Finite Group Databasebeta

← #37571224

Subgroups — #37571224

Order 24448.

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

normal subgroups only

This group has 242 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#27133214C211yesmaximal, characteristic
2#27132719C211yesmaximal, characteristic, complemented
3#27132715C211yesmaximal, characteristic, complemented
4#27132409C211yesmaximal, characteristic, complemented
5#27132406C211yesmaximal, characteristic, complemented
6#27132406C211yesmaximal, characteristic, complemented
7#332×C191C211yesmaximal, characteristic, complemented
8C2×#15690288—21no—
9#20259319C2211yescharacteristic
10#20259311—21no—
11#20259297—21no—
12#20259297C412yes—
13#20259284—21no—
14#20259284C412yes—
15C42×D191C2211yescharacteristic
16C4⋊(C191⋊C8)C2211yescharacteristic
17C4⋊(C191⋊C8)C2211yescharacteristic
18(C2×D191)⋊C8—21no—
19(C2×D191)⋊C8—21no—
20#20259232—21no—
21C8×Dic191—21no—
22C4⋊Q8×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25#15690288—42no—
26#15690288—42no—
27C2×D764—22no—
28C2×D764—21no—
29C2×D764—21no—
30C2×D764C2×C412yes—
31C2×C4×D191—22no—
32C2×C4×D191C2×C411yescharacteristic
33C2×C4×D191D411yescharacteristic
34C2×C4×D191D411yescharacteristic
35C2×C191⋊C8—21no—
36C2×C191⋊C8—21no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—22no—
39C4×Dic191C2×C411yescharacteristic
40C4×Dic191D411yescharacteristic
41C4×Dic191D411yescharacteristic
42C2×Q8×C191—21no—
43C2×C8×C191—21no—
44C2×C8×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191C2×C412yes—
47C42×C191C2311yescharacteristic
48C22×D191—22no—
49C22×D191C22⋊C411yescharacteristic
50D764—42no—
51D764—44no—
52D764—41no—
53D764—41no—
54D764—41no—
55D764—41no—
56C2×Dic191C22⋊C411yescharacteristic
57C2×Dic191C22⋊C411yescharacteristic
58C2×Dic191C22⋊C411yescharacteristic
59C4×D191—21no—
60C4×D191—21no—
61C4×D191—21no—
62C4×D191—21no—
63C4×D191—41no—
64C4×D191—44no—
65C191⋊C8—41no—
66C191⋊C8—41no—
67Q8×C191—41no—
68Q8×C191—41no—
69C2×C4×C191—22no—
70C2×C4×C191C22×C411yescharacteristic
71C2×C4×C191C2×D411yescharacteristic
72C2×C4×C191C2×D411yescharacteristic
73C8×C191—41no—
74C8×C191—41no—
75C22×C191C2×C22⋊C411yescharacteristic
76C2×D191—21no—
77C2×D191—21no—
78C2×D191—21no—
79C2×D191—42no—
80C2×D191—42no—
81C2×D191—42no—
82C4×C191—21no—
83C4×C191—42no—
84C4×C191#13511yescharacteristic
85C4×C191#13511yescharacteristic
86C4×C191AGL(1,ℤ/8)11yescharacteristic
87C4×C191AGL(1,ℤ/8)11yescharacteristic
88Dic191—21no—
89Dic191—21no—
90Dic191—21no—
91C2×C191#41311yescharacteristic
92C2×C191#41611yescharacteristic
93C2×C191#41811yescharacteristic
94D191—41no—
95D191—82no—
96C191#155311yescharacteristic, complemented, Sylow
97#1553—1911nomaximal, Sylow
98#532—1911no—
99#432—1911no—
100#431—1911no—
101#332—1911no—
102#332D19111yescharacteristic, complemented
103#331—1911no—
104#331—1911no—
105C2×#41—3821no—
106C4⋊Q8C2×D19111yescharacteristic
107#127—1911no—
108#124—3821no—
109#121—3821no—
110#121—1912no—
111C4×D4—3821no—
112C4×D4—1912no—
113C2×C42—1911no—
114C4⋊C8—1911no—
115C4⋊C8—1911no—
116C4⋊C8C2×D19111yescharacteristic
117C4⋊C8C2×D19111yescharacteristic
118C22⋊C8—3821no—
119C22⋊C8—3821no—
120#97—3821no—
121C4×C8—3821no—
122#41—7642no—
123#41—7642no—
124C2×Q8—21no—
125C2×D4—3822no—
126C2×D4—3821no—
127C2×D4—3821no—
128C2×D4—1912no—
129C22×C4—3822no—
130C22×C4—1911no—
131C22×C4—1911no—
132C22×C4—1911no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3821no—
136C2×C8—3821no—
137C4⋊C4—21no—
138C4⋊C4C4×D19112yes—
139C22⋊C4—3822no—
140C22⋊C4—3822no—
141C42—1911no—
142C42—1911no—
143C42—1911no—
144C42C22×D19111yescharacteristic
145C23—3822no—
146C23—1911no—
147Q8—41no—
148Q8—41no—
149D4—7642no—
150D4—7644no—
151D4—7641no—
152D4—7641no—
153D4—7641no—
154D4—7641no—
155C2×C4—22no—
156C2×C4—3821no—
157C2×C4—3821no—
158C2×C4—3821no—
159C2×C4—3821no—
160C2×C4—1911no—
161C2×C4—1911no—
162C2×C4—1911no—
163C2×C4—7641no—
164C2×C4—7644no—
165C2×C4C2×C4×D19111yescharacteristic
166C2×C4D4×D19111yescharacteristic
167C2×C4D4×D19111yescharacteristic
168C8—41no—
169C8—41no—
170C8—7641no—
171C8—7641no—
172C22—3821no—
173C22—3821no—
174C22—3821no—
175C22—7642no—
176C22—7642no—
177C22—7642no—
178C22C22⋊C4×D19111yescharacteristic
179C4—21no—
180C4—42no—
181C4—3821no—
182C4—3821no—
183C4—3821no—
184C4#2025934311yescharacteristic
185C4#2025934511yescharacteristic
186C4#2025934611yescharacteristic
187C4#2025934911yescharacteristic
188C2—15282no—
189C2—7641no—
190C2#2713264611yescharacteristic
191C2#2713265811yescharacteristic
192C2#2713266611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D191
commutatorC2×C4×C191C22×C4
FrattiniC42C22×D191
Fitting#332×C191C2
socleC22×C191C2×C22⋊C4
prime radical#27133214C2
cosocleC42×C191C23
hypercenter#332D191
hypocenterC191#1553
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27132719C2semidirect
3. #27132715C2semidirect
4. #27132409C2semidirect
5. #27132406C2semidirect
6. #27132406C2semidirect
7. #332×C191C2semidirect
96. C191#1553semidirect
102. #332D191semidirect

Maximal subgroups

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

subgroupindex
#271332142
#271327192
#271327152
#271324092
#271324062
#332×C1912
#1553191