Finite Group Databasebeta

← #32997502

Subgroups — #32997502

Order 18688.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#22620581C211yesmaximal, characteristic
2#22611697C211yesmaximal, characteristic, complemented
3#22607991C212yesmaximal, complemented
4#22607967C212yesmaximal, complemented
5C73×#2166C211yesmaximal, characteristic, complemented
6#17989192C412yes—
7C2×#14556531—21no—
8#17988989C2211yescharacteristic
9#17988964C2211yescharacteristic
10#17988952C412yes—
11#17988840—21no—
12#17988823—21no—
13#17988700—21no—
14C4×D584—21no—
15#17988408—21no—
16#17988408C2212yes—
17#405×C73C2211yescharacteristic
18#364×C73C2212yes—
19#349×C73—21no—
20#347×C73—21no—
21C2×D584—21no—
22C2×D584—21no—
23#14556588C2×C412yes—
24#14556583C2×C412yes—
25#14556576—42no—
26#14556565—22no—
27#14556564—42no—
28C4×D292—22no—
29#14556531—22no—
30#14556531—41no—
31C2×#11924003—41no—
32#14556525—21no—
33#14556523—21no—
34#14556513—22no—
35#14556513—22no—
36C2×M16×C73D411yescharacteristic
37C22×C8×C73D411yescharacteristic
38#117×C73C2×C411yescharacteristic
39M32×C73—22no—
40#97×C73C2×C411yescharacteristic
41#97×C73D411yescharacteristic
42C4×C8×C73D411yescharacteristic
43C4×C8×C73C2311yescharacteristic
44C2×D292—22no—
45C2×D292—42no—
46C2×C22⋊D73—42no—
47C2×C4×D73—42no—
48D584—42no—
49D584—41no—
50D584—41no—
51C4⋊Dic73—22no—
52#11924007—42no—
53#11924005—42no—
54#11924005—42no—
55#11924003—42no—
56C22×C4×C73C22⋊C411yescharacteristic
57M16×C73—22no—
58C2×C8×C73—21no—
59C2×C8×C73—21no—
60C2×C8×C73C22⋊C411yescharacteristic
61C2×C8×C73C22⋊C411yescharacteristic
62C2×C8×C73C2×D411yescharacteristic
63C2×C8×C73C2×D411yescharacteristic
64C4⋊C4×C73—21no—
65C22⋊C4×C73—21no—
66C42×C73C22⋊C411yescharacteristic
67C42×C73C22×C411yescharacteristic
68C16×C73—42no—
69C23×C73#10011yescharacteristic
70C22×D73—42no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73#10011yescharacteristic
76C2×C4×C73C2×C22⋊C411yescharacteristic
77C22⋊D73—162no—
78C2×Dic73—42no—
79D292—42no—
80D292—42no—
81D292—162no—
82C4×D73—162no—
83C8×C73—21no—
84C8×C73—21no—
85C8×C73#13512yes—
86C8×C73AGL(1,ℤ/8)11yescharacteristic
87C8×C73#13711yescharacteristic
88C22×C73—21no—
89C22×C73—41no—
90C22×C73C2×#10011yescharacteristic
91C2×D73—82no—
92C2×D73—82no—
93C2×D73—82no—
94C4×C73—21no—
95C4×C73—41no—
96C4×C73—41no—
97C4×C73#41611yescharacteristic
98C4×C73#41711yescharacteristic
99Dic73—82no—
100#11989—731nomaximal, Sylow
101C2×C73—21no—
102C2×C73—41no—
103C2×C73#155111yescharacteristic
104D73—162no—
105#3176—731no—
106#2166D7311yescharacteristic, complemented
107#2153—731no—
108#1366—732no—
109#1357—732no—
110C73#1198911yescharacteristic, complemented, Sylow
111#540—732no—
112C2×M32—1461no—
113#499—731no—
114#493—731no—
115#490—732no—
116#469—1461no—
117#466—1461no—
118#442—1461no—
119C4×D8—1461no—
120#405C2×D7311yescharacteristic
121#364—732no—
122#364—1461no—
123#364C2×D7312yes—
124#349—21no—
125#347—21no—
126C2×D8—1461no—
127C2×D8—1461no—
128C2×M16C22⋊D7311yescharacteristic
129C22×C8D29211yescharacteristic
130#127—732no—
131#125—732no—
132#123—2922no—
133#121—2922no—
134#121—1462no—
135C4×D4—1462no—
136#117C4×D7311yescharacteristic
137M32—22no—
138M32—2921no—
139M32—1462no—
140C2×C16—2921no—
141#107—1461no—
142#106—1461no—
143#102—1462no—
144#102—1462no—
145#97C4×D7311yescharacteristic
146#97C22⋊D7311yescharacteristic
147C4×C8D29211yescharacteristic
148C4×C8C22×D7311yescharacteristic
149C2×D4—2922no—
150C2×D4—2922no—
151C2×D4—1462no—
152C22×C4—2922no—
153C22×C4#1192400511yescharacteristic
154D8—2922no—
155D8—2921no—
156D8—2921no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8#1192400511yescharacteristic
161C2×C8#1192400511yescharacteristic
162C2×C8C2×C22⋊D7311yescharacteristic
163C2×C8C2×D29211yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—2922no—
166C4⋊C4—1462no—
167C22⋊C4—21no—
168C22⋊C4—2922no—
169C22⋊C4—2922no—
170C42#1192400511yescharacteristic
171C42C2×C4×D7311yescharacteristic
172C16—42no—
173C16—2922no—
174C23—2922no—
175C23#1455650711yescharacteristic
176D4—2922no—
177D4—2922no—
178D4—11682no—
179D4—11682no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—2922no—
185C2×C4—11682no—
186C2×C4#1455650711yescharacteristic
187C2×C4C2×#1192400511yescharacteristic
188C8—21no—
189C8—21no—
190C8#1455661112yes—
191C8#1455661911yescharacteristic
192C8#1455662411yescharacteristic
193C22—21no—
194C22—41no—
195C22—5842no—
196C22—5842no—
197C22—5842no—
198C22C2×#1455650711yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—5842no—
203C4#1798858111yescharacteristic
204C4#1798858911yescharacteristic
205C2—21no—
206C2—41no—
207C2—11682no—
208C2#2260862711yescharacteristic
subgroupG/H
centerC2#22608627
commutatorC42×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#2166C2
socleC2×C73#1551
prime radical#22620581C2
cosocleC4×C8×C73C23
hypercenter#2166D73
hypocenterC73#11989
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #22611697C2semidirect
3. #22607991C2semidirect
4. #22607967C2semidirect
5. C73×#2166C2semidirect
106. #2166D73semidirect
110. C73#11989semidirect

Maximal subgroups

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

subgroupindex
#226205812
#226116972
#226079912
#226079672
C73×#21662
#1198973