Finite Group Databasebeta

← #37571153

Subgroups — #37571153

Order 24448.

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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#27133212C211yesmaximal, characteristic
2#27132718C212yesmaximal, complemented
3#27132403C211yesmaximal, characteristic, complemented
4#27132402C212yesmaximal, complemented
5#329×C191C211yesmaximal, characteristic, complemented
6C2×#15690288C411yescharacteristic
7C2×#15690286C411yescharacteristic
8#20259320C411yescharacteristic
9#20259318C411yescharacteristic
10#20259310C2211yescharacteristic
11#20259295—22no—
12(C22⋊D191)⋊C4—22no—
13C42×D191C2211yescharacteristic
14Dic191⋊C8—22no—
15(C2×D191)⋊C8—22no—
16#20259233C2212yes—
17#124×C191C2211yescharacteristic
18#97×C191C2212yes—
19#15690288—42no—
20#15690286—42no—
21C2×Dic382C2×C411yescharacteristic
22C2×D764—21no—
23C2×D764C2×C411yescharacteristic
24C2×C22⋊D191—21no—
25C2×C22⋊D191—22no—
26C22×Dic191—21no—
27C2×C4×D191—21no—
28C2×C4×D191C2×C411yescharacteristic
29C2×C4×D191D412yes—
30C2×C191⋊C8—22no—
31Dic191⋊C4—22no—
32(C2×D191)⋊C4—22no—
33C4×Dic191C2×C411yescharacteristic
34C4×Dic191D412yes—
35C2×Q8×C191C2×C411yescharacteristic
36C2×D4×C191C2×C411yescharacteristic
37C2×C8×C191—22no—
38C22⋊C4×C191—22no—
39C42×C191C2311yescharacteristic
40C22×D191—21no—
41C22×D191C22⋊C411yescharacteristic
42Dic382—41no—
43D764—41no—
44D764—42no—
45C22⋊D191—44no—
46C22⋊D191—42no—
47C2×Dic191—21no—
48C2×Dic191—42no—
49C2×Dic191C22⋊C412yes—
50C2×Dic191C22⋊C411yescharacteristic
51C4×D191—22no—
52C4×D191—42no—
53C4×D191—42no—
54C191⋊C8—42no—
55C23×C191—21no—
56Q8×C191—41no—
57D4×C191—41no—
58C2×C4×C191—21no—
59C2×C4×C191C22×C411yescharacteristic
60C2×C4×C191C2×D412yes—
61C8×C191—42no—
62C22×C191—41no—
63C22×C191—42no—
64C22×C191C2×C22⋊C411yescharacteristic
65C2×D191—21no—
66C2×D191—22no—
67C2×D191—41no—
68C2×D191—42no—
69C4×C191—21no—
70C4×C191—22no—
71C4×C191—41no—
72Dic191—22no—
73Dic191—41no—
74Dic191#10012yes—
75C2×C191—81no—
76C2×C191C2×#10011yescharacteristic
77C2×C191#42112yes—
78D191—41no—
79D191—81no—
80C191#153611yescharacteristic, complemented, Sylow
81#1536—1911nomaximal, Sylow
82#532—1911no—
83#432—1912no—
84#329—1911no—
85#329—1912no—
86#329D19111yescharacteristic, complemented
87C2×#41—1911no—
88C2×#41—1911no—
89C4⋊Q8—1911no—
90#127—1911no—
91#124—1911no—
92#124C2×D19111yescharacteristic
93#121—3822no—
94C4×D4—3822no—
95C2×C42—1911no—
96C4⋊C8—3822no—
97C22⋊C8—3822no—
98#97—1912no—
99#97C2×D19112yes—
100#41—7642no—
101#41—7642no—
102C2×Q8—1911no—
103C2×Q8C4×D19111yescharacteristic
104C2×D4—3821no—
105C2×D4—3821no—
106C2×D4—3822no—
107C2×D4—1911no—
108C2×D4C4×D19111yescharacteristic
109C22×C4—3821no—
110C22×C4—3821no—
111C22×C4—1911no—
112C22×C4—1912no—
113C2×C8—22no—
114C2×C8—3822no—
115C4⋊C4—3822no—
116C22⋊C4—22no—
117C22⋊C4—3822no—
118C42—1911no—
119C42—1912no—
120C42C22×D19111yescharacteristic
121C23—21no—
122C23—3821no—
123C23—1911no—
124Q8—41no—
125Q8—7641no—
126D4—41no—
127D4—7641no—
128D4—7642no—
129D4—7644no—
130D4—7642no—
131C2×C4—21no—
132C2×C4—3822no—
133C2×C4—3821no—
134C2×C4—1912no—
135C2×C4—1911no—
136C2×C4—7642no—
137C2×C4—7642no—
138C2×C4—7642no—
139C2×C4C2×C4×D19111yescharacteristic
140C2×C4D4×D19112yes—
141C8—42no—
142C8—7642no—
143C22—42no—
144C22—41no—
145C22—3822no—
146C22—3821no—
147C22—7642no—
148C22—7641no—
149C22C22⋊C4×D19111yescharacteristic
150C4—22no—
151C4—21no—
152C4—41no—
153C4—3822no—
154C4—1912no—
155C4—7641no—
156C2—81no—
157C2—15281no—
158C2—7641no—
159C2#100×D19111yescharacteristic
160C2#2713267812yes—
subgroupG/H
centerC22C22⋊C4×D191
commutatorC2×C4×C191C22×C4
FrattiniC42C22×D191
Fitting#329×C191C2
socleC22×C191C2×C22⋊C4
prime radical#27133212C2
cosocleC42×C191C23
hypercenter#329D191
hypocenterC191#1536
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27132718C2semidirect
3. #27132403C2semidirect
4. #27132402C2semidirect
5. #329×C191C2semidirect
80. C191#1536semidirect
86. #329D191semidirect

Maximal subgroups

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

subgroupindex
#271332122
#271327182
#271324032
#271324022
#329×C1912
#1536191