Finite Group Databasebeta

← #32999758

Subgroups — #32999758

Order 18688.

336 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 67 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#22612102C212yesmaximal, complemented
2#22609562C211yesmaximal, characteristic, complemented
3#22607970C211yesmaximal, characteristic, complemented
4#22607907C212yesmaximal, complemented
5C73×#3293C211yesmaximal, characteristic
6#17988985C2211yescharacteristic
7#17988983—21no—
8#17988939—21no—
9#17988411—22no—
10#17988408C2212yes—
11#17988397—22no—
12#17988308C2211yescharacteristic
13#578×C73—22no—
14#494×C73C2211yescharacteristic
15#473×C73—22no—
16#456×C73—21no—
17#456×C73—21no—
18#443×C73C2212yes—
19#14556591—21no—
20#14556531—22no—
21#14556523—22no—
22#14556522—42no—
23#14556514—42no—
24C4×#10596831—21no—
252−1+4×C73—21no—
262−1+4×C73—21no—
27#137×C73—22no—
28#137×C73—22no—
29#137×C73—42no—
30#137×C73—42no—
31#135×C73—42no—
32#135×C73D412yes—
33C2×Q16×C73—22no—
34C2×Q16×C73—42no—
35#131×C73—22no—
36C4⋊Q8×C73D411yescharacteristic
37C4⋊Q8×C73D411yescharacteristic
38#108×C73D412yes—
39#104×C73—22no—
40#101×C73—21no—
41#101×C73—21no—
42C4×C8×C73C2311yescharacteristic
43C2×Dic146—41no—
44C2×#10596831—22no—
45C4⋊Dic73—41no—
46C4⋊Dic73—41no—
47#11924003—42no—
48#41×C73—22no—
49#41×C73—42no—
50#41×C73—42no—
51#41×C73—81no—
52#41×C73—81no—
53C2×Q8×C73—21no—
54C2×Q8×C73—21no—
55C2×Q8×C73—42no—
56C2×Q8×C73—42no—
57Q16×C73—42no—
58Q16×C73—42no—
59Q16×C73—42no—
60Q16×C73—42no—
61Q16×C73SD1612yes—
62SD16×C73—22no—
63SD16×C73—42no—
64SD16×C73—42no—
65D8×C73SD1612yes—
66M16×C73—22no—
67M16×C73—42no—
68C2×C8×C73—42no—
69C2×C8×C73C2×D412yes—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C42×C73C2×D411yescharacteristic
73Q8×C73—22no—
74Q8×C73—42no—
75Q8×C73—42no—
76Q8×C73—81no—
77Q8×C73—81no—
78D4×C73—22no—
79D4×C73—42no—
80D4×C73—81no—
81D4×C73—81no—
82C2×C4×C73—21no—
83C2×C4×C73—42no—
84C2×C4×C73—41no—
85C2×C4×C73—41no—
86C2×C4×C73—82no—
87C2×C4×C73—82no—
88C2×C4×C73#12011yescharacteristic
89C2×Dic73—81no—
90Dic146—82no—
91C8×C73—42no—
92C8×C73C2×SD1612yes—
93C8×C73AGL(1,ℤ/8)12yes—
94#10596831—42no—
95C22×C73—42no—
96C22×C73#45311yescharacteristic
97C4×C73—22no—
98C4×C73—42no—
99C4×C73—81no—
100C4×C73—81no—
101C4×C73#45012yes—
102Dic73—161no—
103#12367—731nomaximal, Sylow
104C2×C73—21no—
105C2×C73—82no—
106C2×C73#162911yescharacteristic
107#3293D7311yescharacteristic
108#2230—732no—
109#1716—731no—
110#1358—731no—
111#1336—732no—
112C73#1236711yescharacteristic, complemented, Sylow
113#578—22no—
114#498—731no—
115#498—1461no—
116#494C2×D7311yescharacteristic
117#487—1461no—
118#473—22no—
119#456—21no—
120#456—21no—
121#443C2×D7312yes—
122#365—1462no—
123#364—732no—
124#360—1462no—
125#322—731no—
1262−1+4—21no—
1272−1+4—21no—
128#137—22no—
129#137—22no—
130#137—42no—
131#137—42no—
132#135—42no—
133#135C22⋊D7312yes—
134C2×Q16—22no—
135C2×Q16—42no—
136#131—22no—
137C4⋊Q8—1461no—
138C4⋊Q8C22⋊D7311yescharacteristic
139C4⋊Q8C22⋊D7311yescharacteristic
140M32—1462no—
141#108C22⋊D7312yes—
142#106—2922no—
143#106—1462no—
144#104—22no—
145#103—2922no—
146#101—21no—
147#101—21no—
148C4×C8—1461no—
149C4×C8C22×D7311yescharacteristic
150#41—22no—
151#41—42no—
152#41—42no—
153#41—81no—
154#41—81no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—2921no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163Q16—42no—
164Q16#1192401612yes—
165SD16—22no—
166SD16—42no—
167SD16—42no—
168D8#1192401612yes—
169M16—22no—
170M16—42no—
171C2×C8—42no—
172C2×C8—1462no—
173C2×C8C2×C22⋊D7312yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2921no—
177C4⋊C4—2921no—
178C42C2×C22⋊D7311yescharacteristic
179C16—2922no—
180Q8—22no—
181Q8—42no—
182Q8—42no—
183Q8—81no—
184Q8—81no—
185Q8—5842no—
186D4—22no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—5841no—
197C2×C4#1455656011yescharacteristic
198C8—42no—
199C8—2922no—
200C8C2×#1192401612yes—
201C8#1455661712yes—
202C22—42no—
203C22#1798875511yescharacteristic
204C4—22no—
205C4—42no—
206C4—81no—
207C4—81no—
208C4—11681no—
209C4#1798873812yes—
210C2—21no—
211C2—82no—
212C2#2260904711yescharacteristic
subgroupG/H
centerC2#22609047
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
Fitting, prime radicalC73×#3293C2
socleC2×C73#1629
hypercenter#3293D73
hypocenterC73#12367
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #22612102C2semidirect
2. #22609562C2semidirect
3. #22607970C2semidirect
4. #22607907C2semidirect
112. C73#12367semidirect

Maximal subgroups

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

subgroupindex
#226121022
#226095622
#226079702
#226079072
C73×#32932
#1236773