Finite Group Databasebeta

← #37571505

Subgroups — #37571505

Order 24448.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#27133225C211yesmaximal, characteristic
2#27132976C212yesmaximal, complemented
3#27132396C212yesmaximal, complemented
4#27132387C211yesmaximal, characteristic, complemented
5#397×C191C211yesmaximal, characteristic, complemented
6C22×C22⋊D191—21no—
7#20259306—22no—
8#20259296—21no—
9#20259296C2212yes—
10#20259292—22no—
11#20259255—22no—
12#20259242—22no—
13C22⋊(C191⋊C8)C2212yescomplemented
14C2×C4⋊C4×C191C2211yescharacteristic, complemented
15C2×C22⋊C4×C191—21no—
16C2×C22⋊C4×C191C2211yescharacteristic
17#95×C191—21no—
18#95×C191C2211yescharacteristic, complemented
19C23×D191—21no—
20C2×D764—21no—
21C2×D764D412yes—
22C2×C22⋊D191—22no—
23C2×C22⋊D191—22no—
24C2×C22⋊D191—42no—
25C2×C22⋊D191—42no—
26C22×Dic191—21no—
27C2×C191⋊C8—22no—
28C4⋊Dic191—21no—
29C4⋊Dic191D412yes—
30(C2×D191)⋊C4—22no—
31(C2×D191)⋊C4—22no—
32C24×C191D411yescharacteristic
33C22×C4×C191—21no—
34C22×C4×C191—21no—
35C22×C4×C191D411yescharacteristic
36C22×C4×C191C2311yescharacteristic
37C4⋊C4×C191—22no—
38C22⋊C4×C191—22no—
39C22⋊C4×C191—42no—
40C22×D191—22no—
41C22×D191—42no—
42C22×D191—81no—
43C22×D191—81no—
44D764—42no—
45D764—81no—
46C22⋊D191—44no—
47C22⋊D191—42no—
48C22⋊D191—42no—
49C2×Dic191—22no—
50C2×Dic191—42no—
51C191⋊C8—42no—
52C23×C191—21no—
53C23×C191—21no—
54C23×C191—41no—
55C23×C191—41no—
56C23×C191D812yes—
57C23×C191C2×D411yescharacteristic
58C2×C4×C191—21no—
59C2×C4×C191—41no—
60C2×C4×C191—41no—
61C2×C4×C191—41no—
62C2×C4×C191—41no—
63C2×C4×C191—41no—
64C2×C4×C191—41no—
65C2×C4×C191—41no—
66C2×C4×C191—41no—
67C2×C4×C191C2×D412yes—
68C22×C191—21no—
69C22×C191—21no—
70C22×C191—22no—
71C22×C191—42no—
72C22×C191—41no—
73C22×C191—41no—
74C22×C191—81no—
75C22×C191#12011yescharacteristic
76C22×C191C2×D811yescharacteristic
77C22×C191AGL(1,ℤ/8)11yescharacteristic
78C2×D191—42no—
79C2×D191—42no—
80C2×D191—42no—
81C2×D191—84no—
82C4×C191—22no—
83C4×C191—41no—
84C4×C191—81no—
85C4×C191—81no—
86Dic191—42no—
87C2×C191—21no—
88C2×C191—21no—
89C2×C191—42no—
90C2×C191#44711yescharacteristic
91C2×C191#45511yescharacteristic
92C2×C191#45811yescharacteristic
93D191—82no—
94C191#160711yescharacteristic, complemented, Sylow
95#1607—1911nomaximal, Sylow
96#534—1911no—
97#480—1912no—
98#397D19111yescharacteristic, complemented
99#327—1912no—
100#323—1911no—
101C22×D4—3821no—
102#123—3822no—
103#121—3821no—
104#121—1912no—
105#120—3822no—
106C2×C4⋊C4C2×D19111yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D19111yescharacteristic
109#107—3822no—
110#102—3822no—
111C22⋊C8—1912no—
112#95—21no—
113#95C2×D19111yescharacteristic, complemented
114C24—3821no—
115C24C22⋊D19111yescharacteristic
116C2×D4—3821no—
117C2×D4—3822no—
118C2×D4—3822no—
119C2×D4—1912no—
120C2×D4—7642no—
121C2×D4—7642no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—3821no—
125C22×C4C22⋊D19111yescharacteristic
126C22×C4C22×D19111yescharacteristic
127C2×C8—3822no—
128C4⋊C4—22no—
129C4⋊C4—3821no—
130C4⋊C4—1912no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—3822no—
134C22⋊C4—3822no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—3822no—
140C23—15281no—
141C23—15281no—
142C23—7642no—
143C23#1569027212yes—
144C23C2×C22⋊D19111yescharacteristic
145D4—15281no—
146D4—7642no—
147D4—7644no—
148D4—7642no—
149D4—7642no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3822no—
160C2×C4—7642no—
161C2×C4D4×D19112yes—
162C8—7642no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—15284no—
171C22—7642no—
172C22—7642no—
173C22—7642no—
174C22#2025929011yescharacteristic
175C22C2×#1569027211yescharacteristic
176C22#2025934711yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—7642no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—15282no—
186C2#2713279611yescharacteristic
187C2#2713284011yescharacteristic
188C2#2713285211yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#397×C191C2
socleC22×C191#120
prime radical#27133225C2
hypercenter#397D191
hypocenterC191#1607
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #27132976C2semidirect
3. #27132396C2semidirect
4. #27132387C2semidirect
5. #397×C191C2semidirect
13. C22⋊(C191⋊C8)C22semidirect
14. C2×C4⋊C4×C191C22semidirect
18. #95×C191C22semidirect
94. C191#1607semidirect
98. #397D191semidirect
106. C2×C4⋊C4C2×D191semidirect
113. #95C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332252
#271329762
#271323962
#271323872
#397×C1912
#1607191