Finite Group Databasebeta

← #34013240

Subgroups — #34013240

Order 18688.

342 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 68 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#22625840C212yesmaximal
2#22625032C212yesmaximal
3#22625023C211yesmaximal, characteristic
4#22625020C211yesmaximal, characteristic
5#22618576C211yesmaximal, characteristic
6#17989573C2212yes—
7#17989566C2212yes—
8#17989458—22no—
9#17989454—22no—
10C4×#11924028C2211yescharacteristic
11#17988649—22no—
12#17988638—22no—
13#17988615C411yescharacteristic, complemented
14#17988615C411yescharacteristic, complemented
15#17988555C2211yescharacteristic
16#17988555C2211yescharacteristic
17C2×C4×#10596831C411yescharacteristic, complemented
18C4×M16×C73C411yescharacteristic, complemented
19D73⋊M16—22no—
20#14556670—22no—
21#14556652—22no—
22#14556651—22no—
23#14556650—22no—
24#14556649—22no—
25C2×#11924028C2311yescharacteristic
26C22×#10596831C2×C411yescharacteristic
27C22×#10596831C2×C411yescharacteristic
28C4×Dic146—21no—
29C4×D292—21no—
30C4×C22⋊D73—22no—
31#14556550C2×C411yescharacteristic
32#14556548D412yes—
33C42×D73D412yes—
34#14556520—22no—
35#14556520—22no—
36#14556503—22no—
37#14556503—22no—
38C4×#10596831—21no—
39C4×#10596831—21no—
40C2×M16×C73C2×C411yescharacteristic
41C2×M16×C73C2×C411yescharacteristic
42C2×C42×C73C2×C411yescharacteristic
43#97×C73—21no—
44C4×C8×C73—21no—
45#11924038—42no—
46#11924037—42no—
47#11924036—42no—
48C4⋊#64408—42no—
49C22⋊#64408—42no—
50C4×#64408—42no—
51#11924028—41no—
52#11924028—41no—
53C2×Dic146#4111yescharacteristic
54C2×D292#4111yescharacteristic
55C2×C22⋊D73#4112yes—
56C2×C4×D73—22no—
57C2×C4×D73C2×D412yes—
58C2×#10596831—21no—
59C2×#10596831—21no—
60C2×#10596831—41no—
61C2×#10596831—41no—
62C4⋊Dic73—21no—
63#11924007—22no—
64C22⋊Dic73—21no—
65#11924005—22no—
66C4×Dic73—22no—
67C4×Dic73C22⋊C412yes—
68C22×C4×C73—21no—
69C22×C4×C73C22×C411yescharacteristic
70M16×C73—41no—
71M16×C73—41no—
72C2×C8×C73—21no—
73C2×C8×C73—21no—
74C42×C73—21no—
75C42×C73C22⋊C411yescharacteristic
76C42×C73C22⋊C411yescharacteristic
77C23×C73#11711yescharacteristic
78C22×D73#12312yes—
79C2×#64408—82no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—42no—
85C2×C4×C73C2×C22⋊C411yescharacteristic
86C2×C4×C73#11711yescharacteristic
87C22⋊D73—42no—
88C2×Dic73—22no—
89C2×Dic73#12312yes—
90D292—41no—
91C4×D73—22no—
92C4×D73—22no—
93C4×D73—44no—
94Dic146—41no—
95#10596833—82no—
96C8×C73—41no—
97C8×C73—41no—
98#10596831—41no—
99#10596831—41no—
100C22×C73—21no—
101C22×C73—41no—
102C22×C73#38111yescharacteristic
103C2×D73—22no—
104C2×D73—42no—
105#64408—162no—
106C4×C73—22no—
107C4×C73—21no—
108C4×C73—41no—
109C4×C73#41011yescharacteristic
110C4×C73#41011yescharacteristic
111Dic73—22no—
112Dic73—42no—
113#12864—731nomaximal, Sylow
114C2×C73—21no—
115C2×C73—41no—
116C2×C73#182011yescharacteristic
117D73—42no—
118#2922—731no—
119#1827—732no—
120#1395—732no—
121#1392—731no—
122#1391—731no—
123C73#1286411yescharacteristic, complemented, Sylow
124C4×#41—731no—
125#433—1462no—
126#431—1462no—
127#424—731no—
128#424—731no—
129#413—732no—
130#410—732no—
131#408—731no—
132#408—731no—
133C4×M16#6440811yescharacteristic, complemented
134C2×C4×C8—731no—
135#339—1462no—
136H(3,ℤ/4)—1462no—
137C2×#41—731no—
138C2×M16—1462no—
139C2×M16C2×#6440811yescharacteristic
140C2×M16C2×#6440811yescharacteristic
141C22×C8—731no—
142C22×C8—731no—
143C4×Q8—1461no—
144C4×D4—1461no—
145C4×D4—1462no—
146#117—731no—
147#117—732no—
148#117—1462no—
149C2×C42—732no—
150C2×C42C2×#6440811yescharacteristic
151C4⋊C8—1462no—
152C4⋊C8—1462no—
153#101—1462no—
154#100—1462no—
155#99—1462no—
156#99—1462no—
157C22⋊C8—1462no—
158C22⋊C8—1462no—
159#97—21no—
160C4×C8—21no—
161C4×C8—1461no—
162C4×C8—1461no—
163#41—2921no—
164#41—2921no—
165C2×Q8—731no—
166C2×D4—732no—
167C2×D4—731no—
168C22×C4—21no—
169C22×C4—732no—
170C22×C4—1462no—
171C22×C4C22×#6440811yescharacteristic
172M16—41no—
173M16—41no—
174M16—2922no—
175M16—2922no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—2921no—
179C2×C8—2921no—
180C2×C8—2922no—
181C2×C8—1461no—
182C2×C8—1461no—
183C4⋊C4—2922no—
184C4⋊C4—1462no—
185C4⋊C4—1461no—
186C22⋊C4—2922no—
187C22⋊C4—1462no—
188C22⋊C4—1461no—
189C42—21no—
190C42—2922no—
191C42—732no—
192C42—1462no—
193C42C22⋊#6440811yescharacteristic
194C42C22⋊#6440811yescharacteristic
195C23—732no—
196C23#1455667011yescharacteristic
197Q8—2921no—
198D4—2921no—
199D4—2922no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—2924no—
206C2×C4—5842no—
207C2×C4—732no—
208C2×C4—1462no—
209C2×C4—1462no—
210C2×C4—1462no—
211C2×C4C2×C22⋊#6440811yescharacteristic
212C2×C4#1455667011yescharacteristic
213C8—41no—
214C8—41no—
215C8—2921no—
216C8—2921no—
217C8—5842no—
218C22—21no—
219C22—41no—
220C22—2922no—
221C22—1462no—
222C22#1798953011yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—2922no—
227C4—11682no—
228C4—1462no—
229C4#1798956611yescharacteristic
230C4#1798956611yescharacteristic
231C2—21no—
232C2—41no—
233C2—2922no—
234C2#2262582811yescharacteristic
subgroupG/H
centerC4#17989566
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC4×M16×C73C4
socleC2×C73#1820
prime radical, cosocleC2×#11924028C23
hypercenterC4×M16#64408
hypocenterC73#12864
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
13. #17988615C4semidirect
14. #17988615C4semidirect
17. C2×C4×#10596831C4semidirect
18. C4×M16×C73C4semidirect
123. C73#12864semidirect
133. C4×M16#64408semidirect

Maximal subgroups

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

subgroupindex
#226258402
#226250322
#226250232
#226250202
#226185762
#1286473