Finite Group Databasebeta

← #32999005

Subgroups — #32999005

Order 18688.

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

normal subgroups only

This group has 342 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#22617999C211yesmaximal, characteristic
2#22608939C212yesmaximal
3#22607936C212yesmaximal
4#22607875C211yesmaximal, characteristic
5C73×#2061C211yesmaximal, characteristic
6C2×#14556583—21no—
7#17988915—22no—
8#17988527—22no—
9#17988519—22no—
10#17988517—21no—
11#17988517C2212yes—
12#17988324—22no—
13#17988313C2212yes—
14#544×C73C2211yescharacteristic
15#428×C73—21no—
16#382×C73C2211yescharacteristic
17#343×C73C2211yescharacteristic
18#339×C73—21no—
19#14556583—42no—
20#14556583—42no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73—21no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73D412yes—
25C2×#11924007—22no—
26C2×#11924007—22no—
27#14556522—42no—
28#14556515—42no—
29C22⋊#10596831—22no—
30#14556498—22no—
31#14556498—21no—
32#14556498—22no—
33#14556498D412yes—
34C2×M16×C73—21no—
35C4⋊Q8×C73—41no—
36#125×C73—41no—
37C22⋊Q8×C73—22no—
38#117×C73—21no—
39C2×C4⋊C4×C73D411yescharacteristic
40C2×C4⋊C4×C73C2311yescharacteristic
41C2×C42×C73D411yescharacteristic
42#107×C73—41no—
43#106×C73—41no—
44#95×C73—22no—
45C22×Dic73—22no—
46C22×Dic73—22no—
47C2×#10596831—42no—
48C4⋊Dic73—42no—
49C4⋊Dic73—42no—
50C4⋊Dic73—42no—
51C4⋊Dic73—81no—
52#11924007—44no—
53#11924007—42no—
54#11924007—42no—
55C2×Q8×C73—22no—
56C22×C4×C73—21no—
57C22×C4×C73—21no—
58C22×C4×C73C2×D412yes—
59C22×C4×C73C2×D411yescharacteristic
60M16×C73—22no—
61M16×C73—41no—
62C2×C8×C73—41no—
63C4⋊C4×C73—21no—
64C4⋊C4×C73—21no—
65C4⋊C4×C73—22no—
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—42no—
69C22⋊C4×C73—41no—
70C42×C73—21no—
71C42×C73—41no—
72C42×C73SD1612yes—
73C23×C73#12011yescharacteristic
74Q8×C73—82no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73—41no—
80C2×C4×C73—41no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73—41no—
86C2×C4×C73C2×SD1611yescharacteristic
87C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
88C2×Dic73—42no—
89C2×Dic73—42no—
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C8×C73—42no—
95#10596831—82no—
96C22×C73—21no—
97C22×C73—21no—
98C22×C73#45011yescharacteristic
99C22×C73#45611yescharacteristic
100C22×C73B(4,𝔽2)11yescharacteristic
101C4×C73—21no—
102C4×C73—21no—
103C4×C73—42no—
104C4×C73—42no—
105C4×C73—81no—
106C4×C73—81no—
107C4×C73—81no—
108Dic73—82no—
109Dic73—82no—
110#12251—731nomaximal, Sylow
111C2×C73—21no—
112C2×C73—21no—
113C2×C73#160411yescharacteristic
114C2×C73#220111yescharacteristic
115C2×C73#220311yescharacteristic
116#2822—731no—
117#2061D7311yescharacteristic
118#1611—732no—
119#1346—732no—
120#1324—731no—
121C73#1225111yescharacteristic, complemented, Sylow
122#544C2×D7311yescharacteristic
123C2×#125—1461no—
124#483—1462no—
125#428—21no—
126#400—1462no—
127#398—732no—
128#398—1462no—
129#398—1461no—
130#382C2×D7311yescharacteristic
131#343C2×D7311yescharacteristic
132#339—21no—
133#328—1462no—
134#324—732no—
135C2×M16—21no—
136C4⋊Q8—41no—
137#125—41no—
138#125—2922no—
139#125—2922no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—732no—
143C2×C4⋊C4—1462no—
144C2×C4⋊C4—1461no—
145C2×C4⋊C4—1461no—
146C2×C4⋊C4—1462no—
147C2×C4⋊C4—1461no—
148C2×C4⋊C4C22⋊D7311yescharacteristic
149C2×C4⋊C4C22×D7311yescharacteristic
150C2×C42C22⋊D7311yescharacteristic
151#107—41no—
152#106—41no—
153#106—2922no—
154#103—2922no—
155C22⋊C8—1462no—
156#95—22no—
157#95—732no—
158#95—1462no—
159#95—1461no—
160#95—1462no—
161C2×Q8—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1462no—
165C22×C4—1462no—
166C22×C4D4×D7312yes—
167C22×C4C2×C22⋊D7311yescharacteristic
168M16—22no—
169M16—41no—
170C2×C8—41no—
171C2×C8—2922no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—2922no—
179C4⋊C4—2924no—
180C4⋊C4—2922no—
181C4⋊C4—2922no—
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C4⋊C4—5841no—
185C22⋊C4—41no—
186C42—21no—
187C42—41no—
188C42#1192401512yes—
189C23#1455655911yescharacteristic
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2922no—
203C2×C4—2922no—
204C2×C4—2922no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4C2×#1192401511yescharacteristic
209C2×C4#1455661611yescharacteristic
210C8—42no—
211C8—5842no—
212C22—21no—
213C22—21no—
214C22#1798873711yescharacteristic
215C22#1798876911yescharacteristic
216C22#1798877311yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—5842no—
225C4—5842no—
226C2—21no—
227C2—21no—
228C2#2260889911yescharacteristic
229C2#2261194311yescharacteristic
230C2#2261195111yescharacteristic
subgroupG/H
centerC22#17988773
commutator, cosocleC2×C4⋊C4×C73C23
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2061C2
socleC22×C73B(4,𝔽2)
prime radicalC23×C73#120
hypercenter#2061D73
hypocenterC73#12251
LCSG ▷ C2×C4⋊C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
121. C73#12251semidirect

Maximal subgroups

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

subgroupindex
#226179992
#226089392
#226079362
#226078752
C73×#20612
#1225173