Finite Group Databasebeta

← #37571222

Subgroups — #37571222

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#27133211C211yesmaximal, characteristic
2#27132719C211yesmaximal, characteristic, complemented
3#27132715C211yesmaximal, characteristic, complemented
4#27132410C211yesmaximal, characteristic, complemented
5#27132408C211yesmaximal, characteristic, complemented
6#27132407C211yesmaximal, characteristic, complemented
7#331×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9#20259322C2211yescharacteristic
10#20259309—21no—
11#20259293—21no—
12#20259293C412yes—
13D4×Dic191—21no—
14D4×Dic191C412yes—
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—
22#127×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25#15690286—42no—
26#15690286—42no—
27C2×Dic382—21no—
28C2×C22⋊D191—22no—
29C22×Dic191—22no—
30C2×C4×D191C2×C411yescharacteristic
31C2×C4×D191D411yescharacteristic
32C2×C4×D191D411yescharacteristic
33C2×C191⋊C8—21no—
34C2×C191⋊C8—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191C2×C412yes—
37C22⋊Dic191—22no—
38C22⋊Dic191—22no—
39C4×Dic191C2×C411yescharacteristic
40C4×Dic191D411yescharacteristic
41C4×Dic191D411yescharacteristic
42C2×D4×C191—21no—
43C2×D4×C191—21no—
44C2×D4×C191C2×C412yes—
45C2×C8×C191—21no—
46C2×C8×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191C22⋊C411yescharacteristic
49Dic382—41no—
50Dic382—41no—
51C22⋊D191—44no—
52C2×Dic191—22no—
53C2×Dic191—44no—
54C2×Dic191C22⋊C411yescharacteristic
55C2×Dic191C22⋊C411yescharacteristic
56C2×Dic191C22⋊C411yescharacteristic
57C4×D191—21no—
58C4×D191—21no—
59C4×D191—21no—
60C4×D191—21no—
61C4×D191—41no—
62C191⋊C8—41no—
63C191⋊C8—41no—
64C23×C191—22no—
65D4×C191—42no—
66D4×C191—41no—
67D4×C191—41no—
68D4×C191—41no—
69D4×C191—41no—
70C2×C4×C191C22×C411yescharacteristic
71C2×C4×C191C2×D411yescharacteristic
72C2×C4×C191C2×D411yescharacteristic
73C8×C191—41no—
74C8×C191—41no—
75C22×C191—42no—
76C22×C191—42no—
77C22×C191—42no—
78C22×C191C2×C22⋊C411yescharacteristic
79C2×D191—21no—
80C2×D191—21no—
81C2×D191—21no—
82C4×C191—21no—
83C4×C191#13511yescharacteristic
84C4×C191#13511yescharacteristic
85C4×C191AGL(1,ℤ/8)11yescharacteristic
86C4×C191AGL(1,ℤ/8)11yescharacteristic
87Dic191—21no—
88Dic191—21no—
89Dic191—21no—
90Dic191—42no—
91C2×C191—82no—
92C2×C191#41311yescharacteristic
93C2×C191#41611yescharacteristic
94C2×C191#41811yescharacteristic
95D191—41no—
96C191#155311yescharacteristic, complemented, Sylow
97#1553—1911nomaximal, Sylow
98#532—1911no—
99#432—1911no—
100#431—1911no—
101#332—1911no—
102#332—1911no—
103#331—1911no—
104#331D19111yescharacteristic, complemented
105C2×#41—3821no—
106C4⋊Q8—1911no—
107#127C2×D19111yescharacteristic
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—3821no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—3822no—
128C2×D4C4×D19112yes—
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—3821no—
138C4⋊C4—1912no—
139C22⋊C4—3822no—
140C22⋊C4—3822no—
141C42—1911no—
142C42—1911no—
143C42—1911no—
144C42C22×D19111yescharacteristic
145C23—22no—
146C23—1911no—
147Q8—7641no—
148Q8—7641no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—7644no—
155C2×C4—3821no—
156C2×C4—3821no—
157C2×C4—3821no—
158C2×C4—3821no—
159C2×C4—3822no—
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—42no—
173C22—42no—
174C22—42no—
175C22—3821no—
176C22—3821no—
177C22—3821no—
178C22C22⋊C4×D19111yescharacteristic
179C4—21no—
180C4—3821no—
181C4—3821no—
182C4—3821no—
183C4—7642no—
184C4#2025934111yescharacteristic
185C4#2025934311yescharacteristic
186C4#2025934611yescharacteristic
187C4#2025934911yescharacteristic
188C2—82no—
189C2—7641no—
190C2#2713264411yescharacteristic
191C2#2713265611yescharacteristic
192C2#2713266611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D191
commutatorC2×C4×C191C22×C4
FrattiniC42C22×D191
Fitting#331×C191C2
socleC22×C191C2×C22⋊C4
prime radical#27133211C2
cosocleC42×C191C23
hypercenter#331D191
hypocenterC191#1553
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27132719C2semidirect
3. #27132715C2semidirect
4. #27132410C2semidirect
5. #27132408C2semidirect
6. #27132407C2semidirect
7. #331×C191C2semidirect
96. C191#1553semidirect
104. #331D191semidirect

Maximal subgroups

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

subgroupindex
#271332112
#271327192
#271327152
#271324102
#271324082
#271324072
#331×C1912
#1553191