Finite Group Databasebeta

← #32975267

Subgroups — #32975267

Order 18688.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#22616303C211yesmaximal, characteristic
2#22611495C212yesmaximal, complemented
3#22611464C211yesmaximal, characteristic
4#22607755C212yesmaximal, complemented
5C73×#1270C211yesmaximal, characteristic
6C2×#14556568—22no—
7C2×#14556522—21no—
8C2×#14556512—21no—
9#17988510—22no—
10#17988507—21no—
11#17988485C2212yes—
12#17988467—21no—
13#17988457C2211yescharacteristic
14#17988350—22no—
15#17988350—22no—
16#17988342C2211yescharacteristic
17#390×C73C2211yescharacteristic
18#336×C73C2212yes—
19C22×Dic146—21no—
20C23×Dic73—21no—
21C22×#10596831—21no—
22#14556568—44no—
23#14556568—44no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73D412yes—
28C2×#11924007—22no—
29C2×#11924007—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—21no—
32#14556522—42no—
33#14556512—42no—
34#14556498—21no—
35#14556498—22no—
36C22×D4×C73D411yescharacteristic
37C22×C8×C73—22no—
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C22⋊C4×C73—22no—
40C2×C42×C73C2311yescharacteristic
41C2×Dic146—44no—
42C2×Dic146—81no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C22×Dic73—44no—
46C22×Dic73—41no—
47C22×Dic73—81no—
48C2×#10596831—22no—
49C2×#10596831—41no—
50C2×#10596831—41no—
51C4⋊Dic73—22no—
52C4⋊Dic73—42no—
53C4⋊Dic73—41no—
54C4⋊Dic73—41no—
55C4⋊Dic73—44no—
56C4⋊Dic73SD1614yes—
57#11924007—44no—
58#11924007—42no—
59#11924007—42no—
60C22⋊Dic73—44no—
61C22⋊Dic73—81no—
62C24×C73—21no—
63C2×D4×C73—21no—
64C2×D4×C73—41no—
65C2×D4×C73—41no—
66C2×D4×C73Q1612yes—
67C22×C4×C73—21no—
68C22×C4×C73C2×D412yes—
69C22×C4×C73C2×D411yescharacteristic
70C2×C8×C73—42no—
71C2×C8×C73—42no—
72C2×C8×C73—42no—
73C4⋊C4×C73—41no—
74C22⋊C4×C73—42no—
75C22⋊C4×C73—42no—
76C42×C73—41no—
77C23×C73—22no—
78C23×C73—42no—
79C23×C73—41no—
80C23×C73—42no—
81C23×C73—41no—
82C23×C73#12011yescharacteristic
83D4×C73—42no—
84D4×C73—81no—
85C2×C4×C73—21no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—21no—
89C2×C4×C73—42no—
90C2×C4×C73—41no—
91C2×C4×C73C2×SD1612yes—
92C2×C4×C73C2×Q1611yescharacteristic
93C2×C4×C73AGL(1,ℤ/8)12yes—
94C2×C4×C73#13711yescharacteristic
95C2×Dic73—24no—
96C2×Dic73—42no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73—84no—
102C2×Dic73—81no—
103C2×Dic73—82no—
104Dic146—84no—
105C8×C73—82no—
106#10596831—42no—
107C22×C73—42no—
108C22×C73—44no—
109C22×C73—82no—
110C22×C73—81no—
111C22×C73—81no—
112C22×C73#45012yes—
113C22×C73#45111yescharacteristic
114C22×C73#45311yescharacteristic
115C22×C73#45412yes—
116C22×C73#45811yescharacteristic
117C4×C73—21no—
118C4×C73—21no—
119C4×C73—22no—
120C4×C73—22no—
121C4×C73—81no—
122Dic73—44no—
123Dic73—82no—
124Dic73—161no—
125#7826—731nomaximal, Sylow
126C2×C73—82no—
127C2×C73#161411yescharacteristic
128C2×C73#161612yes—
129C2×C73#162911yescharacteristic
130C2×C73#163712yes—
131C2×C73#165711yescharacteristic
132#2683—731no—
133#2100—732no—
134#2094—731no—
135#1271—732no—
136#1270D7311yescharacteristic
137C73#782611yescharacteristic, complemented, Sylow
138C2×C22⋊Q8—1462no—
139C2×#106—1461no—
140C2×#102—1461no—
141#396—1462no—
142#395—1461no—
143#390C2×D7311yescharacteristic
144#389—732no—
145#386—1461no—
146#384—731no—
147#340—1462no—
148#340—1462no—
149#336—731no—
150#336C2×D7312yes—
151C22×Q8—1461no—
152C22×D4D29211yescharacteristic
153C23×C4—1461no—
154C22×C8—22no—
155C22×C8—1461no—
156C22⋊Q8—2924no—
157C22⋊Q8—2924no—
158C2×C4⋊C4—732no—
159C2×C4⋊C4—732no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4—1462no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4—1462no—
164C2×C4⋊C4D29211yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—1462no—
167C2×C22⋊C4—1461no—
168C2×C42C22×D7311yescharacteristic
169#106—2922no—
170#102—2922no—
171#95—1461no—
172#95—1462no—
173C24—21no—
174C2×Q8—2924no—
175C2×Q8—5841no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4Dic29212yes—
180C22×C4—21no—
181C22×C4—2924no—
182C22×C4—2921no—
183C22×C4—5841no—
184C22×C4—1462no—
185C22×C4—1462no—
186C22×C4D4×D7312yes—
187C22×C4C2×D29211yescharacteristic
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—2921no—
192C2×C8—2921no—
193C2×C8—1462no—
194C4⋊C4—41no—
195C4⋊C4—2922no—
196C4⋊C4—2921no—
197C4⋊C4—2921no—
198C4⋊C4—2924no—
199C4⋊C4—2924no—
200C4⋊C4—2922no—
201C4⋊C4—2922no—
202C4⋊C4—734no—
203C4⋊C4—1462no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—2924no—
207C22⋊C4—5841no—
208C42—41no—
209C23—22no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23#1455656111yescharacteristic
215Q8—5844no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—2922no—
225C2×C4—2922no—
226C2×C4—2922no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—5844no—
230C2×C4—5841no—
231C2×C4—5842no—
232C2×C4—1464no—
233C2×C4SD16×D7312yes—
234C2×C4C2×Dic29211yescharacteristic
235C2×C4#1455661512yes—
236C2×C4#1455662411yescharacteristic
237C8—82no—
238C8—2922no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1798873912yes—
245C22#1798874511yescharacteristic
246C22#1798875711yescharacteristic
247C22#1798876112yes—
248C22#1798877811yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—2924no—
255C4—5842no—
256C4—11681no—
257C2—82no—
258C2#2260895811yescharacteristic
259C2#2260896912yes—
260C2#2260904811yescharacteristic
261C2#2260909912yes—
262C2#2260923011yescharacteristic
subgroupG/H
centerC23#14556561
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1270C2
socleC23×C73#120
prime radicalC22×D4×C73D4
hypercenter#1270D73
hypocenterC73#7826
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #22611495C2semidirect
4. #22607755C2semidirect
137. C73#7826semidirect

Maximal subgroups

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

subgroupindex
#226163032
#226114952
#226114642
#226077552
C73×#12702
#782673