Finite Group Databasebeta

← #33000145

Subgroups — #33000145

Order 18688.

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

normal subgroups only

This group has 234 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#22612065C211yesmaximal, characteristic
2#22611977C211yesmaximal, characteristic
3#22609529C211yesmaximal, characteristic, complemented
4#22608725C211yesmaximal, characteristic
5#22607994C211yesmaximal, characteristic, complemented
6#22607979C211yesmaximal, characteristic, complemented
7C73×#1364C211yesmaximal, characteristic, complemented
8C2×#14556536—21no—
9#17988983—21no—
10#17988967C2211yescharacteristic
11#17988937—21no—
12C4×D584C2211yescharacteristic
13#17988652—21no—
14#17988412—21no—
15C4⋊#11924003C2211yescharacteristic
16#17988392—21no—
17#17988387—21no—
18#17988314—21no—
19C8×#10596831C2211yescharacteristic
20#500×C73C2211yescharacteristic
21C4⋊C16×C73C2211yescharacteristic
22#335×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×D584D411yescharacteristic
26C2×C8×D73—41no—
27#14556591—21no—
28C4×D292—21no—
29#14556536—42no—
30C2×#11924003—21no—
31#14556525D411yescharacteristic
32#14556522—42no—
33#14556515—41no—
34#14556515—41no—
35#14556513—21no—
36#14556503—41no—
37C4×#10596831D411yescharacteristic
38C4×#10596831D411yescharacteristic
39C4⋊Q8×C73—21no—
40C2×C16×C73—21no—
41#107×C73—21no—
42#107×C73—21no—
43C4⋊C8×C73—21no—
44C4×C8×C73C2311yescharacteristic
45C2×Dic146—41no—
46C2×D292—21no—
47C2×C4×D73—41no—
48Dic292—41no—
49Dic292—41no—
50Dic292—81no—
51D584—21no—
52D584SD1612yes—
53C2×#10596831—21no—
54C2×#10596831D811yescharacteristic
55C2×#10596831D811yescharacteristic
56C8×D73—82no—
57C4⋊Dic73—21no—
58C4⋊Dic73—41no—
59C4⋊Dic73—41no—
60#11924005—41no—
61#11924003—41no—
62C2×Q8×C73—41no—
63C2×C8×C73—41no—
64C2×C8×C73C2×D411yescharacteristic
65C2×C8×C73#4111yescharacteristic
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C42×C73C2×D411yescharacteristic
69C16×C73—41no—
70C22×D73—41no—
71Q8×C73—81no—
72Q8×C73—81no—
73C2×C4×C73—81no—
74C2×C4×C73#12111yescharacteristic
75C2×C4×C73C2×D811yescharacteristic
76C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
77C2×Dic73—41no—
78C2×Dic73—81no—
79D292—22no—
80D292—41no—
81C4×D73—81no—
82C4×D73—81no—
83Dic146—81no—
84Dic146—81no—
85C8×C73—21no—
86C8×C73—81no—
87C8×C73C2×SD1611yescharacteristic
88C8×C73#13511yescharacteristic
89#10596831—22no—
90#10596831—21no—
91#10596831—21no—
92C22×C73#45911yescharacteristic
93C2×D73—42no—
94C2×D73—81no—
95C2×D73—81no—
96C4×C73—21no—
97C4×C73—161no—
98C4×C73#46511yescharacteristic
99C4×C73#47111yescharacteristic
100C4×C73#50811yescharacteristic
101C4×C73#50911yescharacteristic
102Dic73—81no—
103Dic73—161no—
104#12422—731nomaximal, Sylow
105C2×C73#166511yescharacteristic
106C2×C73#221011yescharacteristic
107C2×C73#221411yescharacteristic
108D73—82no—
109#2224—731no—
110#2208—731no—
111#1707—731no—
112C8×D8—731no—
113#1367—731no—
114#1364D7311yescharacteristic, complemented
115#1362—731no—
116C73#1242211yescharacteristic, complemented, Sylow
117C2×SD32—1461no—
118#500C2×D7311yescharacteristic
119#498—1461no—
120#494—731no—
121#487—1461no—
122C4×D8—731no—
123C8×D4—1461no—
124#366—1461no—
125C4⋊C16—731no—
126C4⋊C16C2×D7311yescharacteristic
127#358—1461no—
128#357—1461no—
129#335C2×D7311yescharacteristic
130D4⋊C8—1461no—
131C82—731no—
132C2×Q16—1461no—
133C2×Q16—1461no—
134C2×D8—731no—
135C22×C8—2921no—
136C4⋊Q8—21no—
137C4⋊Q8—1461no—
138C4×D4—1461no—
139SD32—2922no—
140C2×C16—21no—
141C2×C16—1461no—
142#107—21no—
143#107—21no—
144#107—731no—
145#106—2922no—
146C4⋊C8—21no—
147#103—2921no—
148#103—2921no—
149#102—1461no—
150C22⋊C8—2921no—
151C4×C8—731no—
152C4×C8—731no—
153C4×C8C22×D7311yescharacteristic
154C2×Q8—41no—
155C2×Q8—2921no—
156C2×D4—1461no—
157C22×C4—2921no—
158Q16—2921no—
159Q16—2921no—
160Q16—5841no—
161D8—732no—
162D8—1461no—
163C2×C8—41no—
164C2×C8—5842no—
165C2×C8—731no—
166C2×C8—731no—
167C2×C8—1461no—
168C2×C8D4×D7311yescharacteristic
169C2×C8#1192402811yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—2921no—
173C4⋊C4—2921no—
174C4⋊C4—1461no—
175C22⋊C4—2921no—
176C42D4×D7311yescharacteristic
177C16—41no—
178C16—2921no—
179C23—2921no—
180Q8—81no—
181Q8—81no—
182Q8—5841no—
183Q8—5841no—
184D4—2921no—
185D4—1462no—
186C2×C4—81no—
187C2×C4—2921no—
188C2×C4—5841no—
189C2×C4—5841no—
190C2×C4—5841no—
191C2×C4#1455656411yescharacteristic
192C2×C4D8×D7311yescharacteristic
193C2×C4#1455661811yescharacteristic
194C8—21no—
195C8—81no—
196C8—1462no—
197C8—1461no—
198C8—1461no—
199C8SD16×D7311yescharacteristic
200C8#1455661411yescharacteristic
201C22—2922no—
202C22—5841no—
203C22—5841no—
204C22#1798878111yescharacteristic
205C4—21no—
206C4—161no—
207C4—5841no—
208C4—11681no—
209C4#1798881911yescharacteristic
210C4#1798885311yescharacteristic
211C4#1798901811yescharacteristic
212C4#1798902111yescharacteristic
213C2—5842no—
214C2#2260927511yescharacteristic
215C2#2261198511yescharacteristic
216C2#2261201111yescharacteristic
subgroupG/H
centerC22#17988781
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1364C2
socleC22×C73#459
prime radicalC2×D584D4
hypercenter#1364D73
hypocenterC73#12422
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
3. #22609529C2semidirect
5. #22607994C2semidirect
6. #22607979C2semidirect
7. C73×#1364C2semidirect
114. #1364D73semidirect
116. C73#12422semidirect

Maximal subgroups

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

subgroupindex
#226120652
#226119772
#226095292
#226087252
#226079942
#226079792
C73×#13642
#1242273