Finite Group Databasebeta

← #34010180

Subgroups — #34010180

Order 18688.

348 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 70 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#22626304C212yesmaximal, complemented
2#22624865C212yesmaximal, complemented
3#22624863C212yesmaximal, complemented
4#22618626C211yesmaximal, characteristic
5#17989510—22no—
6#17989483C2214yes—
7#17989479—22no—
8C2×#14556598C411yescharacteristic
9C23×#10596831C411yescharacteristic
10#17988659—22no—
11D4×#10596831—22no—
12#17988614C411yescharacteristic
13#17988554C2211yescharacteristic
14#17988551C2211yescharacteristic
15#17988543C2211yescharacteristic
16#525×C73C411yescharacteristic
17#14556665—24no—
18#14556598—42no—
19C2×#11924012—21no—
20C2×#11924012C2×C411yescharacteristic
21C2×#11924012C2311yescharacteristic
22C22×#10596831—21no—
23C22×#10596831—21no—
24C22×#10596831—21no—
25C22×#10596831—42no—
26C22×#10596831C2×C411yescharacteristic
27C22×#10596831C2×C411yescharacteristic
28C4⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31#14556502D412yes—
32C4×#10596831D412yes—
33C2×#41×C73C2×C411yescharacteristic
34C23×C4×C73C2×C411yescharacteristic
35#123×C73—21no—
36C22⋊Q8×C73—21no—
37#121×C73—21no—
38#120×C73—21no—
39C4×D4×C73—22no—
40#117×C73C2×C411yescharacteristic
41#11924031—44no—
42#11924012—41no—
43#11924012—42no—
44#11924012—41no—
45C2×#10596831—22no—
46C2×#10596831—21no—
47C2×#10596831—21no—
48C2×#10596831—42no—
49C2×#10596831—42no—
50C2×#10596831—42no—
51C2×#10596831—41no—
52C2×#10596831C22⋊C411yescharacteristic
53C2×#10596831C22⋊C411yescharacteristic
54C2×#10596831C2×D412yes—
55C24×C73M1611yescharacteristic
56#41×C73—41no—
57#41×C73—41no—
58C2×Q8×C73M1611yescharacteristic
59C2×D4×C73—22no—
60C2×D4×C73—21no—
61C2×D4×C73M1611yescharacteristic
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73M1611yescharacteristic
68C22×C4×C73C22×C411yescharacteristic
69C4⋊C4×C73—22no—
70C4⋊C4×C73—21no—
71C22⋊C4×C73—22no—
72C22⋊C4×C73—22no—
73C22⋊C4×C73—21no—
74C42×C73C22⋊C412yes—
75C23×C73—21no—
76C23×C73—21no—
77C23×C73—41no—
78C23×C73—42no—
79C23×C73C2×M1611yescharacteristic
80Q8×C73—41no—
81D4×C73—41no—
82D4×C73—44no—
83D4×C73—42no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—22no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—42no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73C2×C22⋊C411yescharacteristic
96C2×C4×C73C2×M1611yescharacteristic
97#10596831—22no—
98#10596831—21no—
99#10596831—41no—
100#10596831—41no—
101#10596831#10012yes—
102C22×C73—22no—
103C22×C73—21no—
104C22×C73—41no—
105C22×C73—41no—
106C22×C73—42no—
107C22×C73—42no—
108C22×C73—42no—
109C22×C73—41no—
110C22×C73—41no—
111C22×C73#40711yescharacteristic
112C4×C73—21no—
113C4×C73—42no—
114C4×C73—41no—
115C4×C73—41no—
116C4×C73#41011yescharacteristic
117C4×C73C2×#10011yescharacteristic
118#11537—731nomaximal, Sylow
119C2×C73—21no—
120C2×C73—42no—
121C2×C73—41no—
122C2×C73—81no—
123C2×C73#146611yescharacteristic
124#2929—731no—
125#2166—732no—
126#1321—732no—
127#1320—732no—
128C73#1153711yescharacteristic, complemented, Sylow
129C2×#131—731no—
130C23×C8—731no—
131#525#6440811yescharacteristic
132#435—1462no—
133C8×D4—1462no—
134#424—731no—
135#408—731no—
136#407—731no—
137#405—731no—
138#364—1462no—
139#349—734no—
140#347—1462no—
141C2×#41C2×#6440811yescharacteristic
142C23×C4C2×#6440811yescharacteristic
143#131—2922no—
144C2×M16—731no—
145C2×M16—731no—
146C2×M16—1461no—
147C22×C8—2922no—
148C22×C8—731no—
149C22×C8—731no—
150C22×C8—1461no—
151C22×C8—1461no—
152C22×C8—1461no—
153#123—21no—
154C22⋊Q8—21no—
155#121—21no—
156#120—21no—
157C4×D4—22no—
158#117C2×#6440811yescharacteristic
159M32—1464no—
160C4⋊C8—1462no—
161C22⋊C8—1462no—
162C22⋊C8—1462no—
163#97—732no—
164C4×C8—732no—
165C24#1192403711yescharacteristic
166#41—41no—
167#41—41no—
168C2×Q8#1192403711yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4#1192403711yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#1192403711yescharacteristic
178C22×C4C22×#6440811yescharacteristic
179M16—2922no—
180M16—2921no—
181M16—2921no—
182C2×C8—2922no—
183C2×C8—2922no—
184C2×C8—2922no—
185C2×C8—2921no—
186C2×C8—731no—
187C2×C8—731no—
188C2×C8—732no—
189C2×C8—1461no—
190C2×C8—1461no—
191C2×C8—1462no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C42C22⋊#6440812yes—
198C16—2924no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×#1192403711yescharacteristic
204Q8—41no—
205D4—41no—
206D4—42no—
207D4—44no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C22⋊#6440811yescharacteristic
220C2×C4C2×#1192403711yescharacteristic
221C8—2921no—
222C8—2921no—
223C8—732no—
224C8—1462no—
225C8—1461no—
226C22—22no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22#1798955711yescharacteristic
236C4—21no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4#1798956711yescharacteristic
241C4D73⋊#10011yescharacteristic
242C2—21no—
243C2—42no—
244C2—41no—
245C2—81no—
246C2#2262519111yescharacteristic
subgroupG/H
centerC4#17989567
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting, prime radical#525×C73C4
socleC2×C73#1466
cosocleC2×#11924012C23
hypercenter#525#64408
hypocenterC73#11537
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ #117 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #22626304C2semidirect
2. #22624865C2semidirect
3. #22624863C2semidirect
128. C73#11537semidirect

Maximal subgroups

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

subgroupindex
#226263042
#226248652
#226248632
#226186262
#1153773