Finite Group Databasebeta

← #37560174

Subgroups — #37560174

Order 23872.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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#26045792C211yesmaximal, characteristic, complemented
2#26045791C211yesmaximal, characteristic, complemented
3#26045787C211yesmaximal, characteristic, complemented
4#26045786C211yesmaximal, characteristic, complemented
5#26045785C211yesmaximal, characteristic, complemented
6#26045783C211yesmaximal, characteristic, complemented
7#26045782C211yesmaximal, characteristic, complemented
8#26045778C211yesmaximal, characteristic
9#26045776C211yesmaximal, characteristic
10#26045774C211yesmaximal, characteristic
11#26045772C211yesmaximal, characteristic, complemented
12#26045770C211yesmaximal, characteristic, complemented
13#26045769C211yesmaximal, characteristic, complemented
14#26045769C211yesmaximal, characteristic, complemented
15#126×C373C211yesmaximal, characteristic, complemented
16C2×Dic746C2211yescharacteristic
17C2×Dic746C2211yescharacteristic
18C2×Dic746C2211yescharacteristic
19C2×C22⋊D373C2211yescharacteristic
20C22×Dic373C2211yescharacteristic
21C2×C4×D373C2211yescharacteristic
22C4⋊Dic373C2211yescharacteristic
23C4⋊Dic373C2211yescharacteristic
24C4⋊Dic373C2211yescharacteristic
25C4⋊Dic373C2211yescharacteristic
26C4⋊Dic373C2211yescharacteristic
27C4⋊Dic373C2211yescharacteristic
28#20238589C2211yescharacteristic
29#20238589C2211yescharacteristic
30#20238589C2211yescharacteristic
31#20238589C2211yescharacteristic
32#20238589C2211yescharacteristic
33#20238589C2211yescharacteristic
34#20238589C2211yescharacteristic
35C22⋊Dic373C2211yescharacteristic
36C22⋊Dic373C2211yescharacteristic
37#20238587C2211yescharacteristic
38#20238587C2211yescharacteristic
39#20238587C2211yescharacteristic
40#20238587C2211yescharacteristic
41#20238587C2211yescharacteristic
42C4×Dic373C2211yescharacteristic
43C4×Dic373C2211yescharacteristic
44C4⋊C4×C373C2211yescharacteristic
45C4⋊C4×C373C2211yescharacteristic
46C4⋊C4×C373C2211yescharacteristic
47C22⋊C4×C373C2211yescharacteristic
48C22⋊C4×C373C2211yescharacteristic
49C22⋊C4×C373C2211yescharacteristic
50C42×C373C2211yescharacteristic
51C22×D373C2311yescharacteristic
52Dic746—41no—
53Dic746—41no—
54Dic746—41no—
55C22⋊D373—41no—
56C2×Dic373—41no—
57C2×Dic373C2311yescharacteristic
58C2×Dic373C2311yescharacteristic
59C2×Dic373C2311yescharacteristic
60C2×Dic373C2311yescharacteristic
61C2×Dic373C2311yescharacteristic
62C2×Dic373C2311yescharacteristic
63C2×Dic373C2311yescharacteristic
64C4×D373—41no—
65C23×C373C2311yescharacteristic
66C2×C4×C373C2311yescharacteristic
67C2×C4×C373C2311yescharacteristic
68C2×C4×C373C2311yescharacteristic
69C2×C4×C373C2311yescharacteristic
70C2×C4×C373C2311yescharacteristic
71C2×C4×C373C2311yescharacteristic
72C22×C373—21no—
73C22×C373—21no—
74C22×C373—21no—
75C22×C373C2411yescharacteristic
76C2×D373—21no—
77C2×D373—21no—
78C2×D373—21no—
79C4×C373—21no—
80C4×C373—21no—
81C4×C373—21no—
82C4×C373—21no—
