Finite Group Databasebeta

← #32998006

Subgroups — #32998006

Order 18688.

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

normal subgroups only

This group has 226 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#22611805C211yesmaximal, characteristic
2#22608845C211yesmaximal, characteristic, complemented
3#22608844C211yesmaximal, characteristic, complemented
4#22607996C211yesmaximal, characteristic, complemented
5#22607899C211yesmaximal, characteristic, complemented
6#22607853C211yesmaximal, characteristic, complemented
7C73×#1333C211yesmaximal, characteristic, complemented
8C2×#14556532C411yescharacteristic
9C4×D584C2211yescharacteristic
10#17988665—21no—
11#17988664—21no—
12#17988414C411yescharacteristic
13#17988388C411yescharacteristic
14#17988387C411yescharacteristic
15#17988363C2211yescharacteristic
16#17988338C2211yescharacteristic
17D4⋊#10596831—21no—
18#17988314—21no—
19#17988308C2211yescharacteristic
20C4×D8×C73C2211yescharacteristic
21#346×C73C2211yescharacteristic
22#334×C73C2211yescharacteristic
23C2×D584C2×C411yescharacteristic
24C2×#11924012—41no—
25C2×#11924011—41no—
26C4×D292—21no—
27#14556532—22no—
28C2×#11924003C2×C411yescharacteristic
29C2×#11924003C2×C411yescharacteristic
30#14556525C2×C411yescharacteristic
31C4⋊#10596831—21no—
32#14556513—21no—
33C22⋊#10596831—41no—
34#14556503—41no—
35C4×#10596831D411yescharacteristic
36C4×#10596831D411yescharacteristic
37C2×D8×C73C2×C411yescharacteristic
38C4×D4×C73—21no—
39C2×C16×C73—21no—
40#107×C73C2×C411yescharacteristic
41C4⋊C8×C73—21no—
42#102×C73—21no—
43C4×C8×C73C2311yescharacteristic
44C2×D292—21no—
45C2×C4×D73—41no—
46D584—21no—
47D584—21no—
48#11924012—82no—
49#11924011—82no—
50C2×#10596831—21no—
51C2×#10596831—21no—
52C2×#10596831—41no—
53C4⋊Dic73—21no—
54#11924005—41no—
55#11924003—21no—
56#11924003M1612yescomplemented
57C2×D4×C73—21no—
58C22×C4×C73—41no—
59D8×C73—21no—
60D8×C73—21no—
61C2×C8×C73—41no—
62C2×C8×C73C22×C411yescharacteristic
63C2×C8×C73#4111yescharacteristic
64C4⋊C4×C73—21no—
65C22⋊C4×C73—41no—
66C42×C73C2×D411yescharacteristic
67C16×C73—41no—
68C23×C73—41no—
69C22×D73—41no—
70D4×C73—41no—
71D4×C73—41no—
72C2×C4×C73—41no—
73C2×C4×C73—81no—
74C2×C4×C73—81no—
75C2×C4×C73C4×D411yescharacteristic
76C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
78C2×Dic73—41no—
79D292—41no—
80D292—41no—
81C4×D73—81no—
82C4×D73—81no—
83C8×C73—21no—
84C8×C73—81no—
85C8×C73C2×M1611yescharacteristic
86C8×C73#13111yescharacteristic
87#10596831—41no—
88#10596831—41no—
89#10596831—81no—
90C22×C73—81no—
91C22×C73—81no—
92C22×C73—81no—
93C22×C73#44211yescharacteristic
94C2×D73—81no—
95C2×D73—81no—
96C2×D73—81no—
97C4×C73—21no—
98C4×C73—21no—
99C4×C73—81no—
100C4×C73#43611yescharacteristic
101C4×C73#44411yescharacteristic
102Dic73—81no—
103#12089—731nomaximal, Sylow
104C2×C73—161no—
105C2×C73#159411yescharacteristic
106C2×C73AGL(1,ℤ/16)11yescharacteristic
107C2×C73AGL(1,ℤ/16)11yescharacteristic
108D73—161no—
109#2177—731no—
110#1594—731no—
111#1594—731no—
112#1368—731no—
113#1333—731no—
114#1333D7311yescharacteristic, complemented
115#1313—731no—
116C73#1208911yescharacteristic, complemented, Sylow
117C2×D16—731no—
118C4×D8—731no—
119C4×D8C2×D7311yescharacteristic
120#436—1461no—
121#436—1461no—
122#367—731no—
123#357—731no—
124#357—731no—
125#346—731no—
126#346C2×D7311yescharacteristic
127#334—731no—
128#334C2×D7311yescharacteristic
129D4⋊C8—1461no—
130D4⋊C8—1461no—
131#322—731no—
132C2×D8—731no—
133C2×D8C4×D7311yescharacteristic
134C2×M16—2921no—
135C2×M16—2921no—
136C4×D4—21no—
137C4×D4—1461no—
138D16—1462no—
139C2×C16—21no—
140C2×C16—731no—
141C2×C16—731no—
142#107—731no—
143#107C4×D7311yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1461no—
146#102—21no—
147#102—1461no—
148C22⋊C8—2921no—
149C22⋊C8—2921no—
150C4×C8—731no—
151C4×C8—731no—
152C4×C8C22×D7311yescharacteristic
153C2×D4—21no—
154C2×D4—1461no—
155C22×C4—41no—
156C22×C4—2921no—
157D8—21no—
158D8—21no—
159D8—1461no—
160D8—1461no—
161M16—5842no—
162M16—5842no—
163C2×C8—41no—
164C2×C8—2921no—
165C2×C8—1461no—
166C2×C8—1461no—
167C2×C8C2×C4×D7311yescharacteristic
168C2×C8#1192402711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1461no—
171C22⋊C4—41no—
172C22⋊C4—2921no—
173C42D4×D7311yescharacteristic
174C16—41no—
175C16—732no—
176C16—1461no—
177C23—41no—
178C23—2921no—
179D4—41no—
180D4—41no—
181D4—2921no—
182D4—2921no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—2921no—
187C2×C4—5841no—
188C2×C4—5841no—
189C2×C4#1455655111yescharacteristic
190C2×C4#1455661511yescharacteristic
191C2×C4#1455661811yescharacteristic
192C8—21no—
193C8—81no—
194C8—2921no—
195C8—2921no—
196C8—5841no—
197C8M16×D7311yescharacteristic
198C8#1455659711yescharacteristic
199C22—81no—
200C22—81no—
201C22—81no—
202C22—5841no—
203C22—5841no—
204C22—5841no—
205C22#1798869611yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—5841no—
210C4#1798866211yescharacteristic
211C4#1798870611yescharacteristic
212C2—161no—
213C2—11681no—
214C2#2260884211yescharacteristic
215C2#2261182611yescharacteristic
216C2#2261183211yescharacteristic
subgroupG/H
centerC22#17988696
commutatorC2×C8×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#1333C2
socleC22×C73#442
prime radicalC2×#14556532C4
cosocleC4×C8×C73C23
hypercenter#1333D73
hypocenterC73#12089
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #22608845C2semidirect
3. #22608844C2semidirect
4. #22607996C2semidirect
5. #22607899C2semidirect
6. #22607853C2semidirect
7. C73×#1333C2semidirect
56. #11924003M16semidirect
114. #1333D73semidirect
116. C73#12089semidirect

Maximal subgroups

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

subgroupindex
#226118052
#226088452
#226088442
#226079962
#226078992
#226078532
C73×#13332
#1208973