Finite Group Databasebeta

← #37583961

Subgroups — #37583961

Order 24448.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#27133268C211yesmaximal, characteristic
2#27132970C212yesmaximal, complemented
3#27132960C212yesmaximal, complemented
4#27132911C212yesmaximal, complemented
5#27132894C212yesmaximal, complemented
6#27132865C212yesmaximal, complemented
7#27132863C212yesmaximal, complemented
8#27132690C211yesmaximal, characteristic, complemented
9#541×C191C211yesmaximal, characteristic, complemented
10C2×#15690274C2212yes—
11C2×#15690274C2212yes—
12C2×#15690271C2211yescharacteristic
13C22×C191⋊C8C2211yescharacteristic
14#20259319C2211yescharacteristic
15#20259314C2211yescharacteristic
16#20259307—22no—
17#20259296C2212yes—
18#20259295—22no—
19C4×D764C2212yes—
20#20259255C2212yes—
21#20259253C2212yes—
22C4⋊(C191⋊C8)C2212yes—
23C4⋊(C191⋊C8)C2212yes—
24#20259246C2212yes—
25#20259246C2212yes—
26#20259242C2212yes—
27#20259242C2212yes—
28C4⋊Q8×C191C2211yescharacteristic
29#126×C191—22no—
30#125×C191—22no—
31#125×C191C2211yescharacteristic
32C22⋊Q8×C191C2212yes—
33C4×Q8×C191C2212yes—
34#117×C191C2211yescharacteristic
35C2×D764—22no—
36C2×D764C2312yes—
37C2×C22⋊D191—22no—
38C2×C4×D191—22no—
39#15690274—42no—
40#15690274—42no—
41#15690271—41no—
42C2×C191⋊C8—41no—
43C2×C191⋊C8C2312yes—
44C2×C191⋊C8C2312yes—
45C4⋊Dic191C2312yes—
46Dic191⋊C4—22no—
47(C2×D191)⋊C4—22no—
48(C2×D191)⋊C4—22no—
49C2×Q8×C191C2312yes—
50C22×C4×C191C2311yescharacteristic
51C4⋊C4×C191—22no—
52C4⋊C4×C191—22no—
53C4⋊C4×C191—22no—
54C4⋊C4×C191—22no—
55C4⋊C4×C191D412yes—
56C4⋊C4×C191C2312yes—
57C22⋊C4×C191—22no—
58C22⋊C4×C191D412yes—
59C42×C191—22no—
60C42×C191C2311yescharacteristic
61C42×C191C2311yescharacteristic
62C22×D191—22no—
63D764—22no—
64D764—22no—
65D764—82no—
66C22⋊D191—82no—
67C2×Dic191—22no—
68C4×D191—82no—
69C191⋊C8—22no—
70C191⋊C8—22no—
71C23×C191C2×D411yescharacteristic
72Q8×C191—22no—
73Q8×C191—22no—
74C2×C4×C191—21no—
75C2×C4×C191—21no—
76C2×C4×C191—22no—
77C2×C4×C191—22no—
78C2×C4×C191C2×D412yes—
79C2×C4×C191C2×D412yes—
80C2×C4×C191C2×D411yescharacteristic
81C2×C4×C191C2411yescharacteristic
82C22×C191—21no—
83C22×C191—22no—
84C22×C191C22×D411yescharacteristic
85C2×D191—42no—
86C2×D191—42no—
87C2×D191—42no—
88C4×C191—21no—
89C4×C191—22no—
90C4×C191—22no—
91C4×C191—42no—
92C4×C191—42no—
93C4×C191H(5,𝔽2)11yescharacteristic
94C4×C1912−1+411yescharacteristic
95Dic191—42no—
96C2×C191—41no—
97C2×C191#53711yescharacteristic
98C2×C191#57712yes—
99D191—82no—
100C191#326011yescharacteristic, complemented, Sylow
101#3260—1911nomaximal, Sylow
102#541D19111yescharacteristic, complemented
103#540—1911no—
104#479—1912no—
105#478—1912no—
106#468—1912no—
107#465—1912no—
108#461—1912no—
109#460—1912no—
110#424—1911no—
111C2×SD16—1912no—
112C2×SD16—1912no—
113C2×M16—1911no—
114C22×C8—1911no—
115C4⋊Q8C2×D19111yescharacteristic
116#127—1911no—
117#126—22no—
118#125—22no—
119#125—1911no—
120#125C2×D19111yescharacteristic
121#123—3822no—
122C22⋊Q8C2×D19112yes—
123#121—3822no—
124#121—1912no—
125C4×Q8C2×D19112yes—
126C4×D4—1912no—
127#117C2×D19111yescharacteristic
128#107—1912no—
129#106—1912no—
130C4⋊C8—1912no—
131C4⋊C8—1912no—
132#103—1912no—
133#103—1912no—
134#102—1912no—
135#102—1912no—
136C2×Q8C22×D19112yes—
137C2×D4—3822no—
138C2×D4—3822no—
139C2×D4—1912no—
140C22×C4—3822no—
141C22×C4C22×D19111yescharacteristic
142SD16—7642no—
143SD16—7642no—
144M16—7641no—
145C2×C8—1912no—
146C2×C8—1912no—
147C2×C8—7641no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3822no—
153C4⋊C4—1912no—
154C4⋊C4C22⋊D19112yes—
155C4⋊C4C22×D19112yes—
156C22⋊C4—22no—
157C22⋊C4—3822no—
158C22⋊C4—3822no—
159C22⋊C4C22⋊D19112yes—
160C42—22no—
161C42C22×D19111yescharacteristic
162C42C22×D19111yescharacteristic
163C23—3822no—
164C23C2×C22⋊D19111yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—3822no—
168D4—3822no—
169D4—15282no—
170D4—15282no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—3822no—
176C2×C4—15282no—
177C2×C4C2×C22⋊D19112yes—
178C2×C4C2×C22⋊D19112yes—
179C2×C4C2×C22⋊D19111yescharacteristic
180C2×C4C23×D19111yescharacteristic
181C8—3822no—
182C8—3822no—
183C22—22no—
184C22—21no—
185C22—7642no—
186C22—7642no—
187C22—7642no—
188C22C22×C22⋊D19111yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—7642no—
195C4#2025936811yescharacteristic
196C4#2025936911yescharacteristic
197C2—41no—
198C2—15282no—
199C2#2713324111yescharacteristic
200C2#2713347112yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#541×C191C2
socleC22×C191C22×D4
prime radical#27133268C2
hypercenter#541D191
hypocenterC191#3260
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #27132970C2semidirect
3. #27132960C2semidirect
4. #27132911C2semidirect
5. #27132894C2semidirect
6. #27132865C2semidirect
7. #27132863C2semidirect
8. #27132690C2semidirect
9. #541×C191C2semidirect
100. C191#3260semidirect
102. #541D191semidirect

Maximal subgroups

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

subgroupindex
#271332682
#271329702
#271329602
#271329112
#271328942
#271328652
#271328632
#271326902
#541×C1912
#3260191