Finite Group Databasebeta

← #33000271

Subgroups — #33000271

Order 18688.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#22612188C211yesmaximal, characteristic
2#22611836C211yesmaximal, characteristic, complemented
3#22609298C211yesmaximal, characteristic, complemented
4#22609292C211yesmaximal, characteristic
5#22608002C211yesmaximal, characteristic, complemented
6#22607978C211yesmaximal, characteristic
7C73×#1362C211yesmaximal, characteristic, complemented
8C2×#14556534—21no—
9C2×#14556534—21no—
10#17988964C2211yescharacteristic
11#17988869—21no—
12#17988788—21no—
13C8×Dic146C2211yescharacteristic
14#17988405—21no—
15#17988362—21no—
16C4×#11924003C2211yescharacteristic
17#17988340C2211yescharacteristic
18#17988333—21no—
19#17988330—21no—
20#494×C73C2211yescharacteristic
21C4⋊C16×C73C2211yescharacteristic
22#335×C73C2211yescharacteristic
23C2×#11924017—41no—
24C2×D584—21no—
25C2×D584—21no—
26#14556588—21no—
27C4×Dic146D411yescharacteristic
28#14556534—42no—
29#14556534—42no—
30C2×#11924003D411yescharacteristic
31C2×#11924003D411yescharacteristic
32#14556522—41no—
33C4⋊#10596831—21no—
34C4⋊#10596831D411yescharacteristic
35#14556520—21no—
36#14556514—41no—
37#14556513—41no—
38C8×Dic73—21no—
39C2×Q16×C73—21no—
40C2×Q16×C73—21no—
41C4⋊Q8×C73—21no—
42C2×C16×C73—21no—
43C4⋊C8×C73—21no—
44C4×C8×C73C2311yescharacteristic
45C2×Dic146D811yescharacteristic
46C2×D292—41no—
47C2×D292—41no—
48C2×D292—41no—
49#11924017—82no—
50D584—41no—
51D584—41no—
52D584—81no—
53C2×#10596831—21no—
54C2×#10596831—41no—
55C4⋊Dic73D811yescharacteristic
56#11924007—21no—
57C4×Dic73—21no—
58#11924003—21no—
59#11924003SD1612yescomplemented
60C2×Q8×C73—41no—
61Q16×C73—41no—
62Q16×C73—41no—
63Q16×C73—81no—
64C2×C8×C73—41no—
65C2×C8×C73C2×D411yescharacteristic
66C2×C8×C73#4111yescharacteristic
67C4⋊C4×C73—41no—
68C4⋊C4×C73—41no—
69C42×C73C2×D411yescharacteristic
70C16×C73—41no—
71C22×D73—81no—
72Q8×C73—81no—
73Q8×C73—81no—
74C2×C4×C73—81no—
75C2×C4×C73#12111yescharacteristic
76C2×C4×C73C2×D811yescharacteristic
77C2×C4×C73#13511yescharacteristic
78C2×Dic73—21no—
79C2×Dic73—21no—
80D292—81no—
81D292—81no—
82D292—81no—
83D292—81no—
84D292—161no—
85Dic146—21no—
86Dic146—21no—
87C8×C73—21no—
88C8×C73—81no—
89C8×C73C2×SD1611yescharacteristic
90C8×C73AGL(1,ℤ/8)11yescharacteristic
91#10596831—41no—
92#10596831—81no—
93C22×C73#46611yescharacteristic
94C2×D73—161no—
95C2×D73—161no—
96C2×D73—161no—
97C4×C73—21no—
98C4×C73—161no—
99C4×C73#46011yescharacteristic
100C4×C73#47211yescharacteristic
101C4×C73#50911yescharacteristic
102C4×C73#51011yescharacteristic
103Dic73—41no—
104Dic73—41no—
105#12444—731nomaximal, Sylow
106C2×C73#166811yescharacteristic
107C2×C73#222011yescharacteristic
108C2×C73#222311yescharacteristic
109D73—321no—
110#2248—731no—
111#2183—731no—
112#1670—731no—
113#1668—731no—
114#1371—731no—
115#1362D7311yescharacteristic, complemented
116#1361—731no—
117C73#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1461no—
119C2×SD32—1461no—
120#494C2×D7311yescharacteristic
121#493—731no—
122#475—1461no—
123#460—1461no—
124C8×Q8—731no—
125C4⋊C16—1461no—
126C4⋊C16C2×D7311yescharacteristic
127#346—1461no—
128C4×C16—731no—
129#335—731no—
130#335C2×D7311yescharacteristic
131#332—1461no—
132#331—1461no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—2921no—
136C2×D8—1461no—
137C2×D8—1461no—
138C4⋊Q8—21no—
139#127—1461no—
140C4×Q8—731no—
141SD32—2922no—
142SD32—2922no—
143C2×C16—21no—
144C2×C16—731no—
145C2×C16—731no—
146#106—2921no—
147C4⋊C8—21no—
148C4⋊C8—731no—
149C4⋊C8—1461no—
150C4⋊C8—1461no—
151#103—2921no—
152#102—2921no—
153C4×C8—1461no—
154C4×C8C22×D7311yescharacteristic
155C2×Q8—41no—
156C2×Q8—731no—
157C2×D4—2921no—
158C2×D4—2921no—
159C2×D4—2921no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—5842no—
164D8—2921no—
165D8—2921no—
166D8—5841no—
167C2×C8—41no—
168C2×C8—2921no—
169C2×C8—1461no—
170C2×C8D4×D7311yescharacteristic
171C2×C8#1192402811yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—731no—
175C4⋊C4—1461no—
176C42—1461no—
177C42D4×D7311yescharacteristic
178C16—41no—
179C16—732no—
180C16—1461no—
181C23—5841no—
182Q8—81no—
183Q8—81no—
184Q8—1461no—
185Q8—1461no—
186D4—5841no—
187D4—5841no—
188D4—5841no—
189D4—5841no—
190D4—11681no—
191C2×C4—81no—
192C2×C4—1461no—
193C2×C4—1461no—
194C2×C4#1455656411yescharacteristic
195C2×C4D8×D7311yescharacteristic
196C2×C4#1455661211yescharacteristic
197C8—21no—
198C8—81no—
199C8—2921no—
200C8—5841no—
201C8SD16×D7311yescharacteristic
202C8#1455661511yescharacteristic
203C22—11681no—
204C22—11681no—
205C22—11681no—
206C22#1798882211yescharacteristic
207C4—21no—
208C4—161no—
209C4—2921no—
210C4—2921no—
211C4#1798878611yescharacteristic
212C4#1798885611yescharacteristic
213C4#1798902411yescharacteristic
214C4#1798902911yescharacteristic
215C2—23361no—
216C2#2260929011yescharacteristic
217C2#2261204711yescharacteristic
218C2#2261206411yescharacteristic
subgroupG/H
centerC22#17988822
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1362C2
socleC22×C73#466
prime radical#17988964C22
hypercenter#1362D73
hypocenterC73#12444
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
2. #22611836C2semidirect
3. #22609298C2semidirect
5. #22608002C2semidirect
7. C73×#1362C2semidirect
59. #11924003SD16semidirect
115. #1362D73semidirect
117. C73#12444semidirect

Maximal subgroups

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

subgroupindex
#226121882
#226118362
#226092982
#226092922
#226080022
#226079782
C73×#13622
#1244473