Finite Group Databasebeta

← #34014362

Subgroups — #34014362

Order 18688.

388 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#17989492C212yesmaximal
2#22625071C212yesmaximal, complemented
3#22625070C212yesmaximal, complemented
4#22620307C211yesmaximal, characteristic
5C2×#14556650—22no—
6#17989492—22no—
7#17989492C2214yes—
8C2×#14556588C2211yescharacteristic
9#17988953C411yescharacteristic, complemented
10#17988953C411yescharacteristic, complemented
11#17988953C2211yescharacteristic
12#17988953C2211yescharacteristic
13#17988935—22no—
14#17988591—22no—
15C4×#11924012C411yescharacteristic, complemented
16#533×C73C411yescharacteristic, complemented
17C2×C22⋊#64408—42no—
18#14556650—24no—
19#14556650—42no—
20C22×D292—41no—
21C22×D292D412yes—
22C2×#11924012—21no—
23#14556588—41no—
24#14556588C2×C411yescharacteristic
25#14556588D412yes—
26#14556588C2311yescharacteristic
27#14556511—22no—
28#14556511—22no—
29#14556511—22no—
30#14556511—22no—
31#14556502C2×C411yescharacteristic, complemented
32#14556502C2×C411yescharacteristic, complemented
33C4×#10596831—21no—
34C4⋊Q8×C73—21no—
35#125×C73C2×C411yescharacteristic, complemented
36#125×C73C2×C411yescharacteristic, complemented
37C22⋊Q8×C73—41no—
38C4×Q8×C73—41no—
39#117×C73—21no—
40C2×C42×C73C2×C411yescharacteristic
41C22×#64408—82no—
42C22⋊#64408—44no—
43C22⋊#64408—82no—
44C23×D73—22no—
45C2×D292—22no—
46C2×D292—42no—
47C2×D292—42no—
48C2×D292—41no—
49C2×D292—42no—
50C2×D292—41no—
51C2×D292—84no—
52C2×D292C22⋊C412yes—
53C2×D292C2×D412yes—
54#11924012—42no—
55C2×#10596831—21no—
56C2×#10596831—21no—
57C2×Q8×C73—41no—
58C22×C4×C73—21no—
59C22×C4×C73C22⋊C411yescharacteristic
60C4⋊C4×C73—21no—
61C4⋊C4×C73—21no—
62C4⋊C4×C73—41no—
63C4⋊C4×C73—41no—
64C4⋊C4×C73—41no—
65C22⋊C4×C73—41no—
66C42×C73—21no—
67C42×C73—41no—
68C42×C73C22⋊C411yescharacteristic
69C42×C73C22×C411yescharacteristic
70C23×C73#9911yescharacteristic
71C22×D73—22no—
72C22×D73—22no—
73C22×D73—42no—
74C22×D73—42no—
75C22×D73—84no—
76C2×#64408—84no—
77C2×#64408—164no—
78Q8×C73—82no—
79C2×C4×C73—22no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—42no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73#9911yescharacteristic
86C2×C4×C73C2×C22⋊C411yescharacteristic
87D292—42no—
88D292—42no—
89D292—82no—
90D292—84no—
91D292—82no—
92D292—82no—
93#10596831—41no—
94#10596831—41no—
95C22×C73—41no—
96C22×C73#35412yes—
97C22×C73C2×#9911yescharacteristic
98C2×D73—42no—
99C2×D73—42no—
100C2×D73—84no—
101C2×D73—82no—
102C2×D73—82no—
103C2×D73—162no—
104#64408—164no—
105C4×C73—22no—
106C4×C73—21no—
107C4×C73—21no—
108C4×C73—41no—
109C4×C73—81no—
110C4×C73—81no—
111#13595—731nomaximal, Sylow
112C2×C73—21no—
113C2×C73—22no—
114C2×C73C2×#35411yescharacteristic
115D73—82no—
116D73—82no—
117#3138—731no—
118C2×#353—732no—
119#1412—732no—
120#1412—732no—
121C73#1359511yescharacteristic, complemented, Sylow
122#533#6440811yescharacteristic, complemented
123C2×#127—731no—
124#490—731no—
125#490—731no—
126#490—731no—
127#490—731no—
128#486—1462no—
129#418—1462no—
130C2×#99—1462no—
131C4×M16—731no—
132#353—734no—
133#353—1462no—
134C22×D4—2921no—
135C22×D4—732no—
136C2×M16—1461no—
137C4⋊Q8—21no—
138#127—2921no—
139#127—732no—
140#127—731no—
141#127—731no—
142#125C2×#6440811yescharacteristic, complemented
143#125C2×#6440811yescharacteristic, complemented
144C22⋊Q8—41no—
145C4×Q8—41no—
146#117—21no—
147C2×C22⋊C4—2922no—
148C2×C42C2×#6440811yescharacteristic
149#102—1462no—
150#102—1462no—
151#102—1462no—
152#102—1462no—
153#99—2922no—
154#99—1464no—
155#97—731no—
156#97—731no—
157C4×C8—1461no—
158C24—1462no—
159C2×Q8—41no—
160C2×D4—2922no—
161C2×D4—2922no—
162C2×D4—2922no—
163C2×D4—2921no—
164C2×D4—2921no—
165C2×D4—5844no—
166C2×D4—732no—
167C2×D4—732no—
168C2×D4—1462no—
169C22×C4—21no—
170C22×C4—5842no—
171C22×C4C22⋊#6440811yescharacteristic
172M16—2922no—
173C2×C8—1461no—
174C2×C8—1461no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2924no—
182C22⋊C4—5842no—
183C42—21no—
184C42—41no—
185C42C22⋊#6440811yescharacteristic
186C42C22×#6440811yescharacteristic
187C23—2922no—
188C23—2922no—
189C23—5844no—
190C23—1462no—
191C23—1462no—
192C23#1455664911yescharacteristic
193Q8—82no—
194D4—2922no—
195D4—2922no—
196D4—5842no—
197D4—5844no—
198D4—5842no—
199D4—5842no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—5844no—
207C2×C4—11684no—
208C2×C4#1455664911yescharacteristic
209C2×C4C2×C22⋊#6440811yescharacteristic
210C8—2921no—
211C8—2921no—
212C22—41no—
213C22—2922no—
214C22—2922no—
215C22—5842no—
216C22—5842no—
217C22—5844no—
218C22—11682no—
219C22#1798949312yes—
220C22C2×#1455664911yescharacteristic
221C4—22no—
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—81no—
227C4—11684no—
228C2—21no—
229C2—22no—
230C2—5842no—
231C2—5842no—
232C2C2×#1798949311yescharacteristic
subgroupG/H
centerC2C2×#17989493
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
Fitting#533×C73C4
socleC2×C73C2×#354
prime radicalC2×#14556588C22
cosocle#14556588C23
hypercenter#533#64408
hypocenterC73#13595
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22625071C2semidirect
3. #22625070C2semidirect
9. #17988953C4semidirect
10. #17988953C4semidirect
15. C4×#11924012C4semidirect
16. #533×C73C4semidirect
31. #14556502C2×C4semidirect
32. #14556502C2×C4semidirect
35. #125×C73C2×C4semidirect
36. #125×C73C2×C4semidirect
121. C73#13595semidirect
122. #533#64408semidirect
142. #125C2×#64408semidirect
143. #125C2×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894922
#226250712
#226250702
#226203072
#1359573