Finite Group Databasebeta

← #34009818

Subgroups — #34009818

Order 18688.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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#22625162C212yesmaximal
2#22624861C214yesmaximal, complemented
3#22612517C211yesmaximal, characteristic
4#17989603—22no—
5#17989432C2214yes—
6#17989430—22no—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988494—22no—
10#17988494C412yes—
11#17988494C2212yes—
12#17988450—22no—
13C23×Q8×C73C411yescharacteristic
14C4⋊#10596833—42no—
15C22⋊#10596833—24no—
16C4×#10596833—42no—
17#14556550—42no—
18#14556550—42no—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73C2×C411yescharacteristic
21C2×C4⋊Dic73C2311yescharacteristic
22C2×C22⋊Dic73—21no—
23C2×C22⋊Dic73D412yes—
24C2×C4×Dic73C2×C412yes—
25C2×C4×Dic73D412yes—
26#14556498—24no—
27#14556498—24no—
28C22×Q8×C73—21no—
29C22×Q8×C73—22no—
30C22×Q8×C73—42no—
31C22×Q8×C73—42no—
32C22×Q8×C73C814yescomplemented
33C22×Q8×C73C2×C412yes—
34C23×C4×C73—21no—
35C23×C4×C73C2×C411yescharacteristic
36C2×#10596833—44no—
37C22×Dic73—22no—
38C22×Dic73C22⋊C412yes—
39C22×Dic73C2×D412yes—
40C4⋊Dic73—41no—
41C4⋊Dic73—41no—
42C4⋊Dic73—42no—
43C22⋊Dic73—42no—
44C22⋊Dic73—42no—
45C4×Dic73—22no—
46C4×Dic73—22no—
47C4×Dic73—42no—
48C24×C73C22⋊C411yescharacteristic
49C2×Q8×C73—24no—
50C2×Q8×C73—22no—
51C2×Q8×C73—48no—
52C2×Q8×C73—48no—
53C2×Q8×C73—42no—
54C2×Q8×C73—42no—
55C2×Q8×C73—42no—
56C2×Q8×C73—42no—
57C2×Q8×C73C2×C814yes—
58C22×C4×C73—22no—
59C22×C4×C73—22no—
60C22×C4×C73—24no—
61C22×C4×C73—22no—
62C22×C4×C73—42no—
63C22×C4×C73—42no—
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C2×C812yes—
66C22×C4×C73M1612yes—
67C22×C4×C73C22×C411yescharacteristic
68C23×C73—21no—
69C23×C73—21no—
70C23×C73—41no—
71C23×C73—41no—
72C23×C73C22⋊C812yes—
73C23×C73C2×C22⋊C411yescharacteristic
74Q8×C73—44no—
75Q8×C73—44no—
76Q8×C73—48no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—24no—
80C2×C4×C73—24no—
81C2×C4×C73—24no—
82C2×C4×C73—48no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73C22⋊C812yes—
86C2×C4×C73#9912yes—
87C2×C4×C73#10012yes—
88C2×C4×C73C22×C811yescharacteristic
89C2×C4×C73C2×M1611yescharacteristic
90C2×Dic73—22no—
91C2×Dic73—22no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—42no—
97#10596833—84no—
98C22×C73—21no—
99C22×C73—21no—
100C22×C73—22no—
101C22×C73—42no—
102C22×C73—42no—
103C22×C73—41no—
104C22×C73—41no—
105C22×C73C2×C22⋊C811yescharacteristic
106C22×C73C2×#9911yescharacteristic
107C22×C73C2×#10011yescharacteristic
108C4×C73—21no—
109C4×C73—21no—
110C4×C73—44no—
111C4×C73#32314yes—
112Dic73—42no—
113Dic73—42no—
114Dic73—82no—
115#11434—731nomaximal, Sylow
116C2×C73—21no—
117C2×C73—21no—
118C2×C73—42no—
119C2×C73C2×#32311yescharacteristic
120C2×C73#212911yescharacteristic
121C2×C73#213511yescharacteristic
122#2310—731no—
123#1458—732no—
124#1319—734no—
125C73#1143411yescharacteristic, complemented, Sylow
126C23×Q8#6440811yescharacteristic
127C2×#117—731no—
128C2×#117—731no—
129#431—1462no—
130#391—732no—
131#391—732no—
132#391—1462no—
133#381—1462no—
134#324—734no—
135#323—1462no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8#1059683314yescomplemented
141C22×Q8C2×#6440812yes—
142C23×C4—21no—
143C23×C4C2×#6440811yescharacteristic
144#117—2922no—
145#117—2922no—
146C2×C4⋊C4—731no—
147C2×C4⋊C4—731no—
148C2×C4⋊C4—1461no—
149C2×C22⋊C4—732no—
150C2×C22⋊C4—1461no—
151C2×C42—732no—
152C2×C42—732no—
153C4⋊C8—2922no—
154C22⋊C8—1464no—
155C4×C8—2922no—
156#95—1464no—
157#95—1464no—
158C24C22⋊#6440811yescharacteristic
159C2×Q8—22no—
160C2×Q8—24no—
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C2×#1059683314yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—732no—
175C22×C4—732no—
176C22×C4—1462no—
177C22×C4C22⋊#6440811yescharacteristic
178C22×C4C2×#1059683312yes—
179C22×C4#1192403712yes—
180C22×C4C22×#6440811yescharacteristic
181C2×C8—2924no—
182C4⋊C4—2921no—
183C4⋊C4—2921no—
184C4⋊C4—2922no—
185C22⋊C4—2922no—
186C22⋊C4—2922no—
187C42—2922no—
188C42—1462no—
189C42—1462no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊#1059683312yes—
195C23C2×C22⋊#6440811yescharacteristic
196Q8—44no—
197Q8—44no—
198Q8—48no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—2922no—
208C2×C4—2922no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—2922no—
212C2×C4—1462no—
213C2×C4—1462no—
214C2×C4C22⋊#1059683312yes—
215C2×C4#1455664912yes—
216C2×C4#1455665112yes—
217C2×C4C22×#1059683311yescharacteristic
218C2×C4C2×#1192403711yescharacteristic
219C8—5844no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×C22⋊#1059683311yescharacteristic
228C22C2×#1455664911yescharacteristic
229C22C2×#1455665111yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—2922no—
234C4—2922no—
235C4—5842no—
236C4#1798943014yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1798943011yescharacteristic
241C2#2262621311yescharacteristic
242C2#2262623311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC23×Q8×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC23×Q8#64408
hypocenterC73#11434
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #22624861C2semidirect
32. C22×Q8×C73C8semidirect
125. C73#11434semidirect
140. C22×Q8#10596833semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248612
#226125172
#1143473