Finite Group Databasebeta

← #37574175

Subgroups — #37574175

Order 24448.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#27133234C211yesmaximal, characteristic
2#27132637C211yesmaximal, characteristic
3#27132454C214yesmaximal, complemented
4#413×C191C211yesmaximal, characteristic
5C22×Dic382C411yescharacteristic, complemented
6#20259320—21no—
7#20259309—21no—
8#20259303C412yescomplemented
9#20259303C2212yes—
10#20259301—21no—
11#20259301—21no—
12#20259281—21no—
13#20259281C411yescharacteristic, complemented
14#20259237—21no—
15#20259237C2212yes—
16C2×#41×C191C2211yescharacteristic
17C2×M16×C191—21no—
18#101×C191—21no—
19#100×C191C2212yes—
20C2×Dic382—21no—
21C2×Dic382—21no—
22C2×Dic382—41no—
23C2×Dic382—41no—
24C22×Dic191C2×C412yes—
25C4⋊Dic191—21no—
26C4⋊Dic191—41no—
27Dic191⋊C4—22no—
28Dic191⋊C4—22no—
29C22⋊Dic191—22no—
30C22⋊Dic191—22no—
31C22⋊Dic191—22no—
32C22⋊Dic191C2×C412yes—
33C4×Dic191—21no—
34C4×Dic191—41no—
35#41×C191—41no—
36#41×C191—41no—
37C2×Q8×C191D411yescharacteristic
38C2×D4×C191C2×C411yescharacteristic
39C2×D4×C191D411yescharacteristic
40C2×D4×C191C2311yescharacteristic
41C22×C4×C191C2×C411yescharacteristic
42C22×C4×C191D411yescharacteristic
43C22×C4×C191D411yescharacteristic
44M16×C191—22no—
45M16×C191—41no—
46C2×C8×C191—41no—
47Dic382—42no—
48Dic382—41no—
49Dic382—41no—
50C2×Dic191—22no—
51C2×Dic191—22no—
52C2×Dic191—42no—
53C2×Dic191—42no—
54C23×C191C22⋊C411yescharacteristic
55C23×C191C22×C411yescharacteristic
56C23×C191C2×D411yescharacteristic
57Q8×C191—41no—
58D4×C191—41no—
59D4×C191—41no—
60D4×C191—41no—
61C2×C4×C191—21no—
62C2×C4×C191—21no—
63C2×C4×C191—21no—
64C2×C4×C191—21no—
65C2×C4×C191—21no—
66C2×C4×C191—21no—
67C2×C4×C191C22⋊C411yescharacteristic
68C2×C4×C191C22⋊C411yescharacteristic
69C2×C4×C191C22⋊C411yescharacteristic
70C2×C4×C191C2×D411yescharacteristic
71C8×C191—42no—
72C22×C191—21no—
73C22×C191—21no—
74C22×C191—21no—
75C22×C191—41no—
76C22×C191—41no—
77C22×C191—41no—
78C22×C191C2×C22⋊C411yescharacteristic
79C4×C191—21no—
80C4×C191—21no—
81C4×C191—21no—
82C4×C191#9911yescharacteristic
83C4×C191#9911yescharacteristic
84Dic191—42no—
85Dic191—42no—
86Dic191—82no—
87C2×C191—21no—
88C2×C191—41no—
89C2×C191—41no—
90C2×C191—41no—
91C2×C191C2×#9911yescharacteristic
92C191#212311yescharacteristic, complemented, Sylow
93#2123—1911nomaximal, Sylow
94#536—1911no—
95#413D19111yescharacteristic
96#410—1911no—
97#352—1914no—
98C2×#41C2×D19111yescharacteristic
99C22×Q8—1911no—
100C2×M16—21no—
101C4⋊Q8—3821no—
102#124—3821no—
103#123—1912no—
104#123—1912no—
105C22⋊Q8—3821no—
106C22⋊Q8—3821no—
107#117—3821no—
108#117—1911no—
109#101—21no—
110#100C2×D19112yes—
111#99—3821no—
112#99—1912no—
113#41—41no—
114#41—41no—
115C2×Q8—3821no—
116C2×Q8—3821no—
117C2×Q8—7641no—
118C2×Q8—7641no—
119C2×Q8C22⋊D19111yescharacteristic
120C2×D4C4×D19111yescharacteristic
121C2×D4D76411yescharacteristic
122C2×D4C22×D19111yescharacteristic
123C22×C4—1912no—
124C22×C4C4×D19111yescharacteristic
125C22×C4C22⋊D19111yescharacteristic
126C22×C4D76411yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—3822no—
131C4⋊C4—3821no—
132C4⋊C4—3822no—
133C4⋊C4—7641no—
134C22⋊C4—3822no—
135C22⋊C4—3822no—
136C22⋊C4—3822no—
137C22⋊C4—1912no—
138C42—3821no—
139C42—7641no—
140C23(C2×D191)⋊C411yescharacteristic
141C23C2×C4×D19111yescharacteristic
142C23C2×D76411yescharacteristic
143Q8—41no—
144Q8—7642no—
145Q8—7641no—
146Q8—7641no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—3822no—
157C2×C4—3822no—
158C2×C4—7642no—
159C2×C4—7642no—
160C2×C4(C2×D191)⋊C411yescharacteristic
161C2×C4(C2×D191)⋊C411yescharacteristic
162C2×C4(C2×D191)⋊C411yescharacteristic
163C2×C4C2×C22⋊D19111yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D191)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—15282no—
176C4—7642no—
177C4—7642no—
178C4#2025923611yescharacteristic
179C4#2025923611yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2025923611yescharacteristic
subgroupG/H
centerC2C2×#20259236
commutatorC23×C191C22×C4
FrattiniC2×D4C22×D191
Fitting#413×C191C2
socleC2×C191C2×#99
prime radicalC2×#41×C191C22
cosocleC2×D4×C191C23
hypercenter#413D191
hypocenterC191#2123
LCSG ▷ C23×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #27132454C2semidirect
5. C22×Dic382C4semidirect
8. #20259303C4semidirect
13. #20259281C4semidirect
92. C191#2123semidirect

Maximal subgroups

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

subgroupindex
#271332342
#271326372
#271324542
#413×C1912
#2123191