Finite Group Databasebeta

← #33005070

Subgroups — #33005070

Order 18688.

370 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 101 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#22611650C212yesmaximal
2#22610802C211yesmaximal, characteristic
3#22608036C212yesmaximal
4#22608015C211yesmaximal, characteristic
5C73×#3162C211yesmaximal, characteristic
6C22×Dic292C2211yescharacteristic
7C2×#14556531C2212yes—
8C2×#14556523C2211yescharacteristic
9C2×#14556516—21no—
10#17988493—21no—
11#17988390—24no—
12#533×C73—21no—
13#501×C73—24no—
14#427×C73—21no—
15#427×C73C2212yes—
16C2×#97×C73C2211yescharacteristic
17C22×Dic146—21no—
18C2×Dic292—22no—
19C2×Dic292—22no—
20C2×Dic292—21no—
21C2×Dic292Q812yescomplemented
22C2×C4⋊Dic73—21no—
23#14556531—22no—
24#14556531—22no—
25C2×#11924003—22no—
26#14556523—22no—
27#14556516—42no—
28#14556498—42no—
29C22×C8×C73D411yescharacteristic
30C22×C8×C73C2311yescharacteristic
31C4⋊Q8×C73—22no—
32#125×C73—22no—
33C22⋊Q8×C73—42no—
34C4×Q8×C73—42no—
35#117×C73—22no—
36C2×C42×C73D411yescharacteristic
37#107×C73—22no—
38#107×C73D414yes—
39#106×C73—22no—
40#106×C73—22no—
41#97×C73—22no—
42C2×Dic146—22no—
43C2×Dic146—41no—
44C2×Dic146—41no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146—41no—
48C22×Dic73—41no—
49C22×Dic73—41no—
50Dic292—24no—
51Dic292—42no—
52C4⋊Dic73—41no—
53C4⋊Dic73—41no—
54#11924003—42no—
55C2×Q8×C73—42no—
56C22×C4×C73D812yes—
57C22×C4×C73C2×D411yescharacteristic
58C2×C8×C73—21no—
59C2×C8×C73—21no—
60C2×C8×C73SD1612yes—
61C2×C8×C73C2×D412yes—
62C2×C8×C73C2×Q811yescharacteristic
63C2×C8×C73#4112yes—
64C2×C8×C73#4111yescharacteristic
65C4⋊C4×C73—24no—
66C4⋊C4×C73—44no—
67C4⋊C4×C73—42no—
68C22⋊C4×C73—42no—
69C42×C73—21no—
70C42×C73—21no—
71C42×C73—42no—
72C23×C73C2×D811yescharacteristic
73Q8×C73—84no—
74C2×C4×C73—22no—
75C2×C4×C73—42no—
76C2×C4×C73—44no—
77C2×C4×C73#12011yescharacteristic
78C2×C4×C73C22⋊Q811yescharacteristic
79C2×C4×C73#12411yescharacteristic
80C2×C4×C73C2×SD1611yescharacteristic
81C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C73#13711yescharacteristic
83C2×Dic73—42no—
84C2×Dic73—81no—
85C2×Dic73—81no—
86C2×Dic73—82no—
87C2×Dic73—81no—
88Dic146—42no—
89Dic146—42no—
90Dic146—81no—
91Dic146—81no—
92C8×C73—22no—
93C8×C73C22⋊Q812yes—
94C8×C73#12412yes—
95C22×C73—22no—
96C22×C73#44711yescharacteristic
97C22×C73#47411yescharacteristic
98C22×C73#48611yescharacteristic
99C4×C73—42no—
100C4×C73—84no—
101C4×C73#39311yescharacteristic
102C4×C73#44811yescharacteristic
103C4×C73#47611yescharacteristic
104C4×C73#48911yescharacteristic
105Dic73—82no—
106Dic73—161no—
107#13368—731nomaximal, Sylow
108C2×C73—21no—
109C2×C73—21no—
110C2×C73#202311yescharacteristic
111C2×C73#219112yes—
112#3162D7311yescharacteristic
113#2141—732no—
114#1971—731no—
115#1386—732no—
116#1377—731no—
117C73#1336811yescharacteristic, complemented, Sylow
118C22×Q16—731no—
119#533—21no—
120C2×M32—732no—
121#501—24no—
122#427—21no—
123#427C2×D7312yes—
124C2×#106—731no—
125C2×#103—1461no—
126C2×#97C2×D7311yescharacteristic
127#391—1461no—
128#358—1464no—
129C22×Q8—1461no—
130C2×Q16—732no—
131C2×Q16—1462no—
132C2×Q16—1461no—
133C2×Q16—1462no—
134C22×C8C22⋊D7311yescharacteristic
135C22×C8C22×D7311yescharacteristic
136C4⋊Q8—22no—
137#125—22no—
138C22⋊Q8—42no—
139C4×Q8—42no—
140#117—22no—
141C2×C4⋊C4—1461no—
142C2×C42C22⋊D7311yescharacteristic
143M32—1462no—
144M32—1462no—
145C2×C16—1462no—
146#107—22no—
147#107C22⋊D7314yes—
148#106—22no—
149#106—22no—
150#106—1462no—
151#103—2922no—
152#97—22no—
153#95—2922no—
154C2×Q8—42no—
155C2×Q8—2921no—
156C2×Q8—2921no—
157C2×Q8—2921no—
158C2×Q8—2921no—
159C2×Q8—2921no—
160C2×Q8—1462no—
161C22×C4—2921no—
162C22×C4—2921no—
163C22×C4#1192401312yes—
164C22×C4C2×C22⋊D7311yescharacteristic
165Q16—2922no—
166Q16—1464no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1192401512yes—
170C2×C8C2×C22⋊D7312yes—
171C2×C8Q8×D7311yescharacteristic
172C2×C8#1192402912yes—
173C2×C8#1192402911yescharacteristic
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—2921no—
178C4⋊C4—2921no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—2922no—
184C23C2×#1192401311yescharacteristic
185Q8—84no—
186Q8—2922no—
187Q8—2922no—
188Q8—5841no—
189Q8—5841no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—2922no—
194C2×C4—5841no—
195C2×C4—5841no—
196C2×C4—5842no—
197C2×C4—5841no—
198C2×C4#1455656011yescharacteristic
199C2×C4#1455656711yescharacteristic
200C2×C4#1455658011yescharacteristic
201C2×C4C2×#1192401511yescharacteristic
202C2×C4#1455661711yescharacteristic
203C2×C4#1455662211yescharacteristic
204C8—22no—
205C8#1455656712yes—
206C8#1455658012yes—
207C22—22no—
208C22#1798872111yescharacteristic
209C22#1798886411yescharacteristic
210C22#1798893511yescharacteristic
211C4—42no—
212C4—84no—
213C4—5842no—
214C4—11681no—
215C4#1798849811yescharacteristic
216C4#1798872611yescharacteristic
217C4#1798887411yescharacteristic
218C4#1798894911yescharacteristic
219C2—21no—
220C2—21no—
221C2#2261106411yescharacteristic
222C2#2261188212yes—
subgroupG/H
centerC22#17988721
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#3162C2
socleC22×C73#447
prime radicalC23×C73C2×D8
hypercenter#3162D73
hypocenterC73#13368
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #3162

Complements

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

NH ≅ G/Ntype
21. C2×Dic292Q8semidirect
117. C73#13368semidirect

Maximal subgroups

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

subgroupindex
#226116502
#226108022
#226080362
#226080152
C73×#31622
#1336873