Finite Group Databasebeta

← #32998831

Subgroups — #32998831

Order 18688.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#22617919C211yesmaximal, characteristic
2#22608973C212yesmaximal
3#22607957C212yesmaximal
4#22607865C211yesmaximal, characteristic
5C73×#2028C211yesmaximal, characteristic
6#17988920—22no—
7#17988526C2212yes—
8#17988519—22no—
9#17988494—21no—
10#17988485—21no—
11#17988474—22no—
12#17988324—22no—
13#17988313C2212yes—
14#544×C73C2211yescharacteristic
15#419×C73—21no—
16#391×C73C2211yescharacteristic
17#343×C73C2211yescharacteristic
18#339×C73—21no—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73D412yes—
23C2×#11924007—22no—
24C2×C4×Dic73—21no—
25#14556524—42no—
26#14556515—42no—
27C22⋊#10596831—22no—
28#14556498—22no—
29#14556498—22no—
30#14556498—22no—
31#14556498D412yes—
32C22×Q8×C73D411yescharacteristic
33C2×M16×C73—21no—
34C4⋊Q8×C73—41no—
35#125×C73—41no—
36C22⋊Q8×C73—22no—
37#117×C73—21no—
38C2×C4⋊C4×C73C2311yescharacteristic
39C2×C42×C73D411yescharacteristic
40#103×C73—41no—
41#103×C73—41no—
42#95×C73—22no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C2×#10596831—42no—
46C4⋊Dic73—42no—
47C4⋊Dic73—42no—
48C4⋊Dic73—81no—
49C4⋊Dic73—81no—
50#11924007—44no—
51C4×Dic73—81no—
52C2×Q8×C73—21no—
53C2×Q8×C73—22no—
54C2×Q8×C73—41no—
55C2×Q8×C73—41no—
56C2×Q8×C73Q1612yes—
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73C2×D412yes—
60C22×C4×C73C2×D411yescharacteristic
61M16×C73—41no—
62M16×C73—41no—
63C2×C8×C73—41no—
64C4⋊C4×C73—21no—
65C4⋊C4×C73—21no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—42no—
69C22⋊C4×C73—41no—
70C42×C73—41no—
71C42×C73—41no—
72C23×C73#12011yescharacteristic
73Q8×C73—42no—
74Q8×C73—81no—
75Q8×C73—82no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—22no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73—42no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73C2×Q1611yescharacteristic
89C2×C4×C73#13711yescharacteristic
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C8×C73—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—81no—
109C4×C73—81no—
110Dic73—82no—
111Dic73—82no—
112#12225—731nomaximal, Sylow
113C2×C73—21no—
114C2×C73—21no—
115C2×C73#160211yescharacteristic
116C2×C73#220111yescharacteristic
117C2×C73#220411yescharacteristic
118#2809—731no—
119#2028D7311yescharacteristic
120#1617—732no—
121#1353—732no—
122#1319—731no—
123C73#1222511yescharacteristic, complemented, Sylow
124#544C2×D7311yescharacteristic
125#484—1462no—
126#419—21no—
127#400—732no—
128#398—1462no—
129#391—1461no—
130#391C2×D7311yescharacteristic
131#389—1461no—
132#387—1462no—
133#343C2×D7311yescharacteristic
134#339—21no—
135#328—1462no—
136#324—732no—
137C22×Q8C22⋊D7311yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—732no—
144C2×C4⋊C4—1462no—
145C2×C4⋊C4—1461no—
146C2×C4⋊C4—1461no—
147C2×C4⋊C4—1461no—
148C2×C4⋊C4C22×D7311yescharacteristic
149C2×C42—1461no—
150C2×C42C22⋊D7311yescharacteristic
151#107—2922no—
152#103—41no—
153#103—41no—
154#103—2922no—
155C22⋊C8—1462no—
156#95—22no—
157#95—732no—
158#95—1462no—
159#95—1462no—
160#95—1462no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1192401812yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1462no—
169C22×C4—1462no—
170C22×C4D4×D7312yes—
171C22×C4C2×C22⋊D7311yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—2922no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—2922no—
182C4⋊C4—2924no—
183C4⋊C4—2922no—
184C4⋊C4—5841no—
185C4⋊C4—5841no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—5841no—
190C23#1455655911yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
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—41no—
205C2×C4—41no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—2922no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—2922no—
212C2×C4C2×#1192401811yescharacteristic
213C2×C4#1455662111yescharacteristic
214C8—81no—
215C8—5842no—
216C22—21no—
217C22—21no—
218C22#1798874211yescharacteristic
219C22#1798876411yescharacteristic
220C22#1798877311yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—5842no—
229C4—5842no—
230C2—21no—
231C2—21no—
232C2#2260888711yescharacteristic
233C2#2261194311yescharacteristic
234C2#2261195611yescharacteristic
subgroupG/H
centerC22#17988773
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2028C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2028D73
hypocenterC73#12225
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

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
#226179192
#226089732
#226079572
#226078652
C73×#20282
#1222573