Finite Group Databasebeta

← #34000617

Subgroups — #34000617

Order 18688.

388 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C4×#14556651C211yesmaximal, characteristic
2C4×#14556649C211yesmaximal, characteristic
3#22610228C211yesmaximal, characteristic
4C2×#14556651—21no—
5C2×#14556649—21no—
6C4×#11924037—21no—
7C4×C22⋊#64408—21no—
8#17989467C412yescomplemented
9#17989465C412yes—
10C2×C4×C22⋊D73C2211yescharacteristic
11C2×C22⋊#10596831C411yescharacteristic, complemented
12C2×#14556503—21no—
13#17988341—21no—
14C22×M16×C73C411yescharacteristic, complemented
15C2×#11924037—22no—
16C2×C22⋊#64408—22no—
17C2×C4×#64408—41no—
18#14556651—24no—
19#14556649—24no—
20#14556646—41no—
21C4×#10596833—41no—
22(C2×#64408)⋊C4—41no—
23C22×C22⋊D73D411yescharacteristic
24C22×C4×D73D411yescharacteristic
25C22×#10596831—21no—
26C4×C22⋊D73—41no—
27C4×C22⋊D73—41no—
28C2×#11924007C2×C411yescharacteristic
29C2×C22⋊Dic73Q811yescharacteristic
30C2×#11924005C2×C411yescharacteristic
31C2×C4×Dic73D411yescharacteristic
32C22⋊#10596831—41no—
33#14556503—42no—
34C23×C4×C73C2×C411yescharacteristic
35C2×M16×C73—42no—
36C2×M16×C73—41no—
37C22×C8×C73—21no—
38C22×#64408—42no—
39#11924037—24no—
40#11924037—42no—
41C2×#10596833—42no—
42C22⋊#64408—24no—
43C22⋊#64408—42no—
44C4×#64408—81no—
45C4×#64408—81no—
46C23×D73C22⋊C411yescharacteristic
47C2×C22⋊D73—22no—
48C2×C22⋊D73—22no—
49C2×C22⋊D73—41no—
50C22×Dic73C22⋊C411yescharacteristic
51C22×Dic73C4⋊C411yescharacteristic
52C22×Dic73C4⋊C411yescharacteristic
53C2×C4×D73—21no—
54C2×C4×D73—21no—
55C2×C4×D73—41no—
56C2×C4×D73—41no—
57C2×#10596831—41no—
58C2×#10596831—41no—
59C2×#10596831—41no—
60#11924007—41no—
61C22⋊Dic73—41no—
62#11924005—41no—
63C4×Dic73—21no—
64C4×Dic73—21no—
65C24×C73C4⋊C411yescharacteristic
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—41no—
69C22×C4×C73—41no—
70C22×C4×C73C4211yescharacteristic
71C22×C4×C73C22⋊C411yescharacteristic
72M16×C73—42no—
73M16×C73—42no—
74C2×C8×C73—22no—
75C2×C8×C73—41no—
76C2×C8×C73—41no—
77C23×C73—21no—
78C23×C73—21no—
79C23×C73—21no—
80C23×C73—41no—
81C23×C73—41no—
82C23×C73#9511yescharacteristic
83C22×D73—21no—
84C22×D73—21no—
85C22×D73—41no—
86C22×D73—41no—
87C22×D73#10412yes—
88C2×#64408—44no—
89C2×#64408—82no—
90C2×#64408—82no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—42no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73#9911yescharacteristic
99C2×C4×C73#9911yescharacteristic
100C22⋊D73—44no—
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—21no—
104C2×Dic73—21no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73#10412yes—
108C4×D73—41no—
109C4×D73—41no—
110C4×D73—81no—
111#10596833—44no—
112C8×C73—42no—
113#10596831—81no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73—41no—
117C22×C73—41no—
118C22×C73—41no—
119C22×C73—81no—
120C22×C73—81no—
121C22×C73#33911yescharacteristic
122C22×C73#33911yescharacteristic
123C22×C73#34211yescharacteristic
124C2×D73—22no—
125C2×D73—41no—
126C2×D73—41no—
127C2×D73—42no—
128C2×D73—81no—
129#64408—84no—
130C4×C73—21no—
131C4×C73—41no—
132C4×C73AΣL(1,𝔽16)11yescharacteristic
133C4×C73#35211yescharacteristic
134Dic73—22no—
135Dic73—41no—
136Dic73—41no—
137#7282—731nomaximal, Sylow
