Finite Group Databasebeta

← #32989056

Subgroups — #32989056

Order 18688.

400 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 93 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#22610903C212yesmaximal
2#22610750C211yesmaximal, characteristic
3#22607812C211yesmaximal, characteristic
4#22607812C212yesmaximal
5C73×#2844C211yesmaximal, characteristic
6C2×#14556591C2211yescharacteristic
7C2×#14556514—21no—
8C2×#14556514—21no—
9C2×#14556502C2211yescharacteristic
10C2×#14556502C2212yes—
11#17988349—22no—
12#17988349—22no—
13#400×C73—21no—
14#400×C73—22no—
15#395×C73—21no—
16#395×C73—22no—
17#382×C73C2211yescharacteristic
18#382×C73C2212yes—
19C22×Dic146—21no—
20C22×#10596831—21no—
21C22×#10596831—22no—
22#14556591—22no—
23#14556591—21no—
24#14556591Q812yescomplemented
25C2×C4⋊Dic73—22no—
26#14556514—42no—
27#14556514—42no—
28#14556502—22no—
29#14556502—22no—
30#14556502—22no—
31C22×Q8×C73—21no—
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73D411yescharacteristic
35C2×C4⋊C4×C73D412yes—
36C2×C4⋊C4×C73D411yescharacteristic
37C2×C4⋊C4×C73D412yes—
38C2×C42×C73C2311yescharacteristic
39#95×C73—22no—
40#95×C73—21no—
41#95×C73—21no—
42#95×C73—22no—
43#95×C73—22no—
44#95×C73—22no—
45C2×Dic146—22no—
46C2×Dic146—41no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C22×Dic73—41no—
52C2×#10596831—22no—
53C2×#10596831—24no—
54C2×#10596831—41no—
55C2×#10596831—42no—
56C2×#10596831—42no—
57C2×#10596831—41no—
58C4⋊Dic73—24no—
59C4⋊Dic73—42no—
60C2×Q8×C73—81no—
61C2×Q8×C73—82no—
62C22×C4×C73—21no—
63C22×C4×C73—22no—
64C22×C4×C73—21no—
65C22×C4×C73—22no—
66C22×C4×C73C2×D412yes—
67C22×C4×C73C2×D411yescharacteristic
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C4⋊C4×C73—81no—
73C4⋊C4×C73—82no—
74C42×C73C2×Q811yescharacteristic
75C42×C73#4112yes—
76C42×C73#4111yescharacteristic
77C23×C73#12011yescharacteristic
78Q8×C73—84no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—22no—
82C2×C4×C73—22no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
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×C73—42no—
92C2×C4×C73—42no—
93C2×C4×C73C22⋊Q812yes—
94C2×C4×C73C22⋊Q811yescharacteristic
95C2×C4×C73#12412yes—
96C2×C4×C73#12411yescharacteristic
97C2×Dic73—42no—
98C2×Dic73—81no—
99C2×Dic73—81no—
100Dic146—42no—
101Dic146—42no—
102Dic146—81no—
103Dic146—81no—
104#10596831—42no—
105#10596831—44no—
106C22×C73#39311yescharacteristic
107C22×C73#45311yescharacteristic
108C22×C73#45411yescharacteristic
109C22×C73#45512yes—
110C22×C73#45612yes—
111C4×C73—21no—
112C4×C73—21no—
113C4×C73—22no—
114C4×C73—22no—
115C4×C73—81no—
116C4×C73—82no—
117C4×C73—81no—
118C4×C73—82no—
119Dic73—82no—
120#10359—731nomaximal, Sylow
121C2×C73#165911yescharacteristic
122C2×C73#166011yescharacteristic
123C2×C73#166412yes—
124C2×C73#202711yescharacteristic
125C2×C73#202812yes—
126#2844D7311yescharacteristic
127#1993—732no—
128#1960—731no—
129#1295—731no—
130#1295—732no—
131C73#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—731no—
133C2×#103—1461no—
134C2×#103—1461no—
135C2×#97—731no—
136C2×#97—732no—
137#400—22no—
138#400—21no—
139#395—22no—
140#395—21no—
141#382C2×D7311yescharacteristic
142#382C2×D7312yes—
143#340—1462no—
144#340—1462no—
145C22×Q8—21no—
146C22×Q8—1461no—
147C22×C8—1462no—
148C22×C8—1461no—
149C4⋊Q8—732no—
150C4⋊Q8—1461no—
151C4⋊Q8—1462no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—1462no—
155C2×C4⋊C4C22⋊D7312yes—
156C2×C4⋊C4C22⋊D7312yes—
157C2×C4⋊C4C22⋊D7311yescharacteristic
158C2×C4⋊C4C22⋊D7311yescharacteristic
159C2×C42C22×D7311yescharacteristic
160#103—2922no—
161#103—2922no—
162#97—1462no—
163#97—1462no—
164#97—1462no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—81no—
172C2×Q8—82no—
173C2×Q8—2921no—
174C2×Q8—2921no—
175C2×Q8—2921no—
176C2×Q8—2921no—
177C2×Q8—2921no—
178C2×Q8—1462no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2921no—
184C22×C4C2×C22⋊D7312yes—
185C22×C4C2×C22⋊D7311yescharacteristic
186C2×C8—2921no—
187C2×C8—2922no—
188C2×C8—2922no—
189C2×C8—2921no—
190C2×C8—1462no—
191C2×C8—1464no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—2922no—
199C4⋊C4—1464no—
200C42Q8×D7311yescharacteristic
201C42#1192402912yes—
202C42#1192402911yescharacteristic
203C23#1455656011yescharacteristic
204Q8—84no—
205Q8—2922no—
206Q8—2922no—
207Q8—5841no—
208Q8—5841no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—2922no—
224C2×C4—5841no—
225C2×C4—5841no—
226C2×C4#1455656712yes—
227C2×C4#1455656711yescharacteristic
228C2×C4#1455658012yes—
229C2×C4#1455658011yescharacteristic
230C8—2922no—
231C8—2924no—
232C22#1798849811yescharacteristic
233C22#1798875511yescharacteristic
234C22#1798876011yescharacteristic
235C22#1798876512yes—
236C22#1798877012yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—5842no—
246C2#2260923711yescharacteristic
247C2#2260924211yescharacteristic
248C2#2260926512yes—
249C2#2261108411yescharacteristic
250C2#2261108912yes—
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2844C2
socle, prime radicalC23×C73#120
hypercenter#2844D73
hypocenterC73#10359
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #14556591Q8semidirect
131. C73#10359semidirect

Maximal subgroups

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

subgroupindex
#226109032
#226107502
#226078122
C73×#28442
#1035973