Finite Group Databasebeta

← #37582811

Subgroups — #37582811

Order 24448.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#27132987C212yesmaximal, complemented
3#27132970C212yesmaximal, complemented
4#27132863C212yesmaximal, complemented
5#27132802C212yesmaximal, complemented
6#27132761C212yesmaximal, complemented
7C4×#15690274C212yesmaximal, complemented
8#27132724C211yesmaximal, characteristic, complemented
9#519×C191C211yesmaximal, characteristic, complemented
10C2×#15690274C2212yes—
11C2×#15690274C2212yes—
12#20259319C2211yescharacteristic
13#20259314C2211yescharacteristic
14#20259307—22no—
15#20259296C2212yes—
16#20259295—22no—
17C4×D764C2212yes—
18#20259255C2212yes—
19#20259253C2212yes—
20C4⋊(C191⋊C8)C2212yes—
21#20259246C2212yes—
22#20259246C2212yes—
23#20259242C2212yes—
24#20259242C2212yes—
25C22⋊(C191⋊C8)C2212yes—
26#20259233C2211yescharacteristic
27C4×C191⋊C8C2211yescharacteristic
28C22×Q8×C191C2211yescharacteristic
29C4×Q8×C191—22no—
30C4×Q8×C191C2212yes—
31C4×Q8×C191C2212yes—
32#117×C191—22no—
33#117×C191C2211yescharacteristic
34#117×C191C2211yescharacteristic
35C2×D764—22no—
36C2×D764C2312yes—
37C2×C22⋊D191—22no—
38C2×C4×D191—22no—
39#15690274—24no—
40#15690274—24no—
41C2×C191⋊C8C2312yes—
42C2×C191⋊C8C2312yes—
43C4⋊Dic191C2312yes—
44Dic191⋊C4—22no—
45(C2×D191)⋊C4—22no—
46(C2×D191)⋊C4—22no—
47C2×Q8×C191—24no—
48C2×Q8×C191—21no—
49C2×Q8×C191C2312yes—
50C22×C4×C191—21no—
51C22×C4×C191C2311yescharacteristic
52C4⋊C4×C191—22no—
53C4⋊C4×C191—22no—
54C4⋊C4×C191D412yes—
55C4⋊C4×C191C2312yes—
56C22⋊C4×C191—21no—
57C22⋊C4×C191D412yes—
58C42×C191—22no—
59C42×C191—22no—
60C42×C191—21no—
61C42×C191C2311yescharacteristic
62C42×C191C2311yescharacteristic
63C22×D191—22no—
64D764—22no—
65D764—22no—
66D764—82no—
67C22⋊D191—82no—
68C2×Dic191—22no—
69C4×D191—82no—
70C191⋊C8—22no—
71C191⋊C8—22no—
72C23×C191C2×D411yescharacteristic
73Q8×C191—24no—
74Q8×C191—22no—
75Q8×C191#4114yes—
76C2×C4×C191—24no—
77C2×C4×C191—22no—
78C2×C4×C191—22no—
79C2×C4×C191—22no—
80C2×C4×C191C2×D412yes—
81C2×C4×C191C2×D412yes—
82C2×C4×C191C2×D411yescharacteristic
83C2×C4×C191C2411yescharacteristic
84C22×C191—21no—
85C22×C191—22no—
86C22×C191C22×D411yescharacteristic
87C2×D191—42no—
88C2×D191—42no—
89C2×D191—42no—
90C4×C191—24no—
91C4×C191—21no—
92C4×C191—22no—
93C4×C191—22no—
94C4×C191—42no—
95C4×C191C2×#4112yes—
96Dic191—42no—
97C2×C191—41no—
98C2×C191#52511yescharacteristic
99C2×C191#57712yes—
100D191—82no—
101C191#312611yescharacteristic, complemented, Sylow
102#3126—1911nomaximal, Sylow
103#540—1911no—
104#519D19111yescharacteristic, complemented
105#482—1912no—
106#479—1912no—
107#460—1912no—
108#448—1912no—
109#440—1912no—
110C4×SD16—1912no—
111#433—1911no—
112C22×Q8C2×D19111yescharacteristic
113C2×SD16—1912no—
114C2×SD16—1912no—
115#127—1911no—
116#125—1911no—
117#123—3822no—
118#121—3822no—
119#121—1912no—
120C4×Q8—22no—
121C4×Q8C2×D19112yes—
122C4×Q8C2×D19112yes—
123C4×D4—1912no—
124#117—22no—
125#117C2×D19111yescharacteristic
126#117C2×D19111yescharacteristic
127#107—1912no—
128#106—1912no—
129C4⋊C8—1912no—
130#103—1912no—
131#103—1912no—
132#102—1912no—
133#102—1912no—
134C22⋊C8—1912no—
135#97—1911no—
136C4×C8—1911no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D19112yes—
140C2×D4—3822no—
141C2×D4—3822no—
142C2×D4—1912no—
143C22×C4—21no—
144C22×C4—3822no—
145C22×C4C22×D19111yescharacteristic
146SD16—3824no—
147SD16—3824no—
148C2×C8—1912no—
149C2×C8—1912no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3822no—
153C4⋊C4—1912no—
154C4⋊C4C22⋊D19112yes—
155C4⋊C4C22×D19112yes—
156C22⋊C4—21no—
157C22⋊C4—3822no—
158C22⋊C4—3822no—
159C22⋊C4C22⋊D19112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D19111yescharacteristic
164C42C22×D19111yescharacteristic
165C23—3822no—
166C23C2×C22⋊D19111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1569028714yes—
170D4—3822no—
171D4—3822no—
172D4—15282no—
173D4—15282no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3822no—
179C2×C4—15282no—
180C2×C4C2×C22⋊D19112yes—
181C2×C4C2×C22⋊D19112yes—
182C2×C4C2×C22⋊D19111yescharacteristic
183C2×C4C23×D19111yescharacteristic
184C8—3822no—
185C8—3822no—
186C22—22no—
187C22—21no—
188C22—7642no—
189C22—7642no—
190C22—7642no—
191C22C22×C22⋊D19111yescharacteristic
192C4—22no—
193C4—22no—
194C4—24no—
195C4—21no—
196C4—42no—
197C4—7642no—
198C4C2×#1569028712yes—
199C2—41no—
200C2—15282no—
201C2#2713317411yescharacteristic
202C2#2713347112yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#519×C191C2
socleC22×C191C22×D4
prime radical#27133268C2
hypercenter#519D191
hypocenterC191#3126
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #27132987C2semidirect
3. #27132970C2semidirect
4. #27132863C2semidirect
5. #27132802C2semidirect
6. #27132761C2semidirect
7. C4×#15690274C2semidirect
8. #27132724C2semidirect
9. #519×C191C2semidirect
101. C191#3126semidirect
104. #519D191semidirect

Maximal subgroups

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

subgroupindex
#271332682
#271329872
#271329702
#271328632
#271328022
#271327612
C4×#156902742
#271327242
#519×C1912
#3126191