Finite Group Databasebeta

← #33006632

Subgroups — #33006632

Order 18688.

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

normal subgroups only

This group has 298 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
1C2×#17989019C211yesmaximal, characteristic, complemented
2C2×#17988425C211yesmaximal, characteristic, complemented
3C2×#17988422C211yesmaximal, characteristic, complemented
4#22608111C211yesmaximal, characteristic, complemented
5#22608108C211yesmaximal, characteristic, complemented
6C22⋊#14556497C211yesmaximal, characteristic, complemented
7C73×#2215C211yesmaximal, characteristic, complemented
8C2×#14556613—21no—
9#17989019—22no—
10#17989019—22no—
11C2×Dic584C2211yescharacteristic
12C2×#14556536—21no—
13C2×D1168C2211yescharacteristic
14#17988425—22no—
15#17988422—22no—
16C2×#14556497C2211yescharacteristic
17C2×#14556497C2211yescharacteristic
18C2×D16×C73C2211yescharacteristic
19C22×C16×C73C2211yescharacteristic
20#466×C73—21no—
21#366×C73C2211yescharacteristic
22#357×C73—21no—
23C2×#11924028—41no—
24#14556613—24no—
25#14556613—42no—
26C2×Dic292—21no—
27C2×#11924017—41no—
28C2×D584—21no—
29Dic584—21no—
30Dic584D412yescomplemented
31#14556536—24no—
32D1168—21no—
33D1168D412yescomplemented
34#14556497—21no—
35#14556497—21no—
36C2×D8×C73—21no—
37C22×C8×C73D411yescharacteristic
38#121×C73—41no—
39D16×C73—21no—
40D16×C73—21no—
41C2×C16×C73—21no—
42C2×C16×C73—21no—
43C2×C16×C73D411yescharacteristic
44C2×C16×C73C2311yescharacteristic
45#107×C73—21no—
46#102×C73—41no—
47#11924028—44no—
48#11924028—82no—
49C2×Dic146—41no—
50C2×D292—41no—
51C2×C22⋊D73—81no—
52C2×C4×D73—81no—
53Dic292—22no—
54Dic292—41no—
55#11924017—44no—
56D584—22no—
57D584—41no—
58C2×D4×C73—41no—
59C2×D4×C73—81no—
60C22×C4×C73D811yescharacteristic
61D8×C73—41no—
62D8×C73—41no—
63C2×C8×C73—21no—
64C2×C8×C73—21no—
65C2×C8×C73D811yescharacteristic
66C2×C8×C73C2×D411yescharacteristic
67C4⋊C4×C73—41no—
68C22⋊C4×C73—81no—
69C16×C73—21no—
70C16×C73C2×D411yescharacteristic
71C16×C73C2×D411yescharacteristic
72C23×C73—81no—
73C23×C73D1611yescharacteristic
74C22×D73—81no—
75D4×C73—81no—
76D4×C73—81no—
77D4×C73—162no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—81no—
81C2×C4×C73D1611yescharacteristic
82C2×C4×C73C2×D811yescharacteristic
83C22⋊D73—84no—
84C2×Dic73—81no—
85D292—42no—
86D292—81no—
87C4×D73—84no—
88Dic146—42no—
89Dic146—81no—
90C8×C73—21no—
91C8×C73#12011yescharacteristic
92C8×C73AGL(1,ℤ/8)11yescharacteristic
93C22×C73—21no—
94C22×C73—21no—
95C22×C73—21no—
96C22×C73—161no—
97C22×C73—161no—
98C22×C73—161no—
99C22×C73C2×D1611yescharacteristic
100C2×D73—82no—
101C2×D73—161no—
102C2×D73—161no—
103C4×C73—21no—
104C4×C73—161no—
105C4×C73#44711yescharacteristic
106C4×C73#50911yescharacteristic
107Dic73—82no—
108#13663—731nomaximal, Sylow
109C2×C73—41no—
110C2×C73—321no—
111C2×C73#218411yescharacteristic
112C2×C73#226211yescharacteristic
113C2×C73#226311yescharacteristic
114D73—162no—
115C2×#508—731no—
