Finite Group Databasebeta

← #32998826

Subgroups — #32998826

Order 18688.

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

normal subgroups only

This group has 380 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#22617924C211yesmaximal, characteristic
2#22608905C212yesmaximal
3#22607960C212yesmaximal
4#22607865C211yesmaximal, characteristic
5C73×#2028C211yesmaximal, characteristic
6#17988742—22no—
7#17988508—22no—
8#17988507C2212yes—
9#17988494—21no—
10#17988493—21no—
11#17988491—22no—
12#17988324—22no—
13#17988313C2212yes—
14#544×C73C2211yescharacteristic
15#419×C73—21no—
16#391×C73C2211yescharacteristic
17#343×C73C2211yescharacteristic
18#339×C73—21no—
19C22×Dic146—21no—
20C22×Dic146D412yes—
21C2×#11924018—42no—
22C2×C4⋊Dic73—21no—
23C2×C4⋊Dic73—21no—
24C2×#11924007—22no—
25C2×C4×Dic73—21no—
26#14556514—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—
43C2×Dic146—24no—
44C2×Dic146—42no—
45C2×Dic146—82no—
46C2×Dic146—81no—
47C2×Dic146—81no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50#11924018—84no—
51C2×#10596831—42no—
52C4⋊Dic73—81no—
53C4⋊Dic73—81no—
54#11924007—82no—
55C4×Dic73—81no—
56C2×Q8×C73—21no—
57C2×Q8×C73—22no—
58C2×Q8×C73—41no—
59C2×Q8×C73—41no—
60C2×Q8×C73Q1612yescomplemented
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73C2×D412yes—
64C22×C4×C73C2×D411yescharacteristic
65M16×C73—41no—
66M16×C73—41no—
67C2×C8×C73—41no—
68C4⋊C4×C73—21no—
69C4⋊C4×C73—21no—
70C4⋊C4×C73—22no—
71C4⋊C4×C73—41no—
72C4⋊C4×C73—42no—
73C22⋊C4×C73—41no—
74C42×C73—41no—
75C42×C73—41no—
76C23×C73#12011yescharacteristic
77Q8×C73—42no—
78Q8×C73—81no—
79Q8×C73—82no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—22no—
83C2×C4×C73—22no—
84C2×C4×C73—22no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—42no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73C2×Q1611yescharacteristic
93C2×C4×C73#13711yescharacteristic
94C2×Dic73—24no—
95C2×Dic73—42no—
96C2×Dic73—42no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100Dic146—44no—
101Dic146—84no—
102Dic146—82no—
103C8×C73—81no—
104#10596831—82no—
105C22×C73—21no—
106C22×C73—21no—
107C22×C73#45111yescharacteristic
108C22×C73#45511yescharacteristic
109C22×C73B(4,𝔽2)11yescharacteristic
110C4×C73—21no—
111C4×C73—21no—
112C4×C73—42no—
113C4×C73—42no—
114C4×C73—42no—
115C4×C73—81no—
116C4×C73—81no—
117Dic73—44no—
118Dic73—82no—
119#12224—731nomaximal, Sylow
120C2×C73—21no—
121C2×C73—21no—
122C2×C73#160211yescharacteristic
123C2×C73#219911yescharacteristic
124C2×C73#220511yescharacteristic
125#2811—731no—
126#2028D7311yescharacteristic
127#1605—732no—
128#1354—732no—
129#1319—731no—
130C73#1222411yescharacteristic, complemented, Sylow
131#544C2×D7311yescharacteristic
132#451—1462no—
133#419—21no—
134#395—732no—
135#395—1462no—
136#391—1462no—
137#391—1461no—
138#391—1461no—
139#391C2×D7311yescharacteristic
140#343C2×D7311yescharacteristic
141#339—21no—
142#328—1462no—
143#324—732no—
144C22×Q8—732no—
145C22×Q8—1461no—
146C22×Q8C22⋊D7311yescharacteristic
147C2×Q16—2922no—
148C2×M16—21no—
149C4⋊Q8—41no—
150#125—41no—
151C22⋊Q8—22no—
152#117—21no—
153C2×C4⋊C4—1461no—
154C2×C4⋊C4—1461no—
155C2×C4⋊C4—1462no—
156C2×C4⋊C4C22×D7311yescharacteristic
157C2×C42—1461no—
158C2×C42C22⋊D7311yescharacteristic
159#103—41no—
160#103—41no—
161#103—2922no—
162C22⋊C8—1462no—
163#95—22no—
164#95—732no—
165#95—1462no—
166#95—1462no—
167#95—1462no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—2922no—
173C2×Q8—5842no—
174C2×Q8—5841no—
175C2×Q8—5841no—
176C2×Q8—1464no—
177C2×Q8#1192401812yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—1462no—
181C22×C4—1462no—
182C22×C4D4×D7312yes—
183C22×C4C2×C22⋊D7311yescharacteristic
184Q16—5844no—
185M16—41no—
186M16—41no—
187C2×C8—41no—
188C2×C8—2922no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—5842no—
195C4⋊C4—5841no—
196C4⋊C4—5841no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—5841no—
201C23#1455655911yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—82no—
205Q8—2924no—
206Q8—5844no—
207Q8—5842no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—2922no—
221C2×C4—2922no—
222C2×C4—2922no—
223C2×C4—2922no—
224C2×C4—2922no—
225C2×C4—1464no—
226C2×C4C2×#1192401811yescharacteristic
227C2×C4#1455662111yescharacteristic
228C8—81no—
229C8—5842no—
230C22—21no—
231C22—21no—
232C22#1798874211yescharacteristic
233C22#1798876411yescharacteristic
234C22#1798877311yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—2924no—
243C4—5842no—
244C2—21no—
245C2—21no—
246C2#2260888711yescharacteristic
247C2#2261193111yescharacteristic
248C2#2261196011yescharacteristic
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#12224
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
60. C2×Q8×C73Q16semidirect
130. C73#12224semidirect
177. C2×Q8#11924018semidirect

Maximal subgroups

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

subgroupindex
#226179242
#226089052
#226079602
#226078652
C73×#20282
#1222473