Finite Group Databasebeta

← #34013098

Subgroups — #34013098

Order 18688.

432 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#17989497C212yesmaximal
2#22625797C212yesmaximal
3#22625029C212yesmaximal
4#22612647C211yesmaximal, characteristic
5#17989590—22no—
6C2×#14556652C2214yes—
7#17989497—24no—
8#17989467—22no—
9C2×#14556589C2211yescharacteristic
10C2×C4×Dic146C2211yescharacteristic
11#17988494C411yescharacteristic
12#17988492C2211yescharacteristic
13#17988471—22no—
14#17988468—22no—
15#17988452C411yescharacteristic
16#17988433C411yescharacteristic
17C2×C4×Q8×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556658—42no—
20C4⋊#10596833—42no—
21#14556652—24no—
22#14556652—24no—
23C22×Dic146C2311yescharacteristic
24#14556589—22no—
25#14556589D412yes—
26#14556589Q812yes—
27C4×Dic146—42no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—22no—
30C2×#11924007D412yes—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D412yes—
34#14556498—22no—
35#14556498—22no—
36#14556498—22no—
37#14556498—21no—
38#14556498—21no—
39C22×Q8×C73C2×C411yescharacteristic
40C4×Q8×C73—42no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C42×C73—21no—
44C2×C42×C73C2×C411yescharacteristic
45#11924037—44no—
46#11924037—44no—
47C2×#10596833—44no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146C2×D411yescharacteristic
51C2×Dic146C2×Q811yescharacteristic
52C2×Dic146#4112yes—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73C22⋊C412yes—
56C22×Dic73C2×D412yes—
57C4⋊Dic73—41no—
58#11924007—24no—
59#11924007—24no—
60#11924007—42no—
61C4×Dic73—22no—
62C4×Dic73—22no—
63C4×Dic73—21no—
64C4×Dic73—41no—
65C4×Dic73C4⋊C412yes—
66C2×Q8×C73—21no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×Q8×C73C4⋊C412yes—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22×C411yescharacteristic
75C4⋊C4×C73—41no—
76C4⋊C4×C73—42no—
77C42×C73—22no—
78C42×C73—42no—
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—21no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73—42no—
91C2×C4×C73—41no—
92C2×C4×C73#9912yes—
93C2×C4×C73C2×C4⋊C411yescharacteristic
94C2×C4×C73#11711yescharacteristic
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—41no—
100C2×Dic73—41no—
101C2×Dic73—42no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73#12112yes—
106C2×Dic73C22⋊Q812yes—
107Dic146—41no—
108Dic146—41no—
109Dic146—42no—
110#10596833—84no—
111C22×C73—21no—
112C22×C73—21no—
113C22×C73#38011yescharacteristic
114C22×C73C2×#9911yescharacteristic
115C22×C73#41011yescharacteristic
116C4×C73—21no—
117C4×C73—21no—
118C4×C73—22no—
119C4×C73—42no—
120C4×C73—41no—
121C4×C73—81no—
122Dic73—22no—
123Dic73—22no—
124Dic73—42no—
125Dic73—42no—
126Dic73—81no—
127#12804—731nomaximal, Sylow
128C2×C73—21no—
129C2×C73—21no—
130C2×C73#179911yescharacteristic
131C2×C73#213311yescharacteristic
132C2×C73#213511yescharacteristic
133#2327—731no—
134C2×#356—732no—
135#1801—732no—
136#1394—732no—
137C73#1280411yescharacteristic, complemented, Sylow
138C2×C4⋊Q8—731no—
139C2×C4×Q8—731no—
140C2×C4×Q8#6440811yescharacteristic
141#423—1462no—
142C2×#101—734no—
143#391—731no—
144#391—731no—
145#387—1462no—
146#387—1462no—
147#382—731no—
148#376—731no—
149#356—1464no—
150#343—1462no—
151C22×Q8—731no—
152C22×Q8C2×#6440811yescharacteristic
153C2×M16—1464no—
154C4⋊Q8—732no—
155C4⋊Q8—732no—
156C4⋊Q8—1462no—
157C4×Q8—42no—
158C4×Q8—2922no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—732no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1462no—
163C2×C4⋊C4C2×#6440811yescharacteristic
164C2×C42—21no—
165C2×C42—731no—
166C2×C42—731no—
167C2×C42—732no—
168C2×C42C2×#6440811yescharacteristic
169C4⋊C8—2922no—
170C4⋊C8—2922no—
171#101—1464no—
172#101—1464no—
173#95—1462no—
174#95—1462no—
175#95—1462no—
176#95—1461no—
177#95—1461no—
178C2×Q8—21no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—2921no—
182C2×Q8—2921no—
183C2×Q8—731no—
184C2×Q8—731no—
185C2×Q8—732no—
186C2×Q8C4⋊#6440812yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—732no—
190C22×C4—732no—
191C22×C4—1461no—
192C22×C4—1461no—
193C22×C4C22⋊#6440811yescharacteristic
194C22×C4C22⋊#6440811yescharacteristic
195C22×C4C22×#6440811yescharacteristic
196M16—2924no—
197M16—2924no—
198C2×C8—2924no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—2922no—
202C4⋊C4—2921no—
203C4⋊C4—1464no—
204C4⋊C4—1464no—
205C42—22no—
206C42—42no—
207C42—2921no—
208C42—732no—
209C42—1462no—
210C42—1462no—
211C42—1461no—
212C23C2×C22⋊#6440811yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—2921no—
216Q8—2921no—
217Q8—2922no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—2921no—
229C2×C4—2921no—
230C2×C4—2922no—
231C2×C4—2921no—
232C2×C4—2921no—
233C2×C4—2921no—
234C2×C4—732no—
235C2×C4—732no—
236C2×C4—1462no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4#1455664912yes—
241C2×C4C2×C4⋊#6440811yescharacteristic
242C2×C4#1455667011yescharacteristic
243C8—5844no—
244C22—21no—
245C22—21no—
246C22#1798952911yescharacteristic
247C22C2×#1455664911yescharacteristic
248C22#1798956611yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—42no—
254C4—81no—
255C4—2922no—
256C4—2922no—
257C4—5841no—
258C4—1462no—
259C4—1462no—
260C2—21no—
261C2—21no—
262C2#2262579111yescharacteristic
263C2#2262622611yescharacteristic
264C2#2262623311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Dic146C23
hypercenterC2×C4×Q8#64408
hypocenterC73#12804
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
137. C73#12804semidirect

Maximal subgroups

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

subgroupindex
C2×#179894972
#226257972
#226250292
#226126472
#1280473