83C4×C373—21no—
84C4×C373—21no—
85Dic373—21no—
86Dic373—21no—
87Dic373—21no—
88Dic373—21no—
89Dic373—21no—
90Dic373—21no—
91Dic373—21no—
92C2×C373—41no—
93C2×C373H(5,𝔽2)11yescharacteristic
94C2×C3732−1+411yescharacteristic
95C2×C3732−1+411yescharacteristic
96D373—41no—
97C373#56311yescharacteristic, complemented, Sylow
98#563—3731nomaximal, Sylow
99C4⋊Q8—3731no—
100C4⋊Q8—3731no—
101#126—3731no—
102#126—3731no—
103#126—3731no—
104#126D37311yescharacteristic, complemented
105#125—3731no—
106#125—3731no—
107#124—3731no—
108#123—3731no—
109#123—3731no—
110C22⋊Q8—3731no—
111C22⋊Q8—3731no—
112C22⋊Q8—3731no—
113C22⋊Q8—3731no—
114C2×Q8—3731no—
115C2×Q8—3731no—
116C2×Q8—3731no—
117C2×D4—3731no—
118C22×C4—3731no—
119C22×C4—3731no—
120C4⋊C4—3731no—
121C4⋊C4—3731no—
122C4⋊C4—3731no—
123C4⋊C4—3731no—
124C4⋊C4—3731no—
125C4⋊C4—3731no—
126C4⋊C4—3731no—
127C4⋊C4—3731no—
128C4⋊C4—3731no—
129C4⋊C4—3731no—
130C4⋊C4—3731no—
131C4⋊C4—3731no—
132C4⋊C4—3731no—
133C4⋊C4C2×D37311yescharacteristic
134C4⋊C4C2×D37311yescharacteristic
135C4⋊C4C2×D37311yescharacteristic
136C22⋊C4—3731no—
137C22⋊C4—3731no—
138C22⋊C4—3731no—
139C22⋊C4—3731no—
140C22⋊C4—3731no—
141C22⋊C4—3731no—
142C22⋊C4—3731no—
143C22⋊C4C2×D37311yescharacteristic
144C22⋊C4C2×D37311yescharacteristic
145C22⋊C4C2×D37311yescharacteristic
146C42—3731no—
147C42—3731no—
148C42C2×D37311yescharacteristic
149C23—3731no—
150C23C22×D37311yescharacteristic
151Q8—14921no—
152Q8—14921no—
153Q8—14921no—
154D4—14921no—
155C2×C4—3731no—
156C2×C4—3731no—
157C2×C4—3731no—
158C2×C4—3731no—
159C2×C4—3731no—
160C2×C4—3731no—
161C2×C4—3731no—
162C2×C4—14921no—
163C2×C4—14921no—
164C2×C4C22×D37311yescharacteristic
165C2×C4C22×D37311yescharacteristic
166C2×C4C22×D37311yescharacteristic
167C2×C4C22×D37311yescharacteristic
168C2×C4C22×D37311yescharacteristic
169C2×C4C22×D37311yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—7461no—
174C22—7461no—
175C22—7461no—
176C22C23×D37311yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—7461no—
184C4—7461no—
185C4—7461no—
186C4—7461no—
187C4—7461no—
188C4—7461no—
189C4—7461no—
190C2—41no—
191C2—14921no—
192C2#2604583811yescharacteristic
193C2#2604583911yescharacteristic
194C2#2604583911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#126×C373C2
prime radicalC2×C22⋊D373C22
hypercenter#126D373
hypocenterC373#563
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #26045792C2semidirect
2. #26045791C2semidirect
3. #26045787C2semidirect
4. #26045786C2semidirect
5. #26045785C2semidirect
6. #26045783C2semidirect
7. #26045782C2semidirect
11. #26045772C2semidirect
12. #26045770C2semidirect
13. #26045769C2semidirect
14. #26045769C2semidirect
15. #126×C373C2semidirect
97. C373#563semidirect
104. #126D373semidirect

Maximal subgroups

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

subgroupindex
#260457922
#260457912
#260457872
#260457862
#260457852
#260457832
#260457822
#260457782
#260457762
#260457742
#260457722
#260457702
#260457692
#126×C3732
#563373