Finite Group Databasebeta

← #34003781

Subgroups — #34003781

Order 18688.

516 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 132 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#22625716C212yesmaximal
2C2×#17989447C212yesmaximal
3#22624809C212yesmaximal
4#22612500C211yesmaximal, characteristic
5C2×#14556658C2214yes—
6C2×#14556646—22no—
7#17989453—22no—
8#17989447C418yescomplemented
9C2×#14556589C2211yescharacteristic
10C2×Q8×Dic73C411yescharacteristic
11C2×#14556556C411yescharacteristic
12#17988493C411yescharacteristic
13#17988484—22no—
14#17988456C2211yescharacteristic
15#17988438—22no—
16C42×Dic73C2211yescharacteristic
17#391×C73C411yescharacteristic
18C22×#10596833—24no—
19#14556658—24no—
20#14556658C2×C418yes—
21#14556646—44no—
22C22×Dic146C2×C411yescharacteristic
23#14556589—22no—
24#14556589C2×C414yes—
25Q8×Dic73—42no—
26#14556556—42no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—22no—
30C2×#11924007—21no—
31C2×#11924007—22no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73D412yes—
36C2×C4×Dic73D412yes—
37C2×C4×Dic73C2311yescharacteristic
38#14556498—22no—
39C22×Q8×C73C2×C411yescharacteristic
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C42×C73C2×C411yescharacteristic
42#95×C73—22no—
43C2×#10596833—28no—
44C2×#10596833—44no—
45C2×#10596833—44no—
46C2×Dic146—21no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146C4212yes—
50C22×Dic73—21no—
51C22×Dic73—21no—
52C22×Dic73C22⋊C411yescharacteristic
53C22×Dic73C22⋊C411yescharacteristic
54C22×Dic73C2×D412yes—
55C4⋊Dic73—42no—
56C4⋊Dic73—41no—
57#11924007—28no—
58#11924007—42no—
59#11924007—42no—
60#11924007—42no—
61C4×Dic73—22no—
62C4×Dic73—22no—
63C4×Dic73—24no—
64C4×Dic73—22no—
65C4×Dic73—42no—
66C4×Dic73—42no—
67C4×Dic73C22×C412yes—
68C4×Dic73#4112yes—
69C2×Q8×C73—21no—
70C2×Q8×C73—41no—
71C2×Q8×C73—41no—
72C2×Q8×C73C4212yes—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73C22⋊C412yes—
76C22×C4×C73C22×C411yescharacteristic
77C4⋊C4×C73—41no—
78C42×C73—41no—
79C23×C73C2×C22⋊C411yescharacteristic
80Q8×C73—42no—
81Q8×C73—42no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—22no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73C2×C4211yescharacteristic
93C2×C4×C73#11711yescharacteristic
94C2×Dic73—22no—
95C2×Dic73—22no—
96C2×Dic73—21no—
97C2×Dic73—22no—
98C2×Dic73—41no—
99C2×Dic73—41no—
100C2×Dic73—41no—
101C2×Dic73—42no—
102C2×Dic73#10012yes—
103C2×Dic73C4×D414yes—
104C2×Dic73AGL(1,ℤ/8)12yes—
105C2×Dic73#13714yes—
106C2×Dic73#13712yes—
107Dic146—42no—
108Dic146—42no—
109#10596833—48no—
110C22×C73C4×C22⋊C411yescharacteristic
111C22×C73C2×#10011yescharacteristic
112C22×C73#41311yescharacteristic
113C22×C73#41712yes—
114C22×C73#41912yes—
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—42no—
118C4×C73—42no—
119C4×C73—81no—
120Dic73—22no—
121Dic73—41no—
122Dic73—42no—
123Dic73—81no—
124Dic73#44014yes—
125Dic73#44114yes—
126#8232—731nomaximal, Sylow
127C2×C73#155811yescharacteristic
128C2×C73#155912yes—
129C2×C73#156011yescharacteristic
130C2×C73C4×#10011yescharacteristic
131C2×C73#176312yes—
132C4×C4⋊Q8—731no—
133#1767—732no—
134C2×#333—732no—
135#1296—732no—
136C73#823211yescharacteristic, complemented, Sylow
137C2×C4⋊Q8—731no—
138C2×C4×Q8—731no—
139C2×C4×Q8—731no—
140C2×C4⋊C8—734no—
141C2×#97—1462no—
142#391—731no—
143#391#6440811yescharacteristic
144#389—1462no—
145#384—731no—
146C4×C4⋊C4—1462no—
147C43—731no—
148#336—1462no—
149#333—738no—
150C22×Q8—731no—
151C22×Q8C2×#6440811yescharacteristic
152C22×C8—1464no—
153C4⋊Q8—734no—
154C4⋊Q8—1462no—
155C4×Q8—2922no—
156C4×Q8—2922no—
157C2×C4⋊C4—731no—
158C2×C4⋊C4—1461no—
159C2×C4⋊C4—1461no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4—1462no—
162C2×C4⋊C4C2×#6440811yescharacteristic
163C2×C42—731no—
164C2×C42—732no—
165C2×C42—732no—
166C2×C42—731no—
167C2×C42—1461no—
168C2×C42—1461no—
169C2×C42C2×#6440811yescharacteristic
170C4⋊C8—738no—
171C4⋊C8—1464no—
172#97—2924no—
173#95—22no—
174#95—1462no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—2921no—
179C2×Q8—2921no—
180C2×Q8—732no—
181C2×Q8—1461no—
182C2×Q8C4×#6440812yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—732no—
186C22×C4—731no—
187C22×C4—731no—
188C22×C4—1461no—
189C22×C4—1461no—
190C22×C4C22⋊#6440812yes—
191C22×C4C22×#6440811yescharacteristic
192C2×C8—2924no—
193C2×C8—2924no—
194C2×C8—1468no—
195C4⋊C4—41no—
196C4⋊C4—2921no—
197C4⋊C4—2922no—
198C4⋊C4—2922no—
199C4⋊C4—2922no—
200C4⋊C4—2922no—
201C4⋊C4—1468no—
202C42—41no—
203C42—2922no—
204C42—2922no—
205C42—732no—
206C42—732no—
207C42—1462no—
208C42—1462no—
209C42—1464no—
210C42—1462no—
211C23C2×C22⋊#6440811yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—2922no—
215Q8—2922no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—2921no—
227C2×C4—2921no—
228C2×C4—2921no—
229C2×C4—2922no—
230C2×C4—734no—
231C2×C4—732no—
232C2×C4—732no—
233C2×C4—734no—
234C2×C4—732no—
235C2×C4—1462no—
236C2×C4—1461no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4C2×C4×#6440811yescharacteristic
240C2×C4#1455667011yescharacteristic
241C8—2928no—
242C22C4×C22⋊#6440811yescharacteristic
243C22D73⋊#10011yescharacteristic
244C22#1798957411yescharacteristic
245C22#1798957812yes—
246C22#1798958012yes—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—2922no—
253C4—2921no—
254C4—5841no—
255C4—734no—
256C4—734no—
257C4—1462no—
258C2#2262546511yescharacteristic
259C2#2262546712yes—
260C2#2262546911yescharacteristic
261C2#2262568111yescharacteristic
262C2#2262570312yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#391#64408
hypocenterC73#8232
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
8. #17989447C4semidirect
136. C73#8232semidirect

Maximal subgroups

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

subgroupindex
#226257162
C2×#179894472
#226248092
#226125002
#823273