Finite Group Databasebeta

← #34013501

Subgroups — #34013501

Order 18688.

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 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#22626900C211yesmaximal, characteristic, complemented
2#22626169C211yesmaximal, characteristic, complemented
3C2×#17989483C212yesmaximal, complemented
4#22624982C212yesmaximal, complemented
5#22618630C211yesmaximal, characteristic
6#17989646—24no—
7#17989643—22no—
8C2×#14556665C2212yes—
9C2×#14556665C2212yes—
10#17989484—22no—
11#17989483—24no—
12C2×#14556599C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14#17989098C411yescharacteristic
15#17988662—22no—
16#17988661—22no—
17#17988615C411yescharacteristic
18#17988555C411yescharacteristic
19#17988545C2211yescharacteristic
20#407×C73C411yescharacteristic
21#14556666—24no—
22#14556665—22no—
23#14556665—22no—
24#14556665—22no—
25#14556665D414yescomplemented
26#14556664—24no—
27C2×#11924031—22no—
28C2×#11924031—22no—
29C2×#11924028C2×C411yescharacteristic
30#14556599—22no—
31#14556599—22no—
32C2×#11924012—21no—
33C2×#11924012—21no—
34C2×#11924012—22no—
35C2×#11924012—22no—
36C2×#11924011D412yes—
37C22×#10596831C2×C411yescharacteristic
38C22×#10596831C2311yescharacteristic
39C2×C8×D73D412yes—
40#14556574C812yes—
41#14556570—21no—
42#14556565—21no—
43#14556560C812yes—
44C4×C22⋊D73—22no—
45#14556550C2×C411yescharacteristic
46#14556520—22no—
47#14556503—22no—
48#14556501—21no—
49C8×Dic73—21no—
50C23×C4×C73C2×C411yescharacteristic
51C2×M16×C73C2×C411yescharacteristic
52C2×M16×C73C2×C411yescharacteristic
53C22⋊C8×C73—22no—
54#11924031—24no—
55#11924031—42no—
56#11924028—22no—
57#11924028—22no—
58C2×Dic146M1611yescharacteristic
59C2×D292M1611yescharacteristic
60C2×C22⋊D73—22no—
61C2×C22⋊D73C2×C812yes—
62C2×C4×D73C22⋊C412yes—
63#11924012—41no—
64#11924012—42no—
65#11924012—41no—
66#11924011—24no—
67C2×#10596831—21no—
68C2×#10596831—22no—
69C2×#10596831C2×D412yes—
70C2×#10596831C2×D411yescharacteristic
71C2×#10596831C2×D411yescharacteristic
72C2×#10596831#4112yes—
73C8×D73—22no—
74C8×D73—22no—
75C4⋊Dic73—21no—
76#11924007—22no—
77C22⋊Dic73—22no—
78C22⋊Dic73C2×C812yes—
79#11924005—22no—
80C4×Dic73—21no—
81C24×C73C2×C811yescharacteristic
82C22×C4×C73—21no—
83C22×C4×C73—22no—
84C22×C4×C73—21no—
85C22×C4×C73—22no—
86C22×C4×C73C2×C811yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88M16×C73—22no—
89M16×C73—22no—
90C2×C8×C73—21no—
91C2×C8×C73C22⋊C412yes—
92C23×C73—21no—
93C23×C73—22no—
94C23×C73—81no—
95C23×C73C22×C811yescharacteristic
96C22×D73C22⋊C812yes—
97C2×C4×C73—21no—
98C2×C4×C73—22no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—42no—
102C2×C4×C73C2×C22⋊C411yescharacteristic
103C2×C4×C73C4×D412yes—
104C2×C4×C73C2×M1611yescharacteristic
105C2×C4×C73#13112yes—
106C22⋊D73—24no—
107C22⋊D73—82no—
108C2×Dic73—22no—
109C2×Dic73C22⋊C812yes—
110D292—22no—
111C4×D73—22no—
112C4×D73—22no—
113Dic146—22no—
114C8×C73—22no—
115C8×C73—41no—
116#10596831—41no—
117#10596831#12011yescharacteristic
118#10596831#12112yes—
119#10596831#12311yescharacteristic
120C22×C73—41no—
121C22×C73—41no—
122C22×C73—41no—
123C22×C73—41no—
124C22×C73—82no—
125C22×C73C2×C22⋊C811yescharacteristic
126C22×C73C8×D412yes—
127C2×D73—22no—
128C2×D73—42no—
129C4×C73—41no—
130C4×C73#38611yescharacteristic
