Finite Group Databasebeta

← #32999848

Subgroups — #32999848

Order 18688.

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

normal subgroups only

This group has 344 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#22612154C211yesmaximal, characteristic, complemented
2#22612068C211yesmaximal, characteristic, complemented
3#22609325C211yesmaximal, characteristic, complemented
4#22607977C211yesmaximal, characteristic, complemented
5Q8⋊#11924003C211yesmaximal, characteristic, complemented
6#22607892C211yesmaximal, characteristic, complemented
7C73×#3391C211yesmaximal, characteristic
8#17988992C2211yescharacteristic
9#17988797—21no—
10#17988412—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988388—21no—
13#17988388—22no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988332—21no—
17#550×C73—21no—
18#493×C73—21no—
19#493×C73C2211yescharacteristic
20#459×C73—22no—
21#459×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×D8×C73—21no—
24C4×D8×C73C2211yescharacteristic
25C2×#11924018—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—41no—
34C2×D8×C73—22no—
35C2×D8×C73—21no—
36C2×D8×C73D411yescharacteristic
37C2×D8×C73D412yes—
38#127×C73—22no—
39#127×C73—21no—
40#121×C73—42no—
41#121×C73—41no—
42C4×Q8×C73D411yescharacteristic
43C4×D4×C73—22no—
44C4×D4×C73—21no—
45#107×C73D411yescharacteristic
46C4⋊C8×C73—21no—
47C4⋊C8×C73D411yescharacteristic
48#102×C73—22no—
49#102×C73—21no—
50C4×C8×C73—21no—
51C4×C8×C73C2311yescharacteristic
52C2×Dic146—41no—
53#11924018—82no—
54C2×#10596831—41no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57#11924003—41no—
58#11924003—41no—
59#41×C73—82no—
60#41×C73—82no—
61C2×Q8×C73D811yescharacteristic
62C2×D4×C73—21no—
63C2×D4×C73—22no—
64C2×D4×C73—42no—
65C2×D4×C73—42no—
66C2×D4×C73—41no—
67C2×D4×C73—41no—
68C22×C4×C73—42no—
69C22×C4×C73—41no—
70D8×C73—21no—
71D8×C73—21no—
72D8×C73—42no—
73D8×C73—41no—
74D8×C73—41no—
75D8×C73—82no—
76C2×C8×C73—21no—
77C2×C8×C73C2×D411yescharacteristic
78C2×C8×C73C2×D411yescharacteristic
79C4⋊C4×C73—21no—
80C4⋊C4×C73—21no—
81C4⋊C4×C73D811yescharacteristic
82C22⋊C4×C73—42no—
83C22⋊C4×C73—41no—
84C42×C73—21no—
85C42×C73C2×D411yescharacteristic
86C23×C73—42no—
87C23×C73—41no—
88Q8×C73—21no—
89Q8×C73D1612yes—
90D4×C73—41no—
91D4×C73—42no—
92D4×C73—42no—
93D4×C73—41no—
94D4×C73—84no—
95D4×C73—81no—
96D4×C73—81no—
97D4×C73—161no—
98D4×C73—162no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—41no—
102C2×C4×C73—84no—
103C2×C4×C73—81no—
104C2×C4×C73—81no—
105C2×C4×C73#12011yescharacteristic
106C2×C4×C73C2×D811yescharacteristic
107C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
108C2×Dic73—81no—
109Dic146—81no—
110Dic146—81no—
111C8×C73—21no—
112C8×C73—41no—
113C8×C73#13511yescharacteristic
114C8×C73#13711yescharacteristic
115#10596831—81no—
116C22×C73—81no—
117C22×C73—82no—
118C22×C73—82no—
119C22×C73—81no—
120C22×C73—82no—
121C22×C73—81no—
122C22×C73#44711yescharacteristic
123C4×C73—22no—
124C4×C73—21no—
125C4×C73—41no—
126C4×C73—81no—
127C4×C73#45211yescharacteristic
128C4×C73#45511yescharacteristic
129C4×C73C2×D1611yescharacteristic
130C4×C73#50911yescharacteristic
131Dic73—161no—
