Finite Group Databasebeta

← #37583415

Subgroups — #37583415

Order 24448.

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

normal subgroups only

This group has 378 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#27133234C212yesmaximal
2#27133005C214yesmaximal, complemented
3#27132921C212yesmaximal, complemented
4#27132904C212yesmaximal, complemented
5#27132821C214yesmaximal, complemented
6#408×C191C211yesmaximal, characteristic
7C22×Dic382C2212yes—
8C2×Dic764C2212yes—
9C2×Dic764C2212yes—
10#20259320—22no—
11#20259309—22no—
12#20259303—22no—
13#20259303—22no—
14#20259301C2214yes—
15#20259301C2214yes—
16#20259281C2212yes—
17#20259256C2212yes—
18#20259254C2212yes—
19#20259247C2214yes—
20#20259247C2214yes—
21C2×#41×C191C2211yescharacteristic
22C2×M16×C191C2211yescharacteristic
23C22×C8×C191C2211yescharacteristic
24C22⋊C8×C191C2214yes—
25C2×Dic382—22no—
26C2×Dic382—42no—
27C2×Dic382—42no—
28C2×Dic382C2314yes—
29C22×Dic191—22no—
30Dic764—42no—
31Dic764—42no—
32C4⋊Dic191C2314yes—
33Dic191⋊C4—24no—
34Dic191⋊C4—24no—
35C22⋊Dic191—22no—
36C22⋊Dic191—24no—
37C22⋊Dic191—24no—
38C4×Dic191—22no—
39#41×C191—41no—
40#41×C191—41no—
41C2×Q8×C191D411yescharacteristic
42C2×D4×C191D411yescharacteristic
43C2×D4×C191D412yes—
44C22×C4×C191C2312yes—
45C22×C4×C191C2311yescharacteristic
46M16×C191—41no—
47C2×C8×C191—41no—
48C2×C8×C191C2312yes—
49C2×C8×C191C2312yes—
50Dic382—24no—
51Dic382—24no—
52Dic382—44no—
53C2×Dic191—24no—
54C2×Dic191—24no—
55C2×Dic191—44no—
56C23×C191C2×D411yescharacteristic
57C23×C191C2×D412yes—
58Q8×C191—41no—
59D4×C191—41no—
60D4×C191—42no—
61C2×C4×C191—21no—
62C2×C4×C191—21no—
63C2×C4×C191—22no—
64C2×C4×C191—22no—
65C2×C4×C191C2×D412yes—
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2411yescharacteristic
68C8×C191—22no—
69C8×C191—22no—
70C22×C191—21no—
71C22×C191—22no—
72C22×C191—22no—
73C22×C191—24no—
74C22×C191C22×D411yescharacteristic
75C4×C191—21no—
76C4×C191—22no—
77C4×C191H(5,𝔽2)11yescharacteristic
78C4×C191H(5,𝔽2)11yescharacteristic
79Dic191—44no—
80Dic191—44no—
81C2×C191—41no—
82C2×C191—42no—
83C2×C191#53411yescharacteristic
84C2×C191#57812yes—
85C191#320211yescharacteristic, complemented, Sylow
86#3202—1911nomaximal, Sylow
87#536—1912no—
88#485—1914no—
89#470—1912no—
90#467—1912no—
91#451—1914no—
92#408D19111yescharacteristic
93C2×#41C2×D19111yescharacteristic
94C22×Q8—1912no—
95C2×Q16—1912no—
96C2×Q16—1912no—
97C2×M16C2×D19111yescharacteristic
98C22×C8C2×D19111yescharacteristic
99C4⋊Q8—3822no—
100#124—3822no—
101#123—3822no—
102#123—3822no—
103C22⋊Q8—1914no—
104C22⋊Q8—1914no—
105#117—1912no—
106#107—1912no—
107#106—1912no—
108#103—1914no—
109#103—1914no—
110C22⋊C8C2×D19114yes—
111#41—41no—
112#41—41no—
113C2×Q8—3822no—
114C2×Q8—1914no—
115C2×Q8—7642no—
116C2×Q8—7642no—
117C2×Q8D76411yescharacteristic
118C2×D4D76411yescharacteristic
119C2×D4D76412yes—
120C22×C4—3822no—
121C22×C4C22×D19112yes—
122C22×C4C22×D19111yescharacteristic
123Q16—7642no—
124Q16—7642no—
125M16—41no—
126C2×C8—41no—
127C2×C8C22×D19112yes—
128C2×C8C22×D19112yes—
129C4⋊C4—3824no—
130C4⋊C4—3824no—
131C4⋊C4—1914no—
132C22⋊C4—3822no—
133C22⋊C4—3824no—
134C22⋊C4—3824no—
135C42—3822no—
136C23C2×D76411yescharacteristic
137C23C2×D76412yes—
138Q8—41no—
139Q8—3824no—
140Q8—3824no—
141Q8—7644no—
142D4—42no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—3824no—
149C2×C4—3824no—
150C2×C4—7644no—
151C2×C4C2×D76412yes—
152C2×C4C2×D76411yescharacteristic
153C2×C4C23×D19111yescharacteristic
154C8—22no—
155C8—22no—
156C22—22no—
157C22—24no—
158C22—22no—
159C22—21no—
160C22C22×D76411yescharacteristic
161C4—22no—
162C4—21no—
163C4—7644no—
164C4—7644no—
165C4#2025936711yescharacteristic
166C4#2025936711yescharacteristic
167C2—42no—
168C2—41no—
169C2#2713322511yescharacteristic
170C2#2713348012yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#408×C191C2
socleC22×C191C22×D4
prime radicalC2×#41×C191C22
hypercenter#408D191
hypocenterC191#3202
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #27133005C2semidirect
3. #27132921C2semidirect
4. #27132904C2semidirect
5. #27132821C2semidirect
85. C191#3202semidirect

Maximal subgroups

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

subgroupindex
#271332342
#271330052
#271329212
#271329042
#271328212
#408×C1912
#3202191