Finite Group Databasebeta

← #32990481

Subgroups — #32990481

Order 18688.

406 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 95 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#22618203C211yesmaximal, characteristic
2#22610097C212yesmaximal
3#22607813C212yesmaximal
4#22607812C211yesmaximal, characteristic
5C73×#1847C211yesmaximal, characteristic
6C2×#14556524—22no—
7C2×#14556522—22no—
8C2×#14556502C2212yes—
9#17988529—22no—
10#17988526—21no—
11#17988508—22no—
12#17988507—21no—
13#17988453C2212yes—
14C2×#107×C73—21no—
15C2×#106×C73—21no—
16C2×#97×C73C2211yescharacteristic
17#384×C73C2211yescharacteristic
18#382×C73C2211yescharacteristic
19C22×Dic146—21no—
20C22×#10596831—22no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73D412yes—
23C2×C4⋊Dic73D412yes—
24C2×#11924007—22no—
25#14556524—44no—
26#14556522—44no—
27#14556502—22no—
28#14556502Q814yescomplemented
29#14556498—21no—
30#14556498—22no—
31#14556498—21no—
32#14556498—22no—
33#14556498—22no—
34C22×C8×C73—21no—
35C2×C4⋊C4×C73—22no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C42×C73C2311yescharacteristic
40#107×C73—42no—
41#106×C73—42no—
42#97×C73—21no—
43#97×C73Q812yescomplemented
44#95×C73—22no—
45C2×Dic146—82no—
46C2×Dic146—81no—
47C22×Dic73—22no—
48C22×Dic73—22no—
49C2×#10596831—24no—
50C2×#10596831—24no—
51C2×#10596831—24no—
52C4⋊Dic73—42no—
53C4⋊Dic73—42no—
54C4⋊Dic73—81no—
55#11924007—82no—
56C22×C4×C73—21no—
57C22×C4×C73—21no—
58C22×C4×C73—41no—
59C22×C4×C73C2×D412yes—
60C22×C4×C73C2×D411yescharacteristic
61C2×C8×C73—22no—
62C2×C8×C73—22no—
63C2×C8×C73—22no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—41no—
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C42×C73C2×Q812yes—
71C42×C73C2×Q811yescharacteristic
72C42×C73#4111yescharacteristic
73C23×C73#12011yescharacteristic
74C2×C4×C73—21no—
75C2×C4×C73—22no—
76C2×C4×C73—21no—
77C2×C4×C73—22no—
78C2×C4×C73—41no—
79C2×C4×C73—41no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73—82no—
83C2×C4×C73—81no—
84C2×C4×C73C22⋊Q812yes—
85C2×C4×C73C22⋊Q811yescharacteristic
86C2×C4×C73C4⋊Q811yescharacteristic
87C2×C4×C73C4⋊Q812yes—
88C2×Dic73—42no—
89C2×Dic73—42no—
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94Dic146—84no—
95C8×C73—24no—
96#10596831—28no—
97C22×C73#39511yescharacteristic
98C22×C73#45312yes—
99C22×C73#45311yescharacteristic
100C22×C73#45612yes—
101C22×C73#45611yescharacteristic
102C4×C73—22no—
103C4×C73—21no—
104C4×C73—22no—
105C4×C73—21no—
106C4×C73—81no—
107C4×C73—81no—
108C4×C73—161no—
109Dic73—82no—
110Dic73—82no—
111#10583—731nomaximal, Sylow
112C2×C73#165511yescharacteristic
113C2×C73#166011yescharacteristic
114C2×C73#166012yes—
115C2×C73#206012yes—
116C2×C73#206011yescharacteristic
117#2844—731no—
118#1847—732no—
119#1847D7311yescharacteristic
120#1295—732no—
121#1295—731no—
122C73#1058311yescharacteristic, complemented, Sylow
123C2×#107—21no—
124C2×#107—1462no—
125C2×#106—21no—
126C2×#106—1462no—
127C2×#97—732no—
128C2×#97C2×D7311yescharacteristic
129#400—1462no—
130#400—1461no—
131#395—1462no—
132#395—1461no—
133#384C2×D7311yescharacteristic
134#382—732no—
135#382C2×D7311yescharacteristic
136C22×Q8—1461no—
137C22×C8—21no—
138C22×C8—1462no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—732no—
142C2×C4⋊C4—732no—
143C2×C4⋊C4—1462no—
144C2×C4⋊C4—1461no—
145C2×C4⋊C4C22⋊D7311yescharacteristic
146C2×C4⋊C4C22⋊D7311yescharacteristic
147C2×C42C22×D7311yescharacteristic
148#107—42no—
149#107—2924no—
150#106—42no—
151#106—2924no—
152#97—21no—
153#97—734no—
154#97—1462no—
155#97Dic14612yescomplemented
156#95—22no—
157#95—1462no—
158#95—1461no—
159#95—1461no—
160#95—1462no—
161#95—1462no—
162C2×Q8—5842no—
163C2×Q8—5841no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—1462no—
168C22×C4—1462no—
169C22×C4D4×D7312yes—
170C22×C4C2×C22⋊D7311yescharacteristic
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—1464no—
175C2×C8—1464no—
176C2×C8—1464no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—2922no—
184C4⋊C4—2922no—
185C4⋊C4—5842no—
186C4⋊C4—5841no—
187C42Q8×D7312yes—
188C42C2×Dic14611yescharacteristic
189C42#1192402811yescharacteristic
190C23#1455655911yescharacteristic
191Q8—5844no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—2922no—
203C2×C4—2922no—
204C2×C4—2922no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4#1455656912yes—
209C2×C4#1455657011yescharacteristic
210C2×C4#1455658911yescharacteristic
211C2×C4#1455659012yes—
212C8—24no—
213C8—1468no—
214C22#1798850711yescharacteristic
215C22#1798875312yes—
216C22#1798875411yescharacteristic
217C22#1798876812yes—
218C22#1798876911yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—161no—
226C4—5842no—
227C4—5842no—
228C2#2260921911yescharacteristic
229C2#2260924011yescharacteristic
230C2#2260924112yes—
231C2#2261127412yes—
232C2#2261127611yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1847C2
socle, prime radicalC23×C73#120
hypercenter#1847D73
hypocenterC73#10583
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
28. #14556502Q8semidirect
43. #97×C73Q8semidirect
122. C73#10583semidirect
155. #97Dic146semidirect

Maximal subgroups

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

subgroupindex
#226182032
#226100972
#226078132
#226078122
C73×#18472
#1058373