131C4×C73#40811yescharacteristic
132C4×C73#43512yes—
133Dic73—22no—
134Dic73—42no—
135#12989—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—22no—
138C2×C73—81no—
139C2×C73#186911yescharacteristic
140D73—42no—
141#3407—731no—
142#2930—731no—
143#2115—731no—
144C2×#349—732no—
145#1378—732no—
146C73#1298911yescharacteristic, complemented, Sylow
147C2×#131—731no—
148C22×M16—731no—
149#525—731no—
150#504—1464no—
151C2×M32—732no—
152C2×M32—732no—
153C2×M32—1462no—
154#436—1462no—
155#435—1462no—
156#424—731no—
157#408—731no—
158#407#6440811yescharacteristic
159#405—731no—
160#349—1464no—
161#349—1462no—
162C2×#41—731no—
163C23×C4C2×#6440811yescharacteristic
164#131—1462no—
165#131—1462no—
166C2×M16—732no—
167C2×M16—1462no—
168C2×M16—1461no—
169C2×M16—1461no—
170C2×M16—1462no—
171C2×M16C2×#6440811yescharacteristic
172C2×M16C2×#6440811yescharacteristic
173C22×C8—731no—
174C22×C8—732no—
175C22×C8—731no—
176#123—732no—
177C22⋊Q8—1461no—
178#121—1461no—
179#120—732no—
180C4×D4—1462no—
181#117—731no—
182M32—734no—
183M32—1462no—
184M32—1462no—
185M32—1462no—
186M32—1464no—
187C2×C16—1462no—
188C2×C16—1462no—
189C2×C16—1464no—
190C4⋊C8—1462no—
191C22⋊C8—22no—
192C22⋊C8—1462no—
193#97—1461no—
194C4×C8—1461no—
195C24#1192403611yescharacteristic
196#41—1462no—
197#41—1462no—
198C2×Q8—731no—
199C2×D4—732no—
200C2×D4—731no—
201C2×D4—1462no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—732no—
207C22×C4#1192403611yescharacteristic
208C22×C4C22×#6440811yescharacteristic
209M16—22no—
210M16—22no—
211M16—2922no—
212M16—2921no—
213M16—2921no—
214M16—1464no—
215C2×C8—21no—
216C2×C8—731no—
217C2×C8—731no—
218C2×C8—732no—
219C2×C8—732no—
220C2×C8—1462no—
221C2×C8—1462no—
222C2×C8—1461no—
223C2×C8—1462no—
224C2×C8C22⋊#6440812yes—
225C4⋊C4—1462no—
226C4⋊C4—1461no—
227C22⋊C4—732no—
228C22⋊C4—1462no—
229C22⋊C4—1462no—
230C42—1461no—
231C16—2922no—
232C16—1464no—
233C23—21no—
234C23—22no—
235C23—81no—
236C23—732no—
237C23C2×#1192403611yescharacteristic
238Q8—1462no—
239D4—5842no—
240D4—1464no—
241D4—1462no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—732no—
248C2×C4—1462no—
249C2×C4—1462no—
250C2×C4—1462no—
251C2×C4C2×C22⋊#6440811yescharacteristic
252C2×C4D4×#6440812yes—
253C2×C4D73⋊M1611yescharacteristic
254C2×C4#1455667812yes—
255C8—22no—
256C8—41no—
257C8—2921no—
258C8—732no—
259C8—731no—
260C8—731no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—82no—
266C22—2922no—
267C22—1462no—
268C22#1798955511yescharacteristic
269C22#1798961412yes—
270C4—41no—
271C4—2922no—
272C4—1462no—
273C4#1798953511yescharacteristic
274C4#1798956111yescharacteristic
275C4#1798961912yes—
276C2—21no—
277C2—22no—
278C2—81no—
279C2—2922no—
280C2#2262590511yescharacteristic
subgroupG/H
centerC4#17989535
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#407×C73C4
socleC2×C73#1869
prime radical#17989098C4
cosocleC22×#10596831C23
hypercenter#407#64408
hypocenterC73#12989
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22626900C2semidirect
2. #22626169C2semidirect
3. C2×#17989483C2semidirect
4. #22624982C2semidirect
25. #14556665D4semidirect
146. C73#12989semidirect

Maximal subgroups

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

subgroupindex
#226269002
#226261692
C2×#179894832
#226249822
#226186302
#1298973