132#12380—731nomaximal, Sylow
133C2×C73—161no—
134C2×C73—162no—
135C2×C73#164111yescharacteristic
136C2×C73#218411yescharacteristic
137C2×C73#219411yescharacteristic
138#3391D7311yescharacteristic
139#2240—731no—
140#2224—731no—
141#1674—731no—
142#1361—731no—
143Q8⋊C16—731no—
144#1331—731no—
145C73#1238011yescharacteristic, complemented, Sylow
146#550—21no—
147#500—731no—
148#493—21no—
149#493C2×D7311yescharacteristic
150#462—1461no—
151#459—22no—
152#459—21no—
153C8×Q8C2×D7311yescharacteristic
154C4×D8—21no—
155C4×D8C2×D7311yescharacteristic
156#366—1461no—
157C4⋊C16—731no—
158#357—1461no—
159#357—1462no—
160C4×C16—731no—
161#335—731no—
162#332—1461no—
163C2×#41—41no—
164C2×Q16—2921no—
165C2×D8—22no—
166C2×D8—21no—
167C2×D8C22⋊D7311yescharacteristic
168C2×D8C22⋊D7312yes—
169C4⋊Q8—1461no—
170#127—22no—
171#127—21no—
172#121—42no—
173#121—41no—
174C4×Q8C22⋊D7311yescharacteristic
175C4×D4—22no—
176C4×D4—21no—
177C2×C16—1461no—
178C2×C16—1461no—
179#107—1461no—
180#107—1461no—
181#107C22⋊D7311yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—1461no—
184C4⋊C8C22⋊D7311yescharacteristic
185#103—2921no—
186#102—22no—
187#102—21no—
188C4×C8—21no—
189C4×C8C22×D7311yescharacteristic
190#41—82no—
191#41—82no—
192C2×Q8—2921no—
193C2×Q8#1192401311yescharacteristic
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—41no—
200C22×C4—42no—
201C22×C4—41no—
202Q16—5842no—
203D8—21no—
204D8—21no—
205D8—42no—
206D8—41no—
207D8—41no—
208D8—82no—
209C2×C8—21no—
210C2×C8—2921no—
211C2×C8C2×C22⋊D7311yescharacteristic
212C2×C8C2×C22⋊D7311yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—2921no—
216C4⋊C4—2921no—
217C4⋊C4#1192401311yescharacteristic
218C22⋊C4—42no—
219C22⋊C4—41no—
220C42—21no—
221C42C2×C22⋊D7311yescharacteristic
222C16—2921no—
223C16—2921no—
224C23—42no—
225C23—41no—
226Q8—21no—
227Q8—5841no—
228Q8—5841no—
229Q8#1455653212yes—
230D4—41no—
231D4—42no—
232D4—42no—
233D4—41no—
234D4—84no—
235D4—81no—
236D4—81no—
237D4—161no—
238D4—162no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
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—
252C8#1455661111yescharacteristic
253C8#1455662111yescharacteristic
254C22—82no—
255C22—81no—
256C22—82no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22#1798872111yescharacteristic
261C4—22no—
262C4—21no—
263C4—41no—
264C4—81no—
265C4—11681no—
266C4#1798874911yescharacteristic
267C4#1798876511yescharacteristic
268C4C2×#1455653211yescharacteristic
269C4#1798902311yescharacteristic
270C2—162no—
271C2—161no—
272C2#2260912411yescharacteristic
273C2#2261184311yescharacteristic
274C2#2261190211yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
Fitting, prime radicalC73×#3391C2
socleC22×C73#447
hypercenter#3391D73
hypocenterC73#12380
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. #22612154C2semidirect
2. #22612068C2semidirect
3. #22609325C2semidirect
4. #22607977C2semidirect
5. Q8⋊#11924003C2semidirect
6. #22607892C2semidirect
145. C73#12380semidirect

Maximal subgroups

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

subgroupindex
#226121542
#226120682
#226093252
#226079772
Q8⋊#119240032
#226078922
C73×#33912
#1238073