Finite Group Databasebeta

← #32990485

Subgroups — #32990485

Order 18688.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#22610100C211yesmaximal, characteristic
3#22610065C212yesmaximal
4#22607813C212yesmaximal
5C73×#1295C211yesmaximal, characteristic
6C2×#14556524—21no—
7C2×#14556522—21no—
8C2×#14556502C2211yescharacteristic
9#17988529—22no—
10#17988526—21no—
11#17988508—22no—
12#17988507—21no—
13#17988457C2211yescharacteristic
14#17988453C2212yes—
15#17988350—22no—
16#17988350—22no—
17C2×#97×C73C2212yes—
18#382×C73C2211yescharacteristic
19C22×Dic146—21no—
20C22×#10596831—21no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73—21no—
23C2×C4⋊Dic73—22no—
24C2×C4⋊Dic73D412yes—
25C2×C4⋊Dic73D412yes—
26C2×#11924007—22no—
27#14556524—42no—
28#14556522—42no—
29#14556502—21no—
30#14556502Q812yescomplemented
31#14556498—21no—
32#14556498—22no—
33#14556498—21no—
34#14556498—22no—
35#14556498—22no—
36C22×C8×C73—22no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C42×C73C2311yescharacteristic
40#97×C73—22no—
41#97×C73Q814yescomplemented
42#95×C73—22no—
43C2×Dic146—81no—
44C2×Dic146—82no—
45C22×Dic73—22no—
46C22×Dic73—22no—
47C22×Dic73—41no—
48C2×#10596831—22no—
49C2×#10596831—22no—
50C2×#10596831—22no—
51C4⋊Dic73—42no—
52C4⋊Dic73—42no—
53C4⋊Dic73—41no—
54C4⋊Dic73—41no—
55C4⋊Dic73—44no—
56C4⋊Dic73—81no—
57#11924007—82no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73C2×D412yes—
61C22×C4×C73C2×D411yescharacteristic
62C2×C8×C73—24no—
63C2×C8×C73—42no—
64C2×C8×C73—42no—
65C4⋊C4×C73—41no—
66C4⋊C4×C73—41no—
67C42×C73C2×Q811yescharacteristic
68C42×C73C2×Q812yes—
69C42×C73#4111yescharacteristic
70C23×C73#12011yescharacteristic
71C2×C4×C73—21no—
72C2×C4×C73—22no—
73C2×C4×C73—22no—
74C2×C4×C73—21no—
75C2×C4×C73—41no—
76C2×C4×C73—41no—
77C2×C4×C73—42no—
78C2×C4×C73—42no—
79C2×C4×C73C22⋊Q812yes—
80C2×C4×C73C22⋊Q811yescharacteristic
81C2×C4×C73C4⋊Q812yes—
82C2×C4×C73C4⋊Q811yescharacteristic
83C2×Dic73—42no—
84C2×Dic73—42no—
85C2×Dic73—42no—
86C2×Dic73—42no—
87C2×Dic73—42no—
88C2×Dic73—42no—
89C2×Dic73—81no—
90C2×Dic73—82no—
91Dic146—84no—
92C8×C73—44no—
93#10596831—24no—
94C22×C73#39511yescharacteristic
95C22×C73#45311yescharacteristic
96C22×C73#45412yes—
97C22×C73#45512yes—
98C22×C73#45611yescharacteristic
99C4×C73—21no—
100C4×C73—21no—
101C4×C73—22no—
102C4×C73—22no—
103C4×C73—81no—
104C4×C73—81no—
105Dic73—82no—
106Dic73—82no—
107Dic73—161no—
108#10584—731nomaximal, Sylow
109C2×C73#165711yescharacteristic
110C2×C73#165911yescharacteristic
111C2×C73#166412yes—
112C2×C73#206011yescharacteristic
113C2×C73#206112yes—
114#2844—731no—
115#1847—731no—
116#1836—732no—
117#1295—732no—
118#1295D7311yescharacteristic
119C73#1058411yescharacteristic, complemented, Sylow
120C2×#107—1461no—
121C2×#106—1461no—
122C2×#97—731no—
123C2×#97C2×D7312yes—
124#400—1462no—
125#400—1461no—
126#395—1462no—
127#395—1461no—
128#384—731no—
129#382—732no—
130#382C2×D7311yescharacteristic
131#340—1462no—
132#340—1462no—
133C22×Q8—1461no—
134C22×C8—22no—
135C22×C8—1461no—
136C2×C4⋊C4—732no—
137C2×C4⋊C4—732no—
138C2×C4⋊C4—1462no—
139C2×C4⋊C4—1461no—
140C2×C4⋊C4—1462no—
141C2×C4⋊C4—1461no—
142C2×C4⋊C4D29211yescharacteristic
143C2×C4⋊C4D29211yescharacteristic
144C2×C42C22×D7311yescharacteristic
145#107—2922no—
146#106—2922no—
147#97—22no—
148#97—732no—
149#97—1461no—
150#97Dic14614yescomplemented
151#95—22no—
152#95—1462no—
153#95—1461no—
154#95—1461no—
155#95—1462no—
156#95—1462no—
157C2×Q8—5841no—
158C2×Q8—5842no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—2921no—
162C22×C4—1462no—
163C22×C4—1462no—
164C22×C4D4×D7312yes—
165C22×C4C2×D29211yescharacteristic
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—1462no—
170C2×C8—1462no—
171C2×C8—1462no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—2922no—
175C4⋊C4—2922no—
176C4⋊C4—2921no—
177C4⋊C4—2921no—
178C4⋊C4—2924no—
179C4⋊C4—5842no—
180C4⋊C4—5841no—
181C42Q8×D7311yescharacteristic
182C42C2×Dic14612yes—
183C42#1192402711yescharacteristic
184C23#1455656111yescharacteristic
185Q8—5844no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—2922no—
195C2×C4—2922no—
196C2×C4—2922no—
197C2×C4—2922no—
198C2×C4—2922no—
199C2×C4—2922no—
200C2×C4—5841no—
201C2×C4—5842no—
202C2×C4#1455656812yes—
203C2×C4#1455657111yescharacteristic
204C2×C4#1455659012yes—
205C2×C4#1455659111yescharacteristic
206C8—44no—
207C8—1464no—
208C22#1798850811yescharacteristic
209C22#1798875711yescharacteristic
210C22#1798876112yes—
211C22#1798876612yes—
212C22#1798877211yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—5842no—
220C4—5842no—
221C4—11681no—
222C2#2260923011yescharacteristic
223C2#2260923811yescharacteristic
224C2#2260926712yes—
225C2#2261127711yescharacteristic
226C2#2261128012yes—
subgroupG/H
centerC23#14556561
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1295C2
socle, prime radicalC23×C73#120
hypercenter#1295D73
hypocenterC73#10584
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
30. #14556502Q8semidirect
41. #97×C73Q8semidirect
119. C73#10584semidirect
150. #97Dic146semidirect

Maximal subgroups

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

subgroupindex
#226182032
#226101002
#226100652
#226078132
C73×#12952
#1058473