Finite Group Databasebeta

← #37571569

Subgroups — #37571569

Order 24448.

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

normal subgroups only

This group has 240 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#27133250C211yesmaximal, characteristic
2#27132999C211yesmaximal, characteristic, complemented
3#27132979C211yesmaximal, characteristic, complemented
4#27132600C211yesmaximal, characteristic, complemented
5#27132399C211yesmaximal, characteristic, complemented
6#27132386C211yesmaximal, characteristic, complemented
7#327×C191C211yesmaximal, characteristic, complemented
8#20259317—21no—
9#20259308—21no—
10#20259306—21no—
11#20259301C2211yescharacteristic
12#20259294—21no—
13#20259290—21no—
14D4×Dic191—21no—
15C2×C4⋊Dic191—21no—
16C2×C4⋊Dic191C2211yescharacteristic
17C2×(C2×D191)⋊C4—21no—
18C2×(C2×D191)⋊C4C2211yescharacteristic
19#20259256—21no—
20#20259253—21no—
21#20259247—21no—
22#20259243—21no—
23C22⋊(C191⋊C8)C2211yescharacteristic
24#121×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28C2×Dic382D411yescharacteristic
29C2×C22⋊D191—21no—
30C2×C22⋊D191—21no—
31C22×Dic191—21no—
32C22×Dic191—21no—
33C22×Dic191D411yescharacteristic
34C2×C191⋊C8—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191—21no—
38C4⋊Dic191—42no—
39C4⋊Dic191D411yescharacteristic
40Dic191⋊C4—21no—
41C22⋊Dic191—21no—
42C22⋊Dic191—21no—
43(C2×D191)⋊C4—21no—
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—21no—
46(C2×D191)⋊C4—21no—
47(C2×D191)⋊C4—42no—
48C4×Dic191—21no—
49C2×D4×C191—21no—
50C2×D4×C191D411yescharacteristic
51C22×C4×C191—21no—
52C22×C4×C191C2311yescharacteristic
53C2×C8×C191—21no—
54C4⋊C4×C191D411yescharacteristic
55C22⋊C4×C191—21no—
56C22×D191—21no—
57C22×D191—21no—
58C22×D191—21no—
59C22×D191—41no—
60C22×D191—41no—
61Dic382—41no—
62C22⋊D191—81no—
63C22⋊D191—81no—
64C2×Dic191—21no—
65C2×Dic191—21no—
66C2×Dic191—21no—
67C2×Dic191—41no—
68C2×Dic191—41no—
69C2×Dic191—41no—
70C2×Dic191—41no—
71C2×Dic191—81no—
72C191⋊C8—41no—
73C23×C191—21no—
74C23×C191C2×D411yescharacteristic
75D4×C191—41no—
76D4×C191—41no—
77D4×C191—41no—
78C2×C4×C191—22no—
79C2×C4×C191—21no—
80C2×C4×C191—41no—
81C2×C4×C191—41no—
82C2×C4×C191—41no—
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2×D411yescharacteristic
85C8×C191—41no—
86C22×C191—21no—
87C22×C191—21no—
88C22×C191—41no—
89C22×C191—41no—
90C22×C191—41no—
91C22×C191#12011yescharacteristic
92C22×C191#13511yescharacteristic
93C22×C191#13711yescharacteristic
94C2×D191—42no—
95C2×D191—41no—
96C2×D191—41no—
97C2×D191—41no—
98C2×D191—81no—
99C4×C191—21no—
100C4×C191—21no—
101C4×C191—42no—
102C4×C191—41no—
103Dic191—41no—
104Dic191—41no—
105Dic191—41no—
106Dic191—81no—
107C2×C191—21no—
108C2×C191—21no—
109C2×C191—81no—
110C2×C191#45211yescharacteristic
111C2×C191#45311yescharacteristic
112C2×C191#45811yescharacteristic
113D191—81no—
114C191#161811yescharacteristic, complemented, Sylow
115#1618—1911nomaximal, Sylow
116#538—1911no—
117#484—1911no—
118#481—1911no—
119#399—1911no—
120#328—1911no—
121#327D19111yescharacteristic, complemented
122#323—1911no—
123#126—3821no—
124#124—3821no—
125#123—3821no—
126C22⋊Q8—1911no—
127#121—3821no—
128#121C2×D19111yescharacteristic
129#120—3821no—
130C4×D4—3821no—
131C2×C4⋊C4—3821no—
132C2×C4⋊C4—1911no—
133C2×C22⋊C4—3821no—
134C2×C22⋊C4—1911no—
135#107—3821no—
136#106—3821no—
137#103—3821no—
138#102—3821no—
139C22⋊C8—1911no—
140C22⋊C8C2×D19111yescharacteristic
141#95C2×D19111yescharacteristic
142C24—1911no—
143C2×Q8—1911no—
144C2×D4—21no—
145C2×D4—3821no—
146C2×D4—3821no—
147C2×D4D76411yescharacteristic
148C22×C4—21no—
149C22×C4—3821no—
150C22×C4—3821no—
151C22×C4—1911no—
152C22×C4C22×D19111yescharacteristic
153C2×C8—21no—
154C2×C8—3821no—
155C4⋊C4—3821no—
156C4⋊C4—3821no—
157C4⋊C4—3821no—
158C4⋊C4—3821no—
159C4⋊C4—1911no—
160C4⋊C4—7642no—
161C4⋊C4D76411yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—3821no—
164C22⋊C4—3821no—
165C22⋊C4—3821no—
166C22⋊C4—3821no—
167C22⋊C4—3821no—
168C22⋊C4—3821no—
169C22⋊C4—7642no—
170C42—3821no—
171C23—21no—
172C23—3821no—
173C23—3821no—
174C23—3821no—
175C23—7641no—
176C23—7641no—
177C23D4×D19111yescharacteristic
178Q8—7641no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—15281no—
183D4—15281no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3821no—
190C2×C4—3821no—
191C2×C4—3821no—
192C2×C4—15281no—
193C2×C4—7641no—
194C2×C4—7641no—
195C2×C4—7641no—
196C2×C4—7641no—
197C2×C4D4×D19111yescharacteristic
198C2×C4C2×D76411yescharacteristic
199C8—41no—
200C8—7641no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—15281no—
207C22—7642no—
208C22—7641no—
209C22—7641no—
210C22—7641no—
211C22#2025929211yescharacteristic
212C22#2025934111yescharacteristic
213C22#2025935111yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—15281no—
219C4—7641no—
220C4—7641no—
221C4—7641no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—15281no—
226C2#2713282611yescharacteristic
227C2#2713283211yescharacteristic
228C2#2713285411yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#327×C191C2
socleC22×C191#120
prime radical#27133250C2
hypercenter#327D191
hypocenterC191#1618
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27132999C2semidirect
3. #27132979C2semidirect
4. #27132600C2semidirect
5. #27132399C2semidirect
6. #27132386C2semidirect
7. #327×C191C2semidirect
114. C191#1618semidirect
121. #327D191semidirect

Maximal subgroups

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

subgroupindex
#271332502
#271329992
#271329792
#271326002
#271323992
#271323862
#327×C1912
#1618191