Finite Group Databasebeta

← #32999813

Subgroups — #32999813

Order 18688.

350 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612158C211yesmaximal, characteristic, complemented
2#22612078C211yesmaximal, characteristic, complemented
3#22609330C211yesmaximal, characteristic, complemented
4#22607980C211yesmaximal, characteristic, complemented
5#22607892C211yesmaximal, characteristic, complemented
6D4⋊#11924003C211yesmaximal, characteristic, complemented
7C73×#3283C211yesmaximal, characteristic
8#17988992C2211yescharacteristic
9#17988877—21no—
10#17988414—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988391—22no—
13#17988388—21no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988335—21no—
17C2×#135×C73—21no—
18#547×C73—21no—
19#494×C73C2211yescharacteristic
20#467×C73—21no—
21#461×C73—21no—
22#452×C73—22no—
23C4×D8×C73C2211yescharacteristic
24C8×D4×C73C2211yescharacteristic
25#14556591—21no—
26C2×#11924003—21no—
27C2×#11924003—21no—
28#14556525—21no—
29#14556525—21no—
30#14556522—41no—
31C4⋊#10596831—21no—
32#14556512—41no—
33C2×#41×C73—22no—
34#135×C73—42no—
35#135×C73—42no—
36C2×Q16×C73—21no—
37C2×Q16×C73D412yes—
38C2×SD16×C73—22no—
39C2×D8×C73D411yescharacteristic
40C22×C8×C73—21no—
41C4⋊Q8×C73—21no—
42#123×C73—42no—
43C22⋊Q8×C73—22no—
44#121×C73—41no—
45C4×D4×C73—21no—
46C4×D4×C73D411yescharacteristic
47C2×C4⋊C4×C73—41no—
48#107×C73D411yescharacteristic
49C4⋊C8×C73D411yescharacteristic
50#103×C73—22no—
51#102×C73—21no—
52C22⋊C8×C73—21no—
53C4×C8×C73C2311yescharacteristic
54C2×Dic146—41no—
55C2×#10596831—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924003—41no—
59#11924003—41no—
60#41×C73—44no—
61#41×C73—82no—
62C2×Q8×C73—22no—
63C2×D4×C73—21no—
64C2×D4×C73—42no—
65C2×D4×C73D811yescharacteristic
66C22×C4×C73—21no—
67C22×C4×C73—42no—
68C22×C4×C73—41no—
69Q16×C73—42no—
70Q16×C73—41no—
71Q16×C73—41no—
72SD16×C73—44no—
73D8×C73—21no—
74D8×C73SD1612yes—
75C2×C8×C73—21no—
76C2×C8×C73—41no—
77C2×C8×C73—41no—
78C2×C8×C73C2×D411yescharacteristic
79C2×C8×C73C2×D411yescharacteristic
80C4⋊C4×C73—21no—
81C4⋊C4×C73—42no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73D811yescharacteristic
85C22⋊C4×C73—21no—
86C22⋊C4×C73—42no—
87C22⋊C4×C73—41no—
88C42×C73C2×D411yescharacteristic
89C23×C73—21no—
90C23×C73—41no—
91Q8×C73—42no—
92Q8×C73—42no—
93D4×C73—22no—
94D4×C73—21no—
95D4×C73—21no—
96D4×C73—41no—
97D4×C73—84no—
98C2×C4×C73—21no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—42no—
103C2×C4×C73—84no—
104C2×C4×C73—81no—
105C2×C4×C73—81no—
106C2×C4×C73#12011yescharacteristic
107C2×C4×C73C2×D811yescharacteristic
108C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
109C2×Dic73—81no—
110Dic146—161no—
111C8×C73—21no—
112C8×C73—41no—
113C8×C73C2×SD1611yescharacteristic
114C8×C73AGL(1,ℤ/8)11yescharacteristic
115#10596831—81no—
116C22×C73—41no—
117C22×C73—42no—
118C22×C73—41no—
119C22×C73—41no—
120C22×C73—81no—
121C22×C73—81no—
122C22×C73#44711yescharacteristic
123C4×C73—21no—
124C4×C73—41no—
125C4×C73—82no—
126C4×C73—81no—
127C4×C73#45011yescharacteristic
128C4×C73#45511yescharacteristic
129C4×C73#50811yescharacteristic
130C4×C73#51011yescharacteristic
131Dic73—161no—
132#12375—731nomaximal, Sylow
133C2×C73—81no—
134C2×C73—82no—
135C2×C73#163911yescharacteristic
