Finite Group Databasebeta

← #37576167

Subgroups — #37576167

Order 24448.

1088 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 197 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×#20259309C212yesmaximal, complemented
2C2×#20259301C211yesmaximal, characteristic, complemented
3#27132577C214yesmaximal, complemented
4#27132566C212yesmaximal, complemented
5#27132519C214yesmaximal, complemented
6C4×C22⋊Dic191C211yesmaximal, characteristic, complemented
7C23×D4×C191C211yesmaximal, characteristic
8C22×Dic382C2211yescharacteristic, complemented
9#20259309—28no—
10#20259301—24no—
11C2×C4⋊Dic191C2211yescharacteristic, complemented
12C2×Dic191⋊C4C2212yescomplemented
13C2×C22⋊Dic191C2212yes—
14C2×C22⋊Dic191C2212yes—
15C2×C22⋊Dic191C2218yescomplemented
16C2×C22⋊Dic191C2218yescomplemented
17C2×C4×Dic191C2212yescomplemented
18C191⋊#95C2212yescomplemented
19C25×C191C2212yes—
20C22×D4×C191—28no—
21C22×D4×C191—24no—
22C22×D4×C191C2212yes—
23C22×D4×C191C2212yes—
24C23×C4×C191C2211yescharacteristic
25C2×Dic382—22no—
26C2×Dic382—24no—
27C22×Dic191C2312yes—
28C22×Dic191C2314yes—
29C22×Dic191C2312yes—
30C4⋊Dic191—22no—
31Dic191⋊C4—24no—
32C22⋊Dic191—24no—
33C22⋊Dic191—216no—
34C22⋊Dic191—24no—
35C22⋊Dic191—48no—
36C4×Dic191—24no—
37C24×C191—28no—
38C24×C191—24no—
39C24×C191—28no—
40C24×C191—44no—
41C24×C191C2314yes—
42C24×C191C2311yescharacteristic
43C2×D4×C191—232no—
44C2×D4×C191—28no—
45C2×D4×C191—24no—
46C2×D4×C191—24no—
47C2×D4×C191—24no—
48C2×D4×C191D418yes—
49C22×C4×C191—24no—
50C22×C4×C191D414yes—
51C22×C4×C191C2311yescharacteristic
52C22×C4×C191C2311yescharacteristic
53Dic382—44no—
54C2×Dic191—24no—
55C2×Dic191—24no—
56C2×Dic191—22no—
57C2×Dic191—22no—
58C2×Dic191—24no—
59C2×Dic191—24no—
60C2×Dic191—24no—
61C23×C191—22no—
62C23×C191—22no—
63C23×C191—216no—
64C23×C191—24no—
65C23×C191—24no—
66C23×C191—28no—
67C23×C191—28no—
68C23×C191—432no—
69C23×C191—48no—
70C23×C191—48no—
71C23×C191C2×D418yes—
72C23×C191C2×D412yes—
73C23×C191#4114yes—
74C23×C191C2411yescharacteristic
75D4×C191—216no—
76D4×C191—216no—
77C2×C4×C191—22no—
78C2×C4×C191—28no—
79C2×C4×C191C2×D412yes—
80C2×C4×C191C2×D414yes—
81C2×C4×C191C2×D412yes—
82C22×C191—22no—
83C22×C191—22no—
84C22×C191—28no—
85C22×C191—28no—
86C22×C191—216no—
87C22×C191—432no—
88C22×C191—44no—
89C22×C191—44no—
90C22×C191—48no—
91C22×C191#12414yescomplemented
92C22×C191C22×D411yescharacteristic
93C22×C191C22×D412yes—
94C22×C191C2×#4112yes—
95C22×C191H(5,𝔽2)12yes—
96C4×C191—22no—
97C4×C191#12014yes—
98Dic191—42no—
99Dic191—42no—
100Dic191—44no—
101C2×C191—24no—
102C2×C191—48no—
103C2×C191C2×#12011yescharacteristic
104C2×C191C2×#12411yescharacteristic
105C2×C191#53511yescharacteristic
106C2×C191#54614yes—
107C191#243511yescharacteristic, complemented, Sylow
108#2435—1911nomaximal, Sylow
109C23×D4D19111yescharacteristic
