Finite Group Databasebeta

← #22615752

Subgroups — #22615752

Order 9344.

488 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14556591C211yesmaximal, characteristic, complemented
2#17988526C212yesmaximal, complemented
3#17988523C212yesmaximal
4#17988507C212yesmaximal, complemented
5#17988499C212yesmaximal
6#17988485C212yesmaximal, complemented
7#17988453C211yesmaximal, characteristic, complemented
8#17988447C212yesmaximal
9C2×#117×C73C211yesmaximal, characteristic
10C22×Dic146C2212yes—
11#14556591—24no—
12C2×C4⋊Dic73C2212yes—
13C2×C4⋊Dic73C2212yes—
14C2×C4⋊Dic73C2212yes—
15C2×#11924007C2214yes—
16C2×C22⋊Dic73C2214yes—
17#14556498C2214yes—
18#14556498C2212yes—
19#14556498C2212yes—
20#14556498C2214yes—
21C23×C4×C73C2211yescharacteristic
22#117×C73—24no—
23C2×C4⋊C4×C73C2211yescharacteristic
24C2×C4⋊C4×C73C2211yescharacteristic
25C2×C22⋊C4×C73C2211yescharacteristic
26C2×C22⋊C4×C73C2211yescharacteristic
27C2×C42×C73C2211yescharacteristic
28C2×C42×C73C2211yescharacteristic
29C2×Dic146—24no—
30C2×Dic146—44no—
31C22×Dic73C2314yes—
32C22×Dic73C2314yes—
33C4⋊Dic73—24no—
34C4⋊Dic73—24no—
35C4⋊Dic73—24no—
36#11924007—44no—
37C22⋊Dic73—44no—
38C24×C73C2311yescharacteristic
39C22×C4×C73—22no—
40C22×C4×C73—22no—
41C22×C4×C73Q814yescomplemented
42C22×C4×C73C2311yescharacteristic
43C22×C4×C73C2311yescharacteristic
44C22×C4×C73C2311yescharacteristic
45C22×C4×C73C2311yescharacteristic
46C22×C4×C73C2311yescharacteristic
47C22×C4×C73C2311yescharacteristic
48C4⋊C4×C73—22no—
49C4⋊C4×C73—22no—
50C22⋊C4×C73—22no—
51C22⋊C4×C73—22no—
52C42×C73—22no—
53C42×C73D414yes—
54C23×C73—21no—
55C23×C73—21no—
56C23×C73—22no—
57C23×C73—22no—
58C23×C73C2×Q812yes—
59C23×C73C2411yescharacteristic
60C2×C4×C73—21no—
61C2×C4×C73—21no—
62C2×C4×C73—22no—
63C2×C4×C73—22no—
64C2×C4×C73—28no—
65C2×C4×C73—21no—
66C2×C4×C73—21no—
67C2×C4×C73—22no—
68C2×C4×C73—22no—
69C2×C4×C73C2×D412yes—
70C2×C4×C73C2×D412yes—
71C2×C4×C73C2×D412yes—
72C2×C4×C73C2×Q812yes—
73C2×C4×C73C2×Q812yes—
74C2×C4×C73#4112yes—
75C2×Dic73—24no—
76C2×Dic73—24no—
77C2×Dic73—24no—
78C2×Dic73—24no—
79C2×Dic73—24no—
80C2×Dic73—24no—
81Dic146—48no—
82C22×C73—22no—
83C22×C73—24no—
84C22×C73—41no—
85C22×C73—41no—
86C22×C73—42no—
87C22×C73C22×D411yescharacteristic
88C22×C73C22×Q811yescharacteristic
89C22×C73C2×#4111yescharacteristic
90C22×C73H(5,𝔽2)12yes—
91C22×C732−1+412yes—
92C4×C73—22no—
93C4×C73—22no—
94C4×C73—22no—
95C4×C73—41no—
96C4×C73—41no—
97C4×C73C22⋊Q814yes—
98Dic73—44no—
99Dic73—44no—
100C2×C73—42no—
101C2×C73C2×C22⋊Q811yescharacteristic
102C2×C73#53511yescharacteristic
103C2×C73#53611yescharacteristic
104C2×C73#54212yes—
105C2×C73#54412yes—
106#2628—731nomaximal, Sylow
107C73#262811yescharacteristic, complemented, Sylow
108C2×C4⋊Q8—731no—
109C2×#117D7311yescharacteristic
110#400—732no—
111#399—732no—
112#395—732no—
113#393—732no—
114#389—732no—
115#382—731no—
116#380—732no—
117C22×Q8—732no—
118C23×C4C2×D7311yescharacteristic
119C4⋊Q8—1464no—
120#117—24no—
121C2×C4⋊C4—734no—
122C2×C4⋊C4—732no—
123C2×C4⋊C4—732no—
124C2×C4⋊C4—732no—
125C2×C4⋊C4C2×D7311yescharacteristic
126C2×C4⋊C4C2×D7311yescharacteristic
127C2×C22⋊C4—734no—
128C2×C22⋊C4C2×D7311yescharacteristic
129C2×C22⋊C4C2×D7311yescharacteristic
130C2×C42C2×D7311yescharacteristic
131C2×C42C2×D7311yescharacteristic
132#95—734no—
133#95—732no—
134#95—732no—
135#95—734no—
136C24C22×D7311yescharacteristic
137C2×Q8—2924no—
138C2×Q8—1464no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—734no—
142C22×C4—734no—
143C22×C4Dic14614yescomplemented
144C22×C4C22×D7311yescharacteristic
145C22×C4C22×D7311yescharacteristic
146C22×C4C22×D7311yescharacteristic
147C22×C4C22×D7311yescharacteristic
148C22×C4C22×D7311yescharacteristic
149C22×C4C22×D7311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—2924no—
153C4⋊C4—1464no—
154C4⋊C4—1464no—
155C4⋊C4—1464no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—2924no—
159C42—22no—
160C42C22⋊D7314yes—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—22no—
165C23C2×Dic14612yes—
166C23C23×D7311yescharacteristic
167Q8—2928no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—1464no—
178C2×C4—1464no—
179C2×C4—1464no—
180C2×C4—1464no—
181C2×C4—1464no—
182C2×C4—1464no—
183C2×C4C2×C22⋊D7312yes—
184C2×C4C2×C22⋊D7312yes—
185C2×C4C2×C22⋊D7312yes—
186C2×C4C2×Dic14612yes—
187C2×C4C2×Dic14612yes—
188C2×C4#1192402812yes—
189C22—24no—
190C22—22no—
191C22—41no—
192C22—41no—
193C22—42no—
194C22C22×C22⋊D7311yescharacteristic
195C22C22×Dic14611yescharacteristic
196C22C2×#1192402811yescharacteristic
197C22#1455663712yes—
198C22#1455663812yes—
199C4—22no—
200C4—22no—
201C4—22no—
202C4—41no—
203C4—41no—
204C4—2924no—
205C4—2924no—
206C4#1455657014yes—
207C2—42no—
208C2C2×#1455657011yescharacteristic
209C2#1798915111yescharacteristic
210C2#1798915811yescharacteristic
211C2#1798920812yes—
212C2#1798921512yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×#117×C73C2
hypercenterC2×#117D73
hypocenterC73#2628
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14556591C2semidirect
2. #17988526C2semidirect
4. #17988507C2semidirect
6. #17988485C2semidirect
7. #17988453C2semidirect
41. C22×C4×C73Q8semidirect
107. C73#2628semidirect
143. C22×C4Dic146semidirect