Finite Group Databasebeta

← #32999785

Subgroups — #32999785

Order 18688.

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

normal subgroups only

This group has 356 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#22612068C211yesmaximal, characteristic, complemented
3#22609320C211yesmaximal, characteristic, complemented
4#22607980C211yesmaximal, characteristic, complemented
5#22607892C211yesmaximal, characteristic, complemented
6D4⋊#11924003C211yesmaximal, characteristic, complemented
7C73×#3283C211yesmaximal, characteristic
8#17988992C2211yescharacteristic
9#17988792—21no—
10#17988412—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988391—22no—
13#17988388—21no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988332—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
25C2×#11924016—41no—
26#14556591—21no—
27C2×#11924003—21no—
28C2×#11924003—21no—
29#14556525—21no—
30#14556525—21no—
31C4⋊#10596831—21no—
32#14556514—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—
55#11924016—82no—
56C2×#10596831—41no—
57C4⋊Dic73—41no—
58C4⋊Dic73—41no—
59#11924003—41no—
60#11924003—41no—
61#41×C73—44no—
62#41×C73—82no—
63C2×Q8×C73—22no—
64C2×D4×C73—21no—
65C2×D4×C73—42no—
66C2×D4×C73D811yescharacteristic
67C22×C4×C73—21no—
68C22×C4×C73—42no—
69C22×C4×C73—41no—
70Q16×C73—42no—
71Q16×C73—41no—
72Q16×C73—41no—
73SD16×C73—44no—
74D8×C73—21no—
75D8×C73SD1612yescomplemented
76C2×C8×C73—21no—
77C2×C8×C73—41no—
78C2×C8×C73—41no—
79C2×C8×C73C2×D411yescharacteristic
80C2×C8×C73C2×D411yescharacteristic
81C4⋊C4×C73—21no—
82C4⋊C4×C73—42no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73D811yescharacteristic
86C22⋊C4×C73—21no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—41no—
89C42×C73C2×D411yescharacteristic
90C23×C73—21no—
91C23×C73—41no—
92Q8×C73—42no—
93Q8×C73—42no—
94D4×C73—22no—
95D4×C73—21no—
96D4×C73—21no—
97D4×C73—41no—
98D4×C73—84no—
99C2×C4×C73—21no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73—84no—
105C2×C4×C73—81no—
106C2×C4×C73—81no—
107C2×C4×C73#12011yescharacteristic
108C2×C4×C73C2×D811yescharacteristic
109C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
110C2×Dic73—81no—
111Dic146—81no—
112Dic146—81no—
113C8×C73—21no—
114C8×C73—41no—
115C8×C73C2×SD1611yescharacteristic
116C8×C73AGL(1,ℤ/8)11yescharacteristic
117#10596831—81no—
118C22×C73—41no—
119C22×C73—42no—
120C22×C73—41no—
121C22×C73—41no—
122C22×C73—81no—
123C22×C73—81no—
124C22×C73#44711yescharacteristic
125C4×C73—21no—
126C4×C73—41no—
127C4×C73—82no—
128C4×C73—81no—
129C4×C73#45011yescharacteristic
130C4×C73#45511yescharacteristic
131C4×C73#50811yescharacteristic
132C4×C73#50911yescharacteristic
133Dic73—161no—
134#12371—731nomaximal, Sylow
135C2×C73—81no—
136C2×C73—82no—
137C2×C73#163911yescharacteristic
138C2×C73#218611yescharacteristic
139C2×C73#219511yescharacteristic
140#3283D7311yescharacteristic
141#2241—731no—
142#2224—731no—
143#1673—731no—
144#1362—731no—
145#1331—731no—
146D4⋊C16—731no—
147C73#1237111yescharacteristic, complemented, Sylow
148C2×#135—21no—
149#547—21no—
150#500—731no—
151#494C2×D7311yescharacteristic
152#467—21no—
153#461—21no—
154#461—1461no—
155#452—22no—
156C4×D8C2×D7311yescharacteristic
157C8×D4C2×D7311yescharacteristic
158#366—1461no—
159C4⋊C16—731no—
160#358—1462no—
161#357—1461no—
162C4×C16—731no—
163#335—731no—
164#332—1461no—
165C2×#41—22no—
166#135—42no—
167#135—42no—
168C2×Q16—21no—
169C2×Q16C22⋊D7312yes—
170C2×SD16—22no—
171C2×SD16—2921no—
172C2×D8C22⋊D7311yescharacteristic
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—1461no—
176#123—42no—
177C22⋊Q8—22no—
178#121—41no—
179C4×D4—21no—
180C4×D4C22⋊D7311yescharacteristic
181C2×C4⋊C4—41no—
182C2×C16—1461no—
183C2×C16—1461no—
184#107—1461no—
185#107—1461no—
186#107C22⋊D7311yescharacteristic
187C4⋊C8—1461no—
188C4⋊C8C22⋊D7311yescharacteristic
189#103—22no—
190#103—2921no—
191#102—21no—
192C22⋊C8—21no—
193C4×C8C22×D7311yescharacteristic
194#41—44no—
195#41—82no—
196C2×Q8—22no—
197C2×Q8—2921no—
198C2×D4—21no—
199C2×D4—42no—
200C2×D4#1192401311yescharacteristic
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—41no—
204Q16—42no—
205Q16—41no—
206Q16—41no—
207SD16—44no—
208SD16—5842no—
209D8—21no—
210D8#1192401612yescomplemented
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—2921no—
215C2×C8C2×C22⋊D7311yescharacteristic
216C2×C8C2×C22⋊D7311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—2921no—
222C4⋊C4—2921no—
223C4⋊C4#1192401311yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C42C2×C22⋊D7311yescharacteristic
228C16—2921no—
229C16—2921no—
230C23—21no—
231C23—41no—
232Q8—42no—
233Q8—42no—
234Q8—5841no—
235Q8—5841no—
236D4—22no—
237D4—21no—
238D4—21no—
239D4—41no—
240D4—84no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—5841no—
250C2×C4#1455656011yescharacteristic
251C2×C4C2×#1192401311yescharacteristic
252C2×C4#1455661711yescharacteristic
253C8—21no—
254C8—41no—
255C8—5841no—
256C8C2×#1192401611yescharacteristic
257C8#1455661711yescharacteristic
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—81no—
263C22—81no—
264C22#1798872111yescharacteristic
265C4—21no—
266C4—41no—
267C4—81no—
268C4—82no—
269C4—11681no—
270C4#1798873811yescharacteristic
271C4#1798876511yescharacteristic
272C4#1798901711yescharacteristic
273C4#1798902211yescharacteristic
274C2—81no—
275C2—82no—
276C2#2260911011yescharacteristic
277C2#2261185411yescharacteristic
278C2#2261190911yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
Fitting, prime radicalC73×#3283C2
socleC22×C73#447
hypercenter#3283D73
hypocenterC73#12371
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. #22612068C2semidirect
3. #22609320C2semidirect
4. #22607980C2semidirect
5. #22607892C2semidirect
6. D4⋊#11924003C2semidirect
75. D8×C73SD16semidirect
147. C73#12371semidirect
210. D8#11924016semidirect

Maximal subgroups

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

subgroupindex
#226121582
#226120682
#226093202
#226079802
#226078922
D4⋊#119240032
C73×#32832
#1237173