Finite Group Databasebeta

← #34009971

Subgroups — #34009971

Order 18688.

428 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 90 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#22625173C212yesmaximal
2#22624874C212yesmaximal
3#22624861C212yesmaximal
4C4×#14556589C211yesmaximal, characteristic
5#17989604—22no—
6#17989433—22no—
7#17989432C2214yes—
8C2×#14556589C411yescharacteristic
9C2×C4×Dic146C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988492C411yescharacteristic
12#17988482—22no—
13#17988456C2211yescharacteristic
14C4×#11924007—22no—
15C42×Dic73C411yescharacteristic
16C2×C4×Q8×C73C411yescharacteristic
17#14556658—42no—
18C22⋊#10596833—24no—
19C4×#10596833—42no—
20C22×Dic146C2×C411yescharacteristic
21#14556589—22no—
22#14556589C814yescomplemented
23C4×Dic146—41no—
24C4×Dic146—41no—
25C2×C4⋊Dic73C2311yescharacteristic
26C2×#11924007—22no—
27C2×#11924007—22no—
28C2×#11924007D412yes—
29C2×C4×Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D412yes—
34#14556498—22no—
35#14556498—22no—
36C22×Q8×C73C2×C411yescharacteristic
37C4×Q8×C73—41no—
38C4×Q8×C73—41no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C42×C73—21no—
42C2×C42×C73C2×C411yescharacteristic
43C2×#10596833—44no—
44C2×Dic146—41no—
45C2×Dic146—41no—
46C2×Dic146C2×C812yes—
47C2×Dic146M1612yes—
48C22×Dic73—21no—
49C22×Dic73—21no—
50C22×Dic73C22⋊C412yes—
51C22×Dic73C2×D412yes—
52C4⋊Dic73—41no—
53#11924007—28no—
54#11924007—42no—
55#11924007—44no—
56C4×Dic73—24no—
57C4×Dic73—22no—
58C4×Dic73—22no—
59C4×Dic73—21no—
60C4×Dic73—42no—
61C4×Dic73—41no—
62C4×Dic73C2×C812yes—
63C2×Q8×C73—21no—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×Q8×C73C2×C812yes—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C22×C411yescharacteristic
72C4⋊C4×C73—41no—
73C4⋊C4×C73—42no—
74C42×C73—21no—
75C42×C73—21no—
76C42×C73—41no—
77C42×C73—41no—
78C23×C73C2×C22⋊C411yescharacteristic
79Q8×C73—42no—
80Q8×C73—42no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—22no—
84C2×C4×C73—22no—
85C2×C4×C73—21no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—42no—
90C2×C4×C73—41no—
91C2×C4×C73#9911yescharacteristic
92C2×C4×C73#9911yescharacteristic
93C2×C4×C73C22×C811yescharacteristic
94C2×C4×C73C2×M1611yescharacteristic
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—42no—
105C2×Dic73C22⋊C814yes—
106Dic146—42no—
107Dic146—42no—
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73C2×C22⋊C811yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73#41311yescharacteristic
114C4×C73—21no—
115C4×C73—21no—
116C4×C73—21no—
117C4×C73—41no—
118C4×C73—42no—
119C4×C73—81no—
120C4×C73#35612yes—
121Dic73—24no—
122Dic73—24no—
123Dic73—42no—
124Dic73—42no—
125#11478—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—21no—
128C2×C73#145911yescharacteristic
129C2×C73#212611yescharacteristic
130C2×C73C2×#35611yescharacteristic
131C4×C4⋊Q8—731no—
132#1461—732no—
133#1325—732no—
134#1319—732no—
135C73#1147811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8—731no—
137C2×C4×Q8—731no—
138C2×C4×Q8#6440811yescharacteristic
139#431—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×Q8—731no—
149C22×Q8C2×#6440811yescharacteristic
150C4⋊Q8—734no—
151C4⋊Q8—1462no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—2921no—
155C4×Q8—2921no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—1462no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4C2×#6440811yescharacteristic
162C2×C42—21no—
163C2×C42—731no—
164C2×C42—731no—
165C2×C42—732no—
166C2×C42—1461no—
167C2×C42—1461no—
168C2×C42C2×#6440811yescharacteristic
169C4⋊C8—2922no—
170C22⋊C8—1464no—
171C4×C8—2922no—
172#95—1462no—
173#95—1462no—
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—2921no—
178C2×Q8—2921no—
179C2×Q8—732no—
180C2×Q8—732no—
181C2×Q8#1192403612yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—732no—
185C22×C4—732no—
186C22×C4—1461no—
187C22×C4—1461no—
188C22×C4C22⋊#6440811yescharacteristic
189C22×C4C22⋊#6440811yescharacteristic
190C22×C4C22×#6440811yescharacteristic
191C2×C8—2924no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—2922no—
195C4⋊C4—2921no—
196C4⋊C4—2924no—
197C4⋊C4—1468no—
198C42—21no—
199C42—21no—
200C42—41no—
201C42—41no—
202C42—2921no—
203C42—2922no—
204C42—732no—
205C42—1464no—
206C42—1461no—
207C42—1462no—
208C42—1462no—
209C23C2×C22⋊#6440811yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—2922no—
213Q8—2922no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—2921no—
225C2×C4—2921no—
226C2×C4—2922no—
227C2×C4—734no—
228C2×C4—1462no—
229C2×C4—1462no—
230C2×C4—1462no—
231C2×C4—1462no—
232C2×C4—1462no—
233C2×C4—1462no—
234C2×C4—1462no—
235C2×C4#1455664911yescharacteristic
236C2×C4#1455664911yescharacteristic
237C2×C4C2×#1192403611yescharacteristic
238C2×C4D73⋊M1611yescharacteristic
239C8—5844no—
240C22—21no—
241C22—21no—
242C22#1798955511yescharacteristic
243C22C2×#1455664911yescharacteristic
244C22#1798957311yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—81no—
251C4—2922no—
252C4—2922no—
253C4—1464no—
254C4—1464no—
255C4#1798949712yes—
256C2—21no—
257C2—21no—
258C2#2262516611yescharacteristic
259C2#2262620211yescharacteristic
260C2C2×#1798949711yescharacteristic
subgroupG/H
centerC2×C4#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#11478
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
22. #14556589C8semidirect
135. C73#11478semidirect

Maximal subgroups

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

subgroupindex
#226251732
#226248742
#226248612
C4×#145565892
#1147873