Finite Group Databasebeta

← #34010160

Subgroups — #34010160

Order 18688.

412 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 76 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#22625194C212yesmaximal
2#22624873C212yesmaximal
3#22624861C212yesmaximal
4C4⋊Q8×Dic73C211yesmaximal, characteristic
5#17989607—22no—
6#17989432—22no—
7#17989432C2214yes—
8C2×Q8×Dic73C411yescharacteristic
9C2×Q8×Dic73C2211yescharacteristic
10#17988494C411yescharacteristic
11#17988494C2211yescharacteristic
12#17988486—22no—
13#17988457C2211yescharacteristic
14C4⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic
16C2×C4⋊Q8×C73C411yescharacteristic
17#14556658—42no—
18C22⋊#10596833—24no—
19#14556646—42no—
20Q8×Dic73—41no—
21Q8×Dic73—41no—
22Q8×Dic73—41no—
23Q8×Dic73—41no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73D412yes—
28C2×C4⋊Dic73C2311yescharacteristic
29C2×C4×Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
35#14556498—22no—
36#14556498—22no—
37C22×Q8×C73C2×C411yescharacteristic
38C22×Q8×C73C2×C411yescharacteristic
39C4⋊Q8×C73—22no—
40C4⋊Q8×C73—22no—
41C2×C4⋊C4×C73—22no—
42C2×C42×C73C2×C411yescharacteristic
43C2×#10596833—44no—
44C22×Dic73—21no—
45C22×Dic73—21no—
46C22×Dic73C22⋊C411yescharacteristic
47C22×Dic73C22⋊C411yescharacteristic
48C22×Dic73C2×D412yes—
49C4⋊Dic73—42no—
50C4⋊Dic73—41no—
51C4⋊Dic73—42no—
52C4⋊Dic73—44no—
53C4⋊Dic73—41no—
54C4×Dic73—24no—
55C4×Dic73—21no—
56C4×Dic73—21no—
57C4×Dic73—22no—
58C4×Dic73—22no—
59C4×Dic73—42no—
60C4×Dic73—41no—
61C4×Dic73—41no—
62C4×Dic73—41no—
63C2×Q8×C73—21no—
64C2×Q8×C73—21no—
65C2×Q8×C73—21no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×Q8×C73—41no—
69C2×Q8×C73—41no—
70C2×Q8×C73M1612yes—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73C22⋊C412yes—
74C22×C4×C73C22×C411yescharacteristic
75C4⋊C4×C73—28no—
76C42×C73—21no—
77C42×C73M1612yes—
78C23×C73C2×C22⋊C411yescharacteristic
79Q8×C73—42no—
80Q8×C73—42no—
81Q8×C73—42no—
82Q8×C73—42no—
83C2×C4×C73—22no—
84C2×C4×C73—22no—
85C2×C4×C73—22no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—22no—
89C2×C4×C73—22no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73C2×M1611yescharacteristic
95C2×C4×C73C2×M1611yescharacteristic
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—21no—
99C2×Dic73—22no—
100C2×Dic73—41no—
101C2×Dic73—41no—
102C2×Dic73—41no—
103C2×Dic73—42no—
104C2×Dic73—42no—
105C2×Dic73—41no—
106C2×Dic73#9911yescharacteristic
107C2×Dic73#9911yescharacteristic
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73#40711yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73#41311yescharacteristic
114C4×C73—24no—
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—42no—
118C4×C73—42no—
119Dic73—21no—
120Dic73—42no—
121Dic73—42no—
122Dic73—41no—
123Dic73—81no—
124Dic73#35612yes—
125#11532—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—21no—
128C2×C73#146511yescharacteristic
129C2×C73#212611yescharacteristic
130C2×C73C2×#35611yescharacteristic
131C4×C4⋊Q8—731no—
132#1467—732no—
133#1325—732no—
134#1319—732no—
135C73#1153211yescharacteristic, complemented, Sylow
136C2×C4⋊Q8#6440811yescharacteristic
137C2×C4×Q8—731no—
138C2×C4×Q8—731no—
139#432—1462no—
140#391—731no—
141#391—731no—
142#389—1462no—
143#384—731no—
144C4×C4⋊C4—1462no—
145C43—731no—
146#324—734no—
147#324—1462no—
148C22×Q8C2×#6440811yescharacteristic
149C22×Q8C2×#6440811yescharacteristic
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4×Q8—2921no—
153C4×Q8—2921no—
154C4×Q8—2921no—
155C4×Q8—2921no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—731no—
160C2×C4⋊C4—1461no—
161C2×C4⋊C4—1462no—
162C2×C42—731no—
163C2×C42—732no—
164C2×C42—731no—
165C2×C42—1461no—
166C2×C42—1461no—
167C2×C42—1461no—
168C2×C42C2×#6440811yescharacteristic
169C4⋊C8—2922no—
170C22⋊C8—1464no—
171#97—2922no—
172#95—1462no—
173#95—1462no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8#1192403712yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—731no—
185C22×C4—732no—
186C22×C4—731no—
187C22×C4—1461no—
188C22×C4—1461no—
189C22×C4C22⋊#6440812yes—
190C22×C4C22×#6440811yescharacteristic
191C2×C8—2924no—
192C4⋊C4—28no—
193C4⋊C4—2921no—
194C4⋊C4—2924no—
195C4⋊C4—2921no—
196C4⋊C4—2922no—
197C4⋊C4—2922no—
198C42—21no—
199C42—2921no—
200C42—2922no—
201C42—2921no—
202C42—2921no—
203C42—1462no—
204C42—1462no—
205C42—1461no—
206C42—1461no—
207C42—1464no—
208C42#1192403712yes—
209C23C2×C22⋊#6440811yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2921no—
226C2×C4—2921no—
227C2×C4—2921no—
228C2×C4—2922no—
229C2×C4—2922no—
230C2×C4—2921no—
231C2×C4—731no—
232C2×C4—731no—
233C2×C4—1462no—
234C2×C4—1461no—
235C2×C4—1462no—
236C2×C4—1462no—
237C2×C4C2×#1192403711yescharacteristic
238C2×C4C2×#1192403711yescharacteristic
239C8—5844no—
240C22—21no—
241C22—21no—
242C22#1798955711yescharacteristic
243C22D73⋊#9911yescharacteristic
244C22#1798957411yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—2922no—
251C4—2922no—
252C4—2921no—
253C4—5841no—
254C4—732no—
255C4—1461no—
256C2—21no—
257C2—21no—
258C2#2262518711yescharacteristic
259C2#2262620511yescharacteristic
260C2D73⋊#35611yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#11532
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
135. C73#11532semidirect

Maximal subgroups

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

subgroupindex
#226251942
#226248732
#226248612
C4⋊Q8×Dic732
#1153273