Finite Group Databasebeta

← #34009987

Subgroups — #34009987

Order 18688.

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

normal subgroups only

This group has 392 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
1#22625173C212yesmaximal
2#22624872C212yesmaximal
3#22624861C212yesmaximal
4#22612647C211yesmaximal, characteristic
5#17989604—22no—
6#17989433—22no—
7#17989432C2214yes—
8C2×#14556589C411yescharacteristic
9C2×C4×Dic146C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988492C411yescharacteristic
12#17988471—22no—
13#17988468—22no—
14#17988452C2211yescharacteristic
15#17988433C411yescharacteristic
16C2×C4×Q8×C73C411yescharacteristic
17#14556658—42no—
18C22⋊#10596833—24no—
19C4×#10596833—42no—
20C22×Dic146C2×C411yescharacteristic
21#14556589—22no—
22#14556589C814yescomplemented
23C4×Dic146—42no—
24C2×C4⋊Dic73C2311yescharacteristic
25C2×#11924007—22no—
26C2×#11924007D412yes—
27C2×C4×Dic73C2×C411yescharacteristic
28C2×C4×Dic73C2×C411yescharacteristic
29C2×C4×Dic73D412yes—
30#14556498—22no—
31#14556498—22no—
32#14556498—22no—
33#14556498—21no—
34#14556498—21no—
35C22×Q8×C73C2×C411yescharacteristic
36C4×Q8×C73—42no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C42×C73—21no—
40C2×C42×C73C2×C411yescharacteristic
41C2×#10596833—44no—
42C2×Dic146—41no—
43C2×Dic146—41no—
44C2×Dic146C2×C812yes—
45C2×Dic146M1612yes—
46C22×Dic73—21no—
47C22×Dic73—21no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—41no—
51#11924007—28no—
52#11924007—42no—
53C4×Dic73—22no—
54C4×Dic73—22no—
55C4×Dic73—21no—
56C4×Dic73—41no—
57C4×Dic73C2×C812yes—
58C2×Q8×C73—21no—
59C2×Q8×C73—41no—
60C2×Q8×C73—41no—
61C2×Q8×C73C2×C812yes—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C22⋊C411yescharacteristic
66C22×C4×C73C22×C411yescharacteristic
67C4⋊C4×C73—41no—
68C4⋊C4×C73—42no—
69C42×C73—22no—
70C42×C73—42no—
71C23×C73C2×C22⋊C411yescharacteristic
72Q8×C73—42no—
73Q8×C73—42no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—22no—
77C2×C4×C73—21no—
78C2×C4×C73—41no—
79C2×C4×C73—41no—
80C2×C4×C73—41no—
81C2×C4×C73—41no—
82C2×C4×C73—42no—
83C2×C4×C73—41no—
84C2×C4×C73#9912yes—
85C2×C4×C73C22×C811yescharacteristic
86C2×C4×C73C2×M1611yescharacteristic
87C2×Dic73—22no—
88C2×Dic73—22no—
89C2×Dic73—22no—
90C2×Dic73—22no—
91C2×Dic73—41no—
92C2×Dic73—41no—
93C2×Dic73—42no—
94C2×Dic73—41no—
95C2×Dic73—41no—
96C2×Dic73—41no—
97C2×Dic73C22⋊C814yes—
98Dic146—42no—
99Dic146—42no—
100#10596833—84no—
101C22×C73—21no—
102C22×C73—21no—
103C22×C73C2×C22⋊C811yescharacteristic
104C22×C73C2×#9911yescharacteristic
105C22×C73#41311yescharacteristic
106C4×C73—22no—
107C4×C73—21no—
108C4×C73—21no—
109C4×C73—41no—
110C4×C73—42no—
111C4×C73—81no—
112Dic73—24no—
113Dic73—42no—
114Dic73—42no—
115Dic73—81no—
116#11482—731nomaximal, Sylow
117C2×C73—21no—
118C2×C73—21no—
119C2×C73#145911yescharacteristic
120C2×C73#212911yescharacteristic
121C2×C73#213311yescharacteristic
122#2327—731no—
123#1461—732no—
124#1324—732no—
125#1319—732no—
126C73#1148211yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—731no—
128C2×C4×Q8—731no—
129C2×C4×Q8#6440811yescharacteristic
130#431—1462no—
131#391—731no—
132#391—731no—
133#387—1462no—
134#387—1462no—
135#382—731no—
136#376—731no—
137#324—734no—
138#324—1462no—
139C22×Q8—731no—
140C22×Q8C2×#6440811yescharacteristic
141C4⋊Q8—734no—
142C4⋊Q8—1462no—
143C4×Q8—42no—
144C4×Q8—2922no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—732no—
147C2×C4⋊C4—731no—
148C2×C4⋊C4—1462no—
149C2×C4⋊C4C2×#6440811yescharacteristic
150C2×C42—21no—
151C2×C42—731no—
152C2×C42—731no—
153C2×C42—732no—
154C2×C42C2×#6440811yescharacteristic
155C4⋊C8—2922no—
156C22⋊C8—1464no—
157C4×C8—2922no—
158#95—1462no—
159#95—1462no—
160#95—1462no—
161#95—1461no—
162#95—1461no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—2921no—
167C2×Q8—2921no—
168C2×Q8—732no—
169C2×Q8—732no—
170C2×Q8#1192403612yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—732no—
174C22×C4—732no—
175C22×C4—1461no—
176C22×C4—1461no—
177C22×C4C22⋊#6440811yescharacteristic
178C22×C4C22⋊#6440811yescharacteristic
179C22×C4C22×#6440811yescharacteristic
180C2×C8—2924no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—2922no—
184C4⋊C4—2921no—
185C4⋊C4—1468no—
186C42—22no—
187C42—42no—
188C42—2921no—
189C42—732no—
190C42—1461no—
191C42—1462no—
192C42—1462no—
193C23C2×C22⋊#6440811yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—2922no—
197Q8—2922no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—2921no—
209C2×C4—2921no—
210C2×C4—2922no—
211C2×C4—2921no—
212C2×C4—2921no—
213C2×C4—2921no—
214C2×C4—734no—
215C2×C4—1462no—
216C2×C4—1462no—
217C2×C4—1462no—
218C2×C4—1462no—
219C2×C4#1455664912yes—
220C2×C4C2×#1192403611yescharacteristic
221C2×C4D73⋊M1611yescharacteristic
222C8—5844no—
223C22—21no—
224C22—21no—
225C22#1798955511yescharacteristic
226C22C2×#1455664911yescharacteristic
227C22#1798957311yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—42no—
232C4—41no—
233C4—81no—
234C4—2922no—
235C4—2922no—
236C4—5841no—
237C4—1464no—
238C2—21no—
239C2—21no—
240C2#2262516611yescharacteristic
241C2#2262621311yescharacteristic
242C2#2262622611yescharacteristic
subgroupG/H
centerC22C2×#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#11482
LCSG ▷ C2×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
22. #14556589C8semidirect
126. C73#11482semidirect

Maximal subgroups

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

subgroupindex
#226251732
#226248722
#226248612
#226126472
#1148273