Finite Group Databasebeta

← #33006316

Subgroups — #33006316

Order 18688.

408 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 79 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#22611616C212yesmaximal, complemented
2#22608100C212yesmaximal, complemented
3#22608098C212yesmaximal, complemented
4C73×#3147C211yesmaximal, characteristic
5#17988571—22no—
6#17988386C2214yes—
7#17988384—22no—
8C2×#135×C73—22no—
9#532×C73C411yescharacteristic
10#532×C73C2211yescharacteristic
11#495×C73—22no—
12#494×C73C2211yescharacteristic
13#493×C73C411yescharacteristic
14#492×C73C411yescharacteristic
15#492×C73C2211yescharacteristic
16C2×C4×C8×C73C411yescharacteristic
17C2×#11924012—42no—
18#14556510—24no—
19#14556508—42no—
20C2×#41×C73—21no—
21C2×#41×C73D412yes—
22#135×C73—44no—
23#135×C73—44no—
24C2×Q16×C73—22no—
25C2×SD16×C73—22no—
26C2×SD16×C73—22no—
27C2×D8×C73—22no—
28C22×C8×C73—21no—
29C4⋊Q8×C73C2×C411yescharacteristic
30C4⋊Q8×C73C2311yescharacteristic
31#127×C73C2×C411yescharacteristic
32#127×C73C2×C411yescharacteristic
33#124×C73—21no—
34#124×C73D412yes—
35#121×C73—41no—
36#121×C73—41no—
37C4×D4×C73—41no—
38C4×D4×C73—41no—
39C2×C42×C73C2×C411yescharacteristic
40C4×C8×C73—21no—
41C4×C8×C73C2×C411yescharacteristic
42C4×C8×C73C2×C411yescharacteristic
43#11924012—44no—
44#11924012—82no—
45C2×#10596831—82no—
46#41×C73—42no—
47#41×C73—42no—
48#41×C73—42no—
49#41×C73—42no—
50C2×Q8×C73—21no—
51C2×Q8×C73C2×D412yes—
52C2×D4×C73—22no—
53C2×D4×C73—21no—
54C2×D4×C73—41no—
55C2×D4×C73—41no—
56C2×D4×C73—41no—
57C2×D4×C73C22⋊C412yes—
58C22×C4×C73—21no—
59C22×C4×C73—22no—
60C22×C4×C73—41no—
61C22×C4×C73C22⋊C411yescharacteristic
62Q16×C73—44no—
63SD16×C73—44no—
64SD16×C73—44no—
65D8×C73—44no—
66C2×C8×C73—22no—
67C2×C8×C73—22no—
68C2×C8×C73—21no—
69C2×C8×C73—21no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C22⋊C4×C73—42no—
73C22⋊C4×C73—42no—
74C42×C73—21no—
75C42×C73C22⋊C411yescharacteristic
76C42×C73C22×C411yescharacteristic
77C23×C73—22no—
78C23×C73—41no—
79C23×C73#9911yescharacteristic
80Q8×C73—42no—
81Q8×C73—42no—
82D4×C73—42no—
83D4×C73—42no—
84D4×C73—82no—
85D4×C73—82no—
86D4×C73—82no—
87D4×C73—82no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—22no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73—42no—
96C2×C4×C73—41no—
97C2×C4×C73—81no—
98C2×C4×C73—81no—
99C2×C4×C73#9911yescharacteristic
100C2×C4×C73C2×C22⋊C411yescharacteristic
101C8×C73—22no—
102C8×C73—22no—
103#10596831—84no—
104C22×C73—21no—
105C22×C73—41no—
106C22×C73—42no—
107C22×C73—82no—
108C22×C73—82no—
109C22×C73—81no—
110C22×C73C2×#9911yescharacteristic
111C4×C73—22no—
112C4×C73—21no—
113C4×C73—41no—
114C4×C73—42no—
115C4×C73—81no—
116C4×C73#35311yescharacteristic
117C4×C73#35311yescharacteristic
118#13589—731nomaximal, Sylow
119C2×C73—21no—
120C2×C73—41no—
121C2×C73—82no—
122C2×C73—161no—
123C2×C73C2×#35311yescharacteristic
124#3147D7311yescharacteristic
125#2132—732no—
126#1411—732no—
127#1410—732no—
128C73#1358911yescharacteristic, complemented, Sylow
129C2×#135—22no—
130#532Dic7311yescharacteristic
131#532C2×D7311yescharacteristic
132#495—22no—
133#494C2×D7311yescharacteristic
134#493Dic7311yescharacteristic
135#492Dic7311yescharacteristic
136#492C2×D7311yescharacteristic
137#413—1462no—
138C2×C4×C8Dic7311yescharacteristic
139#356—734no—
140#355—1462no—
141C2×#41—21no—
142C2×#41C22⋊D7312yes—
143#135—44no—
144#135—44no—
145C2×Q16—22no—
146C2×SD16—22no—
147C2×SD16—22no—
148C2×D8—22no—
149C2×M16—2922no—
150C22×C8—21no—
151C4⋊Q8C2×Dic7311yescharacteristic
152C4⋊Q8C22×D7311yescharacteristic
153#127C2×Dic7311yescharacteristic
154#127C2×Dic7311yescharacteristic
155#124—21no—
156#124C22⋊D7312yes—
157#121—41no—
158#121—41no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C42C2×Dic7311yescharacteristic
162#101—1464no—
163#100—2922no—
164C4×C8—21no—
165C4×C8C2×Dic7311yescharacteristic
166C4×C8C2×Dic7311yescharacteristic
167#41—42no—
168#41—42no—
169#41—42no—
170#41—42no—
171C2×Q8—21no—
172C2×Q8C2×C22⋊D7312yes—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4C22⋊Dic7312yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4C22⋊Dic7311yescharacteristic
183Q16—44no—
184SD16—44no—
185SD16—44no—
186D8—44no—
187M16—2924no—
188M16—5842no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—5842no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C42—21no—
199C42C22⋊Dic7311yescharacteristic
200C42C22×Dic7311yescharacteristic
201C23—22no—
202C23—41no—
203C23#1455650611yescharacteristic
204Q8—42no—
205Q8—42no—
206D4—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—82no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4#1455650611yescharacteristic
224C2×C4C2×C22⋊Dic7311yescharacteristic
225C8—22no—
226C8—22no—
227C8—5844no—
228C22—21no—
229C22—41no—
230C22—42no—
231C22—82no—
232C22—82no—
233C22—81no—
234C22C2×#1455650611yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—81no—
240C4#1798838011yescharacteristic
241C4#1798838011yescharacteristic
242C2—21no—
243C2—41no—
244C2—82no—
245C2—161no—
246C2C2×#1798838011yescharacteristic
subgroupG/H
centerC4#17988380
commutatorC42×C73C22×C4
FrattiniC4⋊Q8C22×D73
Fitting, prime radicalC73×#3147C2
socleC2×C73C2×#353
cosocleC4⋊Q8×C73C23
hypercenter#3147D73
hypocenterC73#13589
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #22611616C2semidirect
2. #22608100C2semidirect
3. #22608098C2semidirect
128. C73#13589semidirect

Maximal subgroups

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

subgroupindex
#226116162
#226081002
#226080982
C73×#31472
#1358973