Finite Group Databasebeta

← #33292407

Subgroups — #33292407

Order 18688.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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#22618195C213yesmaximal
2#22618191C2112yesmaximal, complemented
3#22613198C2112yesmaximal, complemented
4#22612309C213yesmaximal
5C73×#2840C211yesmaximal, characteristic
6C24×Dic73C2211yescharacteristic
7C2×Q8×Dic73C2214yescomplemented
8C2×#14556547C2216yes—
9#17988527C22112yescomplemented
10#17988523C22112yes—
11#17988521C22124yescomplemented
12#17988510C22124yes—
13#17988506C22112yescomplemented
14#17988448C2216yes—
15#17988445C22112yes—
16#17988443C2213yes—
17#17988438C22112yescomplemented
18C22⋊C4×Dic73C22112yes—
19#393×C73C22112yes—
20#379×C73C2213yes—
21C23×Dic73—44no—
22C23×Dic73Q818yescomplemented
23C23×Dic73C2313yes—
24C23×Dic73C2313yes—
25Q8×Dic73—48no—
26#14556547—412no—
27C2×C4⋊Dic73C23112yes—
28C2×#11924007C23124yes—
29C2×#11924007C23112yes—
30C2×C22⋊Dic73C23112yes—
31C2×C22⋊Dic73C2316yes—
32C2×C4×Dic73C23112yes—
33#14556498C23124yes—
34#14556498C23112yes—
35C25×C73C2311yescharacteristic
36C22×Q8×C73C2314yes—
37C2×C22⋊C4×C73C23112yes—
38C2×C22⋊C4×C73C2316yes—
39#95×C73C23112yes—
40C22×Dic73—212no—
41C22×Dic73—416no—
42C22×Dic73—46no—
43C22×Dic73C2×Q8112yes—
44C22×Dic73C2×Q8112yes—
45C22×Dic73C24112yes—
46C22×Dic73C2413yes—
47C22×Dic73C2411yescharacteristic
48C4⋊Dic73—412no—
49#11924007—424no—
50#11924007—412no—
51C22⋊Dic73—412no—
52C22⋊Dic73—46no—
53C4×Dic73—224no—
54C24×C73—44no—
55C24×C73—42no—
56C24×C73C2×Q814yes—
57C24×C73C2413yes—
58C2×Q8×C73—48no—
59C2×Q8×C73—44no—
60C22×C4×C73C24112yes—
61C22⋊C4×C73—412no—
62C22⋊C4×C73—46no—
63C23×C73—26no—
64C23×C73—212no—
65C23×C73—48no—
66C23×C73—44no—
67C23×C73—416no—
68C23×C73C22×Q816yes—
69C23×C73C2511yescharacteristic
70Q8×C73—416no—
71C2×C4×C73—212no—
72C2×C4×C73—224no—
73C2×Dic73—26no—
74C2×Dic73—224no—
75C2×Dic73—212no—
76C2×Dic73—224no—
77C2×Dic73—412no—
78C2×Dic73C22×Q816yes—
79C2×Dic73C22×Q812yes—
80C2×Dic73H(5,𝔽2)14yes—
81C22×C73—26no—
82C22×C73—212no—
83C22×C73—416no—
84C22×C73—412no—
85C22×C73C23×Q811yescharacteristic
86C22×C73C2×H(5,𝔽2)12yes—
87C22×C73#58514yes—
88C4×C73—412no—
89Dic73—26no—
90Dic73—412no—
91Dic73#54314yes—
92#36674—731nomaximal, Sylow
93C2×C73—46no—
94C2×C73C2×#54311yescharacteristic
95C2×C73#345812yes—
96C2×C73#352914yes—
97#2842—733no—
98#2841—7312no—
99#2840D7311yescharacteristic
100#2396—7312no—
101#2280—733no—
102C73#3667411yescharacteristic, complemented, Sylow
103C24×C4—731no—
104C2×C4×Q8—734no—
105C2×#117—736no—
106#400—7312no—
107#399—7312no—
108#398—7324no—
109#396—7324no—
110#395—7312no—
111#393C2×D73112yes—
112#381—736no—
113#380—7312no—
114#379—733no—
115#379C2×D7313yes—
116C4×C4⋊C4—7312no—
117C4×C22⋊C4—7312no—
118C25C22×D7311yescharacteristic
119C22×Q8C22×D7314yes—
120C23×C4—2924no—
121C23×C4—738no—
122C23×C4—733no—
123C23×C4—733no—
124C4×Q8—2928no—
125#117—29212no—
126C2×C4⋊C4—7312no—
127C2×C4⋊C4—7312no—
128C2×C4⋊C4—7324no—
129C2×C22⋊C4—736no—
130C2×C22⋊C4—7312no—
131C2×C22⋊C4C22×D7316yes—
132C2×C22⋊C4C22×D73112yes—
133C2×C42—7312no—
134#95—7324no—
135#95—7312no—
136#95C22×D73112yes—
137C24—42no—
138C24—44no—
139C24Q8×D7314yes—
140C24C23×D7313yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—2926no—
144C22×C4—29216no—
145C22×C4—7312no—
146C22×C4—731no—
147C22×C4—7312no—
148C22×C4—733no—
149C22×C4—7312no—
150C22×C4—14612no—
151C22×C4C23×D73112yes—
152C4⋊C4—29224no—
153C4⋊C4—29212no—
154C4⋊C4—29212no—
155C22⋊C4—412no—
156C22⋊C4—46no—
157C22⋊C4—29212no—
158C22⋊C4—2926no—
159C42—14624no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D7316yes—
166C23C24×D7311yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—29212no—
171C2×C4—732no—
172C2×C4—734no—
173C2×C4—736no—
174C2×C4—14612no—
175C2×C4—14624no—
176C2×C4—14624no—
177C2×C4—1466no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D7311yescharacteristic
183C22H(5,𝔽2)×D7312yes—
184C22#1798942214yes—
185C4—412no—
186C4—29212no—
187C4—734no—
188C4—1466no—
189C2—46no—
190C2#543×D7311yescharacteristic
191C2#2262333312yes—
192C2#2262407114yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC73×#2840C2
prime radicalC25×C73C23
hypercenter#2840D73
hypocenterC73#36674
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #22618191C2semidirect
3. #22613198C2semidirect
7. C2×Q8×Dic73C22semidirect
9. #17988527C22semidirect
11. #17988521C22semidirect
13. #17988506C22semidirect
17. #17988438C22semidirect
22. C23×Dic73Q8semidirect
102. C73#36674semidirect

Maximal subgroups

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

subgroupindex
#226181952
#226181912
#226131982
#226123092
C73×#28402
#3667473