Finite Group Databasebeta

← #32998446

Subgroups — #32998446

Order 18688.

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

normal subgroups only

This group has 354 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#22621276C211yesmaximal, characteristic
2#22611865C212yesmaximal
3#22607942C212yesmaximal
4#22607871C211yesmaximal, characteristic
5C73×#2056C211yesmaximal, characteristic
6#17989144—21no—
7C2×#14556537—22no—
8#17988883—22no—
9#17988828—21no—
10#17988828C2212yes—
11#17988798—22no—
12#17988391—22no—
13C22⋊#11924003C2212yes—
14C22×Q16×C73C2211yescharacteristic
15C2×C4⋊C8×C73C2211yescharacteristic
16C2×#103×C73—21no—
17#391×C73—21no—
18#340×C73C2211yescharacteristic
19C2×Dic292—21no—
20C2×Dic292D412yes—
21#14556591—21no—
22#14556591—21no—
23#14556583—41no—
24#14556570—22no—
25#14556570—41no—
26C4×Dic146—22no—
27C4×Dic146—41no—
28#14556550—41no—
29#14556537—44no—
30C2×#11924003—22no—
31#14556525—21no—
32#14556525D412yes—
33#14556516—22no—
34#14556516—22no—
35C22×Q8×C73—21no—
36C2×Q16×C73—22no—
37C2×Q16×C73—42no—
38C22×C8×C73D411yescharacteristic
39C22×C8×C73C2311yescharacteristic
40C2×C4⋊C4×C73—21no—
41C2×C42×C73D411yescharacteristic
42C4⋊C8×C73—22no—
43#103×C73—42no—
44#95×C73—41no—
45#95×C73—41no—
46C2×Dic146—22no—
47C2×Dic146—41no—
48Dic292—42no—
49Dic292—81no—
50C4⋊Dic73—22no—
51C4⋊Dic73—41no—
52C4⋊Dic73—41no—
53C4⋊Dic73—41no—
54#11924007—42no—
55#11924007—42no—
56C22⋊Dic73—42no—
57C4×Dic73—42no—
58#11924003—42no—
59C2×Q8×C73—22no—
60C2×Q8×C73—41no—
61C2×Q8×C73—41no—
62C2×Q8×C73—41no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73D811yescharacteristic
68C22×C4×C73D811yescharacteristic
69C22×C4×C73C2×D411yescharacteristic
70Q16×C73—24no—
71Q16×C73—42no—
72C2×C8×C73—21no—
73C2×C8×C73—41no—
74C2×C8×C73—41no—
75C2×C8×C73C2×D412yes—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C42×C73—21no—
79C42×C73—21no—
80C23×C73C2×D811yescharacteristic
81Q8×C73—42no—
82Q8×C73—42no—
83Q8×C73—81no—
84Q8×C73—81no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73—81no—
93C2×C4×C73—81no—
94C2×C4×C73—81no—
95C2×C4×C73—81no—
96C2×C4×C73—81no—
97C2×C4×C73Q3212yescomplemented
98C2×C4×C73#12011yescharacteristic
99C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
100C2×Dic73—42no—
101C2×Dic73—42no—
102Dic146—42no—
103Dic146—42no—
104Dic146—82no—
105C8×C73—22no—
106C8×C73—41no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73#44711yescharacteristic
110C22×C73C2×Q3211yescharacteristic
111C22×C73#51011yescharacteristic
112C4×C73—21no—
113C4×C73—22no—
114C4×C73—41no—
115C4×C73—82no—
116C4×C73—161no—
117C4×C73#45411yescharacteristic
118C4×C73B(4,𝔽2)11yescharacteristic
119Dic73—82no—
120Dic73—82no—
121#12162—731nomaximal, Sylow
122C2×C73—21no—
123C2×C73—21no—
124C2×C73#159811yescharacteristic
125C2×C73#218811yescharacteristic
126C2×C73#219111yescharacteristic
127#3249—731no—
128#2188—732no—
129#2056D7311yescharacteristic
130#1348—732no—
131#1322—731no—
132C73#1216211yescharacteristic, complemented, Sylow
133C22×Q16C2×D7311yescharacteristic
134#533—1461no—
135C2×Q32—1462no—
136#477—1462no—
137#467—732no—
138#467—1461no—
139#462—1462no—
140C2×C4⋊C8C2×D7311yescharacteristic
141C2×#103—21no—
142#391—21no—
143#358—1462no—
144C22⋊C16—732no—
145#340C2×D7311yescharacteristic
146C22×Q8—21no—
147C2×Q16—22no—
148C2×Q16—42no—
149C2×Q16—732no—
150C2×Q16—1461no—
151C22×C8C22⋊D7311yescharacteristic
152C22×C8C22×D7311yescharacteristic
153C4⋊Q8—1461no—
154C4⋊Q8—1461no—
155#125—2921no—
156C22⋊Q8—2921no—
157C22⋊Q8—1462no—
158C4×Q8—2921no—
159C4×Q8—1462no—
160#117—2921no—
161C2×C4⋊C4—21no—
162C2×C42C22⋊D7311yescharacteristic
163Q32—2924no—
164C2×C16—1462no—
165#107—732no—
166#107—1461no—
167C4⋊C8—22no—
168#103—42no—
169#103—1462no—
170#103—1462no—
171#95—41no—
172#95—41no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—2921no—
180C2×Q8—1462no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#1192401311yescharacteristic
184C22×C4#1192401311yescharacteristic
185C22×C4C2×C22⋊D7311yescharacteristic
186Q16—24no—
187Q16—42no—
188Q16—2922no—
189Q16—5841no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8D4×D7312yes—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—2922no—
197C4⋊C4—2921no—
198C4⋊C4—2921no—
199C4⋊C4—2921no—
200C4⋊C4—2922no—
201C4⋊C4—1462no—
202C22⋊C4—2922no—
203C42—21no—
204C42—21no—
205C42—2922no—
206C16—2922no—
207C23C2×#1192401311yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—81no—
211Q8—81no—
212Q8—2922no—
213Q8—2922no—
214Q8—5842no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4#1455653712yescomplemented
230C2×C4#1455655911yescharacteristic
231C2×C4#1455661611yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#1798872011yescharacteristic
237C22C2×#1455653711yescharacteristic
238C22#1798902711yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—82no—
243C4—161no—
244C4—5842no—
245C4—5842no—
246C4#1798875911yescharacteristic
247C4#1798877311yescharacteristic
248C2—21no—
249C2—21no—
250C2#2260886511yescharacteristic
251C2#2261186411yescharacteristic
252C2#2261188111yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#2056C2
socleC22×C73#447
prime radicalC23×C73C2×D8
hypercenter#2056D73
hypocenterC73#12162
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
97. C2×C4×C73Q32semidirect
132. C73#12162semidirect
229. C2×C4#14556537semidirect

Maximal subgroups

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

subgroupindex
#226212762
#226118652
#226079422
#226078712
C73×#20562
#1216273