Finite Group Databasebeta

← #32975237

Subgroups — #32975237

Order 18688.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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#22617877C211yesmaximal, characteristic
2#22611340C212yesmaximal, complemented
3#22607756C212yesmaximal, complemented
4#22607752C211yesmaximal, characteristic
5C73×#2093C211yesmaximal, characteristic
6C2×#14556514—22no—
7#17988526—21no—
8#17988499—22no—
9#17988493C2212yes—
10#17988467—21no—
11#17988462—22no—
12#17988349—22no—
13#17988342C2212yes—
14C2×#107×C73—21no—
15C2×#102×C73—21no—
16#390×C73C2211yescharacteristic
17#384×C73C2211yescharacteristic
18#336×C73C2211yescharacteristic
19C22×Dic146D412yes—
20C23×Dic73—21no—
21C22×#10596831—22no—
22C2×C4⋊Dic73—21no—
23C2×C4⋊Dic73D412yes—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—21no—
26#14556514—44no—
27#14556498—22no—
28#14556498—21no—
29#14556498—22no—
30#14556498—22no—
31C22×D4×C73D411yescharacteristic
32C22×C8×C73—21no—
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73D411yescharacteristic
36C2×C22⋊C4×C73—22no—
37C2×C42×C73C2311yescharacteristic
38#107×C73—42no—
39#102×C73—42no—
40C2×Dic146—22no—
41C2×Dic146—22no—
42C2×Dic146—42no—
43C2×Dic146—42no—
44C22×Dic73—22no—
45C22×Dic73—22no—
46C22×Dic73—81no—
47C22×Dic73—82no—
48C2×#10596831—42no—
49C2×#10596831—42no—
50C2×#10596831—42no—
51C4⋊Dic73—42no—
52C4⋊Dic73—81no—
53C22⋊Dic73—82no—
54C22⋊Dic73—81no—
55C24×C73—21no—
56C2×D4×C73—21no—
57C2×D4×C73—41no—
58C2×D4×C73—41no—
59C2×D4×C73Q1612yes—
60C22×C4×C73—21no—
61C22×C4×C73—41no—
62C22×C4×C73C2×D412yes—
63C22×C4×C73C2×D411yescharacteristic
64C2×C8×C73—22no—
65C2×C8×C73—41no—
66C2×C8×C73—41no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—41no—
69C4⋊C4×C73—41no—
70C4⋊C4×C73—42no—
71C4⋊C4×C73—42no—
72C22⋊C4×C73—42no—
73C22⋊C4×C73—42no—
74C42×C73—41no—
75C23×C73—22no—
76C23×C73—42no—
77C23×C73—41no—
78C23×C73—42no—
79C23×C73—41no—
80C23×C73#12011yescharacteristic
81D4×C73—42no—
82D4×C73—81no—
83C2×C4×C73—21no—
84C2×C4×C73—22no—
85C2×C4×C73—21no—
86C2×C4×C73—22no—
87C2×C4×C73—42no—
88C2×C4×C73—41no—
89C2×C4×C73—82no—
90C2×C4×C73—81no—
91C2×C4×C73C2×Q1611yescharacteristic
92C2×C4×C73#13512yes—
93C2×C4×C73AGL(1,ℤ/8)12yes—
94C2×C4×C73#13711yescharacteristic
95C2×Dic73—42no—
96C2×Dic73—42no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—84no—
102Dic146—44no—
103Dic146—82no—
104C8×C73—42no—
105#10596831—82no—
106C22×C73—42no—
107C22×C73—44no—
108C22×C73—82no—
109C22×C73—81no—
110C22×C73—81no—
111C22×C73#44912yes—
112C22×C73#45111yescharacteristic
113C22×C73#45311yescharacteristic
114C22×C73#45512yes—
115C22×C73#45811yescharacteristic
116C4×C73—22no—
117C4×C73—22no—
118C4×C73—21no—
119C4×C73—21no—
120C4×C73—81no—
121C4×C73—161no—
122Dic73—82no—
123Dic73—82no—
124#7821—731nomaximal, Sylow
125C2×C73—82no—
126C2×C73#161212yes—
127C2×C73#161411yescharacteristic
128C2×C73#162011yescharacteristic
129C2×C73#164412yes—
130C2×C73#165911yescharacteristic
131#2804—731no—
132#2093D7311yescharacteristic
133#2072—732no—
134#1272—732no—
135#1270—731no—
136C73#782111yescharacteristic, complemented, Sylow
137C2×#107—21no—
138C2×#103—1462no—
139C2×#102—21no—
140#400—1461no—
141#393—1462no—
142#391—732no—
143#390C2×D7311yescharacteristic
144#386—1461no—
145#385—1462no—
146#384C2×D7311yescharacteristic
147#340—1462no—
148#336—732no—
149#336C2×D7311yescharacteristic
150C22×Q8—732no—
151C22×D4C22⋊D7311yescharacteristic
152C23×C4—1461no—
153C22×C8—21no—
154C22×C8—1462no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—1461no—
159C2×C4⋊C4C22⋊D7311yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—1462no—
162C2×C22⋊C4—1461no—
163C2×C42C22×D7311yescharacteristic
164#107—42no—
165#103—2924no—
166#102—42no—
167#95—1462no—
168#95—1461no—
169#95—1462no—
170#95—1462no—
171C24—21no—
172C2×Q8—2922no—
173C2×Q8—2922no—
174C2×Q8—1462no—
175C2×Q8—1462no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1192401812yes—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—5841no—
183C22×C4—5842no—
184C22×C4—1462no—
185C22×C4—1462no—
186C22×C4D4×D7312yes—
187C22×C4C2×C22⋊D7311yescharacteristic
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—2922no—
192C2×C8—2922no—
193C2×C8—2922no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—2922no—
200C4⋊C4—5841no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—5842no—
204C22⋊C4—5841no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1455655911yescharacteristic
212Q8—2924no—
213Q8—5842no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—2922no—
225C2×C4—2922no—
226C2×C4—2922no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—2922no—
230C2×C4—5844no—
231C2×C4C2×#1192401811yescharacteristic
232C2×C4#1455661012yes—
233C2×C4#1455661512yes—
234C2×C4#1455662111yescharacteristic
235C8—42no—
236C8—5842no—
237C22—42no—
238C22—44no—
239C22—82no—
240C22—81no—
241C22—81no—
242C22#1798872912yes—
243C22#1798874211yescharacteristic
244C22#1798875411yescharacteristic
245C22#1798876312yes—
246C22#1798877611yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—161no—
253C4—5842no—
254C4—5842no—
255C2—82no—
256C2#2260894212yes—
257C2#2260895511yescharacteristic
258C2#2260898811yescharacteristic
259C2#2260914412yes—
260C2#2260923511yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2093C2
socleC23×C73#120
prime radicalC22×D4×C73D4
hypercenter#2093D73
hypocenterC73#7821
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #22611340C2semidirect
3. #22607756C2semidirect
136. C73#7821semidirect

Maximal subgroups

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

subgroupindex
#226178772
#226113402
#226077562
#226077522
C73×#20932
#782173