136C2×C73#218911yescharacteristic
137C2×C73#219311yescharacteristic
138#3283D7311yescharacteristic
139#2241—731no—
140#2226—731no—
141#1675—731no—
142#1362—731no—
143#1331—731no—
144D4⋊C16—731no—
145C73#1237511yescharacteristic, complemented, Sylow
146C2×#135—21no—
147#547—21no—
148#500—731no—
149#494C2×D7311yescharacteristic
150#476—1461no—
151#467—21no—
152#461—21no—
153#452—22no—
154C4×D8C2×D7311yescharacteristic
155C8×D4C2×D7311yescharacteristic
156#367—1461no—
157C4⋊C16—731no—
158#358—1462no—
159#357—1461no—
160C4×C16—731no—
161#335—731no—
162#333—1461no—
163C2×#41—22no—
164#135—42no—
165#135—42no—
166C2×Q16—21no—
167C2×Q16C22⋊D7312yes—
168C2×SD16—22no—
169C2×D8C22⋊D7311yescharacteristic
170C22×C8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—1461no—
173#123—42no—
174C22⋊Q8—22no—
175#121—41no—
176C4×D4—21no—
177C4×D4C22⋊D7311yescharacteristic
178C2×C4⋊C4—41no—
179C2×C16—1461no—
180C2×C16—1461no—
181#107—1461no—
182#107—1461no—
183#107C22⋊D7311yescharacteristic
184#106—2921no—
185C4⋊C8—1461no—
186C4⋊C8C22⋊D7311yescharacteristic
187#103—22no—
188#102—21no—
189#102—2921no—
190C22⋊C8—21no—
191C4×C8C22×D7311yescharacteristic
192#41—44no—
193#41—82no—
194C2×Q8—22no—
195C2×Q8—2921no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4#1192401311yescharacteristic
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—41no—
202Q16—42no—
203Q16—41no—
204Q16—41no—
205SD16—44no—
206D8—21no—
207D8#1192401612yes—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—2921no—
212C2×C8C2×C22⋊D7311yescharacteristic
213C2×C8C2×C22⋊D7311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—2921no—
219C4⋊C4—2921no—
220C4⋊C4#1192401311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—42no—
223C22⋊C4—41no—
224C42C2×C22⋊D7311yescharacteristic
225C16—2921no—
226C16—2921no—
227C23—21no—
228C23—41no—
229Q8—42no—
230Q8—42no—
231Q8—11681no—
232D4—22no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—84no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—84no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—5841no—
246C2×C4#1455656011yescharacteristic
247C2×C4C2×#1192401311yescharacteristic
248C2×C4#1455661711yescharacteristic
249C8—21no—
250C8—41no—
251C8—5841no—
252C8C2×#1192401611yescharacteristic
253C8#1455661711yescharacteristic
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22#1798872111yescharacteristic
261C4—21no—
262C4—41no—
263C4—81no—
264C4—82no—
265C4—11681no—
266C4#1798873811yescharacteristic
267C4#1798876511yescharacteristic
268C4#1798901711yescharacteristic
269C4#1798902711yescharacteristic
270C2—81no—
271C2—82no—
272C2#2260911011yescharacteristic
273C2#2261187111yescharacteristic
274C2#2261189511yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
Fitting, prime radicalC73×#3283C2
socleC22×C73#447
hypercenter#3283D73
hypocenterC73#12375
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #22612158C2semidirect
2. #22612078C2semidirect
3. #22609330C2semidirect
4. #22607980C2semidirect
5. #22607892C2semidirect
6. D4⋊#11924003C2semidirect
145. C73#12375semidirect

Maximal subgroups

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

subgroupindex
#226121582
#226120782
#226093302
#226079802
#226078922
D4⋊#119240032
C73×#32832
#1237573