Finite Group Databasebeta

← #32989118

Subgroups — #32989118

Order 18688.

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

normal subgroups only

This group has 416 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#22613224C211yesmaximal, characteristic
2#22610912C211yesmaximal, characteristic
3#22608431C212yesmaximal
4#22607833C212yesmaximal
5C73×#1274C211yesmaximal, characteristic
6#17989144C2211yescharacteristic
7C2×C4×Dic146—21no—
8#17988611C2211yescharacteristic
9#17988519—22no—
10#17988508—22no—
11C4×C4⋊Dic73—21no—
12C4×C4⋊Dic73C2212yes—
13#17988347—21no—
14#17988347—21no—
15#17988325—22no—
16#17988325—22no—
17#431×C73C2212yes—
18#376×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×#11924012—21no—
21#14556591Q812yescomplemented
22#14556583—21no—
23#14556570—22no—
24C4×Dic146—22no—
25C4×Dic146—42no—
26C4×Dic146—42no—
27#14556550—21no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73—22no—
31C2×#11924007—22no—
32C2×C4×Dic73—22no—
33C4⋊#10596831—22no—
34#14556498—21no—
35#14556498—21no—
36#14556498D412yes—
37#14556498D412yes—
38C2×C42×C73D411yescharacteristic
39C2×C42×C73D411yescharacteristic
40C2×C42×C73C2311yescharacteristic
41C4⋊C8×C73—22no—
42C22⋊C8×C73—22no—
43C4×C8×C73—22no—
44#95×C73—22no—
45C2×Dic146—22no—
46C2×Dic146—44no—
47C2×Dic146—42no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50#11924012—41no—
51#11924012—41no—
52C2×#10596831—41no—
53C4⋊Dic73—22no—
54C4⋊Dic73—22no—
55C4⋊Dic73—44no—
56C4⋊Dic73—44no—
57C4⋊Dic73—42no—
58#11924007—44no—
59#11924007—42no—
60C22⋊Dic73—41no—
61C4×Dic73—41no—
62C4×Dic73—42no—
63C4×Dic73—42no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73C2×D412yes—
67C22×C4×C73C2×D411yescharacteristic
68C2×C8×C73—42no—
69C42×C73—21no—
70C42×C73—21no—
71C42×C73SD1612yes—
72C42×C73Q1612yes—
73C42×C73C2×Q811yescharacteristic
74C42×C73#4111yescharacteristic
75C42×C73#4112yes—
76C23×C73#12011yescharacteristic
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73—42no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73C22⋊Q812yes—
83C2×C4×C73C22⋊Q811yescharacteristic
84C2×C4×C73#12412yes—
85C2×C4×C73#12411yescharacteristic
86C2×C4×C73C2×SD1611yescharacteristic
87C2×C4×C73C2×Q1611yescharacteristic
88C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C73#13711yescharacteristic
90C2×Dic73—24no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—42no—
97Dic146—48no—
98Dic146—82no—
99C8×C73—82no—
100#10596831—81no—
101C22×C73—22no—
102C22×C73#39311yescharacteristic
103C22×C73#45011yescharacteristic
104C22×C73#45111yescharacteristic
105C4×C73—22no—
106C4×C73—22no—
107C4×C73—42no—
108C4×C73—42no—
109C4×C73#47511yescharacteristic
110C4×C73#47711yescharacteristic
111C4×C73#48711yescharacteristic
112C4×C73#48911yescharacteristic
113Dic73—44no—
114Dic73—82no—
115Dic73—82no—
116#10371—731nomaximal, Sylow
117C2×C73—21no—
118C2×C73—21no—
119C2×C73#202411yescharacteristic
120C2×C73#203112yes—
121#2399—731no—
122#1995—731no—
123#1503—732no—
124#1303—732no—
125#1274D7311yescharacteristic
126C73#1037111yescharacteristic, complemented, Sylow
127#533—731no—
128C2×C4×Q8—1461no—
129#431C2×D7312yes—
130#423—731no—
131#398—1462no—
132#395—1462no—
133C4×C4⋊C4—732no—
134C4×C4⋊C4—1461no—
135#376C2×D7311yescharacteristic
136#339—1461no—
137#339—1461no—
138#328—1462no—
139#328—1462no—
140C22×Q8—1461no—
141C2×M16—1461no—
142C4⋊Q8—732no—
143#125—1461no—
144C22⋊Q8—1462no—
145C4×Q8—2922no—
146C4×Q8—2922no—
147C4×Q8—1462no—
148#117—1461no—
149C2×C4⋊C4—1462no—
150C2×C4⋊C4—1461no—
151C2×C4⋊C4—1462no—
152C2×C4⋊C4—1462no—
153C2×C42—1462no—
154C2×C42D29211yescharacteristic
155C2×C42D29211yescharacteristic
156C2×C42C22×D7311yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1462no—
159C22⋊C8—22no—
160C4×C8—22no—
161#95—22no—
162#95—732no—
163#95—732no—
164#95—1461no—
165#95—1461no—
166C2×Q8—2922no—
167C2×Q8—2924no—
168C2×Q8—1462no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1462no—
172C22×C4—1462no—
173C22×C4D4×D7312yes—
174C22×C4C2×D29211yescharacteristic
175M16—2921no—
176M16—2921no—
177C2×C8—42no—
178C2×C8—2921no—
179C4⋊C4—2924no—
180C4⋊C4—2922no—
181C4⋊C4—2924no—
182C4⋊C4—2924no—
183C4⋊C4—2922no—
184C4⋊C4—1462no—
185C4⋊C4—1462no—
186C22⋊C4—2921no—
187C42—21no—
188C42—21no—
189C42—2921no—
190C42—2922no—
191C42—2922no—
192C42#1192401712yes—
193C42Dic29212yes—
194C42Q8×D7311yescharacteristic
195C42#1192402711yescharacteristic
196C42#1192402812yes—
197C23#1455656111yescharacteristic
198Q8—2928no—
199Q8—5842no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—2922no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—1464no—
212C2×C4#1455656912yes—
213C2×C4#1455657111yescharacteristic
214C2×C4#1455657712yes—
215C2×C4#1455657911yescharacteristic
216C2×C4C2×#1192401711yescharacteristic
217C2×C4C2×Dic29211yescharacteristic
218C2×C4#1455661911yescharacteristic
219C2×C4#1455662411yescharacteristic
220C8—82no—
221C8—5841no—
222C22—22no—
223C22#1798850011yescharacteristic
224C22#1798874011yescharacteristic
225C22#1798874511yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—2924no—
231C4—5842no—
232C4—5842no—
233C4#1798887311yescharacteristic
234C4#1798888311yescharacteristic
235C4#1798893811yescharacteristic
236C4#1798894711yescharacteristic
237C2—21no—
238C2—21no—
239C2#2261107311yescharacteristic
240C2#2261110212yes—
subgroupG/H
centerC2×C4#14556579
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1274C2
socleC22×C73#393
prime radicalC23×C73#120
hypercenter#1274D73
hypocenterC73#10371
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
21. #14556591Q8semidirect
126. C73#10371semidirect

Maximal subgroups

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

subgroupindex
#226132242
#226109122
#226084312
#226078332
C73×#12742
#1037173