—
160#329—732no—
161#322—1462no—
162#322—1462no—
163C2×#41—731no—
164C2×#41C2×#6440811yescharacteristic
165#131—2922no—
166#131—2922no—
167C2×M16—731no—
168C2×M16—1461no—
169C2×M16—1461no—
170C2×M16—1462no—
171C2×M16C2×#6440811yescharacteristic
172C22×C8—731no—
173C22×C8—1461no—
174C22×C8—1461no—
175C22×C8C2×#6440811yescharacteristic
176C4⋊Q8—731no—
177#127—731no—
178#124—731no—
179#124—731no—
180#121—1462no—
181C4×D4—1462no—
182C2×C42—731no—
183C4⋊C8—734no—
184C4⋊C8—1462no—
185C22⋊C8—22no—
186C22⋊C8—1462no—
187C22⋊C8—1462no—
188#97—734no—
189C4×C8—732no—
190C4×C8—732no—
191C4×C8—1462no—
192C4×C8—1462no—
193#41—41no—
194#41—41no—
195#41—2921no—
196#41—2921no—
197C2×Q8—731no—
198C2×Q8C4×#6440811yescharacteristic
199C2×D4—21no—
200C2×D4—731no—
201C2×D4—1462no—
202C2×D4—1461no—
203C2×D4C4×#6440811yescharacteristic
204C22×C4—21no—
205C22×C4—732no—
206C22×C4—1461no—
207C22×C4C22×#6440811yescharacteristic
208M16—41no—
209M16—2922no—
210M16—2924no—
211M16—2922no—
212M16—2921no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—2922no—
216C2×C8—2922no—
217C2×C8—731no—
218C2×C8—731no—
219C2×C8—1464no—
220C2×C8—1464no—
221C2×C8—1461no—
222C2×C8—1462no—
223C2×C8C22⋊#6440812yes—
224C4⋊C4—1462no—
225C22⋊C4—1462no—
226C22⋊C4—1462no—
227C42—731no—
228C42—731no—
229C42—732no—
230C23—21no—
231C23—1461no—
232C23#1455667011yescharacteristic
233Q8—41no—
234Q8—2921no—
235D4—41no—
236D4—42no—
237D4—2921no—
238D4—2922no—
239D4—2924no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—2921no—
246C2×C4—2921no—
247C2×C4—732no—
248C2×C4—732no—
249C2×C4—1461no—
250C2×C4—1464no—
251C2×C4C2×C4×#6440811yescharacteristic
252C2×C4C2×C22⋊#6440811yescharacteristic
253C8—22no—
254C8—41no—
255C8—2924no—
256C8—2924no—
257C8—2921no—
258C8—732no—
259C8—1461no—
260C22—21no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—2921no—
265C22—2922no—
266C22C4×C22⋊#6440811yescharacteristic
267C4—21no—
268C4—41no—
269C4—2921no—
270C4—734no—
271C4—1462no—
272C4D73⋊#10011yescharacteristic
273C4#1798957411yescharacteristic
274C2—41no—
275C2—81no—
276C2—5841no—
277C2#2262568111yescharacteristic
278C2#2262570912yes—
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#408×C73C4
socleC22×C73C4×C22⋊C4
prime radical#17989133C22
cosocleC2×C4×Dic73C23
hypercenter#408#64408
hypocenterC73#8243
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #408

Complements

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

NH ≅ G/Ntype
3. #22625140C2semidirect
4. #22624815C2semidirect
145. C73#8243semidirect

Maximal subgroups

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

subgroupindex
#226254342
#226253852
#226251402
#226248152
#226186392
#824373