Finite Group Databasebeta

← #32998832

Subgroups — #32998832

Order 18688.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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#22611092C211yesmaximal, characteristic
2#22608972C212yesmaximal
3#22607958C212yesmaximal
4#22607865C211yesmaximal, characteristic
5C73×#2809C211yesmaximal, characteristic
6#17989218C2211yescharacteristic
7#17988922—22no—
8#17988596—21no—
9#17988356C2211yescharacteristic
10#17988347—21no—
11#17988323—22no—
12#17988313C2212yes—
13#400×C73C2212yes—
14#398×C73—22no—
15#391×C73—21no—
16#391×C73C2211yescharacteristic
17#389×C73—21no—
18#387×C73—22no—
19C2×#11924012—21no—
20#14556590—41no—
21#14556582—41no—
22#14556570—22no—
23#14556550—21no—
24#14556524—42no—
25#14556515—41no—
26#14556515—41no—
27#14556514—42no—
28C22⋊#10596831—22no—
29C22×Q8×C73D411yescharacteristic
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73—21no—
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73D412yes—
35C2×C4⋊C4×C73C2311yescharacteristic
36C2×C42×C73—21no—
37C2×C42×C73D411yescharacteristic
38#95×C73—22no—
39#95×C73—22no—
40#95×C73—22no—
41#95×C73—22no—
42#95×C73D412yes—
43C2×Dic146—22no—
44#11924012—41no—
45#11924012—41no—
46C2×#10596831—42no—
47C2×#10596831—41no—
48C4⋊Dic73—21no—
49C4⋊Dic73—21no—
50C4⋊Dic73—22no—
51#11924007—42no—
52C22⋊Dic73—41no—
53C4×Dic73—41no—
54C2×Q8×C73—21no—
55C2×Q8×C73—41no—
56C2×Q8×C73—41no—
57C2×Q8×C73Q1612yes—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73C2×D411yescharacteristic
63C22×C4×C73C2×D412yes—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—42no—
66C4⋊C4×C73—44no—
67C4⋊C4×C73—42no—
68C4⋊C4×C73—81no—
69C4⋊C4×C73—81no—
70C42×C73—41no—
71C42×C73—81no—
72C23×C73#12011yescharacteristic
73Q8×C73—42no—
74Q8×C73—81no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—41no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73—41no—
84C2×C4×C73—42no—
85C2×C4×C73—42no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73—42no—
91C2×C4×C73C2×Q1611yescharacteristic
92C2×C4×C73#13711yescharacteristic
93C2×Dic73—41no—
94C2×Dic73—41no—
95Dic146—82no—
96#10596831—81no—
97#10596831—82no—
98C22×C73—21no—
99C22×C73—21no—
100C22×C73#45111yescharacteristic
101C22×C73#45511yescharacteristic
102C22×C73B(4,𝔽2)11yescharacteristic
103C4×C73—21no—
104C4×C73—21no—
105C4×C73—42no—
106C4×C73—42no—
107C4×C73—42no—
108C4×C73—82no—
109C4×C73—82no—
110Dic73—81no—
111Dic73—81no—
112#12225—731nomaximal, Sylow
113C2×C73—21no—
114C2×C73—21no—
115C2×C73#160211yescharacteristic
116C2×C73#220111yescharacteristic
117C2×C73#220411yescharacteristic
118#2809D7311yescharacteristic
119#2028—731no—
120#1617—732no—
121#1353—732no—
122#1319—731no—
123C73#1222511yescharacteristic, complemented, Sylow
124#544—731no—
125#484—1462no—
126#419—1461no—
127#400C2×D7312yes—
128#398—22no—
129#391—21no—
130#391C2×D7311yescharacteristic
131#389—21no—
132#387—22no—
133#343—731no—
134#339—1461no—
135#328—1462no—
136#324—732no—
137C22×Q8C22⋊D7311yescharacteristic
138C2×M16—1461no—
139C4⋊Q8—2921no—
140#125—2921no—
141C22⋊Q8—1462no—
142#117—1461no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D7312yes—
148C2×C4⋊C4C22×D7311yescharacteristic
149C2×C42—21no—
150C2×C42C22⋊D7311yescharacteristic
151#107—2922no—
152#103—2922no—
153#103—2921no—
154#103—2921no—
155C22⋊C8—1462no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95C22⋊D7312yes—
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—1462no—
165C2×Q8#1192401812yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C22⋊D7311yescharacteristic
171C22×C4C2×C22⋊D7312yes—
172M16—2921no—
173M16—2921no—
174C2×C8—2922no—
175C2×C8—2921no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—2922no—
183C4⋊C4—1461no—
184C4⋊C4—1461no—
185C4⋊C4—1462no—
186C22⋊C4—2921no—
187C42—41no—
188C42—81no—
189C42—2921no—
190C23#1455656011yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—5842no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—2921no—
211C2×C4—2921no—
212C2×C4C2×#1192401811yescharacteristic
213C2×C4#1455662211yescharacteristic
214C8—5841no—
215C8—5842no—
216C22—21no—
217C22—21no—
218C22#1798874311yescharacteristic
219C22#1798876511yescharacteristic
220C22#1798877411yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—5841no—
229C4—5841no—
230C2—21no—
231C2—21no—
232C2#2260888811yescharacteristic
233C2#2261194411yescharacteristic
234C2#2261195711yescharacteristic
subgroupG/H
centerC22#17988774
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2809C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2809D73
hypocenterC73#12225
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
123. C73#12225semidirect

Maximal subgroups

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

subgroupindex
#226110922
#226089722
#226079582
#226078652
C73×#28092
#1222573