Finite Group Databasebeta

← #37575273

Subgroups — #37575273

Order 24448.

484 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 177 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×#20259301C211yesmaximal, characteristic
2#27132567C212yesmaximal, complemented
3#27132557C212yesmaximal
4#27132551C212yesmaximal
5#27132546C212yesmaximal, complemented
6#27132545C212yesmaximal, complemented
7#27132514C212yesmaximal, complemented
8C4×C22⋊Dic191C211yesmaximal, characteristic
9#381×C191C211yesmaximal, characteristic
10C22×Dic382C2211yescharacteristic
11#20259301C418yescomplemented
12C2×C4⋊Dic191C2211yescharacteristic
13C2×Dic191⋊C4C2212yes—
14C2×Dic191⋊C4C2212yes—
15C2×Dic191⋊C4C2212yes—
16C2×C22⋊Dic191C2212yes—
17C2×C22⋊Dic191C2212yes—
18C2×C4×Dic191C2212yes—
19C2×C4×Dic191C2211yescharacteristic
20C2×C4×Dic191C2211yescharacteristic
21C2×C4×Dic191C2212yes—
22C191⋊#95C2212yes—
23C191⋊#95C2212yes—
24C191⋊#95C2212yes—
25C191⋊#95C2212yes—
26C191⋊#95C2212yes—
27C23×C4×C191C2211yescharacteristic
28C2×C22⋊C4×C191C2211yescharacteristic
29C2×C22⋊C4×C191C2211yescharacteristic
30#95×C191C2212yes—
31#95×C191C2212yes—
32C2×Dic382—41no—
33C2×Dic382—41no—
34C2×Dic382C2×C414yes—
35C22×Dic191C2312yes—
36C22×Dic191C2312yes—
37C22×Dic191C2312yes—
38C22×Dic191C2312yes—
39C4⋊Dic191C2×C414yes—
40Dic191⋊C4—24no—
41Dic191⋊C4—42no—
42Dic191⋊C4C2×C418yes—
43C22⋊Dic191—24no—
44C22⋊Dic191C2×C418yes—
45C4×Dic191—24no—
46C4×Dic191—41no—
47C4×Dic191—41no—
48C4×Dic191—42no—
49C24×C191C2311yescharacteristic
50C22×C4×C191—41no—
51C22×C4×C191—41no—
52C22×C4×C191C2×C414yes—
53C22×C4×C191C2312yes—
54C22×C4×C191C2312yes—
55C22×C4×C191C2311yescharacteristic
56C22×C4×C191C2311yescharacteristic
57C22⋊C4×C191—41no—
58C22⋊C4×C191—41no—
59Dic382—44no—
60C2×Dic191—22no—
61C2×Dic191—22no—
62C2×Dic191—22no—
63C2×Dic191—22no—
64C2×Dic191—22no—
65C2×Dic191—22no—
66C2×Dic191—22no—
67C2×Dic191—22no—
68C2×Dic191—22no—
69C2×Dic191C22×C414yes—
70C2×Dic191C22×C414yes—
71C2×Dic191#4114yes—
72C23×C191—22no—
73C23×C191—21no—
74C23×C191—21no—
75C23×C191—22no—
76C23×C191C22×C412yes—
77C23×C191C2411yescharacteristic
78C2×C4×C191—21no—
79C2×C4×C191—21no—
80C2×C4×C191—21no—
81C2×C4×C191—21no—
82C2×C4×C191—22no—
83C2×C4×C191—22no—
84C2×C4×C191—22no—
85C2×C4×C191—22no—
86C2×C4×C191—22no—
87C2×C4×C191—22no—
88C2×C4×C191—44no—
89C2×C4×C191C22×C412yes—
90C2×C4×C191C22×C412yes—
91C22×C191—22no—
92C22×C191—22no—
93C22×C191—22no—
94C22×C191—42no—
95C22×C191—42no—
96C22×C191C23×C411yescharacteristic
97C22×C191C2×#4112yes—
98C22×C191H(5,𝔽2)11yescharacteristic
99C22×C191H(5,𝔽2)11yescharacteristic
100C22×C1912−1+412yes—
101C4×C191—22no—
102C4×C191—22no—
103C4×C191—42no—
104C4×C191—42no—
105Dic191—24no—
106Dic191—24no—
107Dic191—24no—
108Dic191—42no—
109C2×C191—42no—
110C2×C191C4×#4111yescharacteristic
