Finite Group Databasebeta

← #32996838

Subgroups — #32996838

Order 18688.

280 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 71 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#22620676C211yesmaximal, characteristic
2#22610080C211yesmaximal, characteristic
3#22607926C212yesmaximal
4#22607920C212yesmaximal
5C73×#2165C211yesmaximal, characteristic
6#17989144—21no—
7#17988985C411yescharacteristic, complemented
8#17988967C411yescharacteristic, complemented
9#17988938—21no—
10#17988845—21no—
11#17988845C2212yes—
12#17988695—21no—
13#17988695C412yescomplemented
14#17988358—21no—
15#17988347—21no—
16#17988345C2211yescharacteristic
17#17988344C2211yescharacteristic
18C4×M16×C73C2211yescharacteristic
19#364×C73—21no—
20#349×C73C2212yes—
21#347×C73—21no—
22C2×Dic292—21no—
23C2×Dic292C2×C412yes—
24C2×#11924012—21no—
25#14556591—21no—
26#14556591—21no—
27#14556583—41no—
28#14556570—22no—
29#14556570—41no—
30C4×Dic146—22no—
31C4×Dic146—41no—
32#14556550—21no—
33#14556550—41no—
34#14556523—21no—
35#14556523C2×C412yes—
36#14556516—22no—
37#14556516—22no—
38C2×M16×C73D411yescharacteristic
39C2×M16×C73C2311yescharacteristic
40C2×C42×C73D411yescharacteristic
41M32×C73—22no—
42#97×C73C2×C411yescharacteristic
43#97×C73D411yescharacteristic
44C4×C8×C73C2×C411yescharacteristic
45C4×C8×C73D411yescharacteristic
46C2×Dic146—22no—
47C2×Dic146—41no—
48Dic292—42no—
49Dic292—42no—
50#11924012—41no—
51#11924012—41no—
52C2×#10596831—41no—
53C4⋊Dic73—22no—
54C4⋊Dic73—41no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58#11924007—42no—
59#11924007—42no—
60C22⋊Dic73—42no—
61C22⋊Dic73—41no—
62C4×Dic73—42no—
63C4×Dic73—41no—
64C22×C4×C73—21no—
65C22×C4×C73C2×D411yescharacteristic
66M16×C73—41no—
67M16×C73—41no—
68C2×C8×C73C22⋊C411yescharacteristic
69C2×C8×C73C22⋊C411yescharacteristic
70C2×C8×C73C22×C411yescharacteristic
71C2×C8×C73C2×D411yescharacteristic
72C42×C73—21no—
73C42×C73C22⋊C411yescharacteristic
74C42×C73C22⋊C411yescharacteristic
75C16×C73—42no—
76C23×C73AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—42no—
82C2×C4×C73C2×C22⋊C411yescharacteristic
83C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
84C2×Dic73—42no—
85C2×Dic73—42no—
86C2×Dic73—81no—
87Dic146—42no—
88Dic146—42no—
89Dic146—82no—
90C8×C73—21no—
91C8×C73—21no—
92C8×C73—21no—
93C8×C73—21no—
94#10596831—81no—
95C22×C73—21no—
96C22×C73—41no—
97C22×C73#41811yescharacteristic
98C4×C73—22no—
99C4×C73—21no—
100C4×C73—41no—
101C4×C73#41011yescharacteristic
102C4×C73#42111yescharacteristic
103Dic73—82no—
104Dic73—82no—
105Dic73—161no—
106#11838—731nomaximal, Sylow
107C2×C73—21no—
108C2×C73—41no—
109C2×C73#151411yescharacteristic
110#3185—731no—
111#2165D7311yescharacteristic
112#1841—731no—
113#1342—732no—
114#1340—732no—
115C73#1183811yescharacteristic, complemented, Sylow
116#533—1461no—
117#498—731no—
118#494—731no—
119#487—1461no—
120#470—732no—
121#470—1461no—
122#441—732no—
123#441—1461no—
124C4×M16C2×D7311yescharacteristic
125#364—21no—
126#349C2×D7312yes—
127#347—21no—
128#344—1461no—
129#339—1461no—
130#338—731no—
131H(3,ℤ/4)—731no—
132C2×Q16—732no—
133C2×Q16—1461no—
134C2×M16—1461no—
135C2×M16D29211yescharacteristic
136C2×M16C22×D7311yescharacteristic
137C4⋊Q8—1461no—
138C4⋊Q8—1461no—
139#125—2921no—
140C22⋊Q8—2921no—
141C22⋊Q8—1462no—
142C4×Q8—2921no—
143C4×Q8—1462no—
144#117—2921no—
145#117—1461no—
146C2×C42D29211yescharacteristic
147M32—22no—
148#106—732no—
149#106—1461no—
150#103—1462no—
151#103—1462no—
152#97C4×D7311yescharacteristic
153#97C22⋊D7311yescharacteristic
154C4×C8C4×D7311yescharacteristic
155C4×C8C22⋊D7311yescharacteristic
156C2×Q8—2921no—
157C2×Q8—1462no—
158C22×C4—21no—
159C22×C4C2×D29211yescharacteristic
160Q16—2922no—
161Q16—2922no—
162M16—41no—
163M16—41no—
164M16—2921no—
165M16—2921no—
166C2×C8—2921no—
167C2×C8#1192400511yescharacteristic
168C2×C8#1192400511yescharacteristic
169C2×C8C2×C4×D7311yescharacteristic
170C2×C8C2×C22⋊D7311yescharacteristic
171C4⋊C4—2922no—
172C4⋊C4—2921no—
173C4⋊C4—2921no—
174C4⋊C4—2921no—
175C4⋊C4—2921no—
176C4⋊C4—2922no—
177C4⋊C4—1462no—
178C22⋊C4—2922no—
179C22⋊C4—2921no—
180C42—21no—
181C42—2922no—
182C42—2921no—
183C42#1192400511yescharacteristic
184C42#1192400511yescharacteristic
185C16—42no—
186C23#1455661911yescharacteristic
187Q8—2922no—
188Q8—2922no—
189Q8—5842no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—2922no—
196C2×C4—2922no—
197C2×C4—5841no—
198C2×C4C2×#1192400511yescharacteristic
199C2×C4#1455661911yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—5841no—
205C22—21no—
206C22—41no—
207C22#1798859311yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—5842no—
212C4—5842no—
213C4—11681no—
214C4#1798856011yescharacteristic
215C4#1798860311yescharacteristic
216C2—21no—
217C2—41no—
218C2#2260847911yescharacteristic
subgroupG/H
centerC2#22608479
commutatorC2×C8×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#2165C2
socleC2×C73#1514
prime radicalC23×C73AGL(1,ℤ/8)
cosocleC2×M16×C73C23
hypercenter#2165D73
hypocenterC73#11838
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
7. #17988985C4semidirect
8. #17988967C4semidirect
13. #17988695C4semidirect
115. C73#11838semidirect

Maximal subgroups

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

subgroupindex
#226206762
#226100802
#226079262
#226079202
C73×#21652
#1183873