138C2×C73—21no—
139C2×C73—21no—
140C2×C73—81no—
141C2×C73#129711yescharacteristic
142C2×C73#139011yescharacteristic
143C2×C73#139511yescharacteristic
144D73—42no—
145#1870—731no—
146C4×#100—731no—
147C4×#99—731no—
148C73#728211yescharacteristic, complemented, Sylow
149C22×M16#6440811yescharacteristic, complemented
150C2×C4×D4—731no—
151C2×#100—1461no—
152C2×#99—1461no—
153C2×C22⋊C8—731no—
154C2×C22⋊C8—1461no—
155C4×M16—1461no—
156C4×C22⋊C4—1461no—
157#343—732no—
158#342—732no—
159#336—1461no—
160C22×D4—731no—
161C23×C4—731no—
162C23×C4C2×#6440811yescharacteristic
163C2×M16—42no—
164C2×M16—41no—
165C2×M16—1462no—
166C22×C8—21no—
167C22×C8—1461no—
168C4×D4—2921no—
169C4×D4—2921no—
170C2×C4⋊C4—731no—
171C2×C22⋊C4—731no—
172C2×C22⋊C4—731no—
173C2×C22⋊C4—1462no—
174C2×C42—2921no—
175C2×C42—731no—
176#100—1464no—
177#99—1464no—
178C22⋊C8—2922no—
179C22⋊C8—2921no—
180#97—2921no—
181C4×C8—2921no—
182#95—2921no—
183C24—731no—
184C24C4⋊#6440811yescharacteristic
185C2×D4—2921no—
186C2×D4—1462no—
187C2×D4—1462no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—2921no—
193C22×C4—2921no—
194C22×C4—2922no—
195C22×C4—731no—
196C22×C4—731no—
197C22×C4—731no—
198C22×C4—1461no—
199C22×C4—1461no—
200C22×C4C4×#6440811yescharacteristic
201C22×C4C22⋊#6440811yescharacteristic
202M16—42no—
203M16—42no—
204M16—2922no—
205M16—1464no—
206C2×C8—22no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—2922no—
210C2×C8—2921no—
211C2×C8—2921no—
212C2×C8—2921no—
213C4⋊C4—2921no—
214C22⋊C4—2921no—
215C22⋊C4—2922no—
216C22⋊C4—2921no—
217C22⋊C4—1464no—
218C42—5841no—
219C42—5841no—
220C42—1461no—
221C42—1461no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—2921no—
228C23—2921no—
229C23—732no—
230C23—1461no—
231C23—1461no—
232C23(C2×#64408)⋊C411yescharacteristic
233D4—2924no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—2921no—
245C2×C4—2924no—
246C2×C4—5841no—
247C2×C4—5842no—
248C2×C4—5842no—
249C2×C4—732no—
250C2×C4—1461no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4—1461no—
254C2×C4#1455664911yescharacteristic
255C2×C4#1455665011yescharacteristic
256C8—42no—
257C8—2924no—
258C8—5841no—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—2922no—
267C22—2921no—
268C22—2921no—
269C22—5841no—
270C22—1462no—
271C22#1798945911yescharacteristic
272C22#1798945911yescharacteristic
273C22#1798946511yescharacteristic
274C4—21no—
275C4—41no—
276C4—2921no—
277C4—2921no—
278C4—5844no—
279C4—1462no—
280C4#1798948711yescharacteristic
281C4#1798948911yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—2922no—
286C2#2262481111yescharacteristic
287C2#2262501511yescharacteristic
288C2#2262503211yescharacteristic
subgroupG/H
centerC2×C4#14556649
commutatorC22×C4×C73C42
FrattiniC23×C4C2×#64408
FittingC22×M16×C73C4
socleC22×C73#342
prime radicalC22×C22⋊D73D4
cosocleC2×C4×C22⋊D73C22
hypercenterC22×M16#64408
hypocenterC73#7282
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
8. #17989467C4semidirect
11. C2×C22⋊#10596831C4semidirect
14. C22×M16×C73C4semidirect
148. C73#7282semidirect
149. C22×M16#64408semidirect

Maximal subgroups

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

subgroupindex
C4×#145566512
C4×#145566492
#226102282
#728273