110C2×#124—1912no—
111C2×C22⋊Q8—1911no—
112#394—1914no—
113#390—1912no—
114#379—1914no—
115C4×C22⋊C4—1911no—
116C25C2×D19112yes—
117C22×Q8—1911no—
118C22×D4—28no—
119C22×D4—24no—
120C22×D4C2×D19112yes—
121C22×D4C2×D19112yes—
122C23×C4C2×D19111yescharacteristic
123#124—3828no—
124C22⋊Q8—3824no—
125C2×C4⋊C4—1911no—
126C2×C4⋊C4—1912no—
127C2×C22⋊C4—1912no—
128C2×C22⋊C4—1918no—
129C2×C22⋊C4—1912no—
130C2×C22⋊C4—1918no—
131C2×C42—1912no—
132#95—1912no—
133C24—28no—
134C24—24no—
135C24—28no—
136C24—44no—
137C24C22×D19114yes—
138C24C22×D19111yescharacteristic
139C2×Q8—3822no—
140C2×Q8—3824no—
141C2×D4—232no—
142C2×D4—28no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4C22⋊D19118yes—
147C22×C4—24no—
148C22×C4—1914no—
149C22×C4—1912no—
150C22×C4—1912no—
151C22×C4C22⋊D19114yes—
152C22×C4C22×D19111yescharacteristic
153C22×C4C22×D19111yescharacteristic
154C4⋊C4—3824no—
155C4⋊C4—3822no—
156C22⋊C4—3824no—
157C22⋊C4—38216no—
158C22⋊C4—3824no—
159C22⋊C4—7648no—
160C42—3824no—
161C23—22no—
162C23—22no—
163C23—216no—
164C23—24no—
165C23—24no—
166C23—28no—
167C23—28no—
168C23—432no—
169C23—48no—
170C23—48no—
171C23C2×C22⋊D19118yes—
172C23C2×C22⋊D19112yes—
173C23#1569028614yes—
174C23C23×D19111yescharacteristic
175Q8—7644no—
176D4—216no—
177D4—216no—
178C2×C4—22no—
179C2×C4—28no—
180C2×C4—3824no—
181C2×C4—3824no—
182C2×C4—3822no—
183C2×C4—3822no—
184C2×C4—3824no—
185C2×C4—3824no—
186C2×C4—3824no—
187C2×C4C2×C22⋊D19112yes—
188C2×C4C2×C22⋊D19114yes—
189C2×C4C2×C22⋊D19112yes—
190C22—22no—
191C22—28no—
192C22—22no—
193C22—216no—
194C22—28no—
195C22—432no—
196C22—44no—
197C22—48no—
198C22—44no—
199C22#2025930914yescomplemented
200C22C22×C22⋊D19111yescharacteristic
201C22C22×C22⋊D19112yes—
202C22C2×#1569028612yes—
203C22#2025936712yes—
204C4—22no—
205C4—7642no—
206C4—7644no—
207C4—7642no—
208C4#2025929114yes—
209C2—24no—
210C2—48no—
211C2C2×#2025929111yescharacteristic
212C2C2×#2025930911yescharacteristic
213C2#2713323011yescharacteristic
214C2#2713330714yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting, prime radicalC23×D4×C191C2
hypercenterC23×D4D191
hypocenterC191#2435
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20259309C2semidirect
2. C2×#20259301C2semidirect
3. #27132577C2semidirect
4. #27132566C2semidirect
5. #27132519C2semidirect
6. C4×C22⋊Dic191C2semidirect
8. C22×Dic382C22semidirect
11. C2×C4⋊Dic191C22semidirect
12. C2×Dic191⋊C4C22semidirect
15. C2×C22⋊Dic191C22semidirect
16. C2×C22⋊Dic191C22semidirect
17. C2×C4×Dic191C22semidirect
18. C191⋊#95C22semidirect
91. C22×C191#124semidirect
107. C191#2435semidirect
199. C22#20259309semidirect

Maximal subgroups

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

subgroupindex
C2×#202593092
C2×#202593012
#271325772
#271325662
#271325192
C4×C22⋊Dic1912
C23×D4×C1912
#2435191