Finite Group Databasebeta

← #37574473

Subgroups — #37574473

Order 24448.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133463C211yesmaximal, characteristic
2#27133106C212yesmaximal, complemented
3#27132921C211yesmaximal, characteristic, complemented
4#27132474C212yesmaximal, complemented
5#349×C191C211yesmaximal, characteristic, complemented
6C2×#15690287—21no—
7#20259355—22no—
8#20259350—22no—
9#20259344—21no—
10#20259344C2212yes—
11C2×Dic764C2211yescharacteristic, complemented
12#20259301—21no—
13Dic1528—22no—
14#20259267—22no—
15#20259254C2211yescharacteristic, complemented
16#20259247—21no—
17C2×M16×C191C2211yescharacteristic
18M32×C191C2212yescomplemented
19#15690287—22no—
20#15690287—22no—
21#15690287—41no—
22#15690287—41no—
23C2×Dic382—21no—
24C2×Dic382—21no—
25C2×D764—21no—
26C2×C22⋊D191—41no—
27C2×C4×D191—41no—
28Dic764—21no—
29Dic764—22no—
30Dic764D412yes—
31#15690276—22no—
32#15690276—22no—
33D1528—21no—
34D1528D412yes—
35C4⋊Dic191—21no—
36Dic191⋊C4—41no—
37C22⋊Dic191—41no—
38C22×C4×C191D411yescharacteristic
39M16×C191—22no—
40C2×C8×C191D411yescharacteristic
41C2×C8×C191C2311yescharacteristic
42C16×C191—22no—
43C22×D191—41no—
44Dic382—22no—
45Dic382—41no—
46Dic382—41no—
47Dic382—41no—
48D764—22no—
49D764—41no—
50C22⋊D191—42no—
51C22⋊D191—42no—
52C2×Dic191—41no—
53C2×Dic191—41no—
54C2×Dic191—41no—
55C4×D191—42no—
56C4×D191—42no—
57C23×C191D811yescharacteristic
58C2×C4×C191—21no—
59C2×C4×C191—21no—
60C2×C4×C191D811yescharacteristic
61C2×C4×C191C2×D411yescharacteristic
62C8×C191—21no—
63C8×C191C2×D412yes—
64C22×C191—21no—
65C22×C191—41no—
66C22×C191C2×D811yescharacteristic
67C2×D191—42no—
68C2×D191—82no—
69C4×C191—21no—
70C4×C191#12011yescharacteristic
71C4×C191AGL(1,ℤ/8)11yescharacteristic
72Dic191—42no—
73Dic191—81no—
74Dic191—81no—
75C2×C191—21no—
76C2×C191—41no—
77C2×C191#44711yescharacteristic
78D191—82no—
79C191#219111yescharacteristic, complemented, Sylow
80#2191—1911nomaximal, Sylow
81#575—1911no—
82#510—1912no—
83#470—1911no—
84#360—1912no—
85#349D19111yescharacteristic, complemented
86C2×#41—3821no—
87#137—3822no—
88AGL(1,ℤ/8)—3822no—
89#135—3821no—
90#135—1912no—
91C2×Q16—1911no—
92C2×M16C2×D19111yescharacteristic
93C22⋊Q8—3821no—
94Q32—3822no—
95SD32—3822no—
96M32C2×D19112yescomplemented
97#106—1911no—
98#103—3821no—
99#41—3822no—
100#41—3822no—
101#41—7641no—
102#41—7641no—
103C2×Q8—3821no—
104C2×Q8—3821no—
105C2×D4—3821no—
106C2×D4—7641no—
107C22×C4—7641no—
108C22×C4D76411yescharacteristic
109Q16—3821no—
110Q16—3822no—
111Q16—1912no—
112SD16—3822no—
113SD16—3822no—
114D8—3821no—
115D8—1912no—
116M16—22no—
117C2×C8D76411yescharacteristic
118C2×C8C22×D19111yescharacteristic
119C4⋊C4—3821no—
120C4⋊C4—7641no—
121C22⋊C4—7641no—
122C16—22no—
123C23—7641no—
124C23D152811yescharacteristic
125Q8—3822no—
126Q8—7641no—
127Q8—7641no—
128Q8—7641no—
129D4—3822no—
130D4—7641no—
131D4—7642no—
132D4—7642no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—7642no—
136C2×C4—7641no—
137C2×C4—7641no—
138C2×C4—7641no—
139C2×C4—7642no—
140C2×C4D152811yescharacteristic
141C2×C4C2×D76411yescharacteristic
142C8—21no—
143C8D4×D19112yes—
144C22—21no—
145C22—41no—
146C22—15282no—
147C22—7642no—
148C22C2×D152811yescharacteristic
149C4—21no—
150C4—15281no—
151C4—15281no—
152C4—7642no—
153C4#2025929211yescharacteristic
154C4#2025935011yescharacteristic
155C2—21no—
156C2—41no—
157C2—15282no—
158C2#2713279911yescharacteristic
subgroupG/H
centerC2#27132799
commutator, cosocleC2×C8×C191C23
FrattiniC2×C8C22×D191
Fitting#349×C191C2
socleC2×C191#447
prime radical#27133463C2
hypercenter#349D191
hypocenterC191#2191
LCSG ▷ C2×C8×C191 ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #27133106C2semidirect
3. #27132921C2semidirect
4. #27132474C2semidirect
5. #349×C191C2semidirect
11. C2×Dic764C22semidirect
15. #20259254C22semidirect
18. M32×C191C22semidirect
79. C191#2191semidirect
85. #349D191semidirect
96. M32C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334632
#271331062
#271329212
#271324742
#349×C1912
#2191191