Finite Group Databasebeta

← #32999192

Subgroups — #32999192

Order 18688.

344 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 67 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#22620706C211yesmaximal, characteristic
2#22612129C212yesmaximal, complemented
3#22611354C211yesmaximal, characteristic
4#22607944C212yesmaximal, complemented
5C73×#1314C211yesmaximal, characteristic
6C2×#14556570—21no—
7#17988923—22no—
8#17988828—21no—
9#17988828C2212yes—
10#17988745—22no—
11C2×#14556525C2211yescharacteristic
12#17988447—21no—
13#17988415—22no—
14#17988392—22no—
15#17988350—21no—
16#17988350C2211yescharacteristic
17C2×C22⋊C8×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C22×Dic146—21no—
20C2×Dic292—21no—
21C2×Dic292D412yes—
22#14556570—22no—
23#14556570—22no—
24#14556570—41no—
25#14556570—41no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73—21no—
29C2×#11924007—41no—
30C2×C22⋊Dic73—41no—
31C2×C22⋊Dic73—41no—
32#14556525—21no—
33#14556525—22no—
34#14556525D412yes—
35#14556516—22no—
36#14556516—22no—
37#14556498—41no—
38C23×C4×C73D411yescharacteristic
39C22×C8×C73D411yescharacteristic
40C22×C8×C73C2311yescharacteristic
41C2×C16×C73—22no—
42C22⋊C8×C73—22no—
43C2×Dic146—22no—
44C2×Dic146—41no—
45C2×Dic146—81no—
46C2×Dic146—81no—
47C22×Dic73—41no—
48C22×Dic73—41no—
49C22×Dic73—41no—
50C22×Dic73—41no—
51Dic292—42no—
52Dic292—81no—
53C4⋊Dic73—22no—
54C4⋊Dic73—41no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58C4⋊Dic73—41no—
59#11924007—42no—
60#11924007—42no—
61C22⋊Dic73—42no—
62C22⋊Dic73—42no—
63C22⋊Dic73—82no—
64C24×C73D811yescharacteristic
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—41no—
68C22×C4×C73—41no—
69C22×C4×C73D811yescharacteristic
70C22×C4×C73C2×D411yescharacteristic
71C2×C8×C73—21no—
72C2×C8×C73—41no—
73C2×C8×C73—41no—
74C2×C8×C73C2×D412yes—
75C16×C73—42no—
76C23×C73—21no—
77C23×C73—21no—
78C23×C73—41no—
79C23×C73—41no—
80C23×C73SD3212yes—
81C23×C73C2×D811yescharacteristic
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—81no—
89C2×C4×C73#12011yescharacteristic
90C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—82no—
94C2×Dic73—82no—
95C2×Dic73—81no—
96C2×Dic73—81no—
97C2×Dic73—81no—
98C2×Dic73—81no—
99C2×Dic73—81no—
100C2×Dic73—81no—
101Dic146—42no—
102Dic146—42no—
103Dic146—82no—
104C8×C73—22no—
105C8×C73—41no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73—22no—
109C22×C73—42no—
110C22×C73—41no—
111C22×C73—41no—
112C22×C73—81no—
113C22×C73#44711yescharacteristic
114C22×C73C2×SD3211yescharacteristic
115C22×C73#51011yescharacteristic
116C4×C73—21no—
117C4×C73—41no—
118C4×C73#45511yescharacteristic
119C4×C73#45811yescharacteristic
120Dic73—82no—
121Dic73—82no—
122Dic73—161no—
123Dic73—161no—
124#12279—731nomaximal, Sylow
125C2×C73—21no—
126C2×C73—21no—
127C2×C73—42no—
128C2×C73#160711yescharacteristic
129C2×C73#218511yescharacteristic
130C2×C73#219111yescharacteristic
131#3189—731no—
132#2235—732no—
133#2075—731no—
134#1348—732no—
135#1314D7311yescharacteristic
136C73#1227911yescharacteristic, complemented, Sylow
137C2×C22⋊Q8—1461no—
138#484—1462no—
139#467—732no—
140#467—1461no—
141#451—1462no—
142C2×#107—731no—
143C2×C22⋊C8C2×D7311yescharacteristic
144#380—1461no—
145#367—1462no—
146#358—1462no—
147C22⋊C16C2×D7312yes—
148#340—731no—
149#340—1461no—
150C22×Q8—1461no—
151C23×C4D29211yescharacteristic
152C2×Q16—732no—
153C2×Q16—1461no—
154C22×C8D29211yescharacteristic
155C22×C8C22×D7311yescharacteristic
156C22⋊Q8—2921no—
157C22⋊Q8—2921no—
158C22⋊Q8—1462no—
159C22⋊Q8—1462no—
160C2×C4⋊C4—2921no—
161C2×C4⋊C4—1461no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4—1461no—
164C2×C22⋊C4—2921no—
165C2×C22⋊C4—2921no—
166C2×C16—22no—
167#107—732no—
168#107—1462no—
169#107—1461no—
170#103—1462no—
171#103—1462no—
172C22⋊C8—22no—
173#95—2921no—
174C24D58411yescharacteristic
175C2×Q8—2921no—
176C2×Q8—5841no—
177C2×Q8—5841no—
178C2×Q8—1462no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—2921no—
184C22×C4—2921no—
185C22×C4—2921no—
186C22×C4—2921no—
187C22×C4D58411yescharacteristic
188C22×C4C2×D29211yescharacteristic
189Q16—2922no—
190Q16—5841no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8D4×D7312yes—
195C4⋊C4—2922no—
196C4⋊C4—2921no—
197C4⋊C4—2921no—
198C4⋊C4—2921no—
199C4⋊C4—2922no—
200C4⋊C4—2921no—
201C4⋊C4—2921no—
202C4⋊C4—1462no—
203C22⋊C4—2922no—
204C22⋊C4—2922no—
205C22⋊C4—5842no—
206C16—42no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1455653612yes—
212C23C2×D58411yescharacteristic
213Q8—2922no—
214Q8—2922no—
215Q8—5842no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—2922no—
224C2×C4—2922no—
225C2×C4—5842no—
226C2×C4—5842no—
227C2×C4—5841no—
228C2×C4—5841no—
229C2×C4—5841no—
230C2×C4—5841no—
231C2×C4—5841no—
232C2×C4—5841no—
233C2×C4#1455656111yescharacteristic
234C2×C4#1455661911yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1798872311yescharacteristic
245C22C2×#1455653611yescharacteristic
246C22#1798903011yescharacteristic
247C4—21no—
248C4—41no—
249C4—5842no—
250C4—5842no—
251C4—11681no—
252C4—11681no—
253C4#1798876711yescharacteristic
254C4#1798877811yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2260891911yescharacteristic
259C2#2261185011yescharacteristic
260C2#2261188411yescharacteristic
subgroupG/H
centerC22#17988723
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#1314C2
socleC22×C73#447
prime radicalC24×C73D8
hypercenter#1314D73
hypocenterC73#12279
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #22612129C2semidirect
4. #22607944C2semidirect
136. C73#12279semidirect

Maximal subgroups

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

subgroupindex
#226207062
#226121292
#226113542
#226079442
C73×#13142
#1227973