111C2×C191#51811yescharacteristic
112C2×C191#51911yescharacteristic
113C2×C191#53912yes—
114C2×C191#54212yes—
115C191#233211yescharacteristic, complemented, Sylow
116#2332—1911nomaximal, Sylow
117C2×C22⋊Q8—1911no—
118#391—1912no—
119#388—1912no—
120#388—1912no—
121#387—1912no—
122#387—1912no—
123#381D19111yescharacteristic
124C4×C4⋊C4—1912no—
125C4×C22⋊C4—1911no—
126C22×Q8—1911no—
127C23×C4C2×D19111yescharacteristic
128C22⋊Q8—1918no—
129C2×C4⋊C4—1912no—
130C2×C4⋊C4—1912no—
131C2×C4⋊C4—1911no—
132C2×C4⋊C4—1912no—
133C2×C22⋊C4—1912no—
134C2×C22⋊C4—1912no—
135C2×C22⋊C4C2×D19111yescharacteristic
136C2×C22⋊C4C2×D19111yescharacteristic
137C2×C42—1912no—
138C2×C42—1911no—
139C2×C42—1911no—
140C2×C42—1912no—
141#95—1912no—
142#95—1912no—
143#95—1912no—
144#95—1912no—
145#95—1912no—
146#95C2×D19112yes—
147#95C2×D19112yes—
148C24C22×D19111yescharacteristic
149C2×Q8—1914no—
150C2×Q8—7641no—
151C2×Q8—7641no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—1912no—
155C22×C4—1912no—
156C22×C4—1912no—
157C22×C4—1912no—
158C22×C4C4×D19114yes—
159C22×C4C22×D19112yes—
160C22×C4C22×D19112yes—
161C22×C4C22×D19111yescharacteristic
162C22×C4C22×D19111yescharacteristic
163C4⋊C4—3824no—
164C4⋊C4—1918no—
165C4⋊C4—1914no—
166C4⋊C4—7642no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—3824no—
170C22⋊C4—1918no—
171C42—3824no—
172C42—7641no—
173C42—7641no—
174C42—7642no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C2×C4×D19112yes—
180C23C23×D19111yescharacteristic
181Q8—7644no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—3822no—
194C2×C4—3822no—
195C2×C4—3822no—
196C2×C4—3822no—
197C2×C4—3822no—
198C2×C4—3822no—
199C2×C4—3822no—
200C2×C4—3822no—
201C2×C4—3822no—
202C2×C4—1914no—
203C2×C4—1914no—
204C2×C4—1914no—
205C2×C4C2×C4×D19112yes—
206C2×C4C2×C4×D19112yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C22×C4×D19111yescharacteristic
213C22#41×D19112yes—
214C22#2025936711yescharacteristic
215C22#2025936711yescharacteristic
216C22#2025936912yes—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—3824no—
222C4—3824no—
223C4—3824no—
224C4—7642no—
225C2—42no—
226C2#2713313111yescharacteristic
227C2#2713313611yescharacteristic
228C2#2713314011yescharacteristic
229C2#2713325612yes—
230C2#2713328112yes—
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#381×C191C2
socle, cosocleC23×C191C24
prime radicalC24×C191C23
hypercenter#381D191
hypocenterC191#2332
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #27132567C2semidirect
5. #27132546C2semidirect
6. #27132545C2semidirect
7. #27132514C2semidirect
11. #20259301C4semidirect
115. C191#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#202593012
#271325672
#271325572
#271325512
#271325462
#271325452
#271325142
C4×C22⋊Dic1912
#381×C1912
#2332191