116C2×SD64—731no—
117C2×D32—731no—
118#2215D7311yescharacteristic, complemented
119#1416—731no—
120#1415—731no—
121C22⋊C32—731no—
122C73#1366311yescharacteristic, complemented, Sylow
123C2×#135—1461no—
124#508—1462no—
125#508—1462no—
126C2×Q32—731no—
127C2×SD32—1461no—
128C2×D16—731no—
129C2×D16C2×D7311yescharacteristic
130C22×C16C2×D7311yescharacteristic
131#466—21no—
132SD64—1462no—
133D32—1462no—
134C2×C32—731no—
135C2×C32—731no—
136#366C2×D7311yescharacteristic
137#357—21no—
138C2×#41—2921no—
139#135—2922no—
140#135—1464no—
141C2×Q16—1461no—
142C2×SD16—2921no—
143C2×D8—21no—
144C2×D8—1461no—
145C22×C8C22⋊D7311yescharacteristic
146#121—41no—
147Q32—732no—
148Q32—1461no—
149SD32—1464no—
150D16—21no—
151D16—21no—
152D16—732no—
153D16—1461no—
154C2×C16—21no—
155C2×C16—21no—
156C2×C16C22⋊D7311yescharacteristic
157C2×C16C22×D7311yescharacteristic
158#107—21no—
159#102—41no—
160C32—1461no—
161C32—1461no—
162#41—2924no—
163#41—5842no—
164C2×Q8—2921no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—2921no—
168C2×D4—5841no—
169C22×C4—5841no—
170C22×C4#1192401311yescharacteristic
171Q16—2921no—
172Q16—1462no—
173SD16—2924no—
174D8—41no—
175D8—41no—
176D8—2921no—
177D8—1462no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8#1192401311yescharacteristic
181C2×C8C2×C22⋊D7311yescharacteristic
182C4⋊C4—41no—
183C22⋊C4—81no—
184C16—21no—
185C16D4×D7311yescharacteristic
186C16D4×D7311yescharacteristic
187C23—81no—
188C23—5841no—
189C23#1455653211yescharacteristic
190Q8—2922no—
191Q8—5841no—
192D4—81no—
193D4—81no—
194D4—162no—
195D4—2922no—
196D4—5844no—
197D4—5841no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—81no—
201C2×C4—5844no—
202C2×C4—5841no—
203C2×C4#1455653211yescharacteristic
204C2×C4C2×#1192401311yescharacteristic
205C8—21no—
206C8#1455655911yescharacteristic
207C8#1455661611yescharacteristic
208C22—21no—
209C22—21no—
210C22—21no—
211C22—161no—
212C22—161no—
213C22—161no—
214C22—5842no—
215C22—11681no—
216C22—11681no—
217C22C2×#1455653211yescharacteristic
218C4—21no—
219C4—161no—
220C4—5842no—
221C4#1798872011yescharacteristic
222C4#1798902211yescharacteristic
223C2—41no—
224C2—321no—
225C2—11682no—
226C2#2261184211yescharacteristic
227C2#2261223811yescharacteristic
228C2#2261224411yescharacteristic
subgroupG/H
centerC22C2×#14556532
commutator, cosocleC2×C16×C73C23
FrattiniC2×C16C22×D73
FittingC73×#2215C2
socleC22×C73C2×D16
hypercenter#2215D73
hypocenterC73#13663
LCSG ▷ C2×C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. C2×#17989019C2semidirect
2. C2×#17988425C2semidirect
3. C2×#17988422C2semidirect
4. #22608111C2semidirect
5. #22608108C2semidirect
6. C22⋊#14556497C2semidirect
7. C73×#2215C2semidirect
30. Dic584D4semidirect
33. D1168D4semidirect
118. #2215D73semidirect
122. C73#13663semidirect

Maximal subgroups

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

subgroupindex
C2×#179890192
C2×#179884252
C2×#179884222
#226081112
#226081082
C22⋊#145564972
C73×#22152
#1366373