Finite Group Databasebeta

← #32998103

Subgroups — #32998103

Order 18688.

424 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 71 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#22611864C212yesmaximal, complemented
2#22610941C211yesmaximal, characteristic
3#22607943C212yesmaximal, complemented
4#22607855C211yesmaximal, characteristic
5C73×#3189C211yesmaximal, characteristic
6C22×Dic292C2211yescharacteristic
7C2×#14556570—21no—
8C2×#14556537—22no—
9C2×#14556516—21no—
10#17988390—22no—
11C22⋊#11924003C2212yes—
12#17988350C2211yescharacteristic
13C2×C22⋊Q8×C73—21no—
14#484×C73—22no—
15#467×C73—21no—
16#467×C73C2212yes—
17#451×C73—22no—
18C2×C22⋊C8×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×Dic292—22no—
21C2×Dic292—42no—
22#14556570—42no—
23#14556570—42no—
24C2×C4⋊Dic73—21no—
25C2×#11924007—41no—
26C2×C22⋊Dic73—41no—
27#14556537—44no—
28C2×#11924003—22no—
29#14556516—42no—
30C22×Q8×C73—21no—
31C23×C4×C73D411yescharacteristic
32C2×Q16×C73—21no—
33C2×Q16×C73D412yes—
34C22×C8×C73D411yescharacteristic
35C22×C8×C73C2311yescharacteristic
36C22⋊Q8×C73—22no—
37C22⋊Q8×C73—22no—
38C22⋊Q8×C73—41no—
39C22⋊Q8×C73—41no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—41no—
42C2×C22⋊C4×C73—41no—
43#107×C73—21no—
44#107×C73D412yes—
45#103×C73—22no—
46#103×C73—22no—
47C22⋊C8×C73—22no—
48C2×Dic146—22no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C2×Dic146—41no—
53C2×Dic146—41no—
54C22×Dic73—41no—
55C22×Dic73—41no—
56Dic292—24no—
57Dic292—42no—
58C4⋊Dic73—22no—
59C4⋊Dic73—41no—
60#11924007—44no—
61C22⋊Dic73—44no—
62#11924003—42no—
63C24×C73D811yescharacteristic
64C2×Q8×C73—22no—
65C2×Q8×C73—41no—
66C2×Q8×C73—81no—
67C2×Q8×C73—81no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—41no—
71C22×C4×C73—41no—
72C22×C4×C73—41no—
73C22×C4×C73—41no—
74C22×C4×C73D811yescharacteristic
75C22×C4×C73C2×D411yescharacteristic
76Q16×C73—42no—
77Q16×C73—81no—
78C2×C8×C73—21no—
79C2×C8×C73—41no—
80C2×C8×C73—41no—
81C2×C8×C73C2×D412yes—
82C4⋊C4×C73—22no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—42no—
85C4⋊C4×C73—42no—
86C22⋊C4×C73—42no—
87C22⋊C4×C73—42no—
88C23×C73—21no—
89C23×C73—21no—
90C23×C73—41no—
91C23×C73—41no—
92C23×C73Q3212yescomplemented
93C23×C73C2×D811yescharacteristic
94Q8×C73—42no—
95Q8×C73—42no—
96Q8×C73—82no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—81no—
106C2×C4×C73—82no—
107C2×C4×C73—82no—
108C2×C4×C73#12011yescharacteristic
109C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
110C2×Dic73—42no—
111C2×Dic73—42no—
112C2×Dic73—81no—
113C2×Dic73—81no—
114C2×Dic73—81no—
115C2×Dic73—81no—
116Dic146—42no—
117Dic146—42no—
118Dic146—81no—
119Dic146—81no—
120C8×C73—22no—
121C8×C73—41no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—22no—
125C22×C73—42no—
126C22×C73—41no—
127C22×C73—41no—
128C22×C73—81no—
129C22×C73#44711yescharacteristic
130C22×C73C2×Q3211yescharacteristic
131C22×C73#51011yescharacteristic
132C4×C73—21no—
133C4×C73—41no—
134C4×C73—82no—
135C4×C73—82no—
136C4×C73#45311yescharacteristic
137C4×C73B(4,𝔽2)11yescharacteristic
138Dic73—82no—
139Dic73—82no—
140#12106—731nomaximal, Sylow
141C2×C73—21no—
142C2×C73—21no—
143C2×C73—42no—
144C2×C73#159511yescharacteristic
145C2×C73#218811yescharacteristic
146C2×C73#219111yescharacteristic
147#3189D7311yescharacteristic
148#2188—732no—
149#2001—731no—
150#1348—732no—
151#1314—731no—
152C73#1210611yescharacteristic, complemented, Sylow
153C22×Q16—731no—
154C2×C22⋊Q8—21no—
155C2×C22⋊Q8—1461no—
156C2×Q32—1462no—
157#484—22no—
158#467—21no—
159#467C2×D7312yes—
160#451—22no—
161C2×#103—1461no—
162C2×C22⋊C8C2×D7311yescharacteristic
163#358—1462no—
164C22⋊C16—732no—
165#340—731no—
166C22×Q8—21no—
167C22×Q8—1461no—
168C23×C4C22⋊D7311yescharacteristic
169C2×Q16—21no—
170C2×Q16—2922no—
171C2×Q16—1462no—
172C2×Q16C22⋊D7312yes—
173C22×C8C22⋊D7311yescharacteristic
174C22×C8C22×D7311yescharacteristic
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—2922no—
180C22⋊Q8—2922no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—2921no—
184C2×C4⋊C4—1461no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—2921no—
187Q32—2924no—
188C2×C16—1462no—
189#107—21no—
190#107C22⋊D7312yes—
191#103—22no—
192#103—22no—
193#103—2922no—
194C22⋊C8—22no—
195C24#1192401311yescharacteristic
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×Q8—2921no—
201C2×Q8—2921no—
202C2×Q8—2921no—
203C2×Q8—2921no—
204C2×Q8—2921no—
205C2×Q8—1462no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—2921no—
213C22×C4—2921no—
214C22×C4#1192401311yescharacteristic
215C22×C4C2×C22⋊D7311yescharacteristic
216Q16—42no—
217Q16—81no—
218Q16—2922no—
219Q16—1464no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8C2×C22⋊D7312yes—
224C4⋊C4—22no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—2921no—
229C4⋊C4—2924no—
230C4⋊C4—1462no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—2924no—
234C16—2922no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1455653712yescomplemented
240C23C2×#1192401311yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—82no—
244Q8—2922no—
245Q8—2922no—
246Q8—5841no—
247Q8—5841no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—2922no—
260C2×C4—2922no—
261C2×C4—5841no—
262C2×C4—5841no—
263C2×C4—5841no—
264C2×C4—5841no—
265C2×C4#1455656011yescharacteristic
266C2×C4#1455661711yescharacteristic
267C8—22no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22#1798872111yescharacteristic
277C22C2×#1455653711yescharacteristic
278C22#1798902811yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—5842no—
284C4—5842no—
285C4#1798875511yescharacteristic
286C4#1798877411yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2#2260884911yescharacteristic
291C2#2261186511yescharacteristic
292C2#2261188211yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#3189C2
socleC22×C73#447
prime radicalC24×C73D8
hypercenter#3189D73
hypocenterC73#12106
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #22611864C2semidirect
3. #22607943C2semidirect
92. C23×C73Q32semidirect
152. C73#12106semidirect
239. C23#14556537semidirect

Maximal subgroups

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

subgroupindex
#226118642
#226109412
#226079432
#226078552
C73×#31892
#1210673