Finite Group Databasebeta

← #37574193

Subgroups — #37574193

Order 24448.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#27133233C211yesmaximal, characteristic
2#27132635C212yesmaximal
3#27132458C211yesmaximal, characteristic, complemented
4#27132457C212yesmaximal, complemented
5#354×C191C211yesmaximal, characteristic, complemented
6C2×#15690286C2211yescharacteristic
7C2×Q8×D191C411yescharacteristic, complemented
8#20259322C411yescharacteristic, complemented
9#20259320C411yescharacteristic, complemented
10#20259309C2211yescharacteristic
11#20259304—22no—
12#20259299—22no—
13#20259279C411yescharacteristic, complemented
14#20259278—22no—
15#20259237C2212yes—
16#20259236—22no—
17#124×C191C2211yescharacteristic
18#99×C191C2212yes—
19#15690286—42no—
20C2×Dic382—21no—
21C2×Dic382C2×C411yescharacteristic
22C2×Dic382C2×C411yescharacteristic
23Q8×D191—42no—
24C2×C22⋊D191D412yes—
25C22×Dic191D412yes—
26C2×C4×D191—21no—
27C2×C4×D191C2×C411yescharacteristic
28C4⋊Dic191—22no—
29Dic191⋊C4—21no—
30Dic191⋊C4—22no—
31Dic191⋊C4—42no—
32C22⋊Dic191—22no—
33C22⋊Dic191—22no—
34(C2×D191)⋊C4—21no—
35C4×Dic191—42no—
36C4×Dic191C2×C411yescharacteristic
37C2×Q8×C191C2×C411yescharacteristic
38C2×D4×C191C2311yescharacteristic
39C22⋊C4×C191—22no—
40C22⋊C4×C191—22no—
41C42×C191C2×C411yescharacteristic
42C22×D191C22⋊C411yescharacteristic
43Dic382—41no—
44Dic382—42no—
45Dic382—41no—
46C22⋊D191—42no—
47C2×Dic191—22no—
48C2×Dic191—22no—
49C2×Dic191—21no—
50C2×Dic191—21no—
51C2×Dic191—42no—
52C2×Dic191C22⋊C412yes—
53C2×Dic191C22⋊C411yescharacteristic
54C4×D191—22no—
55C4×D191—42no—
56C23×C191C2×D412yes—
57Q8×C191—41no—
58D4×C191—41no—
59C2×C4×C191—21no—
60C2×C4×C191—21no—
61C2×C4×C191—42no—
62C2×C4×C191C22×C411yescharacteristic
63C22×C191—22no—
64C22×C191—42no—
65C22×C191C2×C22⋊C411yescharacteristic
66C2×D191—21no—
67C2×D191—41no—
68C4×C191—21no—
69C4×C191—41no—
70C4×C191—41no—
71C4×C191—82no—
72Dic191—22no—
73Dic191—41no—
74Dic191—41no—
75Dic191—82no—
76Dic191#9912yes—
77C2×C191—21no—
78C2×C191—42no—
79C2×C191C2×#9911yescharacteristic
80D191—41no—
81C191#212911yescharacteristic, complemented, Sylow
82#2129—1911nomaximal, Sylow
83#536—1911no—
84#410—1912no—
85#354—1911no—
86#354—1912no—
87#354D19111yescharacteristic, complemented
88C2×#41—1911no—
89C22×Q8—1911no—
90C4⋊Q8—1911no—
91C4⋊Q8—1911no—
92#124—1911no—
93#124C2×D19111yescharacteristic
94#123—3822no—
95C22⋊Q8—3822no—
96#117—3822no—
97#117—1911no—
98#99—3822no—
99#99—1912no—
100#99C2×D19112yes—
101#41—7642no—
102C2×Q8—3821no—
103C2×Q8—1911no—
104C2×Q8—1911no—
105C2×Q8—7642no—
106C2×Q8C4×D19111yescharacteristic
107C2×D4—1912no—
108C2×D4C22×D19111yescharacteristic
109C22×C4—3821no—
110C22×C4—1912no—
111C22×C4—1911no—
112C4⋊C4—3821no—
113C4⋊C4—3822no—
114C4⋊C4—3822no—
115C4⋊C4—7642no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—3821no—
119C22⋊C4—3822no—
120C22⋊C4—3822no—
121C42—1911no—
122C42—7642no—
123C42C4×D19111yescharacteristic
124C23—1911no—
125C23D4×D19112yes—
126Q8—41no—
127Q8—7641no—
128Q8—7641no—
129Q8—7642no—
130D4—41no—
131D4—7642no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—3822no—
136C2×C4—3822no—
137C2×C4—3822no—
138C2×C4—3821no—
139C2×C4—3821no—
140C2×C4—1912no—
141C2×C4—1911no—
142C2×C4—7642no—
143C2×C4—7642no—
144C2×C4C2×C4×D19111yescharacteristic
145C22—22no—
146C22—42no—
147C22—3821no—
148C22—7641no—
149C22C22⋊C4×D19111yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—3822no—
155C4—1912no—
156C4—15282no—
157C4—7641no—
158C4—7641no—
159C2—21no—
160C2—42no—
161C2—7641no—
162C2#99×D19111yescharacteristic
subgroupG/H
centerC2#99×D191
commutatorC2×C4×C191C22×C4
FrattiniC2×D4C22×D191
Fitting#354×C191C2
socleC2×C191C2×#99
prime radicalC2×#15690286C22
cosocleC2×D4×C191C23
hypercenter#354D191
hypocenterC191#2129
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #27132458C2semidirect
4. #27132457C2semidirect
5. #354×C191C2semidirect
7. C2×Q8×D191C4semidirect
8. #20259322C4semidirect
9. #20259320C4semidirect
13. #20259279C4semidirect
81. C191#2129semidirect
87. #354D191semidirect

Maximal subgroups

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

subgroupindex
#271332332
#271326352
#271324582
#271324572
#354×C1912
#2129191