Finite Group Databasebeta

← #37527022

Subgroups — #37527022

Order 22976.

294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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×#19153341C211yesmaximal, characteristic, complemented
2C2×#19153340C211yesmaximal, characteristic, complemented
3#26039995C211yesmaximal, characteristic, complemented
4#26039994C211yesmaximal, characteristic, complemented
5#26039992C212yesmaximal, complemented
6#26039988C212yesmaximal, complemented
7#26039987C212yesmaximal, complemented
8#26039986C211yesmaximal, characteristic, complemented
9#26039986C212yesmaximal, complemented
10C4⋊C4×D359C211yesmaximal, characteristic, complemented
11#121×C359C211yesmaximal, characteristic, complemented
12#19153341—24no—
13#19153340—24no—
14C2×Dic718C2211yescharacteristic
15C2×Dic718C2211yescharacteristic
16C2×D1436C2211yescharacteristic
17C2×C22⋊D359C2211yescharacteristic
18C2×C22⋊D359C2212yes—
19C2×C22⋊D359C2211yescharacteristic
20C2×C22⋊D359C2212yes—
21C22×Dic359C2212yes—
22C2×C4×D359C2211yescharacteristic
23C2×C4×D359C2212yes—
24C2×C4×D359C2211yescharacteristic
25C4⋊Dic359C2211yescharacteristic
26C4⋊Dic359C2212yes—
27Dic359⋊C4C2212yes—
28Dic359⋊C4C2212yes—
29C22⋊Dic359C2212yes—
30C22⋊Dic359C2212yes—
31(C2×D359)⋊C4C2212yes—
32C2×D4×C359C2211yescharacteristic
33C2×D4×C359C2212yes—
34C22×C4×C359C2211yescharacteristic
35C4⋊C4×C359C2211yescharacteristic
36C22⋊C4×C359C2212yes—
37C22×D359C2311yescharacteristic
38C22×D359C2311yescharacteristic
39Dic718—22no—
40Dic718—22no—
41D1436—22no—
42C22⋊D359—24no—
43C22⋊D359—22no—
44C22⋊D359—22no—
45C22⋊D359—42no—
46C2×Dic359—24no—
47C2×Dic359C2312yes—
48C2×Dic359C2312yes—
49C2×Dic359C2311yescharacteristic
50C2×Dic359C2311yescharacteristic
51C4×D359—22no—
52C4×D359—42no—
53C4×D359D414yescomplemented
54C23×C359C2312yes—
55C23×C359C2311yescharacteristic
56D4×C359—22no—
57D4×C359—42no—
58C2×C4×C359—22no—
59C2×C4×C359C2311yescharacteristic
60C2×C4×C359C2312yes—
61C2×C4×C359C2311yescharacteristic
62C22×C359—21no—
63C22×C359—22no—
64C22×C359—21no—
65C22×C359—22no—
66C22×C359—22no—
67C22×C359—21no—
68C22×C359C2411yescharacteristic
69C2×D359—21no—
70C2×D359—21no—
71C2×D359—21no—
72C2×D359—21no—
73C2×D359—21no—
74C2×D359C2×D412yes—
75C4×C359—21no—
76C4×C359—22no—
77C4×C359C2×D412yes—
78Dic359—21no—
79Dic359—22no—
80Dic359—22no—
81Dic359C2×D412yes—
82C2×C359—41no—
83C2×C359—42no—
84C2×C359C22×D411yescharacteristic
85C2×C359H(5,𝔽2)11yescharacteristic
86C2×C3592−1+411yescharacteristic
87D359—22no—
88D359—41no—
89C359#53711yescharacteristic, complemented, Sylow
90#537—3591nomaximal, Sylow
91C2×#41—3591no—
92C2×#41—3591no—
93C22⋊Q8—3591no—
94C22⋊Q8—3591no—
95C22⋊Q8—3592no—
96#121—3591no—
97#121—3592no—
98#121—3592no—
99#121—3592no—
100#121D35911yescharacteristic, complemented
101C2×C4⋊C4—3591no—
102#41—7184no—
103#41—7184no—
104C2×Q8—3591no—
105C2×Q8—3591no—
106C2×D4—3592no—
107C2×D4—3591no—
108C2×D4—3591no—
109C2×D4—3592no—
110C2×D4—3591no—
111C2×D4C2×D35911yescharacteristic
112C2×D4C2×D35912yes—
113C22×C4—3591no—
114C22×C4—3592no—
115C22×C4—3592no—
116C22×C4—3591no—
117C22×C4C2×D35911yescharacteristic
118C4⋊C4—3592no—
119C4⋊C4—3591no—
120C4⋊C4—3592no—
121C4⋊C4—3592no—
122C4⋊C4C2×D35911yescharacteristic
123C22⋊C4—3592no—
124C22⋊C4—3592no—
125C22⋊C4—3592no—
126C22⋊C4C2×D35912yes—
127C23—3591no—
128C23—3591no—
129C23C22×D35911yescharacteristic
130C23C22×D35912yes—
131Q8—7182no—
132Q8—7182no—
133D4—22no—
134D4—42no—
135D4—14362no—
136D4—7182no—
137D4—7182no—
138D4—7182no—
139D4—7184no—
140C2×C4—22no—
141C2×C4—3591no—
142C2×C4—3594no—
143C2×C4—3592no—
144C2×C4—3591no—
145C2×C4—3592no—
146C2×C4—14362no—
147C2×C4—7184no—
148C2×C4—7182no—
149C2×C4C22×D35912yes—
150C2×C4C22×D35911yescharacteristic
151C2×C4C22×D35911yescharacteristic
152C22—21no—
153C22—22no—
154C22—21no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—3592no—
159C22—7181no—
160C22—7181no—
161C22—7181no—
162C22—7181no—
163C22—7181no—
164C22C23×D35911yescharacteristic
165C4—21no—
166C4—22no—
167C4—3592no—
168C4—7182no—
169C4—7181no—
170C4—7182no—
171C4D4×D35912yes—
172C2—41no—
173C2—42no—
174C2—14361no—
175C2—7182no—
176C2C2×D4×D35911yescharacteristic
177C2#2604006011yescharacteristic
178C2#2604006211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#121×C359C2
hypercenter#121D359
hypocenterC359#537
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19153341C2semidirect
2. C2×#19153340C2semidirect
3. #26039995C2semidirect
4. #26039994C2semidirect
5. #26039992C2semidirect
6. #26039988C2semidirect
7. #26039987C2semidirect
8. #26039986C2semidirect
9. #26039986C2semidirect
10. C4⋊C4×D359C2semidirect
11. #121×C359C2semidirect
53. C4×D359D4semidirect
89. C359#537semidirect
100. #121D359semidirect

Maximal subgroups

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

subgroupindex
C2×#191533412
C2×#191533402
#260399952
#260399942
#260399922
#260399882
#260399872
#260399862
C4⋊C4×D3592
#121×C3592
#537359